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The quantum volume is typically taken as a popular metric to determine the capability of a quantum computer to solve complex problems. Since the end goal is to develop quantum computers that can solve problems that classical computers cannot, I am wondering if one can calculate a quantum volume associated to a classical computer. If one did this for the largest supercomputers, then this could give, for example, a value of the quantum volume that must be surpassed for quantum computers to be advantageous over classical computers.

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The log quantum volume of Quirk ( https://algassert.com/quirk ) is 16, because it has a maximum of 16 qubits and can easily do the ~200 gates that are part of the test circuit. The slowest part will be drawing the result.

For state vector simulators that don't impose maximum qubit counts, like qsim, the limit is basically available memory and the size of floating point numbers. So, for a desktop machine with 8 gigabytes of memory the log quantum volume will be somewhere around 30 and for a supercomputer with tens of petabytes of storage the log quantum volume will be somewhere around 50.

According to the wikipedia page on quantum volume, IBM has reported a log quantum volume of 6 and Honeywell has reported a log quantum volume of 9. So qubit counts need to get between 5x and 10x better, and fidelities need to get between 25x and 100x better, before physical quantum machines start beating super computers at this metric.

(If you're wondering why I keep saying log quantum volume instead of quantum volume, it's because measuring not-log quantum volume is like saying that adding 20 bits of storage to a terabyte hard drive makes it a million times better.)

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    $\begingroup$ Here quantum-computing.ibm.com/catalog you can find a system with quantum volume 128 (log = 7) $\endgroup$ May 13 at 8:09
  • $\begingroup$ Well to be fair, QV is equal to the dimension of the Hilbert space over which the algorithm was successful. This need not be the full Hilbert space of the system. Taking the log is what we did in the paper, and just amounts to measuring Hilbert space dimension in terms of number of qubits. This is not the same as your hard drive analogy. $\endgroup$ May 15 at 16:14

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