What is a qubit?

The simplest way to define a quantum computer, and perhaps the aspect that most often makes it into popular press accounts, is the use of qubits instead of bits for elementary building blocks of computation. While a classical computer’s bits, made of transistors, are binary (meaning two-state e.g. on/off or 1/0), a quantum computer’s equivalent building block, qubits, operates in a quantum superposition of the 1 and 0 states which will collapse to either a value of 1 or 0 upon a measurement.

For this discussion I will mostly abstract out the physical manifestation of the qubit and gate architecture and instead focus on the underlying principles of operation, however it is worth a brief overview of some of the key architectures competing for adoption.

Source: A Blueprint for Building a Quantum Computer by Van Meter and Horsman — Link

The description of a qubit’s state is a little more complicated with a much wider range of values than a classical bit. To describe the quantum superposition of a qubit requires complex numbers, for example a single qubit’s quantum superposition ψ (a common notation for a superposition, the Greek symbol Psi) could be described as ψ=a|0>+b|1>, where a and b are any combination of complex numbers a and b subject to the constraint that |a|²+|b|²=1, with the absolute value of the complex number a squared (|a|²) is equal to the probability that a measurement of the qubit’s state results in the value 0, similarly the absolute value of the complex number b squared (|b|²) is equal to the probability that a measurement of the qubit’s state results in the value 1, and hence the sum of those two probabilities has to sum to 1 (meaning there is exactly a 100% probability of the measured state of a single qubit being either 1 or 0, or using the “bra-ket” notation the states |0> or |1>). Of course we can’t know the actual values of the superposition ψ with a single measurement of a qubit because each measurement will collapse the state to either of the two values under superposition and we lose all of the probabilistic information, but given enough measurements we can attempt to estimate probabilities — or alternatively if we create a known state of ψ prior to a series of transformations through defined quantum gates then we might know the full representation of the resulting state ψ. An easy way to visualize the range of possible values for a qubit’s superposition between the states |0> and |1> is through what is known as a Bloch Sphere.

A few comments about the Bloch Sphere. First the equivalent formation for a classical computer would just be the two points |0> and |1>, which are shown here as to top and bottom points of the sphere along the z axis.

The image on the left also represents the potential states of a qubit after a measurement causes the superposition to collapse.

If the superposition ψ where to fall exactly on the point |0> at the top of the z axis for instance there would be a 100% probability of a measurement finding a qubit in state 0. Note that the positions of |0> and |1> are arbitrary, if we wanted we could change our basis of measurement and notation to any two points on opposite ends of the sphere. Another common basis besides |0> and |1> are the states |+> and |->, related to our original basis by the formulas |+>=(1/√2)|0>+(1/√2)|1> and |->=(1/√2)|0>-(1/√2)|1>. This +/- basis is notable because both represent a 50% probability of a measurement revealing either 0 or 1 (i.e. |(1/√2)|²=1/2=50%). As we start subjecting a qubit’s superposition to transformation around the bloch sphere by applying the various quantum gates we may find the alternate measurement and notation basis of |+> and |-> more appropriate, just keep in mind that these notation basis are arbitrary and while it will change the values of a & b to change the basis of notation it won’t change the location on the bloch sphere. This is worth restating for clarity. When a qubit is subject to transformations by applying a quantum gate, it results in some kind of rotation around the various axis of the bloch sphere, thus changing the probability of a measurements collapse to states of |0> or |1> (or depending on basis of measurement perhaps another such as |+> or |->). Keep in mind that the bloch sphere is a two dimensional representation for the state of only a single qubit (this sphere is two dimensional instead of three dimensional because we are restricted to its surface and it is a sphere and not a circle because of the numbers are complex, note that any point on the sphere can be identified with just two variables such as the two angles θ and φ shown in the graphic). As we introduce additional qubits to the system the resulting combined superposition would require a shape of additional dimensions and hence not really as useful for visualization purposes (for example for a two qubit superposition the combined superposition would be described as ψ=a|00>+b|01>+c|10>+d|11>, subject to the restriction that |a|²+|b|²+|c|²+|d|²=1, or in other words |a|² gives the probability that a measurement finds both qubits in the 0 state, |b|² gives the probability of a measurement finding the first qubit in the 0 state and the second qubit in the 1 state, etc.)

image via The Far Side by Gary Larson

Often a quantum computer vendor will report on the number of qubits that are achieved as a measure of their scale. As the Hilbert space (dimensionality) of a quantum computer’s state climbs exponentially with the number of qubits as 2^n this is an important metric. However in evaluating the number of qubits it is important to keep in mind the difference between a functioning logical qubit and a physical qubit. The superimposed state of a physical qubit is extremely fragile, and depending on the architecture, coherence can likely only be maintained for minute fractions of a second, thus there is a demanding need for error correction, comparable to although not identical as the error correction used in a classical computer. These errors cause the ideal shape of the Bloch Sphere to deform in different ways based on the channels of error (i.e. bit flips, phase flips, depolarizing, amplitude dampening, or phase dampening) and their rate. In order to achieve suitable quantum error correction an architecture may have to expend multiple physical qubits to construct a resulting single functioning logical qubit. A key enabler of future more scalable quantum computing architectures will be their ability to maintain their coherence for extended periods and thus reduce the amount of error correction required.

Quote Source: Scientific American