Nuclear magnetic resonance (NMR)-based ensemble quantum information processing (QIP) devices have provided excellent test beds for controlling non-trivial numbers of qubits12,13,14,15. A solid-state NMR QIP architecture builds on this success by incorporating the essential features of the liquid-state devices while offering the potential to reach unit polarization and thus control more qubits15,16. In this architecture, the abundant nuclear spins with polarization P form a large-heat-capacity spin-bath that can be either coupled to, or decoupled from, a dilute, embedded ensemble of spin-labelled isotopomers that comprise the qubit register. Bulk spin-cooling procedures such as dynamic nuclear polarization are well known and capable of reaching polarizations near unity15,17. This architecture is one realization within a large class of possible solid-state QIP systems in which coherently controlled qubits can be brought into contact with an external system that behaves as a heat bath. The principles and methods applied in solid-state NMR QIP will therefore apply to many other systems. An additional motivation is development of control techniques that future quantum devices will harness. For this experiment, we develop a novel technique to implement the controlled qubit–bath interaction, and also report the first application of strongly modulating pulses18 to solid-state NMR for high-fidelity, coherent qubit control.

The three-qubit quantum information processor used here is formed by the three spin-1/2 13C nuclei of isotopically labelled malonic acid molecules, occupying a dilute fraction of lattice sites in an otherwise unlabelled single crystal of malonic acid (unlabelled, with the exception of naturally occurring 13C isotopes at the rate of 1.1%). The concentration of labelled molecules was 3.2%. Malonic acid also contains abundant spin-1/2 1H nuclei, which comprise the heat-bath. Figure 1 shows the 1H-decoupled, 13C-NMR spectrum for the crystal (and crystal orientation) used in this work. The spectrum shows the NMR absorption peaks of both the qubit spins (quartets) and natural abundance 13C spins (singlets), the latter being inconsequential for QIP purposes. The table in Fig. 1 lists the parameters of the ensemble qubit hamiltonian obtained from fitting the spectrum, and also includes couplings involving the methylene protons calculated for this crystal orientation from the known crystal structure19. Experiments were performed at room temperature at a static magnetic field strength of 7.1 T, where the thermal 1H polarization is P H ≈ 2.4 × 10-5.

Figure 1: Characteristics of the dilute 3-13C malonic acid spin system. Bottom, 1H-decoupled, 13C spectrum near the [010] orientation with respect to the static magnetic field. The blue-dashed line is the experimental NMR absorption spectrum, and the solid red line is a fit. Multiplet assignments are indicated by the labels C 1 , C 2 and C m . The central peaks in each multiplet correspond to natural abundance 13C in the sample, which are inconsequential for QIP purposes. The peak height differences in the 3-13C molecule peaks indicate the strong coupling regime, that is, the 13C–13C intramolecular dipolar couplings are significant compared to the relative chemical shifts. Top, table showing the 13C rotating-frame hamiltonian parameters (chemical shifts along diagonal; dipolar coupling strengths off-diagonal; all values in kHz) obtained from the spectral fit. It also includes calculated dipolar couplings involving the methylene protons based on the atomic coordinates19 and the crystal orientation obtained from the spectral fit. Full size image

In this orientation, the methylene carbon C m has a dipolar coupling of 19 kHz to H m1 of the methylene 1H pair, whereas no other 13C-1H dipolar coupling in the system is larger than 2 kHz (see Fig. 1 for atom nomenclature). Therefore, a spin-exchange hamiltonian of the form that couples the two nuclear species will generate dynamics dominated by the large C m –H m1 coupling at short times (the D jk are 13C–1H dipolar couplings, indices j,k run over 13C,1H nuclei, respectively, and is the β-axis Pauli operator for spin α). Starting from the natural coupling hamiltonian, , we applied a multiple-pulse ‘time-suspension’ sequence20 synchronously to both 13C and 1H spins to create the effective spin-exchange hamiltonian (in the toggling frame), to lowest order in the Magnus expansion of the average hamiltonian21. Application of the sequence for the C m –H m1 exchange period τ = 3/(4 × 19 kHz) ≈ 40 µs results in an approximate swap gate (state exchange) between the C m and H m1 spins. With an initial bulk 1H polarization P H , this procedure yields a selective dynamic transfer of polarization P′ = ηP H to C m , where 0 ≤ |η| ≤ 1 and ideally |η| = 1. We define the effective spin-bath temperature to be that which corresponds to the experimentally obtained P′ under this procedure, and refer to this transfer as a refresh operation. We obtained P′ ≈ 0.83P H experimentally, and found that repeated refresh operations showed no loss in efficiency given at least a 6 ms delay for 1H–1H equilibration. However, we observed a decay of P H as a function of the number of repetitions, due to accumulated control errors, which lead to an identical loss in the refresh polarization.

The experiment consists of the first six operations of the partner-pairing algorithm (PPA) on three qubits: three refresh operations, and three permutation gates that operate on the qubit register. This is described in the quantum circuit diagram of Fig. 2. During the register operations, the 1H polarization is first rotated into the transverse plane, and then ‘spin-locked’ by a strong, phase-matched radio frequency (r.f.) field that both preserves the bulk 1H polarization and decouples the 1H–13C dipolar interactions. As 1H–1H dipolar interactions are merely scaled by a factor -1/2 under spin-locking, H m1 is allowed to equilibrate with the bulk 1H nuclei via spin diffusion. This occurs on a timescale longer than the transverse dephasing time (T 2 (H m ) ≈ 100 µs), but much shorter than the spin–lattice relaxation time of H m1 . Hence, H m1 plays the role of the fast-relaxing qubit described in the protocol of ref. 11. The first two register operations are swap gates; the third is a three-bit compression (3BC) gate8,9,10 that boosts the polarization of the first qubit, C 1 , at the expense of the polarizations of the other two qubits. Ideally, the protocol builds a uniform polarization on all three qubits corresponding to the bath polarization (first five steps), then selectively transfers as much entropy as possible from the first qubit to the other two (last step). The last step (3BC) leads to a polarization boost by a factor of 3/2 on the first qubit. Subsequently, the heated qubits can be re-cooled to the spin-bath temperature, and the compression step repeated, iteratively, until the asymptotic value of the first-bit polarization is reached. This limiting polarization depends only on the number of qubits and the bath polarization11, and is ideally P(C 1 ) = 2P′ for three qubits (for n qubits it is 2n-2P′ in the regime P′≪2-n, and 1.0 in the regime P′≫2-n (refs 11, 22)). The first six steps carried out here should yield a polarization of 1.5P′ on C 1 , assuming ideal operations.

Figure 2: Schematic quantum circuit diagram of the implemented protocol. Time flows from left to right. The three-bit compression (3BC) gate is shown here decomposed as control-not gates and a control-control-not (Toffoli) gate. The gate sequence corresponds to the first six steps of the partner-pairing algorithm11 on three qubits. The input state is a collective polarization P H of the bulk 1H. The refresh operation is approximately 40 µs in duration, whereas the register operations are between 0.7 and 1.3 ms in duration. Thermal contact takes place during 1H spin-locking pulses that begin just before the register operations, and extend an additional 12 ms after each operation. H m1 can be thought of as an additional ‘special purpose’ qubit in this experiment; despite non-selective 1H control (due to bulk hydrogenation), the refresh and thermal contact operations could be performed using collective 1H control. Thus, H m1 serves as a fast-relaxing ‘qubit’ and the bulk 1H-bath as a thermal bath of large heat capacity. Full size image

The control operations performed here are quantum control operations: state-independent unitary rotations in the Hilbert space. However, it should be noted that the heat-bath algorithmic cooling gates are all permutations that map computational basis states to other computational basis states. Therefore, gate fidelities were measured with respect to correlation with these known states, rather than the manifold of generic quantum states. We took advantage of this property to further optimize the control parameters of the 13C gates (register operations) for the state-specific transformations of the protocol. These operations were carried out using numerically optimized control sequences referred to as strongly modulating pulses18. Such pulses drive the system strongly at all times, such that the average r.f. amplitude is comparable to, or greater than, the magnitude of the internal hamiltonian. This allows inhomogeneities in the ensemble qubit hamiltonian to be efficiently refocused, so that ensemble coherence is better maintained throughout the gate operations.

In this set of experiments, the 13C qubit spins are initialized to infinite temperature (a preceding broadband 13C π/2 excitation pulse is followed by a dephasing period in which 1H dipolar fields effectively dephase the 13C polarization). Following the fifth step, polarizations (in units of P′) of 0.88, 0.83 and 0.76 (± 0.03) are built up on C 1 , C 2 and C m , respectively. The final 3BC operation yields P(C 1 )/P′ = 1.22 ± 0.03, an increase of 48% compared to the average polarization (0.82) following step five. Despite control imperfections that effectively heat the qubits at each step, we are able to cool the C 1 qubit ensemble well below the effective 1H spin-bath temperature.

The results are summarized in Fig. 3; in Fig. 3a are shown the spectral intensities corresponding to 13C spin polarizations following each of the six steps, and in Fig. 3b the integrated intensities are graphed in comparison with the ideal values. We note that the overall fidelity of the experiment, F = 1.22/1.50 = 0.81, implies an error per step of 3.7%. This error rate is only about a factor of two larger than the average error per two-qubit gate obtained in a benchmark liquid-state NMR QIP experiment12. Furthermore, the state-correlation fidelity of the 3BC gate over the polarizations on all three qubits is 0.96 ± 0.03. From Fig. 3b, it can be seen that the fidelity of the refresh operation drops off roughly quadratically in the number of steps; this is consistent with the loss of bulk 1H polarization due to pulse imperfections both in the multiple-pulse refresh operations and in the spin-locking sequence. As the broadband pulses have been optimized for flip-angle in these sequences, we suspect that the remaining errors are mainly due to switching transients that occur in the tuned r.f. circuitry of the NMR probe head, and to a lesser extent off-resonance and finite pulse-width effects that modify the average hamiltonian20. Similar effects lead to imperfect fidelity of the 13C control. With suitable improvements to the resonant circuit response and by incorporating numerical optimization of the multiple-pulse refresh operations, we expect that several iterations of the protocol could be carried out and that the limiting polarization of 2P′ could be approached in this system. The same methodologies should also be applicable in larger qubit systems with similar architecture. For a six-qubit system using the PPA, a bath polarization P′ > 0.2 would be sufficient, in principle, to reach a pure state on one qubit22. Such bulk nuclear polarizations are well within reach via well-known dynamic nuclear polarization techniques17; for example, unpaired electron spins at defects (g-factor = 2) in a field of 3.4 T and at temperature 4.2 K are polarized to 0.5.

Figure 3: Experimental results in terms of 13C spectra and their integrated intensities. a, Readout spectra obtained following each of the six steps in the protocol. A colour scale indicates peak intensities, which are in arbitrary units. The integrated peak intensities for each multiplet correspond to the ensemble spin polarizations. The natural abundance C m signal that appears at each refresh step (adding to the intensity of the central peaks) should be ignored; we are only interested in the part of the signal arising from the 3-13C qubit molecules, which can be seen clearly in the C 1 and C 2 spectral regions. b, Bars indicate ideal qubit polarizations at each step; experimental values obtained from integration of the above spectra are shown as shaded bands, whose thickness indicates experimental uncertainty. Full size image

This work demonstrates that solid-state NMR QIP devices could be used to implement active error correction. Given a bath polarization near unity, the refresh operation implemented here would constitute the dynamic resetting of a chosen qubit. This would allow a new NMR-based test bed for the ideas of quantum error correction and for controlled open-system quantum dynamics in the regime of high state purity and up to approximately 20 qubits.