Original Definition

The quantum volume of a quantum computer was originally defined in 2018 by Nikolaj Moll et al.^[9] However, since around 2021 that definition has been supplanted by IBM's 2019 redefinition.^[10][11] The original definition depends on the number of qubits N as well as the number of steps that can be executed, the circuit depth d

${\displaystyle {\tilde {V}}_{Q}=\min[N,d(N)]^{2}.}$

The circuit depth depends on the effective error rate ${\displaystyle \epsilon _{\mathrm {eff} }}$ as

${\displaystyle d\simeq {\frac {1}{N\epsilon _{\mathrm {eff} }}}.}$

The effective error rate ${\displaystyle \epsilon _{\mathrm {eff} }}$ is defined as the average error rate of a two-qubit gate. If the physical two-qubit gates do not have all-to-all connectivity, additional SWAP gates may be needed to implement an arbitrary two-qubit gate and ${\displaystyle \epsilon _{\mathrm {eff} }>\epsilon }$, where ${\displaystyle \epsilon }$ is the error rate of the physical two-qubit gates. If more complex hardware gates are available, such as the three-qubit Toffoli gate, it is possible that ${\displaystyle \epsilon _{\mathrm {eff} }<\epsilon }$.

The allowable circuit depth decreases when more qubits with the same effective error rate are added. So with these definitions, as soon as ${\displaystyle d(N)<N}$, the quantum volume goes down if more qubits are added. To run an algorithm that only requires ${\displaystyle n<N}$ qubits on an N-qubit machine, it could be beneficial to select a subset of qubits with good connectivity. For this case, Moll et al. ^[9] give a refined definition of quantum volume.

${\displaystyle V_{Q}=\max _{n<N}\left\{\min \left[n,{\frac {1}{n\epsilon _{\mathrm {eff} }(n)}}\right]^{2}\right\},}$

where the maximum is taken over an arbitrary choice of n qubits.

IBM's redefinition

In 2019, IBM's researchers modified the quantum volume definition to be an exponential of the circuit size, stating that it corresponds to the complexity of simulating the circuit on a classical computer:^[5][12]

${\displaystyle \log _{2}V_{Q}={\underset {n\leq N}{\operatorname {arg\,max} }}\left\{\min \left[n,d(n)\right]\right\}}$