Bigger is better: Quantum volume expresses computer’s limit

The race to build the first useful quantum computer continues apace. And, like all races, there are decisions to be made, including the technology each competitor must choose. But, in science, no one knows the race course, where the finish line is, or even if the race has any sort of prize (financial or intellectual) along the way.

On the other hand, the competitors can take a hand in the outcome by choosing the criteria by which success is judged. And, in this rather cynical spirit, we come to IBM’s introduction (PDF) of “quantum volume” as a single numerical benchmark for quantum computers. In the world of quantum computing, it seems that everyone is choosing their own benchmark. But, on closer inspection, the idea of quantum volume has merit.

Many researchers benchmark using gate speed—how fast a quantum gate can perform an operation—or gate fidelity, which is how reliable a gate operation is. But these single-dimensional characteristics do not really capture the full performance of a quantum processor. For analogy, it would be like comparing CPUs by clock speed or cache size, but ignoring any of the other bazillion features that impact computational performance.

The uselessness of these various individual comparisons were highlighted when researchers compared a slow, but high-fidelity quantum computer to a fast, but low-fidelity quantum computer, and came to the conclusion that the result was pretty much a draw.

It gets even worse when you consider that, unlike classical computers, you need a certain number of qubits to even carry out a calculation of a certain computational size. So, maybe, IBM researchers thought, a benchmark needs to somehow encompass the idea of what a quantum computer is capable of calculating, but not necessarily how fast it will perform a calculation.

How deep is your quantum?

The IBM staff are building on a concept called circuit depth. Circuit depth starts with the idea that, because quantum gates can always introduce an error, there is a maximum number of operations that can be performed before it is unreasonable to expect the qubit state to be correct. Circuit depth is that number, multiplied by the number of qubits. If used honestly, this provides a reasonable idea of what a quantum computer can do.

The problem with depth is that you can keep the total number of qubits constant (and small), while reducing the error rate to very close to zero. That gives you a huge depth, but, only computations that fit within the number of qubits can be calculated. A two-qubit quantum computer with enormous depth is still useless.

The goal, then, is to express computational capability, which must include the number of qubits and the circuit depth. Given an algorithm and problem size, there is a minimum number of qubits required to perform the computation. And, depending on how the qubits are connected to each other, a certain number of operations have to be performed to carry out the algorithm. The researchers express this by comparing the maximum number of qubits involved in a computation to the circuit depth and take the square of the smaller number. So, the maximum possible quantum volume is just the number of qubits squared.

To give you an idea, a 30-qubit system with no gate errors has a quantum volume of 900 (no units for this). To achieve the same quantum volume with imperfect gates, the error rate has to be below 0.1 percent. But, once this is achieved, all computations require 30 or fewer qubits can be performed on that quantum computer.

That seems simple enough, but figuring out the depth takes a bit of work because it depends on how the qubits are interconnected. So, the benchmark indirectly takes into account architecture.

The idea is that the minimum number of operations required to complete an algorithm occurs when every qubit is directly connected to every other qubit. But, in most cases, direct connections like that are not possible, so additional gates or qubits have to be added to connect qubits that are distant from each other. But each gate operation comes with the chance of introducing an error, so the depth changes.