The metric Quantum Volume (QV) has been in the news lately and it deserves a few more words of explanation.  This metric was first created by IBM in 2017 and modified in 2018 as a metric that would allow comparison of different quantum computers.  It has many valuable characteristics.  It is a well-documented, straightforward way of comparing different quantum computers that can be run on any gate level quantum computer, not just ones that use a superconducting technology.  Most important, it considers qubit count, qubit quality and other factors so it does not emphasize the number of qubits alone.  It is a good tool for use by quantum hardware engineers to measure their progress in development.  If they are able to come up with a new generation of hardware that increases this metric, they are going in the right direction.  However, it is a poor tool for end users to use if they want to measure the goodness of a quantum computer for solving their computation problems.

We will explain why Quantum Volume is not really appropriate for end users in this brief.  But first, we will provide a simplified explanation of how this measure is derived.  The quantum volume measurement involves testing a series of circuits with a square configuration.  By that, we mean that the number of qubits equals the gate depth of the circuit.  For example, a test may begin by trying out a 2×2 configuration.  This would mean a 2 qubit circuit with each gate going through 2 gate levels.  In these tests, the gate sequences are alternating sequences of single qubit gates followed by two qubit gates.  The circuit is run multiple times with the results measured and the results are analyzed with a certain statistical test to see how accurate the answer is to what would be the theoretical result.  If the test passes a certain criteria, then the test is repeated for a 3×3 configuration, a 4×4 configuration, etc.  Because qubits are imperfect, each increased level gets harder and harder because the gate errors start stacking up and the test no longer passes the criteria. Once the largest square array is found that will pass the test, the Quantum Volume is calculated as 2n where n is the number of qubits and gate depth.  So a 4×4 circuit would have a QV of 16, a 5×5 circuit would have a QV of 32, a 6×6 circuit would have a QV of 64, etc. For complete details on how this test is conducted, we refer you to the paper that IBM posted on arXiv.

Here are our concerns about Quantum Volume:

1. The test is all based upon a square circuit configuration, but very few quantum programs really have a square configuration.  Some of the algorithms being developed for NISQ computers such as VQE and QAOA, are wide and shallow.  This means that use a larger number of qubits but only a few levels of gate depth.  Others, may have a much larger number of gate operations versus the numbers of qubits.  For example, Shor’s algorithm can theoretically factor a 2048 bit number using about 4100 logical qubits, but it requires about 8.6×109 gate operations.  (Note that is based upon logical or error-free qubits and not physical qubits).

2. We do not agree with calculating the quantum volume by using a formula of 2n.  We think this gives a distorted view of how fast the quantum computer performance is increasing.  Let us explain this by an analogy.  Let’s suppose we were looking for office space and the landlord shows us two offices spaces with dimensions 5×5 or 6×6.  (You can use either feet or meters as the dimension depending upon what country you are in.) Would you expect to pay twice as much for the 6×6 office as the 5×5 office?  No!  You could calculate the worth by looking at the square area and determine that the 6×6 is about 44% more valuable (36/25) than the 5×5, not 100%.  In a quantum computing algorithm, we do not think an end user would be able to increase their problem size by 100% if they were provided a new quantum computer that had just one more qubit and one more gate level.

3. The focus for anyone developing a quantum computer should be how to make it achieve quantum advantage and solve problems better than a classical computer.  Since classical computers are error free, the equivalent quantum volume for a quantum program running on a quantum simulator on a classical computer can be very high.  For example, in 2019, Google ran a quantum benchmark on the Summit supercomputer at the Oak Ridge National Lab that successfully calculated the results of a 49×40 circuit.  So the equivalent QV for Summit would be 240 or about 1.1 x 1012.

So for anyone claiming to have the world’s highest performance quantum computer, a high Quantum Volume figure is helpful, but we would not regard it as definitive proof.  Perhaps a more appropriate challenge would be to replicate or beat Google’s Quantum Supremacy experiment.  At this point in time, anyone who can achieve that would indeed have something noteworthy.

March 5, 2020