\section{\label{sec:level1}First-level heading:\protect\\ The line
break was forced \lowercase{via} \textbackslash\textbackslash}

This sample document demonstrates proper use of REV\TeX~4.2 (and
\LaTeXe) in manuscripts prepared for submission to AAPM
journals. Further information can be found in the documentation included in the distribution or available at
\url{http://www.aapm.org} and in the documentation for 
REV\TeX~4.2 itself.

When commands are referred to in this example file, they are always
shown with their required arguments, using normal \TeX{} format. In
this format, \verb+#1+, \verb+#2+, etc. stand for required
author-supplied arguments to commands. For example, in
\verb+\section{#1}+ the \verb+#1+ stands for the title text of the
author's section heading, and in \verb+\title{#1}+ the \verb+#1+
stands for the title text of the paper.

Line breaks in section headings at all levels can be introduced using
\textbackslash\textbackslash. A blank input line tells \TeX\ that the
paragraph has ended. 

\subsection{\label{sec:level2}Second-level heading: Formatting}

This file may be formatted in both the \texttt{preprint} (the default) and
\texttt{reprint} styles; the latter format may be used to 
mimic final journal output. In addition, there is another
option available, \texttt{lengthcheck}, which formats the document as closely
as possible to an actual journal article, to facilitate the author's
performance of a length check. Either format may be used for submission
purposes; however, for peer review and production, AAPM will format the
article using the \texttt{preprint} class option. Hence, it is
essential that authors check that their manuscripts format acceptably
under \texttt{preprint}. Manuscripts submitted to AAPM that do not
format correctly under the \texttt{preprint} option may be delayed in
both the editorial and production processes.

The \texttt{widetext} environment will make the text the width of the
full page, as on page~\pageref{eq:wideeq}. (Note the use the
\verb+\pageref{#1}+ to get the page number right automatically.) The
width-changing commands only take effect in \texttt{twocolumn}
formatting. It has no effect if \texttt{preprint} formatting is chosen
instead.

\subsubsection{\label{sec:level3}Third-level heading: Citations and Footnotes}

Citations in text refer to entries in the Bibliography;
they use the commands \verb+\cite{#1}+ or \verb+\onlinecite{#1}+. 
Because REV\TeX\ uses the \verb+natbib+ package of Patrick Daly, 
its entire repertoire of commands are available in your document;
see the \verb+natbib+ documentation for further details.
The argument of \verb+\cite+ is a comma-separated list of \emph{keys};
a key may consist of letters and numerals. 

By default, AAPM citations are numerical; \cite{feyn54} 
to give a textual citation, use \verb+\onlinecite{#1}+: (Refs.~\onlinecite{witten2001,epr,Bire82}). 
REV\TeX\ ``collapses'' lists of consecutive numerical citations when appropriate. 
To illustrate, we cite several together \cite{feyn54,witten2001,epr,Berman1983}, 
and once again (Refs.~\onlinecite{epr,feyn54,Bire82,Berman1983}). 
Note that, when numerical citations are used, the references were sorted into the same order they appear in the bibliography. 

A reference within the bibliography is specified with a \verb+\bibitem{#1}+ command,
where the argument is the citation key mentioned above. 
\verb+\bibitem{#1}+ commands may be crafted by hand or, preferably,
generated by using Bib\TeX. 
The AAPM styles for REV\TeX~4 include Bib\TeX\ style file
\verb+aapmrev4-2.bst+, appropriate for
numbered bibliography. 
REV\TeX~4 will automatically choose the style appropriate for 
the document's selected class options: the default is numerical.

This sample file demonstrates a simple use of Bib\TeX\ 
via a \verb+\bibliography+ command referencing the \verb+aapmsamp.bib+ file.
Running Bib\TeX\ (in this case \texttt{bibtex
aapmsamp}) after the first pass of \LaTeX\ produces the file
\verb+aapmsamp.bbl+ which contains the automatically formatted
\verb+\bibitem+ commands (including extra markup information via
\verb+\bibinfo+ commands). If not using Bib\TeX, the
\verb+thebibiliography+ environment should be used instead.

\paragraph{Fourth-level heading is run in.}%
Footnotes are produced using the \verb+\footnote{#1}+ command. 
Numerical style citations put footnotes into the 
bibliography\footnote{Automatically placing footnotes into the bibliography requires using BibTeX to compile the bibliography.}.
Note: due to the method used to place footnotes in the bibliography, \emph{you
must re-run BibTeX every time you change any of your document's
footnotes}. 

\section{Math and Equations}
Inline math may be typeset using the \verb+$+ delimiters. Bold math
symbols may be achieved using the \verb+bm+ package and the
\verb+\bm{#1}+ command it supplies. For instance, a bold $\alpha$ can
be typeset as \verb+$\bm{\alpha}$+ giving $\bm{\alpha}$. Fraktur and
Blackboard (or open face or double struck) characters should be
typeset using the \verb+\mathfrak{#1}+ and \verb+\mathbb{#1}+ commands
respectively. Both are supplied by the \texttt{amssymb} package. For
example, \verb+$\mathbb{R}$+ gives $\mathbb{R}$ and
\verb+$\mathfrak{G}$+ gives $\mathfrak{G}$

In \LaTeX\ there are many different ways to display equations, and a
few preferred ways are noted below. Displayed math will flush left by
default.

Below we have numbered single-line equations, the most common kind: 
\begin{eqnarray}
\chi_+(p)\alt{\bf [}2|{\bf p}|(|{\bf p}|+p_z){\bf ]}^{-1/2}
\left(
\begin{array}{c}
|{\bf p}|+p_z\\
px+ip_y
\end{array}\right)\;,
\\
\left\{%
 \openone234567890abc123\alpha\beta\gamma\delta1234556\alpha\beta
 \frac{1\sum^{a}_{b}}{A^2}%
\right\}%
\label{eq:one}.
\end{eqnarray}
Note the open one in Eq.~(\ref{eq:one}).

Not all numbered equations will fit within a narrow column this
way. The equation number will move down automatically if it cannot fit
on the same line with a one-line equation:
\begin{equation}
\left\{
 ab12345678abc123456abcdef\alpha\beta\gamma\delta1234556\alpha\beta
 \frac{1\sum^{a}_{b}}{A^2}%
\right\}.
\end{equation}

When the \verb+\label{#1}+ command is used [cf. input for
Eq.~(\ref{eq:one})], the equation can be referred to in text without
knowing the equation number that \TeX\ will assign to it. Just
use \verb+\ref{#1}+, where \verb+#1+ is the same name that used in
the \verb+\label{#1}+ command.

Unnumbered single-line equations can be typeset
using the \verb+\[+, \verb+\]+ format:
\[g^+g^+ \rightarrow g^+g^+g^+g^+ \dots ~,~~q^+q^+\rightarrow
q^+g^+g^+ \dots ~. \]

\subsection{Multiline equations}

Multiline equations are obtained by using the \verb+eqnarray+
environment.  Use the \verb+\nonumber+ command at the end of each line
to avoid assigning a number:
\begin{eqnarray}
{\cal M}=&&ig_Z^2(4E_1E_2)^{1/2}(l_i^2)^{-1}
\delta_{\sigma_1,-\sigma_2}
(g_{\sigma_2}^e)^2\chi_{-\sigma_2}(p_2)\nonumber\\
&&\times
[\epsilon_jl_i\epsilon_i]_{\sigma_1}\chi_{\sigma_1}(p_1),
\end{eqnarray}
\begin{eqnarray}
\sum \vert M^{\text{viol}}_g \vert ^2&=&g^{2n-4}_S(Q^2)~N^{n-2}
        (N^2-1)\nonumber \\
 & &\times \left( \sum_{i<j}\right)
  \sum_{\text{perm}}
 \frac{1}{S_{12}}
 \frac{1}{S_{12}}
 \sum_\tau c^f_\tau~.
\end{eqnarray}
\textbf{Note:} Do not use \verb+\label{#1}+ on a line of a multiline
equation if \verb+\nonumber+ is also used on that line. Incorrect
cross-referencing will result. Notice the use \verb+\text{#1}+ for
using a Roman font within a math environment.

To set a multiline equation without \emph{any} equation
numbers, use the \verb+\begin{eqnarray*}+,
\verb+\end{eqnarray*}+ format:
\begin{eqnarray*}
\sum \vert M^{\text{viol}}_g \vert ^2&=&g^{2n-4}_S(Q^2)~N^{n-2}
        (N^2-1)\\
 & &\times \left( \sum_{i<j}\right)
 \left(
  \sum_{\text{perm}}\frac{1}{S_{12}S_{23}S_{n1}}
 \right)
 \frac{1}{S_{12}}~.
\end{eqnarray*}
To obtain numbers not normally produced by the automatic numbering,
use the \verb+\tag{#1}+ command, where \verb+#1+ is the desired
equation number. For example, to get an equation number of
(\ref{eq:mynum}),
\begin{equation}
g^+g^+ \rightarrow g^+g^+g^+g^+ \dots ~,~~q^+q^+\rightarrow
q^+g^+g^+ \dots ~. \tag{2.6$'$}\label{eq:mynum}
\end{equation}

A few notes on \verb=\tag{#1}=. \verb+\tag{#1}+ requires
\texttt{amsmath}. The \verb+\tag{#1}+ must come before the
\verb+\label{#1}+, if any. The numbering set with \verb+\tag{#1}+ is
\textit{transparent} to the automatic numbering in REV\TeX{};
therefore, the number must be known ahead of time, and it must be
manually adjusted if other equations are added. \verb+\tag{#1}+ works
with both single-line and multiline equations. \verb+\tag{#1}+ should
only be used in exceptional case - do not use it to number all
equations in a paper.

Note the equation number gets reset again:
\begin{equation}
g^+g^+g^+ \rightarrow g^+g^+g^+g^+g^+ \dots ~,~~q^+q^+\rightarrow
q^+g^+g^+ \dots ~. 
\end{equation}

Enclosing single-line and multiline equations in
\verb+\begin{subequations}+ and \verb+\end{subequations}+ will produce
a set of equations that are ``numbered'' with letters, as shown in
Eqs.~(\ref{subeq:1}) and (\ref{subeq:2}) below:
\begin{subequations}
\label{eq:whole}
\begin{equation}
\left\{
 abc123456abcdef\alpha\beta\gamma\delta1234556\alpha\beta
 \frac{1\sum^{a}_{b}}{A^2}
\right\},\label{subeq:1}
\end{equation}
\begin{eqnarray}
{\cal M}=&&ig_Z^2(4E_1E_2)^{1/2}(l_i^2)^{-1}
(g_{\sigma_2}^e)^2\chi_{-\sigma_2}(p_2)\nonumber\\
&&\times
[\epsilon_i]_{\sigma_1}\chi_{\sigma_1}(p_1).\label{subeq:2}
\end{eqnarray}
\end{subequations}
Putting a \verb+\label{#1}+ command right after the
\verb+\begin{subequations}+, allows one to
reference all the equations in a subequations environment. For
example, the equations in the preceding subequations environment were
Eqs.~(\ref{eq:whole}).

\subsubsection{Wide equations}
The equation that follows is set in a wide format, i.e., it spans
across the full page. The wide format is reserved for long equations
that cannot be easily broken into four lines or less:
\begin{widetext}
\begin{equation}
{\cal R}^{(\text{d})}=
 g_{\sigma_2}^e
 \left(
   \frac{[\Gamma^Z(3,21)]_{\sigma_1}}{Q_{12}^2-M_W^2}
  +\frac{[\Gamma^Z(13,2)]_{\sigma_1}}{Q_{13}^2-M_W^2}
 \right)
 + x_WQ_e
 \left(
   \frac{[\Gamma^\gamma(3,21)]_{\sigma_1}}{Q_{12}^2-M_W^2}
  +\frac{[\Gamma^\gamma(13,2)]_{\sigma_1}}{Q_{13}^2-M_W^2}
 \right)\;. \label{eq:wideeq}
\end{equation}
\end{widetext}
This is typed to show the output is in wide format.
(Since there is no input line between \verb+\equation+ and
this paragraph, there is no paragraph indent for this paragraph.)
\section{Cross-referencing}
REV\TeX{} will automatically number sections, equations, figure
captions, and tables. In order to reference them in text, use the
\verb+\label{#1}+ and \verb+\ref{#1}+ commands. To reference a
particular page, use the \verb+\pageref{#1}+ command.

The \verb+\label{#1}+ should appear in a section heading, within an
equation, or in a table or figure caption. The \verb+\ref{#1}+ command
is used in the text where the citation is to be displayed.  Some
examples: Section~\ref{sec:level1} on page~\pageref{sec:level1},
Table~\ref{tab:table1},%
\begin{table}
\caption{\label{tab:table1}This is a narrow table which fits into a
text column when using \texttt{twocolumn} formatting. Note that
REV\TeX~4 adjusts the intercolumn spacing so that the table fills the
entire width of the column. Table captions are numbered
automatically. This table illustrates left-aligned, centered, and
right-aligned columns.  }
\begin{ruledtabular}
\begin{tabular}{lcr}
Left\footnote{Note a.}&Centered\footnote{Note b.}&Right\\
\hline
1 & 2 & 3\\
10 & 20 & 30\\
100 & 200 & 300\\
\end{tabular}
\end{ruledtabular}
\end{table}
and Fig.~\ref{fig:epsart}.

\section{Figures and Tables}
Figures and tables are typically ``floats''; \LaTeX\ determines their
final position via placement rules. 
\LaTeX\ isn't always successful in automatically placing floats where you wish them.

Figures are marked up with the \texttt{figure} environment, the content of which
imports the image (\verb+\includegraphics+) followed by the figure caption (\verb+\caption+).
The argument of the latter command should itself contain a \verb+\label+ command if you
wish to refer to your figure with \verb+\ref+.

Import your image using either the \texttt{graphics} or
\texttt{graphix} packages. These packages both define the
\verb+\includegraphics{#1}+ command, but they differ in the optional
arguments for specifying the orientation, scaling, and translation of the figure.
Fig.~\ref{fig:epsart}%
\begin{figure}
\includegraphics{fig_1
\caption{\label{fig:epsart} A figure caption. The figure captions are
automatically numbered.}
\end{figure}
is small enough to fit in a single column, while
Fig.~\ref{fig:wide}%
\begin{figure*}
\includegraphics{fig_2
\caption{\label{fig:wide}Use the \texttt{figure*} environment to get a wide
figure, spanning the page in \texttt{twocolumn} formatting.}
\end{figure*}
is too wide for a single column,
so instead the \texttt{figure*} environment has been used.

The analog of the \texttt{figure} environment is \texttt{table}, which uses
the same \verb+\caption+ command.
However, you should type your caption command first within the \texttt{table}, 
instead of last as you did for \texttt{figure}.

The heart of any table is the \texttt{tabular} environment,
which represents the table content as a (vertical) sequence of table rows,
each containing a (horizontal) sequence of table cells. 
Cells are separated by the \verb+&+ character;
the row terminates with \verb+\\+. 
The required argument for the \texttt{tabular} environment
specifies how data are displayed in each of the columns. 
For instance, a column
may be centered (\verb+c+), left-justified (\verb+l+), right-justified (\verb+r+),
or aligned on a decimal point (\verb+d+). 
(Table~\ref{tab:table4}%
\begin{table}
\caption{\label{tab:table4}Numbers in columns Three--Five have been
aligned by using the ``d'' column specifier (requires the
\texttt{dcolumn} package). 
Non-numeric entries (those entries without
a ``.'') in a ``d'' column are aligned on the decimal point. 
Use the
``D'' specifier for more complex layouts. }
\begin{ruledtabular}
\begin{tabular}{ccddd}
One&Two&\mbox{Three}&\mbox{Four}&\mbox{Five}\\
\hline
one&two&\mbox{three}&\mbox{four}&\mbox{five}\\
He&2& 2.77234 & 45672. & 0.69 \\
C\footnote{Some tables require footnotes.}
  &C\footnote{Some tables need more than one footnote.}
  & 12537.64 & 37.66345 & 86.37 \\
\end{tabular}
\end{ruledtabular}
\end{table}
illustrates the use of decimal column alignment.)

Extra column-spacing may be be specified as well, although
REV\TeX~4 sets this spacing so that the columns fill the width of the
table.
Horizontal rules are typeset using the \verb+\hline+
command.
The doubled (or Scotch) rules that appear at the top and
bottom of a table can be achieved by enclosing the \texttt{tabular}
environment within a \texttt{ruledtabular} environment.
Rows whose columns span multiple columns can be typeset using \LaTeX's
\verb+\multicolumn{#1}{#2}{#3}+ command
(for example, see the first row of Table~\ref{tab:table3}).%
\begin{table*}
\caption{\label{tab:table3}This is a wide table that spans the page
width in \texttt{twocolumn} mode. It is formatted using the
\texttt{table*} environment. It also demonstrates the use of
\textbackslash\texttt{multicolumn} in rows with entries that span
more than one column.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
 &\multicolumn{2}{c}{$D_{4h}^1$}&\multicolumn{2}{c}{$D_{4h}^5$}\\
 Ion&1st alternative&2nd alternative&lst alternative
&2nd alternative\\ \hline
 K&$(2e)+(2f)$&$(4i)$ &$(2c)+(2d)$&$(4f)$ \\
 Mn&$(2g)$\footnote{The $z$ parameter of these positions is $z\sim\frac{1}{4}$.}
 &$(a)+(b)+(c)+(d)$&$(4e)$&$(2a)+(2b)$\\
 Cl&$(a)+(b)+(c)+(d)$&$(2g)$\footnote{This is a footnote in a table that spans the full page
width in \texttt{twocolumn} mode. It is supposed to set on the full width of the page, just as the caption does. }
 &$(4e)^{\text{a}}$\\
 He&$(8r)^{\text{a}}$&$(4j)^{\text{a}}$&$(4g)^{\text{a}}$\\
 Ag& &$(4k)^{\text{a}}$& &$(4h)^{\text{a}}$\\
\end{tabular}
\end{ruledtabular}
\end{table*}

The tables in this document illustrate various effects.
Tables that fit in a narrow column are contained in a \texttt{table}
environment.
Table~\ref{tab:table3} is a wide table, therefore set with the
\texttt{table*} environment.
Lengthy tables may need to break across pages.
A simple way to allow this is to specify
the \verb+[H]+ float placement on the \texttt{table} or
\texttt{table*} environment.
Alternatively, using the standard \LaTeXe\ package \texttt{longtable} 
gives more control over how tables break and allows headers and footers 
to be specified for each page of the table.
An example of the use of \texttt{longtable} can be found
in the file \texttt{summary.tex} that is included with the REV\TeX~4
distribution.

There are two methods for setting footnotes within a table (these
footnotes will be displayed directly below the table rather than at
the bottom of the page or in the bibliography).
The easiest
and preferred method is just to use the \verb+\footnote{#1}+
command. This will automatically enumerate the footnotes with
lowercase roman letters.
However, it is sometimes necessary to have
multiple entries in the table share the same footnote.
In this case,
create the footnotes using
\verb+\footnotemark[#1]+ and \verb+\footnotetext[#1]{#2}+.
\texttt{\#1} is a numeric value.
Each time the same value for \texttt{\#1} is used, 
the same mark is produced in the table. 
The \verb+\footnotetext[#1]{#2}+ commands are placed after the \texttt{tabular}
environment. 
Examine the \LaTeX\ source and output for Tables~\ref{tab:table1} and 
\ref{tab:table2}%
\begin{table}
\caption{\label{tab:table2}A table with more columns still fits
properly in a column. Note that several entries share the same
footnote. Inspect the \LaTeX\ input for this table to see
exactly how it is done.}
\begin{ruledtabular}
\begin{tabular}{cccccccc}
 &$r_c$ (\AA)&$r_0$ (\AA)&$\kappa r_0$&
 &$r_c$ (\AA) &$r_0$ (\AA)&$\kappa r_0$\\
\hline
Cu& 0.800 & 14.10 & 2.550 &Sn\footnotemark[1]
& 0.680 & 1.870 & 3.700 \\
Ag& 0.990 & 15.90 & 2.710 &Pb\footnotemark[2]
& 0.450 & 1.930 & 3.760 \\
Au& 1.150 & 15.90 & 2.710 &Ca\footnotemark[3]
& 0.750 & 2.170 & 3.560 \\
Mg& 0.490 & 17.60 & 3.200 &Sr\footnotemark[4]
& 0.900 & 2.370 & 3.720 \\
Zn& 0.300 & 15.20 & 2.970 &Li\footnotemark[2]
& 0.380 & 1.730 & 2.830 \\
Cd& 0.530 & 17.10 & 3.160 &Na\footnotemark[5]
& 0.760 & 2.110 & 3.120 \\
Hg& 0.550 & 17.80 & 3.220 &K\footnotemark[5]
&  1.120 & 2.620 & 3.480 \\
Al& 0.230 & 15.80 & 3.240 &Rb\footnotemark[3]
& 1.330 & 2.800 & 3.590 \\
Ga& 0.310 & 16.70 & 3.330 &Cs\footnotemark[4]
& 1.420 & 3.030 & 3.740 \\
In& 0.460 & 18.40 & 3.500 &Ba\footnotemark[5]
& 0.960 & 2.460 & 3.780 \\
Tl& 0.480 & 18.90 & 3.550 & & & & \\
\end{tabular}
\end{ruledtabular}
\footnotetext[1]{Here's the first, from Ref.~\onlinecite{feyn54}.}
\footnotetext[2]{Here's the second.}
\footnotetext[3]{Here's the third.}
\footnotetext[4]{Here's the fourth.}
\footnotetext[5]{And etc.}
\end{table}
for an illustration. 

All AAPM journals require that the initial citation of
figures or tables be in numerical order.
\LaTeX's automatic numbering of floats is your friend here:
just put each \texttt{figure} environment immediately following 
its first reference (\verb+\ref+), as we have done in this example file. 

\begin{acknowledgments}
We wish to acknowledge the support of the author community in using
REV\TeX{}, offering suggestions and encouragement, testing new versions,
\dots.
\end{acknowledgments}


\section{\label{sec:level1}First-level heading:\protect\\ The line
break was forced \lowercase{via} \textbackslash\textbackslash}

This sample document demonstrates proper use of REV\TeX~4.2 (and
\LaTeXe) in manuscripts prepared for submission to AIP
journals. Further information can be found in the documentation included in the distribution or available at
\url{http://authors.aip.org} and in the documentation for 
REV\TeX~4.2 itself.

When commands are referred to in this example file, they are always
shown with their required arguments, using normal \TeX{} format. In
this format, \verb+#1+, \verb+#2+, etc. stand for required
author-supplied arguments to commands. For example, in
\verb+\section{#1}+ the \verb+#1+ stands for the title text of the
author's section heading, and in \verb+\title{#1}+ the \verb+#1+
stands for the title text of the paper.

Line breaks in section headings at all levels can be introduced using
\textbackslash\textbackslash. A blank input line tells \TeX\ that the
paragraph has ended. 

\subsection{\label{sec:level2}Second-level heading: Formatting}

This file may be formatted in both the \texttt{preprint} (the default) and
\texttt{reprint} styles; the latter format may be used to 
mimic final journal output. Either format may be used for submission
purposes; however, for peer review and production, AIP will format the
article using the \texttt{preprint} class option. Hence, it is
essential that authors check that their manuscripts format acceptably
under \texttt{preprint}. Manuscripts submitted to AIP that do not
format correctly under the \texttt{preprint} option may be delayed in
both the editorial and production processes.

The \texttt{widetext} environment will make the text the width of the
full page, as on page~\pageref{eq:wideeq}. (Note the use the
\verb+\pageref{#1}+ to get the page number right automatically.) The
width-changing commands only take effect in \texttt{twocolumn}
formatting. It has no effect if \texttt{preprint} formatting is chosen
instead.

\subsubsection{\label{sec:level3}Third-level heading: Citations and Footnotes}

Citations in text refer to entries in the Bibliography;
they use the commands \verb+\cite{#1}+ or \verb+\onlinecite{#1}+. 
Because REV\TeX\ uses the \verb+natbib+ package of Patrick Daly, 
its entire repertoire of commands are available in your document;
see the \verb+natbib+ documentation for further details.
The argument of \verb+\cite+ is a comma-separated list of \emph{keys};
a key may consist of letters and numerals. 

By default, citations are numerical; \cite{feyn54} author-year citations are an option. 
To give a textual citation, use \verb+\onlinecite{#1}+: (Refs.~\onlinecite{witten2001,epr,Bire82}). 
REV\TeX\ ``collapses'' lists of consecutive numerical citations when appropriate. 
REV\TeX\ provides the ability to properly punctuate textual citations in author-year style;
this facility works correctly with numerical citations only with \texttt{natbib}'s compress option turned off. 
To illustrate, we cite several together \cite{feyn54,witten2001,epr,Berman1983}, 
and once again (Refs.~\onlinecite{epr,feyn54,Bire82,Berman1983}). 
Note that, when numerical citations are used, the references were sorted into the same order they appear in the bibliography. 

A reference within the bibliography is specified with a \verb+\bibitem{#1}+ command,
where the argument is the citation key mentioned above. 
\verb+\bibitem{#1}+ commands may be crafted by hand or, preferably,
generated by using Bib\TeX. 
The AIP styles for REV\TeX~4 include Bib\TeX\ style files
\verb+aipnum.bst+ and \verb+aipauth.bst+, appropriate for
numbered and author-year bibliographies,
respectively. 
REV\TeX~4 will automatically choose the style appropriate for 
the document's selected class options: the default is numerical, and
you obtain the author-year style by specifying a class option of \verb+author-year+.

This sample file demonstrates a simple use of Bib\TeX\ 
via a \verb+\bibliography+ command referencing the \verb+aipsamp.bib+ file.
Running Bib\TeX\ (in this case \texttt{bibtex
aipsamp}) after the first pass of \LaTeX\ produces the file
\verb+aipsamp.bbl+ which contains the automatically formatted
\verb+\bibitem+ commands (including extra markup information via
\verb+\bibinfo+ commands). If not using Bib\TeX, the
\verb+thebibiliography+ environment should be used instead.

\paragraph{Fourth-level heading is run in.}%
Footnotes are produced using the \verb+\footnote{#1}+ command. 
Numerical style citations put footnotes into the 
bibliography\footnote{Automatically placing footnotes into the bibliography requires using BibTeX to compile the bibliography.}.
Author-year and numerical author-year citation styles (each for its own reason) cannot use this method. 
Note: due to the method used to place footnotes in the bibliography, \emph{you
must re-run BibTeX every time you change any of your document's
footnotes}. 

\section{Math and Equations}
Inline math may be typeset using the \verb+$+ delimiters. Bold math
symbols may be achieved using the \verb+bm+ package and the
\verb+\bm{#1}+ command it supplies. For instance, a bold $\alpha$ can
be typeset as \verb+$\bm{\alpha}$+ giving $\bm{\alpha}$. Fraktur and
Blackboard (or open face or double struck) characters should be
typeset using the \verb+\mathfrak{#1}+ and \verb+\mathbb{#1}+ commands
respectively. Both are supplied by the \texttt{amssymb} package. For
example, \verb+$\mathbb{R}$+ gives $\mathbb{R}$ and
\verb+$\mathfrak{G}$+ gives $\mathfrak{G}$

In \LaTeX\ there are many different ways to display equations, and a
few preferred ways are noted below. Displayed math will center by
default. Use the class option \verb+fleqn+ to flush equations left.

Below we have numbered single-line equations, the most common kind: 
\begin{eqnarray}
\chi_+(p)\alt{\bf [}2|{\bf p}|(|{\bf p}|+p_z){\bf ]}^{-1/2}
\left(
\begin{array}{c}
|{\bf p}|+p_z\\
px+ip_y
\end{array}\right)\;,
\\
\left\{%
 \openone234567890abc123\alpha\beta\gamma\delta1234556\alpha\beta
 \frac{1\sum^{a}_{b}}{A^2}%
\right\}%
\label{eq:one}.
\end{eqnarray}
Note the open one in Eq.~(\ref{eq:one}).

Not all numbered equations will fit within a narrow column this
way. The equation number will move down automatically if it cannot fit
on the same line with a one-line equation:
\begin{equation}
\left\{
 ab12345678abc123456abcdef\alpha\beta\gamma\delta1234556\alpha\beta
 \frac{1\sum^{a}_{b}}{A^2}%
\right\}.
\end{equation}

When the \verb+\label{#1}+ command is used [cf. input for
Eq.~(\ref{eq:one})], the equation can be referred to in text without
knowing the equation number that \TeX\ will assign to it. Just
use \verb+\ref{#1}+, where \verb+#1+ is the same name that used in
the \verb+\label{#1}+ command.

Unnumbered single-line equations can be typeset
using the \verb+\[+, \verb+\]+ format:
\[g^+g^+ \rightarrow g^+g^+g^+g^+ \dots ~,~~q^+q^+\rightarrow
q^+g^+g^+ \dots ~. \]

\subsection{Multiline equations}

Multiline equations are obtained by using the \verb+eqnarray+
environment.  Use the \verb+\nonumber+ command at the end of each line
to avoid assigning a number:
\begin{eqnarray}
{\cal M}=&&ig_Z^2(4E_1E_2)^{1/2}(l_i^2)^{-1}
\delta_{\sigma_1,-\sigma_2}
(g_{\sigma_2}^e)^2\chi_{-\sigma_2}(p_2)\nonumber\\
&&\times
[\epsilon_jl_i\epsilon_i]_{\sigma_1}\chi_{\sigma_1}(p_1),
\end{eqnarray}
\begin{eqnarray}
\sum \vert M^{\text{viol}}_g \vert ^2&=&g^{2n-4}_S(Q^2)~N^{n-2}
        (N^2-1)\nonumber \\
 & &\times \left( \sum_{i<j}\right)
  \sum_{\text{perm}}
 \frac{1}{S_{12}}
 \frac{1}{S_{12}}
 \sum_\tau c^f_\tau~.
\end{eqnarray}
\textbf{Note:} Do not use \verb+\label{#1}+ on a line of a multiline
equation if \verb+\nonumber+ is also used on that line. Incorrect
cross-referencing will result. Notice the use \verb+\text{#1}+ for
using a Roman font within a math environment.

To set a multiline equation without \emph{any} equation
numbers, use the \verb+\begin{eqnarray*}+,
\verb+\end{eqnarray*}+ format:
\begin{eqnarray*}
\sum \vert M^{\text{viol}}_g \vert ^2&=&g^{2n-4}_S(Q^2)~N^{n-2}
        (N^2-1)\\
 & &\times \left( \sum_{i<j}\right)
 \left(
  \sum_{\text{perm}}\frac{1}{S_{12}S_{23}S_{n1}}
 \right)
 \frac{1}{S_{12}}~.
\end{eqnarray*}
To obtain numbers not normally produced by the automatic numbering,
use the \verb+\tag{#1}+ command, where \verb+#1+ is the desired
equation number. For example, to get an equation number of
(\ref{eq:mynum}),
\begin{equation}
g^+g^+ \rightarrow g^+g^+g^+g^+ \dots ~,~~q^+q^+\rightarrow
q^+g^+g^+ \dots ~. \tag{2.6$'$}\label{eq:mynum}
\end{equation}

A few notes on \verb=\tag{#1}=. \verb+\tag{#1}+ requires
\texttt{amsmath}. The \verb+\tag{#1}+ must come before the
\verb+\label{#1}+, if any. The numbering set with \verb+\tag{#1}+ is
\textit{transparent} to the automatic numbering in REV\TeX{};
therefore, the number must be known ahead of time, and it must be
manually adjusted if other equations are added. \verb+\tag{#1}+ works
with both single-line and multiline equations. \verb+\tag{#1}+ should
only be used in exceptional case - do not use it to number all
equations in a paper.

Enclosing single-line and multiline equations in
\verb+\begin{subequations}+ and \verb+\end{subequations}+ will produce
a set of equations that are ``numbered'' with letters, as shown in
Eqs.~(\ref{subeq:1}) and (\ref{subeq:2}) below:
\begin{subequations}
\label{eq:whole}
\begin{equation}
\left\{
 abc123456abcdef\alpha\beta\gamma\delta1234556\alpha\beta
 \frac{1\sum^{a}_{b}}{A^2}
\right\},\label{subeq:1}
\end{equation}
\begin{eqnarray}
{\cal M}=&&ig_Z^2(4E_1E_2)^{1/2}(l_i^2)^{-1}
(g_{\sigma_2}^e)^2\chi_{-\sigma_2}(p_2)\nonumber\\
&&\times
[\epsilon_i]_{\sigma_1}\chi_{\sigma_1}(p_1).\label{subeq:2}
\end{eqnarray}
\end{subequations}
Putting a \verb+\label{#1}+ command right after the
\verb+\begin{subequations}+, allows one to
reference all the equations in a subequations environment. For
example, the equations in the preceding subequations environment were
Eqs.~(\ref{eq:whole}).

\subsubsection{Wide equations}
The equation that follows is set in a wide format, i.e., it spans
across the full page. The wide format is reserved for long equations
that cannot be easily broken into four lines or less:
\begin{widetext}
\begin{equation}
{\cal R}^{(\text{d})}=
 g_{\sigma_2}^e
 \left(
   \frac{[\Gamma^Z(3,21)]_{\sigma_1}}{Q_{12}^2-M_W^2}
  +\frac{[\Gamma^Z(13,2)]_{\sigma_1}}{Q_{13}^2-M_W^2}
 \right)
 + x_WQ_e
 \left(
   \frac{[\Gamma^\gamma(3,21)]_{\sigma_1}}{Q_{12}^2-M_W^2}
  +\frac{[\Gamma^\gamma(13,2)]_{\sigma_1}}{Q_{13}^2-M_W^2}
 \right)\;. \label{eq:wideeq}
\end{equation}
\end{widetext}
This is typed to show the output is in wide format.
(Since there is no input line between \verb+\equation+ and
this paragraph, there is no paragraph indent for this paragraph.)
\section{Cross-referencing}
REV\TeX{} will automatically number sections, equations, figure
captions, and tables. In order to reference them in text, use the
\verb+\label{#1}+ and \verb+\ref{#1}+ commands. To reference a
particular page, use the \verb+\pageref{#1}+ command.

The \verb+\label{#1}+ should appear in a section heading, within an
equation, or in a table or figure caption. The \verb+\ref{#1}+ command
is used in the text where the citation is to be displayed.  Some
examples: Section~\ref{sec:level1} on page~\pageref{sec:level1},
Table~\ref{tab:table1},%
\begin{table}
\caption{\label{tab:table1}This is a narrow table which fits into a
text column when using \texttt{twocolumn} formatting. Note that
REV\TeX~4 adjusts the intercolumn spacing so that the table fills the
entire width of the column. Table captions are numbered
automatically. This table illustrates left-aligned, centered, and
right-aligned columns.  }
\begin{ruledtabular}
\begin{tabular}{lcr}
Left\footnote{Note a.}&Centered\footnote{Note b.}&Right\\
\hline
1 & 2 & 3\\
10 & 20 & 30\\
100 & 200 & 300\\
\end{tabular}
\end{ruledtabular}
\end{table}
and Fig.~\ref{fig:epsart}.

\section{Figures and Tables}
Figures and tables are typically ``floats''; \LaTeX\ determines their
final position via placement rules. 
\LaTeX\ isn't always successful in automatically placing floats where you wish them.

Figures are marked up with the \texttt{figure} environment, the content of which
imports the image (\verb+\includegraphics+) followed by the figure caption (\verb+\caption+).
The argument of the latter command should itself contain a \verb+\label+ command if you
wish to refer to your figure with \verb+\ref+.

Import your image using either the \texttt{graphics} or
\texttt{graphix} packages. These packages both define the
\verb+\includegraphics{#1}+ command, but they differ in the optional
arguments for specifying the orientation, scaling, and translation of the figure.
Fig.~\ref{fig:epsart}%
\begin{figure}
\includegraphics{fig_1
\caption{\label{fig:epsart} A figure caption. The figure captions are
automatically numbered.}
\end{figure}
is small enough to fit in a single column, while
Fig.~\ref{fig:wide}%
\begin{figure*}
\includegraphics{fig_2
\caption{\label{fig:wide}Use the \texttt{figure*} environment to get a wide
figure, spanning the page in \texttt{twocolumn} formatting.}
\end{figure*}
is too wide for a single column,
so instead the \texttt{figure*} environment has been used.

The analog of the \texttt{figure} environment is \texttt{table}, which uses
the same \verb+\caption+ command.
However, you should type your caption command first within the \texttt{table}, 
instead of last as you did for \texttt{figure}.

The heart of any table is the \texttt{tabular} environment,
which represents the table content as a (vertical) sequence of table rows,
each containing a (horizontal) sequence of table cells. 
Cells are separated by the \verb+&+ character;
the row terminates with \verb+\\+. 
The required argument for the \texttt{tabular} environment
specifies how data are displayed in each of the columns. 
For instance, a column
may be centered (\verb+c+), left-justified (\verb+l+), right-justified (\verb+r+),
or aligned on a decimal point (\verb+d+). 
(Table~\ref{tab:table4}%
\begin{table}
\caption{\label{tab:table4}Numbers in columns Three--Five have been
aligned by using the ``d'' column specifier (requires the
\texttt{dcolumn} package). 
Non-numeric entries (those entries without
a ``.'') in a ``d'' column are aligned on the decimal point. 
Use the
``D'' specifier for more complex layouts. }
\begin{ruledtabular}
\begin{tabular}{ccddd}
One&Two&\mbox{Three}&\mbox{Four}&\mbox{Five}\\
\hline
one&two&\mbox{three}&\mbox{four}&\mbox{five}\\
He&2& 2.77234 & 45672. & 0.69 \\
C\footnote{Some tables require footnotes.}
  &C\footnote{Some tables need more than one footnote.}
  & 12537.64 & 37.66345 & 86.37 \\
\end{tabular}
\end{ruledtabular}
\end{table}
illustrates the use of decimal column alignment.)

Extra column-spacing may be be specified as well, although
REV\TeX~4 sets this spacing so that the columns fill the width of the
table.
Horizontal rules are typeset using the \verb+\hline+
command.
The doubled (or Scotch) rules that appear at the top and
bottom of a table can be achieved by enclosing the \texttt{tabular}
environment within a \texttt{ruledtabular} environment.
Rows whose columns span multiple columns can be typeset using \LaTeX's
\verb+\multicolumn{#1}{#2}{#3}+ command
(for example, see the first row of Table~\ref{tab:table3}).%
\begin{table*}
\caption{\label{tab:table3}This is a wide table that spans the page
width in \texttt{twocolumn} mode. It is formatted using the
\texttt{table*} environment. It also demonstrates the use of
\textbackslash\texttt{multicolumn} in rows with entries that span
more than one column.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
 &\multicolumn{2}{c}{$D_{4h}^1$}&\multicolumn{2}{c}{$D_{4h}^5$}\\
 Ion&1st alternative&2nd alternative&lst alternative
&2nd alternative\\ \hline
 K&$(2e)+(2f)$&$(4i)$ &$(2c)+(2d)$&$(4f)$ \\
 Mn&$(2g)$\footnote{The $z$ parameter of these positions is $z\sim\frac{1}{4}$.}
 &$(a)+(b)+(c)+(d)$&$(4e)$&$(2a)+(2b)$\\
 Cl&$(a)+(b)+(c)+(d)$&$(2g)$\footnote{This is a footnote in a table that spans the full page
width in \texttt{twocolumn} mode. It is supposed to set on the full width of the page, just as the caption does. }
 &$(4e)^{\text{a}}$\\
 He&$(8r)^{\text{a}}$&$(4j)^{\text{a}}$&$(4g)^{\text{a}}$\\
 Ag& &$(4k)^{\text{a}}$& &$(4h)^{\text{a}}$\\
\end{tabular}
\end{ruledtabular}
\end{table*}

The tables in this document illustrate various effects.
Tables that fit in a narrow column are contained in a \texttt{table}
environment.
Table~\ref{tab:table3} is a wide table, therefore set with the
\texttt{table*} environment.
Lengthy tables may need to break across pages.
A simple way to allow this is to specify
the \verb+[H]+ float placement on the \texttt{table} or
\texttt{table*} environment.
Alternatively, using the standard \LaTeXe\ package \texttt{longtable} 
gives more control over how tables break and allows headers and footers 
to be specified for each page of the table.
An example of the use of \texttt{longtable} can be found
in the file \texttt{summary.tex} that is included with the REV\TeX~4
distribution.

There are two methods for setting footnotes within a table (these
footnotes will be displayed directly below the table rather than at
the bottom of the page or in the bibliography).
The easiest
and preferred method is just to use the \verb+\footnote{#1}+
command. This will automatically enumerate the footnotes with
lowercase roman letters.
However, it is sometimes necessary to have
multiple entries in the table share the same footnote.
In this case,
create the footnotes using
\verb+\footnotemark[#1]+ and \verb+\footnotetext[#1]{#2}+.
\texttt{\#1} is a numeric value.
Each time the same value for \texttt{\#1} is used, 
the same mark is produced in the table. 
The \verb+\footnotetext[#1]{#2}+ commands are placed after the \texttt{tabular}
environment. 
Examine the \LaTeX\ source and output for Tables~\ref{tab:table1} and 
\ref{tab:table2}%
\begin{table}
\caption{\label{tab:table2}A table with more columns still fits
properly in a column. Note that several entries share the same
footnote. Inspect the \LaTeX\ input for this table to see
exactly how it is done.}
\begin{ruledtabular}
\begin{tabular}{cccccccc}
 &$r_c$ (\AA)&$r_0$ (\AA)&$\kappa r_0$&
 &$r_c$ (\AA) &$r_0$ (\AA)&$\kappa r_0$\\
\hline
Cu& 0.800 & 14.10 & 2.550 &Sn\footnotemark[1]
& 0.680 & 1.870 & 3.700 \\
Ag& 0.990 & 15.90 & 2.710 &Pb\footnotemark[2]
& 0.450 & 1.930 & 3.760 \\
Au& 1.150 & 15.90 & 2.710 &Ca\footnotemark[3]
& 0.750 & 2.170 & 3.560 \\
Mg& 0.490 & 17.60 & 3.200 &Sr\footnotemark[4]
& 0.900 & 2.370 & 3.720 \\
Zn& 0.300 & 15.20 & 2.970 &Li\footnotemark[2]
& 0.380 & 1.730 & 2.830 \\
Cd& 0.530 & 17.10 & 3.160 &Na\footnotemark[5]
& 0.760 & 2.110 & 3.120 \\
Hg& 0.550 & 17.80 & 3.220 &K\footnotemark[5]
&  1.120 & 2.620 & 3.480 \\
Al& 0.230 & 15.80 & 3.240 &Rb\footnotemark[3]
& 1.330 & 2.800 & 3.590 \\
Ga& 0.310 & 16.70 & 3.330 &Cs\footnotemark[4]
& 1.420 & 3.030 & 3.740 \\
In& 0.460 & 18.40 & 3.500 &Ba\footnotemark[5]
& 0.960 & 2.460 & 3.780 \\
Tl& 0.480 & 18.90 & 3.550 & & & & \\
\end{tabular}
\end{ruledtabular}
\footnotetext[1]{Here's the first, from Ref.~\onlinecite{feyn54}.}
\footnotetext[2]{Here's the second.}
\footnotetext[3]{Here's the third.}
\footnotetext[4]{Here's the fourth.}
\footnotetext[5]{And etc.}
\end{table}
for an illustration. 

All AIP journals require that the initial citation of
figures or tables be in numerical order.
\LaTeX's automatic numbering of floats is your friend here:
just put each \texttt{figure} environment immediately following 
its first reference (\verb+\ref+), as we have done in this example file. 

\begin{acknowledgments}
We wish to acknowledge the support of the author community in using
REV\TeX{}, offering suggestions and encouragement, testing new versions,
\dots.
\end{acknowledgments}


\section{Introduction}
The dynamics of a closed quantum system in complete isolation from its surroundings is governed
by the Schr{\"o}dinger equation of the system degrees of freedom only and, due to the properties 
of the Hilbert space description of quantum many particle systems, displays a level of complexity
that renders its simulation on a classical computer inefficient. Thus, the founding fathers of 
the field of quantum computation posited that a programmable but otherwise closed quantum device, 
a quantum computer, should be able to simulate this dynamics by harnessing the increased complexity 
of controlled quantum systems to our advantage~\cite{yuri1980computable,feynman1982simulating}. 
Thanks to the sustained research and development effort over the last 30 years we are now reaching 
a situation where the construction of ever more complex quantum information processors is becoming 
a reality. However, in practice interactions with uncontrolled environmental degrees of freedom, a.k.a. 
noise, are unavoidable and render present day quantum information processors unsuitable in principle 
for the efficient simulation of the pure state dynamics of isolated quantum systems. In fact, they are 
so noisy that even quantum error correction with an arbitrary overhead in physical resources would 
not yet be able to remove the effect of this noise.

Under the influence of intense noise, classical correlations tend to prevail, allowing for an accurate and efficient classical description.
But for a moderate level of noise, such as that present in today's quantum information processors, the situation is different because the full density matrix formalism needs to be adopted to achieve an accurate description of the open quantum system. 
While the effect of some environments may be captured accurately by Markovian master
equations~\cite{breuer2002theory,rivas2012open}, the situation is even more challenging for complex 
environments that display temporal correlations as now environmental degrees of freedom need to be
accounted for in some detail making the full system extremely large and challenging to simulate~\cite{tanimura1989time,prior2010efficient,strathearn2018efficient,tamascelli2019efficient,somoza2019dissipation,somoza2022driving,guimaraes2022efficient}.

This added complexity suggests that under suitable conditions and control, environmental noise on quantum 
systems may confer additional benefits, thus adding motivation to the development of methods to simulate 
the properties of noisy quantum systems. The effect of environmental degrees of freedom on the static and 
dynamical properties of a quantum system is especially important in cases, where the time and energy scales 
involved are likely to make interactions between the system and the surrounding environment a key actor in
their own right in the physics at play. Indeed, it was recognised early on in the development of quantum 
information processing that noise and dissipation may serve as a resource for robust state preparation~\cite{plenio1999cavity,plenio2002entangled}. More recently, the recognition that quantum dynamics may play
an important role in certain processes of life~\cite{mohseni2014quantum} led to the discovery that the
interplay of coherent quantum dynamics and environmental noise has the potential to impart fundamental
advantages on processes of life~\cite{plenio2008dephasing,mohseni2008environment,caruso2009highly,huelga2013vibrations}.
The electronic properties of bio-molecular complexes are exceedingly difficult to compute and it is 
well-recognized that such quantum chemistry challenges represent a promising application for 
quantum computers~\cite{mcardle2020quantum}. However, these algorithms typically treat closed static systems 
while capturing the properties of environmentally assisted quantum dynamics requires the extension of these
methods to open systems dynamics on a quantum information processor. This implies considerable, potentially 
crippling, overheads when pursued in the standard approaches using fully error corrected and thus noise-free
devices. 

These considerations raise the natural question whether it might be advantageous to subject a quantum 
information processor to carefully controlled internal noise sources in order to increase its efficiency
in simulating a desired open system dynamics. Indeed, very first steps in this direction have been taken 
in analog quantum simulators~\cite{lemmer2018trapped} where added noise increases the efficiency of the 
quantum simulation of open system dynamics compared to standard approaches~\cite{porras2008mesoscopic}.
Nevertheless, until now noise in the field of digital quantum computation and 
simulation has been regarded as \textit{detrimental}, to be fought by quantum error correction, and the 
question whether it can be leveraged for a particular quantum application has not been considered nor have
methods been developed to achieve it. Here we address this challenge by combining the capability of a 
Noisy Intermediate-Scale Quantum (NISQ) computer to execute noisy quantum gate sequences with methods from 
error mitigation and controlled addition of errors to reshape the noise intrinsic to the NISQ device in such a
manner that it models accurately the desired environmental noise of an open quantum system model 
of practical interest (see Fig.~\ref{fig:overview} for an overview of our scheme).

\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{overview.jpg}
\caption{Overview of the noise-assisted technique for quantum digital simulation of open quantum systems proposed in this work. We implemented a $k$th-order Trotter-Suzuki decomposition of the evolution operator $e^{-i\hat{H}t}$ in a quantum circuit, where a Trotter layer $\hat{U}_{k}(\Delta t)$ is repeated $D$ times. We characterized the stochastic Pauli noise channel $\mathcal{E}$ acting on the noisy Trotter layer $\mathcal{U}_{k}(\Delta t)$ when effectively implemented in the quantum circuit. The Hamiltonian $\hat{H}$ is firstly mapped into Pauli operators as per Eq.~\eqref{ham_pauli} and then used in the evolution operator as shown in Eq.~\eqref{trotter_equation}. Cycle Benchmarking and Error Reconstruction were the techniques used that allowed us to obtain the error probabilities of $\mathcal{E}$ acting over a Trotter layer. In the left box, we show an example of the output of the noise benchmarking technique in the form of a histogram (horizontal axis denotes the two-qubit Pauli errors and the vertical axis denotes the error probability of the respective interaction over a Trotter layer). Afterwards, we implement a noise-control step, where the stroboscopic (Markovian) noisy evolution performed in a quantum circuit for $D$ Trotter iterations can be approximately traced by a classically solved Lindblad equation. We demonstrate this in the top plot of the right box, where the time-evolution of the population terms of a two-qubit density matrix is displayed. By using Probabilistic Error Cancellation (PEC) and controlling the Trotter time-step $\Delta t$, one is able to vary the decoherence rates $\gamma_{k}$ emulated by the intrinsic Markovian noise of the quantum computer, given by the Pauli error probabilities $\epsilon_{k}$ (see section \ref{control_decoh}). While a smaller $\Delta t$ allows one to increase the simulated decoherence rate introduced in the open system, the mitigation factor $r_{k}$ (controled by PEC) provides us with the ability to mitigate the probability of a Pauli error $\hat{P}_{k}$ to occur in one Trotter layer for a fixed $\Delta t$, hence reducing the simulated decoherence rate $\gamma_{k}$. This is demonstrated in the bottom plot in the right box, where we mitigate all the noise ($r_{k}=1$ for all $k$) in the time-evolution circuit thus, in contrast to the top plot, decoherence is now suppressed by PEC.}
\label{fig:overview}
\end{figure*}

As a major benefit of this approach, the unavoidable decoherence of the quantum bits and quantum gates in a
NISQ device is used to replace some of the quantum computational resources, i.e. qubits and two-qubit quantum 
gates, that would otherwise be necessary to simulate faithfully the effect of noise in the desired open quantum
system model. Thus, the overall resource requirements for achieving the desired open system simulation are reduced
and the capabilities of the same device for executing more complex applications are extended. 
We exemplify our approach of turning a bug into a feature, with the explicitly worked out example of the 
\textit{simulation of the time-evolution of open quantum systems}. We show that it is possible to leverage 
noise processes, intrinsic and added noise, in a current NISQ device when the action of the desired environment 
to be simulated onto the open system can be approximated by a \emph{Markovian stochastic Pauli noise channel}~\cite{breuer2002theory}, but we stress that our methods are not restricted to this specific case.


This paper is organized as follows. In Sec.~II, we discuss the implementation of unitary time-evolution operator on a quantum circuit via the Trotter-
Suzuki product formula and the intrinsic noise of quantum devices. We briefly explain how the intrinsic noise can be transformed to stochastic Pauli noise channels by using Randomized Compiling~\cite{wallman2016noise,hashim2020randomized} and how the corresponding error probabilities can be estimated by Cycle Benchmarking~\cite{erhard2019characterizing} and Error Reconstruction~\cite{flammia2020efficient} techniques. We demonstrate that the Pauli noise channels with estimated error probabilities enable one to simulate open quantum system dynamics under a Lindblad-type noise on NISQ devices in such a way that only the open-system degrees of freedom are encoded in the qubits. We show that the open-system dynamics computed by NISQ devices can be well-described by classical solutions of the Lindblad equation constructed based on the estimated error probabilities of the Pauli noise channels. In Sec.~III, we discuss the conventional Probabilistic Error Cancellation technique~\cite{endo2018practical,sun2021mitigating}, developed to fully mitigate noise in a quantum circuit, and demonstrate that it can also be employed to partially mitigate the error probabilities of stochastic Pauli noise channels in a controlled manner. We show that our approach can be used to implement various Lindblad models with desired decoherence rates on NISQ devices.

\section{Encoding of the time evolution in a quantum circuit} \label{time_evol_sec}

For simulations of closed system dynamics, various methods have been developed, such as hybrid classical-quantum variational approaches~\cite{gibbs2022dynamical,mizuta2022local,berthusen2022quantum,cirstoiu2020variational}, quantum tensor networks~\cite{niu2022holographic,haghshenas2022variational,foss2021holographic}, and quantum signal processing techniques~\cite{low2019hamiltonian,martyn2021grand}. In this work, we consider the Trotter-Suzuki product formula \cite{childs2021theory,csahinouglu2021hamiltonian} suitable for closed-system simulations on NISQ devices~\cite{clinton2021hamiltonian,childs2018toward}. We demonstrate that it can also be used for efficient quantum simulations of open-system dynamics on a noisy quantum device.

\subsection{Trotter-Suzuki product formula}
The dynamics of a quantum system is governed by a Hamiltonian $\hat{H}$. We consider a Hamiltonian decomposed into $N$ components represented by the tensor products of Pauli matrices $\hat{X},\hat{Y},\hat{Z}$ of multiple qubits
\begin{equation}
    \hat{H} = \sum_{j=1}^{N}\hat{H}_{j},\quad \hat{H}_{j}=\alpha_{j} \hat{P}_{j}, \label{ham_pauli} 
\end{equation}
where $\alpha_{j} \in \mathbb{R}$ and $\hat{P}_{j}=\{\hat{X},\hat{Y},\hat{Z}\}^{\otimes n_j}$ is a Pauli string acting 
on $n_j$ qubits. As an example, we consider the Hamiltonian of a linear chain consisting of $n$ molecules
\begin{align}
    \hat{H} = -&\sum_{m=1}^{n}\frac{E_m}{2}\hat{Z}_m \label{eq:H_PPC}\\
    + &\sum_{m=1}^{n-1}\frac{J_{m,m+1}}{2}(\hat{X}_m \hat{X}_{m+1} + \hat{Y}_m \hat{Y}_{m+1}),
    \nonumber 
\end{align}
where each qubit describes a two-level molecule consisting of ground and excited states, encoded by $\ket{0}$ and $\ket{1}$, respectively, with an energy-gap of $E_m$. The inter-qubit couplings $J_{m,m+1}$ describe a coherent excitation transfer between nearest-neighbour molecules. This model has been widely used to study energy transfer dynamics in various molecular systems, such as photosynthetic complexes and organic solar cells, but a more general form of Hamiltonian can be considered in a noise-assisted digital quantum simulation, including spin and fermionic models, such as the Heisenberg~\cite{guimaraes2022towards,tran2021faster} and Fermi-Hubbard models~\cite{clinton2021hamiltonian,arute2020observation}, respectively.

The product formula describing the time evolution of a quantum system over time $t$ is formally expressed as
\begin{equation}
    e^{-i\hat{H}t} \approx \prod_{d=1}^{D} \hat{U}_{k}(\Delta t), \label{trotter_equation_anyorder}
\end{equation}
where $\hat{U}_{k}(\Delta t)$ represents the unitary time evolution over a finite Trotter time-step $\Delta t$, described by the $k$-th order Trotter-Suzuki product formula, where the first two lowest order examples are given by 
\begin{align}
    \hat{U}_{1}(\Delta t)&=\prod_{j=1}^{N} e^{-i \hat{H}_{j}\Delta t},\label{trotter_equation}\\
    \hat{U}_{2}(\Delta t)&=\left[ \prod_{j=1}^{N} e^{-i \hat{H}_{j}\Delta t/2} \right]\left[\prod_{j'=N}^{1} e^{-i \hat{H}_{j'}\Delta t/2}\right].\label{trotter_equation2}
\end{align}
The total time evolution is then described by $D = t/\Delta t$ Trotter layers where $\hat{U}_{k}(\Delta t)$ is considered in each layer.

An estimate for the number of implemented
Trotter layers $D=t/\Delta t$ required to obtain a Trotter decomposition error $||\hat{U}^{D}_{k}(\Delta t)-e^{-i\hat{H}t}||=O(\varepsilon_{\rm Trot})$ can be given as follows \cite{childs2021theory},
\begin{equation}
    D = O\left(\frac{\alpha_{\rm comm}^{1/k} t^{1+1/k}}{\varepsilon_{\rm Trot}^{1/k}}\right)\label{error_trot}
\end{equation}
for a $k$th-order Trotter-Suzuki product formula, where, $$\alpha_{\rm 
 comm}=\sum_{l_{1},l_{2},\dots,l_{k+1}=1}^{N}||[\hat{H}_{l_{k+1}},\dots [\hat{H}_{l_{2}},\hat{H}_{l_{1}}]\dots]||,$$ and $||\cdot||$ denotes the spectral norm. $\hat{H}_{l_{k}}$ denotes a Pauli string term out of a total of $N$ terms the Hamiltonian is decomposed into (see Eq.~\eqref{ham_pauli}).

We note that variants of Trotter-Suzuki product formulae can also be considered in our approach to reduce simulation error, such as symmetry-protected formulae~\cite{tran2021faster}, random formulae~\cite{chen2021concentration}, and implementation of a specific Trotter sequence of Hamiltonian terms that preserves the locality of the simulated system~\cite{childs2021theory}.

\subsection{Noise characterization}\label{sec: noisy_evol}
On NISQ devices, the implementation of each unitary gate of a Trotterized time-evolution operator, such as $e^{-i\hat{H}_j \Delta t}$ in Eq.~\eqref{trotter_equation}, suffers from noise and dissipation. As a result, the time evolution of a quantum system encoded in the qubits does not only depend on the Hamiltonian implemented on a quantum circuit, but also on the parameters that characterize the intrinsic noise of the quantum device. When its noise can be well-described by a Markovian theory, as demonstrated in previous studies on superconducting quantum computing platforms~\cite{berg2022probabilistic,ferracin2022efficiently}, the dynamics of the density matrix $\hat{\rho}(t)$ of the open system encoded on the qubits is described by a Markovian quantum master equation in the form
\begin{equation}
    \frac{d\hat{\rho}(t)}{dt} = \mathcal{L}[\hat{\rho}(t)] = -i[\hat{H},\hat{\rho}(t)]+\mathcal{D}_{\rm intrinsic}[\hat{\rho}(t)], \label{Lindblad_eq}
\end{equation}
where $\mathcal{D}_{\rm intrinsic}[\hat{\rho}(t)]$ represents a Lindblad dissipator describing the intrinsic noise of NISQ devices.

In this work, we aim to implement the decoherence of open quantum systems modelled by a Lindblad equation by harnessing the intrinsic noise of NISQ devices as a resource, rather than encoding environmental degrees of freedom on qubits. Therefore, to simulate open-system dynamics under various decoherence models of interest, we need to characterize the noise channels present in NISQ devices and control the corresponding noise rates. To that end, we apply a noise characterization technique to the noisy implementation $\mathcal{U}$ of the Trotter layer on a quantum circuit (we omitted the $k$-th order and $\Delta t$ dependence of $\mathcal{U}_{k}(\Delta t)$ for simplicity), prior to the implementation of the digital quantum simulation of open-system dynamics. We employ the Randomized Compiling technique~\cite{wallman2016noise,hashim2020randomized} to transform the intrinsic coherent noise in a NISQ device, described by the Kraus operators in the form of $\sum_{j\neq k}\epsilon_{jk}\hat{P}_{j} \hat{\rho} \hat{P}_{k}$, to stochastic Pauli noise channels $\sum_{k}\epsilon_k \hat{P}_{k} \hat{\rho} \hat{P}_{k}$, where $\hat{P}_{k}$ are Pauli strings, including identity operator, and $\epsilon_k$ are the corresponding error probabilities satisfying $\sum_{k}\epsilon_{k}=1$ and $\epsilon_k\ge 0$. Randomized Compiling is implemented in both noise characterization and digital quantum simulations of open-system dynamics on real IBMQ devices, as explained below.

We employ the Cycle Benchmarking~\cite{erhard2019characterizing}, one of the Randomized Benchmarking methods~\cite{helsen2022general}, together with the Error Reconstruction technique~\cite{flammia2020efficient} to estimate the error probabilities $\epsilon_k$ of quantum circuits on NISQ devices, as recently demonstrated in Ref.~\cite{hashim2020randomized,ferracin2022efficiently,berg2022probabilistic}. This enables the characterization of $K$-qubit Pauli noise channels acting on $K$ nearest-neighbour qubits. For instance, when $K=1$, the estimated stochastic Pauli noise channels acting on a single qubit $m$ are expressed as
\begin{equation}
    {\cal E}_{m}^{(1)}(\hat{\rho})=\epsilon_0 \hat{\rho} + \epsilon_X \hat{X}_{m}\hat{\rho}\hat{X}_{m} + \epsilon_Y \hat{Y}_{m}\hat{\rho}\hat{Y}_{m} + \epsilon_Z \hat{Z}_{m}\hat{\rho}\hat{Z}_{m}. \label{eq:E_K1}
\end{equation}
For $K=2$, the stochastic Pauli noise channels acting on two nearest-neighbour qubits $m$ and $m+1$ consist of the single-qubit noise in Eq.~\eqref{eq:E_K1}, independently acting on each qubit, and the correlated two-qubit noise acting on two qubits at the same time
\begin{align}
    {\cal E}_{m,m+1}^{(2)}(\hat{\rho})&={\cal E}_{m}^{(1)}(\hat{\rho})+{\cal E}_{m+1}^{(1)}(\hat{\rho})\\
    &\quad+\epsilon_{XX}\hat{X}_m\hat{X}_{m+1}\hat{\rho}\hat{X}_m\hat{X}_{m+1},\nonumber\\
    &\quad+\epsilon_{XY}\hat{X}_m\hat{Y}_{m+1}\hat{\rho}\hat{X}_m\hat{Y}_{m+1}+\cdots,\nonumber
\end{align}
where the latter takes into account all possible combinations of the Pauli operators of qubits $m$ and $m+1$. Formally, the $K$-qubit noise channels are expressed as
\begin{equation}
    \mathcal{E}^{(K)}(\rho)=\sum_{k=0}^{4^{K}-1}\epsilon_{k} \hat{P}_{k}\hat{\rho}\hat{P}_{k},\label{stochastic_channel}
\end{equation}
with $\hat{P}_k$ denoting Pauli strings. For a quantum circuit consisting of $n$ qubits, when $K=n$, all possible Pauli strings are considered in the noise characterization. When $K<n$, the $K$-qubit Pauli noise channels can be characterized independently for each subgroup consisting of $K$ nearest-neighbour qubits.

We remark that the Cycle Benchmarking technique can be employed to characterize the noise acting on a quantum circuit $\hat{U}$ satisfying $\hat{U}^{m}=\hat{I}$ for several integer values $m$. The Trotter layer $\hat{U}_{k}(\Delta t)$ constructed based on an open-system Hamiltonian, however, does not satisfy this condition in general. In this work, we modified the parameters of single-qubit gates $\hat{R}_Z(\theta)=e^{-i\theta \hat{Z}/2}$ of $\hat{U}_{k}(\Delta t)$, while maintaining its two-qubit gate structure, yielding a Clifford circuit $\hat{V}$, so that the Cycle-Benchmarking condition $\hat{V}^{m}=\hat{I}$ is satisfied for every even integer $m$. Since the degree of noise of two-qubit gates is approximately two orders of magnitude higher than that of single-qubit gates on superconducting quantum devices, this approach enables one to estimate the error probabilities $\epsilon_k$ of stochastic Pauli noise acting on the original Trotter layer $\hat{U}_{k}(\Delta t)$ in an accurate manner, as demonstrated below.

\subsection{Digital quantum simulation of Lindblad model}

\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{noise_char.jpg}
\caption{(a) Example of a characterized two-qubit stochastic Pauli noise channel for the quantum simulation shown in (b). Vertical and horizontal axes denote error probabilities $\{\epsilon_{k}\}$ and their respective two-qubit Pauli strings $\{\hat{P}_{k}\}$, respectively, and red bars denote standard deviation. Average negative error probabilities within the standard deviation are taken to be $0$. Dark blue bars denote the expected highest errors on IBMQ devices, namely, single-qubit and $\hat{Z}\hat{Z}$ Pauli errors. (b) Measured population terms of the density matrix of a two-qubit quantum simulation under stroboscopic noisy evolution on the quantum computer \textit{ibmq jakarta} shown as circles and under a classically computed Trotterized evolution of equation \eqref{eq:gamma_stochastic_uncontrolled} shown by crosses. (c) Same as (b) but we measured the inter-qubit coherence term on the emulated quantum computer \textit{ibmq jakarta} as explained in the main text. (d) Same as (b), but for a 1D system of 4 qubits, where we measured the population terms of the reduced density matrix of the second and third qubits of a 1D qubit array on the quantum computer \emph{ibmq lagos}. We denote qubit states in the computational basis  as $\ket{g} \leftrightarrow \ket{0}$ and $\ket{e} \leftrightarrow \ket{1}$. The classically computed Trotterized Lindblad equation \eqref{eq:gamma_stochastic_uncontrolled} is constructed using the error probabilities obtained by the noise benchmarking technique. For sake of clarity, we only show the error probabilities in (a) for the quantum simulation in (b). Randomized Compiling was used in the noise characterization routines and quantum simulations for the results obtained on real IBMQ devices, i.e. plots (a) and (d), with $R=30$ and $R=22$ randomized compiled circuits, respectively. We fixed the Hamiltonian parameters in Eq.~\eqref{eq:H_PPC} to be $E_{m}=2\pi(122-0.5m), \quad m=0,1,
\dots,n$ (i.e. rescaled values of the typical photosynthetic systems have values of around $12200-50m$ $cm^{-1}$ 
\cite{mohseni2014quantum}) and 
$J_{mm'}=\pi$ (equivalently to $50$ $cm^{-1}$ after rescaling). The 
initial state of the open system is chosen as $\ket{\Psi(0)}=\ket{1}_{1}\ket{0}_{2}\dots \ket{0}_{n}$.}
\label{fig:evolve_normal}
\end{figure*}

\begin{figure}
\[\Qcircuit @C=0.8em @R=0.6em{ 
\text{(a)} \\
\lstick{\ket{q_{1}}} & \qw & \multigate{1}{e^{-iJ_{12}\hat{X}\hat{X}\Delta t/2}} & \qw & \multigate{1}{e^{-iJ_{12}\hat{Y}\hat{Y}\Delta t/2}} & \qw & \gate{e^{iE_{1}\hat{Z}\Delta t/2}} & \qw\\
\lstick{\ket{q_{2}}}& \qw & \ghost{e^{-iJ_{12}\hat{X}\hat{X}\Delta t/2}} & \qw & \ghost{e^{-iJ_{12}\hat{Y}\hat{Y}\Delta t/2}} & \qw &\gate{e^{iE_{2}\hat{Z}\Delta t/2}}  & \qw\\
}\]

\[\Qcircuit @C=0.8em @R=0.6em{ 
\text{(b)} \\
\lstick{\ket{q_{1}}} & \qw & \multigate{1}{e^{-i\hat{H}_{odd}\Delta t}} & \qw & \qw & \qw & \gate{e^{-i\hat{H}_{sing}\Delta t}} & \qw\\
\lstick{\ket{q_{2}}}& \qw & \ghost{e^{-i\hat{H}_{odd}\Delta t}} & \qw & \multigate{1}{e^{-i\hat{H}_{even}\Delta t}} & \qw & \gate{e^{-i\hat{H}_{sing}\Delta t}} & \qw\\
\lstick{\ket{q_{3}}}& \qw & \multigate{1}{e^{-i\hat{H}_{odd}\Delta t}} & \qw & \ghost{e^{-i\hat{H}_{even}\Delta t}} & \qw& \gate{e^{-i\hat{H}_{sing}\Delta t}} & \qw\\
\lstick{\ket{q_{4}}}& \qw &\ghost{e^{-i\hat{H}_{odd}\Delta t}} & \qw & \qw & \qw & \gate{e^{-i\hat{H}_{sing}\Delta t}} & \qw\\
}\]
\centering
    \caption{(a) Implementation of a Trotter layer circuit with the Hamiltonian shown in Eq.~\eqref{eq:H_PPC} for $n=2$ qubits. (b) Same as (a), but for $n=4$ qubits. $\hat{H}_{odd}$ ($\hat{H}_{even}$) represent terms of the Hamiltonian, which couple the nearest-neighbour qubits $q_{i}$ and $q_{i+1}$ with $i$ odd (even), i.e. $J_{i,i+1}\hat{X}\hat{X}/2$ and $J_{i,i+1}\hat{Y}\hat{Y}/2$ applied in the same order as shown in (a). $\hat{H}_{sing}$ contains the single-qubit terms of the Hamiltonian.}
    \label{fig:trotter_circuit}
\end{figure}

To demonstrate that the estimated error probabilities $\epsilon_k$ of the stochastic Pauli noise channels can be used to investigate open-system dynamics under a Lindblad-type noise, we consider a Markovian quantum master equation in the form
\begin{equation}
    \frac{d\hat{\rho}(t)}{dt} = \mathcal{L}[\hat{\rho}(t)] = -i[\hat{H},\hat{\rho}(t)]+\mathcal{D}_{\rm stochastic}[\hat{\rho}(t)],\label{eq:L_stochastic_uncontrolled}
\end{equation}
where the Lindblad dissipator $\mathcal{D}_{\rm stochastic}[\hat{\rho}(t)]$ is modelled by
\begin{align}
    \mathcal{D}_{\rm stochastic}[\hat{\rho}(t)] &= \sum_{k=0}^{4^K-1}\gamma_k \left(\hat{P}_k\hat{\rho}(t)\hat{P}_k - \hat{\rho}(t)\right),\label{eq:D_stochastic_uncontrolled}\\
    \gamma_k &= \epsilon_k/\Delta t.\label{eq:gamma_stochastic_uncontrolled}
\end{align}
Here the decoherence rates $\gamma_k$ are defined as a function of the error probabilities $\epsilon_k$ and the Trotter time-step $\Delta t$. 

In the following, we demonstrate by means of an example that the dynamics of a reduced system density matrix $\hat{\rho}_{c}(t)$ simulated by using Eq.~\eqref{eq:L_stochastic_uncontrolled}--\eqref{eq:gamma_stochastic_uncontrolled} on classical computers is well-matched to its counterpart $\hat{\rho}_{q}(t)$ obtained from real and emulated quantum computers, where the system Hamiltonian is implemented by using the first-order Trotter-Suzuki formula in Eq.~\eqref{trotter_equation} and the noise characteristics have been obtained as described above. We note that the Randomized Compiling is applied to every Trotter layer in our approach, so that the stochastic Pauli noise channels, identified by the noise characterization scheme presented in Sec.~\ref{sec: noisy_evol}, are maintained during the digital quantum simulation of open-system dynamics (see Appendix~\ref{A} for more details).

In Fig.~\ref{fig:evolve_normal}(a) and (b), we consider a quantum system encoded on two qubits ($n=2$) modelled by the Hamiltonian in Eq.~\eqref{eq:H_PPC}. The structure of the implemented quantum circuit is shown in Fig.~\ref{fig:trotter_circuit}(a). For a real IBMQ computer, we estimated the error probabilities $\epsilon_k$ of the full stochastic Pauli noise channel ($K=2$). As shown in Fig.~\ref{fig:evolve_normal}(a), we found that the error probabilities $\epsilon_k$ are not uniform with relatively higher values for $\hat{P}_{k}\in\{\hat{X}_{1},\hat{Y}_{1},\hat{Z}_{1},\hat{Z}_{2},\hat{X}_1\hat{Y}_2,\hat{Y}_1\hat{Z}_2,\hat{Z}_1\hat{Z}_2\}$, hinting that qubit 1 is more noisy than qubit 2. As shown in Fig.~\ref{fig:evolve_normal}(b), the population dynamics of the reduced system density matrices $\bra{i,j}\hat{\rho}_{q,c}(t)\ket{i,j}$ simulated by quantum and classical computers are well-matched for all possible values of $i,j\in\{0,1\}$. It is found that when only single-qubit Pauli noise channels are characterized ($K=1$), the quantum and classical results are not matched (not shown here), which can be rationalized based on the fact that the error probabilities of two-qubit Pauli noise channels, such as $\hat{P}_{k}\in\{\hat{X}_1\hat{Y}_2,\hat{Y}_1\hat{Z}_2,\hat{Z}_1\hat{Z}_2\}$, are not negligible. Fig.~\ref{fig:evolve_normal}(c) shows that the inter-qubit coherence dynamics $\bra{10}\hat{\rho}_{q,c}(t)\ket{01}$ are also well-matched, where the emulated noisy quantum device \emph{ibmq jakarta} was used instead of the real IBMQ device due to the long queue waiting time on the IBMQ platform. In Fig.~\ref{fig:evolve_normal}(d), we consider a larger quantum system encoded on four qubits ($n=4$) with a circuit structure shown in Fig.~\ref{fig:trotter_circuit}(b). It is found that the reduced system dynamics $\hat{\rho}_{q}(t)$ computed by the real IBMQ computer are well-matched to the classical solutions $\hat{\rho}_{c}(t)$ of the Lindblad equation when also two-qubit stochastic Pauli noise channels are characterized $(K=2)$. This implies that the Pauli noise channels acting on more than two qubits are negligible in our case due to the short depth of the Trotter layer structure shown in Fig.~\ref{fig:trotter_circuit}(b). We note that the first-order Trotter decomposition is considered in both quantum and classical simulations to investigate the accuracy of the noise characterization scheme considered in our work, independent of the Trotter decomposition error induced by a finite Trotter time-step $\Delta t$. We performed all the simulations with a sufficiently small $\Delta t$, for which classical solutions $\hat{\rho}_{c}(t)$ can also be obtained by using a standard RK4-based solver. These results demonstrate that the intrinsic noise of NISQ devices can be transformed to the stochastic Pauli noise channels via Randomized Compiling, which can be used as a platform to simulate open-system dynamics under a Lindblad-type noise.

\section{Digital control of decoherence in a quantum circuit}\label{control_decoh}

So far we have demonstrated that the intrinsic noise of quantum devices can be transformed to stochastic Pauli noise channels and the corresponding error probabilities $\epsilon_k$ can be estimated, enabling one to implement a Lindblad equation on quantum computers with decoherence rates $\gamma_k=\epsilon_k/\Delta t$ (see Eq.~\eqref{eq:L_stochastic_uncontrolled}--\eqref{eq:gamma_stochastic_uncontrolled}). Now we show that the error probabilities of the Pauli noise channels can be controlled independently by generalizing the Probabilistic Error Mitigation (PEC) technique~\cite{endo2018practical,sun2021mitigating}, leading to reduced error probabilities $\epsilon_k (1-r_k)$ with independent partial mitigation factors $r_k \in [0,1]$. This approach makes it possible to implement an arbitrary set of target decoherence rates $\Gamma_k$ on NISQ devices, so that one can investigate open-system dynamics described by a Lindblad equation of interest in the following form
\begin{align}
    \mathcal{D}_{\rm stochastic}^{\rm (controlled)}[\hat{\rho}(t)] &= \sum_{k=0}^{4^K-1}\Gamma_k \left(\hat{P}_k\hat{\rho}(t)\hat{P}_k - \hat{\rho}(t)\right),\label{eq:D_stochastic_controlled}\\
    \Gamma_k &=\epsilon_k (1-r_k)/\Delta t.\label{eq:gamma_stochastic_controlled}
\end{align}
We note that our method can be modified to implement more general Lindblad noise models beyond the stochastic Pauli noise channels in Eq.~\eqref{eq:D_stochastic_controlled}, such as amplitude damping noise as will be discussed later.

\subsection{Probabilistic Error Cancellation}

PEC starts by identifying the noise channel $\mathcal{E}$ acting on 
an ideal, noiseless circuit $\mathcal{C}(\hat{\rho})=\hat{U}_{k}(\Delta t)\hat{\rho}\hat{U}_{k}^{\dagger}(\Delta t)$, for instance, describing the Hamiltonian dynamics of a quantum system over a Trotter time-step $\Delta t$. As discussed in Sec.~\ref{sec: noisy_evol}, one can characterize the noise of a quantum circuit, yielding $K$-qubit stochastic Pauli noise channels, $\mathcal{E}(\hat{\rho})=\sum_{k=0}^{4^{K}-1}\epsilon_{k}\mathcal{P}_{k}(\hat{\rho})=\sum_{k=0}^{4^{K}-1}\epsilon_{k} \hat{P}_{k}\hat{\rho} \hat{P}_{k}$ with $\hat{P}_{0}=\hat{I}^{\otimes K}$ denoting identity operator. To fully mitigate the noise, the conventional PEC has considered the inverted noise channel $\mathcal{E}^{-1}$ acting on the noisy quantum circuit, namely $\mathcal{E}^{-1}$ applied to $\mathcal{U}(\hat{\rho}) = \mathcal{E}\cdot\mathcal{C}(\hat{\rho})$. Since $\mathcal{E}^{-1}$ is not a complete-positive (CP) map~\cite{endo2018practical,sun2021mitigating}, one cannot physically implement it, hence one cannot perfectly cancel the stochastic Pauli noise channels in $\mathcal{U}$. However, when the error probabilities $\epsilon_k$ are sufficiently low, the conventional PEC technique can fully mitigate the noise to first order as $\mathcal{C}(\hat{\rho})\approx \mathcal{E}^{-1}\mathcal{U}(\hat{\rho})$ by implementing the non-physical inverted noise channel in a probabilistic manner.

When $K$-qubit Pauli noise channels act on a quantum circuit consisting of $K$ qubits ($n=K$), the inverted noise channel is written as~\cite{endo2018practical,sun2021mitigating}
\begin{align}  \mathcal{E}^{-1}=\sum_{k=0}^{4^{K}-1}q_{k}\mathcal{P}_{k}=C_{\rm mit}\sum_{k=0}^{4^{K}-1}p^{\rm (PEC)}_{k}\text{sign}(q_{k})\mathcal{P}_{k}, \label{eq:PEC}
\end{align}
where $q_{0}=1+\sum_{k>0}\epsilon_{k}$, $q_{k>0}=-\epsilon_{k}$, $p_{k}^{\rm (PEC)} = |q_{k}|/C_{\rm mit}$ with $C_{\rm mit}=\sum_{k=0}|q_{k}|$ called the mitigation cost. In the probabilistic application of the non-CP map $\mathcal{E}^{-1}$, one of the Pauli operators $\hat{P}_{k}$ is randomly chosen based on the probabilities $p_{k}^{\rm (PEC)}$ and then applied to a Trotter layer. For a quantum circuit with $D=t/\Delta t$ Trotter layers, the probabilistic non-CP map is applied $D$ times with the Pauli operators $\hat{P}_{k}$ independently sampled for each Trotter layer, as schematically shown in Fig.~\ref{fig:PEC}(a).

\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{PEC.jpg}
\caption{(a) Example of PEC applied to a Trotter time-evolution circuit with $n=K$ encoded qubits. A $n$-qubit stochastic Pauli channel is randomly sampled from a probability distribution given by $p_{k}^{\rm (PEC)}, \quad k = {1,\dots,4^{K}-1}$, independently for each $d$-th Trotter layer. The channel is then applied after each $\mathcal{U}$ on the $n$ qubits. (b) Example of PEC applied to a Trotter time-evolution circuit with $n>K$ encoded qubits, where $K=2$. A two-qubit stochastic Pauli channel $\mathcal{P}_{i,k}$ is randomly sampled from a probability distribution given by $\{p_{k}^{\rm (PEC)}\}$, where $k = 1,\dots,15$, independently for the $i$-th pair of nearest-neighbour qubits and the $d$-th Trotter layer. Afterwards, the sampled Pauli operator is applied after each $\mathcal{U}$ on $K=2$ qubits. We considered the open system model to be encoded on an one-dimensional array of qubits for simplicity, however the application of PEC can be easily generalized to higher-dimensional systems.}
\label{fig:PEC}
\end{figure}

PEC can also be applied to a quantum circuit consisting of $n$ qubits where $K$-qubit Pauli noise channels with $K<n$ act on several subgroups consisting of $K$ qubits. Here the noise mitigation can be implemented by considering multiple inverted noise channels acting on different subgroups. Fig.~\ref{fig:PEC}(b) shows an example of $n>K=2$ where the inverted noise channels act on every pair of nearest-neighbour qubits. We note that when different $K$-qubit Pauli noise channels act on the qubits in the intersection of different subgroups, one needs to adjust the mitigation probabilities $p_{k}^{\rm (PEC)}$ in such a way that the noise acting on the shared qubits is not cancelled multiple times. 

In PEC, the outcome of an observable $\hat{O}$ measured on a noise-mitigated quantum circuit is multiplied by a product of mitigation costs and other prefactors in Eq.~\eqref{eq:PEC}, $\prod_{d=1}^{D}\prod_{m}C^{(m)}_{\rm mit}\text{sign}(q_{k}^{(m,d)})$, where $m$ describes different subgroups of qubits under the action of $K$-qubit Pauli noise channels. The value of $\text{sign}(q_{k}^{(m,d)})$ depends on which Pauli operator $\hat{P}_k$ is randomly sampled in the $d$-th Trotter layer. The total mitigation cost of quantum simulation is defined as
\begin{equation}
    C_{\rm tot}=\prod_{d=1}^{D}\prod_{m}C^{(m)}_{\rm mit}.\label{cost_definition}
\end{equation}
For a noise-mitigated density matrix $\hat{\rho}_{\rm mit}(t)$, the expectation value $\text{Tr}[\hat{O}\hat{\rho}_{\rm mit}(t)]$ of the observable $\hat{O}$ is obtained by classically averaging the outcomes of the PEC scheme, requiring multiple copies of quantum circuits.

\subsection{Decoherence rate control scheme}\label{sec:decoherence_control}

So far we have discussed the conventional PEC that aims to fully mitigate the stochastic Pauli noise channels~\cite{endo2018practical,sun2021mitigating}. Contrary to the previous studies on PEC, here we aim to partially cancel the noise, so that the controlled error probabilities of the Pauli noise channels can be used as a resource for open-system simulations. To that end, we consider $q_{0}=1+\sum_{k>0}r_{k}\epsilon_{k}$ and $q_{k>0}=-r_{k}\epsilon_{k}$ with $r_k \in [0,1]$, renormalizing the probabilities of the PEC scheme, namely $p_{k}^{\rm (PEC)} = |q_{k}|/C_{\rm mit}$ with $C_{\rm mit}=\sum_{k=0}|q_{k}|$. The corresponding {\it partially} inverted noise channel enables one to reduce the Pauli error probabilities in a controlled manner, $\epsilon_k\rightarrow (1-r_k)\epsilon_k$. The reduced error probabilities make it possible to implement the Lindblad equation in Eq.~\eqref{eq:D_stochastic_controlled} on NISQ devices with controlled decoherence rates $\Gamma_k=\epsilon_k(1-r_k)/\Delta t$.

For a given set of target decoherence rates $\Gamma_{k}$, our scheme works as follows. Before our PEC scheme is applied to a quantum circuit, for a given Trotter time-step $\Delta t$, the decoherence rates $\gamma_k=\epsilon_k/\Delta t$ of the stochastic Pauli noise channels may satisfy $\Gamma_{k}>\gamma_k$ for some $k$. Since the error probabilities $\epsilon_k$ of the Pauli noise channels can be decreased but not increased by our PEC scheme, namely $\epsilon_k\rightarrow (1-r_k)\epsilon_k$ with $r_k \in [0,1]$, the decoherence rates $\gamma_k$ implemented on the quantum device can be increased only by reducing the Trotter time-step $\Delta t$. Hence we decrease $\Delta t$ until $\Gamma_{k}\le \gamma_{k}=\epsilon_{k}/\Delta t$ is satisfied for all $k$ (see Fig.~\ref{fig:mitigation_diagram}(a)). Here one can find a range of $\Delta t \in (0,\Delta t_{\rm max}]$ satisfying this condition and take the maximum value $\Delta t_{\rm max}$ to minimize the number $D=t/\Delta t_{\rm max}$ of Trotter layers. After that, for some $k$ where the target decoherence rates are lower than the implemented decoherence rates, namely $\Gamma_{k}<\tilde{\Gamma}_{k}=\epsilon_k/\Delta t_{\rm max}$, we apply our PEC scheme to partially mitigate the corresponding error probabilities $\epsilon_{k}$, so that $\Gamma_{k}=\epsilon_k (1-r_k)/\Delta t_{\rm max}$ holds for all $k$ (see Fig.~\ref{fig:mitigation_diagram}(b)). In this way, one can implement arbitrary target decoherence rates $\Gamma_k$, in principle, on NISQ devices.

We note that the error probabilities $\epsilon_k$ of real quantum devices are not uniform, as shown in Fig.~\ref{fig:evolve_normal}(a). This implies that even a simple decoherence model with uniform target decoherence rates $\Gamma_k$ requires our {\it noise-specific} mitigation scheme with non-uniform values of $r_k$. In addition, the control of the Trotter time-step $\Delta t$ alone is not sufficient to implement arbitrary target decoherence rates, especially when $\Gamma_k$ are small, as a larger $\Delta t$ results in a higher Trotter decomposition error. This is indeed the main issue of Ref.~\cite{leppakangas2022quantum}.

\begin{figure}
\centering
\includegraphics[width=0.47\textwidth]{mitigation_diagram.jpg}
\caption{Decoherence control scheme proposed in our work. In step (a), we increase the Trotter time-step from $\Delta t$ to $\Delta t_{max}$, such that all decoherence rates $\gamma_{k}=\epsilon_{k}/\Delta t$, for some initially chosen $\Delta t$, are increased to $\tilde{\Gamma}_{k}=\epsilon_{k}/\Delta t_{max}\geq \Gamma_{k}$, where $\Gamma_{k}$ are the target decoherence rates to be simulated. In step (b), we use PEC to decrease arbitrary decoherence rates $\tilde{\Gamma}_k$ by partially mitigating the error probability of a specific Pauli string $k$, i.e. $\epsilon_{k}$ is reduced to $(1-r_{k})\epsilon_{k}$. By doing this, $\tilde{\Gamma}_{k}$ approaches the target decoherence rates $\Gamma_{k}=(1-r_{k})\epsilon_{k}/\Delta t_{max}$ as desired.}
\label{fig:mitigation_diagram}
\end{figure}

\subsection{Implementation} \label{sec: implementation}

\begin{figure*}
\centering
\includegraphics[width=0.8\textwidth]{implementation.jpg}
\caption{(a,b) Results of the noise-controlled quantum digital simulation of an open system computed on the emulated noisy IBMQ device (dots) and solved by the Lindblad equation given by Eq.~\eqref{eq:gamma_stochastic_controlled}. We chose uniform mitigation factors $r=0.2$ and $r=0.8$ for plots (a) and (b), respectively. For each data point $d$, $160C_{tot}^{2}$ circuits were executed in the PEC scheme with $C_{tot}$ calculated with $r=0.2$ in plot (a) and $r=0.8$ on plot (b) as a function of $d$ (see Eq.~\eqref{Ctot_simple}). (c,d) Error of the quantum digital simulations as a function of time with uniform mitigation factors $r=0.2, 0.8$. (d)  Variance of the results obtained in (c) as a function of time. For each data point $d$ in plots (c,d), $16C_{tot}^{2}$ circuits were classically executed using PEC, where $C_{tot}$ was calculated with $r=0.8$ for both quantum simulations as a function of $d$ (see Eq.~\eqref{Ctot_simple}). The Hamiltonian used is the same as for Fig.~\ref{fig:evolve_normal} and the implemented circuit is shown in Fig.~\ref{fig:trotter_circuit}(a).}
\label{fig:error_implementation}
\end{figure*}

To demonstrate that our partial error mitigation scheme can be used to implement the Lindblad equation in Eq.~\eqref{eq:D_stochastic_controlled} on NISQ devices with controlled decoherence rates in Eq.~\eqref{eq:gamma_stochastic_controlled}, here we consider a quantum system encoded on two qubits ($n=2$). For simplicity, we consider a uniform mitigation factor $r=r_k$ for all $k$. We used the emulated noisy IBMQ device \emph{ibmq lagos} as our testbed.

For two different mitigation factors $r=0.2$ and $r=0.8$, respectively, Fig.~\ref{fig:error_implementation}(a) and (b) show the population dynamics $\bra{i,j}\hat{\rho}_{q}(t)\ket{i,j}$ with $i,j\in\{0,1\}$ simulated by the emulated quantum computer, which are well-matched to the solutions $\hat{\rho}_{c}(t)$ of the Lindblad equation solved on classical computers by a RK4-based solver (coherence dynamics are also well-matched, not shown here). As expected, the open-system dynamics becomes more coherent with a slower decay of oscillations for a larger mitigation factor $r$. Here the number of samples considered in our generalized PEC scheme is increased until the quantum and classical results are well-matched, and it is found that a larger number of samples is required for a higher mitigation factor $r$.

To demonstrate the dependence of the sampling cost of our PEC scheme on the mitigation factor $r$, Fig.~\ref{fig:error_implementation}(c) shows the difference in population dynamics simulated by quantum and classical computers, quantified by
\begin{equation}
  \eta(\hat{\rho}_{q}(t),\hat{\rho}_{c}(t)) = \frac{1}{4}\sum_{i=0}^{1}\sum_{j=0}^{1}|\bra{i,j}\hat{\rho}_{q}(t)-\hat{\rho}_{c}(t)\ket{i,j}|,\label{error_eq_impl}
\end{equation}
for the mitigation factors $r=0.2$ and $r=0.8$. Here the number of samples of our PEC scheme is taken to be independent of the mitigation factor $r$, and the classical solutions $\hat{\rho}_{c}(t)$ of the Lindblad equation were obtained by using the Trotter-Suzuki product formula, instead of the standard RK4-based solver, so that the Trotter decomposition error is identical in quantum and classical simulations. It is notable that the average difference $\eta(\hat{\rho}_{q}(t),\hat{\rho}_{c}(t))$ between quantum and classical results is larger for higher $r$. In addition, as shown in Fig.~\ref{fig:error_implementation}(d), the variance of $\eta(\hat{\rho}_{q}(t),\hat{\rho}_{c}(t))$ is also larger for higher $r$, hinting that the sampling cost of our PEC scheme, required to obtain reliable open-system dynamics, increases as a function of the mitigation factor $r$, as will be analyzed later in more detail.

These results demonstrate that a Lindblad equation with controlled decoherence rates can be implemented on NISQ devices by our partial error mitigation scheme. We note that quantum and classical results are also well-matched when a larger quantum system is considered, encoded on four qubits ($n=4$), together with non-uniform mitigation factors $r_k$, as shown in Appendix~\ref{D}.

\subsection{Resource scaling of partial noise mitigation}\label{scalingPEC}

The application of PEC to large quantum circuits faces a bottleneck due to the enhanced statistical fluctuations. For a given observable $\hat{O}$, when a finite number $M$ of outcomes is measured in experiments without PEC, the uncertainty in expectation value $\langle \hat{O}\rangle_M$ may be described by its variance $\Delta \hat{O}_M\propto M^{-1}$. It has been shown that when PEC is employed, the variance is approximately increased to $\Delta \hat{O}_{M}^{\rm (PEC)}\propto C_{\rm tot}^{2} M^{-1}$ where $C_{\rm tot}\ge 1$ is the total mitigation cost in Eq.~\eqref{cost_definition}~\cite{endo2018practical,sun2021efficient}. Therefore, to maintain the degree of the uncertainty in expectation values, one needs to increase the number of single-shot measurements from $M$ to $M C_{\rm tot}^{2}$ when PEC is employed. For the full mitigation scheme with $r_k=1$ for all $k$, it can be shown that the total mitigation cost $C_{\rm tot}$ increases exponentially as a function of the number $n$ of qubits, the depth of quantum circuits, and the total error probability $\sum_{k>0}\epsilon_k$ of stochastic Pauli noise channels.

\begin{figure*}
\centering
\includegraphics[width=1\textwidth]{scaling_res.jpg}
\caption{(a,b) Total cost $C_{tot}$ scaling with noise obtained from noise benchmarking experiments up to 4 qubits on \textit{ibmq lagos} shown as data points. The error bars denote standard deviation and solid lines denote extrapolated results. The dashed gray lines denote the maximum cost supported by the current quantum devices (defined by the number of circuits that can be executed in $24\,$h~\cite{bravyi2022future}). In (a), the scaling of $C_{tot}$ is shown as a function of the number of Trotter iterations $D$ for different numbers of qubits $n$ and uniform mitigation factor $r=1$. In (b), we show the scaling of $C_{tot}$  with $n=4$ for different choices of $r$. We estimated that the fitted exponents on the exponent of $C_{tot}$, i.e. $\alpha$ in (a) and $\beta$ in (b), have a linear dependence on $n$ and $r$, respectively, as Eq.~\eqref{cost_scaling2} predicts, namely, $\alpha=(0.186 \pm 0.007)n$ and $\beta \approx (0.678 \pm 0.013) r$. (c) Data points denote the scaling of $C_{tot}$ for a quantum circuit with $n=4$ and uniform mitigation factor $r=1$, where the error probabilities $\epsilon_{k}$ of each stochastic Pauli noise channel are sampled over a Gaussian probability distribution with average values $\langle \epsilon_{k}\rangle$ and FWHM $\Delta \epsilon_{k}=\left<\epsilon_{k}\right>/2$. $C_{tot}$ is calculated based on the sampled error probabilities and then averaged over $500$ trajectories. We remark the results in (c) for $\langle \epsilon_{k}\rangle = 1\times 10^{-2}$ closely follow the ones in (a) for $n=4$.}
\label{fig:scaling_res}
\end{figure*}

For the case that the error probabilities $\epsilon_k$ are partially reduced with the mitigation factors $r_k \in [0,1]$, the total mitigation cost $C_{\rm tot}$ in Eq.~\eqref{cost_definition} can be computed analytically
\begin{align}
    C_{\rm tot} = \prod_{d=1}^{D}\prod_{m}(1+2\epsilon_r)= (1+2\epsilon_r)^{g(n) D},\label{Ctot_simple}
\end{align}
where $\epsilon_{r}=\sum_{k>0}\epsilon_{k}r_{k}$ is the sum of the error probabilities $\epsilon_k$ weighted by the corresponding mitigation factors $r_k$, and $g(n)$ is the number of $K$-qubit Pauli noise channels acting on a quantum circuit consisting of $n$ qubits. For simplicity, we assume that $\epsilon_r$ are identical for all the $K$-qubit stochastic Pauli noise channels acting on different subgroups of qubits ($n>K$). Practically we are interested in the limit of $n\gg K$ where a large open-system is encoded in a quantum circuit, and the noise is not strongly correlated amongst many qubits and therefore well-characterized by small $K\approx 2$. In this case, the number of different $K$-qubit Pauli noise channels acting on nearest-neighbour qubits increases linearly as a function of the total number $n$ of qubits, $g(n)\propto n$. In the limit of a small total error probability $\epsilon_r$ weighted by the mitigation factors and a sufficiently large number of qubits and/or Trotter layers, namely $\epsilon_r\rightarrow 0$ and $nD \rightarrow \infty$, the total mitigation cost can be approximately described by an exponential function $C_{\rm tot}\sim e^{2\epsilon_r n D}$. For the finite values of $\epsilon_r$, $n$ and $D$, now we show that the total mitigation factor can also be well-described by an exponential function in the form
\begin{equation}
    C_{\rm tot} \sim e^{\lambda n D \epsilon_{r}}, \label{cost_scaling2}
\end{equation}
with a positive constant $\lambda$ introduced as a fitting parameter of simulated results.

In Fig.~\ref{fig:scaling_res}(a) and (b), we show the total mitigation cost $C_{\rm tot}$ in a logarithmic scale, computed based on its definition in Eq.~\eqref{cost_definition}, as a function of the number $D$ of Trotter layers. The error probabilities $\epsilon_k$, determining the cost function $C_{\rm tot}$, were obtained from a real IBMQ computer (see Sec.~\ref{sec: noisy_evol}). For simplicity, we consider a uniform mitigation factor $r$, satisfying $r=r_k$ for all $k$. The maximum cost supported by current quantum devices is highlighted by horizontal dashed lines. In Fig.~\ref{fig:scaling_res}(a), where $r=1$, $C_{\rm tot}$ is displayed for different sizes of quantum circuits, $n\in \{2,3,4\}$. It is found that the numerical results can be well fitted by an exponential function in the form $e^{\alpha(n) D}$ with $\alpha(n) \approx 0.186\,n$, demonstrating the linear dependence of $\log(C_{\rm tot})$ on the number $n$ of qubits. In Fig.~\ref{fig:scaling_res}(b), where $n=4$, $C_{\rm tot}$ is shown for different uniform mitigation factors $r\in \{0.25,0.50,0.75,1.00\}$, where numerical results can be well-fitted by $e^{\beta(r) D}$ with $\beta\approx 0.678\,r$, revealing the linear dependence of $\log(C_{\rm tot})$ on the partial mitigation factor $r$. These results are in line with the approximate form of the total mitigation cost $C_{\rm tot}$ in Eq.~\eqref{cost_scaling2}.

We note that the total mitigation cost $C_{\rm tot}$ of our generalized PEC scheme decreases exponentially, as the partial mitigation factor $r<1$ is reduced, when compared to the conventional full mitigation scheme with $r=1$. This implies that our PEC scheme can be applied to a larger quantum circuit consisting of an increased number of qubits and/or Trotter layers, hinting that digital quantum simulation of open-system dynamics via our technique may be a promising application to NISQ devices. It is also notable that the total mitigation cost can be exponentially decreased further as the noise probabilities $\epsilon_k$ are reduced in the future quantum devices. To highlight this aspect, in Fig.~\ref{fig:scaling_res}(c), where $n=4$, we computed $C_{\rm tot}$ based on randomly generated error probabilities $\epsilon_k$ from Gaussian distributions with equal average $\langle\epsilon_k\rangle$ and FWHM $\Delta\epsilon_k=\frac{1}{2}\langle\epsilon_k\rangle$ for all $k$. It is found that when $\langle\epsilon_k\rangle=0.01$, this model can quantitatively reproduce the total mitigation cost computed based on the error probabilities of a real IBMQ computer (see the case of $n=4$ in Fig.~\ref{fig:scaling_res}(a)). As the average error probabilities $\langle\epsilon_k\rangle$ are reduced from 0.01, via 0.005, to 0.002, the total mitigation cost decreases exponentially, as shown in Fig.~\ref{fig:scaling_res}(c).

These results demonstrate that our partial noise mitigation scheme can be employed to implement a Lindblad model with arbitrary target decoherence rates $\Gamma_k$ on NISQ devices, but it requires a sufficiently large number of samples that scales with $C_{\rm tot}^{2}$. In real quantum devices, however, the maximum number $M_{\rm max}^{\rm (NISQ)}$ of circuits that can be executed within a given period of time is finite. According to Ref.~\cite{bravyi2022future}, $M_{\rm max}^{\rm (NISQ)}\sim 10^{8}$ for moderate circuit depths within a day. Therefore, the number of samples required for our technique should satisfy $C_{\rm tot}^{2}\sim e^{\lambda n D \epsilon_r}\lesssim M_{\rm max}^{\rm (NISQ)}$. For a given set of target decoherence rates $\Gamma_k$, this inequality can be expressed as
\begin{equation}
    \epsilon \lesssim \frac{\Gamma t}{D} + \frac{\ln(M_{\rm max}^{\rm (NISQ)})}{2\lambda n D^2}, \label{scaling_ctot}
\end{equation}
where $\epsilon=\sum_{k>0}\epsilon_k$ denotes the total Pauli noise probability, $\Gamma=\sum_{k>0}\Gamma_{k}$ the total target decoherence rate, and $t$ the simulation time with a Trotter time-step $\Delta t\le \Delta t_{\rm max}$ (see Sec.~\ref{sec:decoherence_control}). The first term in Eq.~\eqref{scaling_ctot} shows that the total Pauli noise probability $\epsilon$ scales linearly with the total target decoherence rate $\Gamma$. On one hand, this implies that when the total target decoherence rate $\Gamma$ is sufficiently high, the corresponding Lindblad model can be readily implemented on a quantum device even if its total noise probability $\epsilon$ is high. On the other hand, when the total target decoherence rate $\Gamma$ is low, the quantum device should have a sufficiently low total noise probability $\epsilon$, or the Pauli noise channels should be heavily mitigated, requiring a high sampling cost. The second term in Eq.~\eqref{scaling_ctot} shows that the total Pauli noise probability $\epsilon$ allowed by our partial noise mitigation scheme may be linearly increased by exponentially enhancing the capability of a quantum device to run several copies of circuits within a given period of time, quantified by $M_{\rm max}^{\rm (NISQ)}$. Note that the second term in Eq.~\eqref{scaling_ctot} is inversely proportional to the number $n$ of qubits and the square of the number of Trotter layers, $D^2$, implying that a higher $M_{\rm max}^{\rm (NISQ)}$ is required for digital quantum simulations of a larger open quantum system on a longer time scale.

We remark that the number $D$ of Trotter layers, required to achieve a desired Trotter decomposition error $\varepsilon_{\rm Trot}$ in Eq.~\eqref{error_trot}, depends on the structure of the Hamiltonian of a target open quantum system, the simulation time $t$, and the order $k$ of the Trotter-Suzuki product formula considered in simulations. As an example, we may consider the first order product formula ($k=1$), and a linear chain (or square grid) structure of qubits with a uniform nearest-neighbor coupling strength $J$ and on-site energy $E$ (see Eq.~\eqref{eq:H_PPC}). In this case, the number $D$ of Trotter layers can be expressed as
\begin{equation}
    D = O\left(\frac{n^d J (J+E) t^{2}}{\varepsilon_{\rm Trot}}\right),\label{scaling_trot}
\end{equation}
where $d=1$ (or 2) for the linear chain (or two-dimensional square grid) structure (for other classes of Hamiltonian, see Ref.~\cite{childs2021theory}). Using Eq.~\eqref{scaling_trot}, now we can express Eq.~\eqref{scaling_ctot} as a function of the Trotter decomposition error $\varepsilon_{\rm Trot}$
\begin{equation}
    \epsilon \lesssim O\left(\frac{\varepsilon_{\rm Trot}\Gamma}{n^d J(J+E)t} + \frac{\varepsilon_{\rm Trot}^{2}\ln(M_{\rm max}^{\rm (NISQ)})}{2\lambda n^{2d+1}J^2 (J+E)^2 t^4}\right).
\end{equation}
Note that the total Pauli noise probability $\epsilon$ allowed by our scheme increases as a function of the Trotter decomposition error $\varepsilon_{\rm Trot}$, and it is inversely proportional to the parameters $J$ and $E$ of the open-system Hamiltonian, and the simulation time $t$.

\subsection{Amplitude damping noise}

So far we have demonstrated that the stochastic Pauli noise channels in Eq.~\eqref{stochastic_channel} can be implemented by using our partial noise mitigation scheme where the intrinsic noise of quantum devices is transformed to stochastic Pauli noise via Randomized Compiling. Here we discuss how the noise model implemented on quantum devices can be generalized beyond the stochastic Pauli noise, such as the amplitude damping noise model that has been widely considered in classical simulations of open-system dynamics~\cite{breuer2002theory}.

The local amplitude damping of qubit $m$ is described by the Kraus operators in the form
\begin{equation}
    {\cal E}_{m}^{\rm (ad)}(\hat{\rho})= w \hat{\sigma}_{m}^{(-)}\hat{\rho}\hat{\sigma}_{m}^{(+)}+\hat{\sigma}_{m}^{(0)} \hat{\rho} \hat{\sigma}_{m}^{(0)},\label{eq:amplitude_damping}
\end{equation}
with $\hat{\sigma}_{m}^{(-)}=\frac{1}{2}(\hat{X}_{m}+i\hat{Y}_{m})$, $\hat{\sigma}_{m}^{(+)}=\frac{1}{2}(\hat{X}_{m}-i\hat{Y}_{m})$ and $\hat{\sigma}_{0}=\frac{1}{2}(1+\sqrt{1-w})\hat{I}_{m}+\frac{1}{2}(1-\sqrt{1-w})\hat{Z}_{m}$, where $w$ represents an incoherent transition probability from $\ket{1}$ to $\ket{0}$ of qubit $m$. The amplitude damping noise model in Eq.~\eqref{eq:amplitude_damping} cannot be described by the stochastic Pauli noise channels~\cite{blume2022taxonomy} in Eq.~\eqref{stochastic_channel}, thus requiring an approach different from the noise-assisted technique discussed so far. To that end, we introduce an ancilla qubit coupled to qubit $m$ where the interaction between them is described by a circuit shown in Fig.~\ref{fig:amp_damp_circuit}(a), consisting of three CNOT gates and single-qubit rotations $\hat{R}_{Y}(\pm\theta/2)=e^{\mp i\theta\hat{Y}_{a}/4}$ with a Pauli operator $\hat{Y}_{a}$ acting on the ancilla qubit \cite{nielsen2002quantum}. Crucially, the initial state of the ancilla qubit is reset to $\ket{0}$ in each Trotter layer, as schematically shown in Fig.~\ref{fig:amp_damp_circuit}(b). In this way, the amplitude damping probability $w=\sin^{2}(\theta/2)$ can be controlled by the single-qubit rotations $\hat{R}_{Y}(\pm\theta/2)$, resulting in an effective amplitude damping rate given by $w/\Delta t$.

We note that the source of amplitude damping noise is artificial in the sense that it is induced by the added specific sequence of quantum gates and reset operations in each Trotter layer. This is contrary to the stochastic Pauli noise channels where the faulty implementation of CNOT gates, i.e. their intrinsic noise, is transformed to stochastic Pauli noise via Randomized Compiling.

The additional CNOT and measurement operations, however, can increase the overall error probabilities of the Pauli noise channels. By using the emulated quantum device \emph{ibmq oslo}, we found that for a two-qubit case ($n=2$) the amplitude damping and stochastic Pauli noise channels can be implemented at the same time, but the total error probability $\sum_{k>0}\epsilon_k$ of the stochastic Pauli noise channels is increased by several tens of percents. This implies that the total mitigation cost of the Pauli noise channels can be increased (see Eq.~\eqref{cost_scaling2}) by introducing the amplitude damping circuit. Nevertheless, in this scheme, the mitigation cost does not depend on the amplitude damping probability $w$, since it is controlled by the single-qubit rotations $\hat{R}_{Y}(\pm\theta/2)$ and their intrinsic noise is negligible compared to CNOT gates. The amplitude damping channel can be introduced to all the open-system qubits by attaching ancilla qubits individually. The implementation of the amplitude damping noise and its resource scaling deserves a separate investigation and will be present somewhere else in the near future.

\begin{figure}
\centering
\includegraphics[width=0.48\textwidth]{amp_damp.jpg}
 \caption{(a) Decomposition of $\mathcal{E}_{m}^{(ad)}$ into quantum gates. The probability of qubit decay over a Trotter layer is $w = sin^2(\theta/2)$. (b) Example of the application of an amplitude damping channel $\mathcal{E}_{m}^{(ad)}$ to a system qubit $m$ in a quantum circuit. On each Trotter iteration, the ancilla qubit is initialized at $\ket{0}$,  followed by the application of $\mathcal{E}_{m}^{(ad)}$ to the system and ancilla qubits and then the state of the later is measured such that it can be reused in the next Trotter layer.}
 \label{fig:amp_damp_circuit}
\end{figure}




\section{Conclusion} \label{conclusions}
In this work, we have established the concept of intrinsic noise-assisted digital quantum algorithms and demonstrated the 
principle in real and emulated IBM Quantum computers based on superconducting qubits.
We show that the application of intrinsic noise-assisted digital quantum algorithms in real world devices can be 
achieved using three key steps. First, the coherent system dynamics is decomposed via Trotterization into product formulae with
time-step $\Delta t$ of our choice to gain a first level of control over the effective noise realised in the quantum 
simulation. Based on this, the second crucial step is the characterization of the system-dependent 
intrinsic noise in the NISQ device on which the algorithm is run. This can be achieved via Cycle Benchmarking and Error
Reconstruction techniques. Building on this intrinsic noise reconstruction, the third and final step uses Probabilistic Error
Cancellation to control the noise in the quantum circuit, i.e. the effective decoherence in the simulation model. 


The principle of digital quantum simulation assisted by the device's intrinsic noise offers
a number of advantages in the field of NISQ computation. First, the noise-assisted digital 
quantum algorithm does not need to alter the quantum hardware to tune the intrinsic noise but achieves it via
results postprocessing. Secondly, it does not require additional {\it quantum} computational resources, i.e. additional 
qubits and CNOT gates, for simulating open quantum systems compared to more standard approaches~\cite{barreiro2011open,hu2022general,cleve2016efficient,wang2011quantum,kamakari2022digital}.
The quantum resource reduction is achieved via an additional {\it classical} overhead, i.e. an increased number
of runs of the algorithm in the device, due to the use of a quantum error mitigation technique to control the noise acting on the qubits. As a result, our work provides guiding principles for the execution of
quantum digital simulation of open quantum systems on real world devices, where noise is not detrimental, but 
leveraged for a more efficient computation. 

In future work, we expect that the results can be extended in a variety of fruitful directions. First, it will
be interesting to benchmark (and adapt) this technique on (to) different quantum hardware technologies beyond 
superconducting quantum devices which may exhibit particularly useful intrinsic noise. Secondly, in this spirit, 
the addition of modest quantum resources may allow for the extension to the efficient digital simulation of 
non-Markovian environments combining it with techniques exposed in Ref.~\cite{lemmer2018trapped,mascherpa2020optimized}.
Thirdly, we expect these techniques to find fruitful combinations with Quantum TEDOPA~\cite{guimaraes2022efficient}, a technique to simulate non-perturbative dynamics of open quantum systems in a 
quantum computer, by making use of the results of a recent work on Markovian closure~\cite{nusseler2022fingerprint}. 
Finally, we expect further efficiency enhancements of our technique to be possible with improvements in tailormade
characterization and error control techniques to give access to general non-perturbative dynamics of open quantum systems. 

    

\section*{Acknowledgements} The authors acknowledge helpful discussions and support by Benjamin Desef. JDG acknowledges funding from the 
Portuguese Foundation for Science and Technology (FCT) through PhD grant UI/BD/151173/2021. JDG, JL, MBP acknowledge support by the BMBF 
project PhoQuant (grant no. 13N16110). MIV acknowledges support from the FCT through Strategic Funding UIDB/04650/2020. SFH and MBP acknowledge
support by the DFG via the QuantERA project ExtraQt. The authors acknowledge support by the state of Baden-Württemberg through bwHPC and the 
German Research Foundation (DFG) through Grant No. INST 40/575-1 FUGG (JUSTUS 2 cluster) and acknowledge the use of IBMQ devices via the 
Researcher Program.


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In \LaTeX\ there are many different ways to display equations, and a
few preferred ways are noted below. Displayed math will center by
default. Use the class option \verb+fleqn+ to flush equations left.

Below we have numbered single-line equations, the most common kind: 
\begin{eqnarray}
\chi_+(p)\alt{\bf [}2|{\bf p}|(|{\bf p}|+p_z){\bf ]}^{-1/2}
\left(
\begin{array}{c}
|{\bf p}|+p_z\\
px+ip_y
\end{array}\right)\;,
\\
\left\{%
 \openone234567890abc123\alpha\beta\gamma\delta1234556\alpha\beta
 \frac{1\sum^{a}_{b}}{A^2}%
\right\}%
\label{eq:one}.
\end{eqnarray}
Note the open one in Eq.~(\ref{eq:one}).

Not all numbered equations will fit within a narrow column this
way. The equation number will move down automatically if it cannot fit
on the same line with a one-line equation:
\begin{equation}
\left\{
 ab12345678abc123456abcdef\alpha\beta\gamma\delta1234556\alpha\beta
 \frac{1\sum^{a}_{b}}{A^2}%
\right\}.
\end{equation}

When the \verb+\label{#1}+ command is used [cf. input for
Eq.~(\ref{eq:one})], the equation can be referred to in text without
knowing the equation number that \TeX\ will assign to it. Just
use \verb+\ref{#1}+, where \verb+#1+ is the same name that used in
the \verb+\label{#1}+ command.

Unnumbered single-line equations can be typeset
using the \verb+\[+, \verb+\]+ format:
\[g^+g^+ \rightarrow g^+g^+g^+g^+ \dots ~,~~q^+q^+\rightarrow
q^+g^+g^+ \dots ~. \]

\subsection{Multiline equations}

Multiline equations are obtained by using the \verb+eqnarray+
environment.  Use the \verb+\nonumber+ command at the end of each line
to avoid assigning a number:
\begin{eqnarray}
{\cal M}=&&ig_Z^2(4E_1E_2)^{1/2}(l_i^2)^{-1}
\delta_{\sigma_1,-\sigma_2}
(g_{\sigma_2}^e)^2\chi_{-\sigma_2}(p_2)\nonumber\\
&&\times
[\epsilon_jl_i\epsilon_i]_{\sigma_1}\chi_{\sigma_1}(p_1),
\end{eqnarray}
\begin{eqnarray}
\sum \vert M^{\text{viol}}_g \vert ^2&=&g^{2n-4}_S(Q^2)~N^{n-2}
        (N^2-1)\nonumber \\
 & &\times \left( \sum_{i<j}\right)
  \sum_{\text{perm}}
 \frac{1}{S_{12}}
 \frac{1}{S_{12}}
 \sum_\tau c^f_\tau~.
\end{eqnarray}
\textbf{Note:} Do not use \verb+\label{#1}+ on a line of a multiline
equation if \verb+\nonumber+ is also used on that line. Incorrect
cross-referencing will result. Notice the use \verb+\text{#1}+ for
using a Roman font within a math environment.

To set a multiline equation without \emph{any} equation
numbers, use the \verb+\begin{eqnarray*}+,
\verb+\end{eqnarray*}+ format:
\begin{eqnarray*}
\sum \vert M^{\text{viol}}_g \vert ^2&=&g^{2n-4}_S(Q^2)~N^{n-2}
        (N^2-1)\\
 & &\times \left( \sum_{i<j}\right)
 \left(
  \sum_{\text{perm}}\frac{1}{S_{12}S_{23}S_{n1}}
 \right)
 \frac{1}{S_{12}}~.
\end{eqnarray*}
To obtain numbers not normally produced by the automatic numbering,
use the \verb+\tag{#1}+ command, where \verb+#1+ is the desired
equation number. For example, to get an equation number of
(\ref{eq:mynum}),
\begin{equation}
g^+g^+ \rightarrow g^+g^+g^+g^+ \dots ~,~~q^+q^+\rightarrow
q^+g^+g^+ \dots ~. \tag{2.6$'$}\label{eq:mynum}
\end{equation}

A few notes on \verb=\tag{#1}=. \verb+\tag{#1}+ requires
\texttt{amsmath}. The \verb+\tag{#1}+ must come before the
\verb+\label{#1}+, if any. The numbering set with \verb+\tag{#1}+ is
\textit{transparent} to the automatic numbering in REV\TeX{};
therefore, the number must be known ahead of time, and it must be
manually adjusted if other equations are added. \verb+\tag{#1}+ works
with both single-line and multiline equations. \verb+\tag{#1}+ should
only be used in exceptional case - do not use it to number all
equations in a paper.

Enclosing single-line and multiline equations in
\verb+\begin{subequations}+ and \verb+\end{subequations}+ will produce
a set of equations that are ``numbered'' with letters, as shown in
Eqs.~(\ref{subeq:1}) and (\ref{subeq:2}) below:
\begin{subequations}
\label{eq:whole}
\begin{equation}
\left\{
 abc123456abcdef\alpha\beta\gamma\delta1234556\alpha\beta
 \frac{1\sum^{a}_{b}}{A^2}
\right\},\label{subeq:1}
\end{equation}
\begin{eqnarray}
{\cal M}=&&ig_Z^2(4E_1E_2)^{1/2}(l_i^2)^{-1}
(g_{\sigma_2}^e)^2\chi_{-\sigma_2}(p_2)\nonumber\\
&&\times
[\epsilon_i]_{\sigma_1}\chi_{\sigma_1}(p_1).\label{subeq:2}
\end{eqnarray}
\end{subequations}
Putting a \verb+\label{#1}+ command right after the
\verb+\begin{subequations}+, allows one to
reference all the equations in a subequations environment. For
example, the equations in the preceding subequations environment were
Eqs.~(\ref{eq:whole}).

\subsubsection{Wide equations}
The equation that follows is set in a wide format, i.e., it spans
across the full page. The wide format is reserved for long equations
that cannot be easily broken into four lines or less:
\begin{widetext}
\begin{equation}
{\cal R}^{(\text{d})}=
 g_{\sigma_2}^e
 \left(
   \frac{[\Gamma^Z(3,21)]_{\sigma_1}}{Q_{12}^2-M_W^2}
  +\frac{[\Gamma^Z(13,2)]_{\sigma_1}}{Q_{13}^2-M_W^2}
 \right)
 + x_WQ_e
 \left(
   \frac{[\Gamma^\gamma(3,21)]_{\sigma_1}}{Q_{12}^2-M_W^2}
  +\frac{[\Gamma^\gamma(13,2)]_{\sigma_1}}{Q_{13}^2-M_W^2}
 \right)\;. \label{eq:wideeq}
\end{equation}
\end{widetext}
This is typed to show the output is in wide format.
(Since there is no input line between \verb+\equation+ and
this paragraph, there is no paragraph indent for this paragraph.)
\section{Cross-referencing}
REV\TeX{} will automatically number sections, equations, figure
captions, and tables. In order to reference them in text, use the
\verb+\label{#1}+ and \verb+\ref{#1}+ commands. To reference a
particular page, use the \verb+\pageref{#1}+ command.

The \verb+\label{#1}+ should appear in a section heading, within an
equation, or in a table or figure caption. The \verb+\ref{#1}+ command
is used in the text where the citation is to be displayed.  Some
examples: Section~\ref{sec:level1} on page~\pageref{sec:level1},
Table~\ref{tab:table1},%
\begin{table}
\caption{\label{tab:table1}This is a narrow table which fits into a
text column when using \texttt{twocolumn} formatting. Note that
REV\TeX~4 adjusts the intercolumn spacing so that the table fills the
entire width of the column. Table captions are numbered
automatically. This table illustrates left-aligned, centered, and
right-aligned columns.  }
\begin{ruledtabular}
\begin{tabular}{lcr}
Left\footnote{Note a.}&Centered\footnote{Note b.}&Right\\
\hline
1 & 2 & 3\\
10 & 20 & 30\\
100 & 200 & 300\\
\end{tabular}
\end{ruledtabular}
\end{table}
and Fig.~\ref{fig:epsart}.

\section{Figures and Tables}
Figures and tables are typically ``floats''; \LaTeX\ determines their
final position via placement rules. 
\LaTeX\ isn't always successful in automatically placing floats where you wish them.

Figures are marked up with the \texttt{figure} environment, the content of which
imports the image (\verb+\includegraphics+) followed by the figure caption (\verb+\caption+).
The argument of the latter command should itself contain a \verb+\label+ command if you
wish to refer to your figure with \verb+\ref+.

Import your image using either the \texttt{graphics} or
\texttt{graphix} packages. These packages both define the
\verb+\includegraphics{#1}+ command, but they differ in the optional
arguments for specifying the orientation, scaling, and translation of the figure.
Fig.~\ref{fig:epsart}%
\begin{figure}
\includegraphics{fig_1
\caption{\label{fig:epsart} A figure caption. The figure captions are
automatically numbered.}
\end{figure}
is small enough to fit in a single column, while
Fig.~\ref{fig:wide}%
\begin{figure*}
\includegraphics{fig_2
\caption{\label{fig:wide}Use the \texttt{figure*} environment to get a wide
figure, spanning the page in \texttt{twocolumn} formatting.}
\end{figure*}
is too wide for a single column,
so instead the \texttt{figure*} environment has been used.

The analog of the \texttt{figure} environment is \texttt{table}, which uses
the same \verb+\caption+ command.
However, you should type your caption command first within the \texttt{table}, 
instead of last as you did for \texttt{figure}.

The heart of any table is the \texttt{tabular} environment,
which represents the table content as a (vertical) sequence of table rows,
each containing a (horizontal) sequence of table cells. 
Cells are separated by the \verb+&+ character;
the row terminates with \verb+\\+. 
The required argument for the \texttt{tabular} environment
specifies how data are displayed in each of the columns. 
For instance, a column
may be centered (\verb+c+), left-justified (\verb+l+), right-justified (\verb+r+),
or aligned on a decimal point (\verb+d+). 
(Table~\ref{tab:table4}%
\begin{table}
\caption{\label{tab:table4}Numbers in columns Three--Five have been
aligned by using the ``d'' column specifier (requires the
\texttt{dcolumn} package). 
Non-numeric entries (those entries without
a ``.'') in a ``d'' column are aligned on the decimal point. 
Use the
``D'' specifier for more complex layouts. }
\begin{ruledtabular}
\begin{tabular}{ccddd}
One&Two&\mbox{Three}&\mbox{Four}&\mbox{Five}\\
\hline
one&two&\mbox{three}&\mbox{four}&\mbox{five}\\
He&2& 2.77234 & 45672. & 0.69 \\
C\footnote{Some tables require footnotes.}
  &C\footnote{Some tables need more than one footnote.}
  & 12537.64 & 37.66345 & 86.37 \\
\end{tabular}
\end{ruledtabular}
\end{table}
illustrates the use of decimal column alignment.)

Extra column-spacing may be be specified as well, although
REV\TeX~4 sets this spacing so that the columns fill the width of the
table.
Horizontal rules are typeset using the \verb+\hline+
command.
The doubled (or Scotch) rules that appear at the top and
bottom of a table can be achieved by enclosing the \texttt{tabular}
environment within a \texttt{ruledtabular} environment.
Rows whose columns span multiple columns can be typeset using \LaTeX's
\verb+\multicolumn{#1}{#2}{#3}+ command
(for example, see the first row of Table~\ref{tab:table3}).%
\begin{table*}
\caption{\label{tab:table3}This is a wide table that spans the page
width in \texttt{twocolumn} mode. It is formatted using the
\texttt{table*} environment. It also demonstrates the use of
\textbackslash\texttt{multicolumn} in rows with entries that span
more than one column.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
 &\multicolumn{2}{c}{$D_{4h}^1$}&\multicolumn{2}{c}{$D_{4h}^5$}\\
 Ion&1st alternative&2nd alternative&lst alternative
&2nd alternative\\ \hline
 K&$(2e)+(2f)$&$(4i)$ &$(2c)+(2d)$&$(4f)$ \\
 Mn&$(2g)$\footnote{The $z$ parameter of these positions is $z\sim\frac{1}{4}$.}
 &$(a)+(b)+(c)+(d)$&$(4e)$&$(2a)+(2b)$\\
 Cl&$(a)+(b)+(c)+(d)$&$(2g)$\footnote{This is a footnote in a table that spans the full page
width in \texttt{twocolumn} mode. It is supposed to set on the full width of the page, just as the caption does. }
 &$(4e)^{\text{a}}$\\
 He&$(8r)^{\text{a}}$&$(4j)^{\text{a}}$&$(4g)^{\text{a}}$\\
 Ag& &$(4k)^{\text{a}}$& &$(4h)^{\text{a}}$\\
\end{tabular}
\end{ruledtabular}
\end{table*}

The tables in this document illustrate various effects.
Tables that fit in a narrow column are contained in a \texttt{table}
environment.
Table~\ref{tab:table3} is a wide table, therefore set with the
\texttt{table*} environment.
Lengthy tables may need to break across pages.
A simple way to allow this is to specify
the \verb+[H]+ float placement on the \texttt{table} or
\texttt{table*} environment.
Alternatively, using the standard \LaTeXe\ package \texttt{longtable} 
gives more control over how tables break and allows headers and footers 
to be specified for each page of the table.
An example of the use of \texttt{longtable} can be found
in the file \texttt{summary.tex} that is included with the REV\TeX~4
distribution.

There are two methods for setting footnotes within a table (these
footnotes will be displayed directly below the table rather than at
the bottom of the page or in the bibliography).
The easiest
and preferred method is just to use the \verb+\footnote{#1}+
command. This will automatically enumerate the footnotes with
lowercase roman letters.
However, it is sometimes necessary to have
multiple entries in the table share the same footnote.
In this case,
create the footnotes using
\verb+\footnotemark[#1]+ and \verb+\footnotetext[#1]{#2}+.
\texttt{\#1} is a numeric value.
Each time the same value for \texttt{\#1} is used, 
the same mark is produced in the table. 
The \verb+\footnotetext[#1]{#2}+ commands are placed after the \texttt{tabular}
environment. 
Examine the \LaTeX\ source and output for Tables~\ref{tab:table1} and 
\ref{tab:table2}%
\begin{table}
\caption{\label{tab:table2}A table with more columns still fits
properly in a column. Note that several entries share the same
footnote. Inspect the \LaTeX\ input for this table to see
exactly how it is done.}
\begin{ruledtabular}
\begin{tabular}{cccccccc}
 &$r_c$ (\AA)&$r_0$ (\AA)&$\kappa r_0$&
 &$r_c$ (\AA) &$r_0$ (\AA)&$\kappa r_0$\\
\hline
Cu& 0.800 & 14.10 & 2.550 &Sn\footnotemark[1]
& 0.680 & 1.870 & 3.700 \\
Ag& 0.990 & 15.90 & 2.710 &Pb\footnotemark[2]
& 0.450 & 1.930 & 3.760 \\
Au& 1.150 & 15.90 & 2.710 &Ca\footnotemark[3]
& 0.750 & 2.170 & 3.560 \\
Mg& 0.490 & 17.60 & 3.200 &Sr\footnotemark[4]
& 0.900 & 2.370 & 3.720 \\
Zn& 0.300 & 15.20 & 2.970 &Li\footnotemark[2]
& 0.380 & 1.730 & 2.830 \\
Cd& 0.530 & 17.10 & 3.160 &Na\footnotemark[5]
& 0.760 & 2.110 & 3.120 \\
Hg& 0.550 & 17.80 & 3.220 &K\footnotemark[5]
&  1.120 & 2.620 & 3.480 \\
Al& 0.230 & 15.80 & 3.240 &Rb\footnotemark[3]
& 1.330 & 2.800 & 3.590 \\
Ga& 0.310 & 16.70 & 3.330 &Cs\footnotemark[4]
& 1.420 & 3.030 & 3.740 \\
In& 0.460 & 18.40 & 3.500 &Ba\footnotemark[5]
& 0.960 & 2.460 & 3.780 \\
Tl& 0.480 & 18.90 & 3.550 & & & & \\
\end{tabular}
\end{ruledtabular}
\footnotetext[1]{Here's the first, from Ref.~\onlinecite{feyn54}.}
\footnotetext[2]{Here's the second.}
\footnotetext[3]{Here's the third.}
\footnotetext[4]{Here's the fourth.}
\footnotetext[5]{And etc.}
\end{table}
for an illustration. 

All AIP journals require that the initial citation of
figures or tables be in numerical order.
\LaTeX's automatic numbering of floats is your friend here:
just put each \texttt{figure} environment immediately following 
its first reference (\verb+\ref+), as we have done in this example file. 

\begin{acknowledgments}
We wish to acknowledge the support of the author community in using
REV\TeX{}, offering suggestions and encouragement, testing new versions,
\dots.
\end{acknowledgments}

