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According to this paper, quantum volume is defined by the width or number of QV layers of the largest random square circuit (with width equal to the number of layers) that a quantum processor can successfully run. The way I understand this definition is that the circuit must be a square one where on one axis we have the number of circuit layers and on the other the number of qubits. The definition of circuit layer is presented in the same paper: define a QV layer as one layer of permutation among qubits and one layer of pair-wise random SU(4) 2-qubit unitary gates.

From the definitions above, I would say that the quantum volume of a quantum computer cannot be larger than the number of qubits. However, in the same paper, at Figure 4, a 5 qubit device is specified as having a quantum volume of 32. What am I missing here?

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If $N$ is the number of QV layers that a quantum processor can successfully run (and therefore the number of qubits of the QV circuit), the QV is actually given by $2^N$, so a QV of $32=2^5$ means $5$ qubits/layers.

The reasoning behind this is to highlight the dimension of the corresponding Hilbert space and the complexity of simulating these circuits classically.

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