Quantum Volume

Quantum volume (QV) is a single number that summarizes how large a circuit a device can run reliably. Unlike RB, which reports a per-gate error, QV is holistic: it bundles qubit count, connectivity, gate fidelity, crosstalk, and compiler quality into one figure of merit, so two very different machines can be ranked on the same scale.

Square model circuits

QV is defined through model circuits that are deliberately square: a circuit on $m$ qubits with $m$ layers (depth $m$ ). Each layer does two things (Cross et al.):

A random permutation of the $m$ qubits, pairing them up.
On each pair, a Haar-random two-qubit gate drawn from $SU(4)$ .

The circuit is "square" because width and depth grow together — increasing $m$ stresses the device on both axes at once. The largest $m$ a device can pass becomes its quantum volume.

Heavy outputs and the passing criterion

For each model circuit, classically compute the ideal output distribution and find its median probability. A bitstring is a heavy output if its ideal probability exceeds that median. Because the ideal distribution is Porter–Thomas-like, the heavy outputs carry more than half the total probability — an ideal device produces them with probability

p_{\text{heavy}}^{\text{ideal}} \approx \frac{1 + \ln 2}{2} \approx 0.847

in the large- $m$ limit. A noisy device produces them less often. The device passes width $m$ if, averaged over many random model circuits, it generates heavy outputs more than two-thirds of the time with adequate statistical confidence:

p_{\text{heavy}} > \tfrac{2}{3}.

The threshold $2/3$ sits between the ideal $0.847$ and the value $1/2$ that a useless (uniform-output) device would give.

The quantum-volume number

Let $m^\ast$ be the largest width the device passes. The quantum volume is

V_Q = 2^{\,m^\ast}.

The exponential makes the number grow fast — a device that reliably runs square circuits up to $m^\ast = 6$ has $V_Q = 64$ . The exponent $m^\ast$ is the part that actually carries information; the exponentiation is a convention so that bigger is dramatically better.

Building a model-circuit layer

The atom of a QV model circuit is one layer: pair the qubits, then apply an entangling two-qubit block to each pair. For two qubits there is exactly one pair and no nontrivial permutation, so a layer is just a single two-qubit block. A simple, fully-entangling representative is "Hadamard on each qubit, then a controlled- $Z$ " — it spreads single-qubit superposition and then entangles, the two ingredients every $SU(4)$ block contains.

Try it

Implement a single two-qubit model-circuit block on 2 qubits: apply a Hadamard to each qubit and then a controlled- $Z$ between them. The grader checks the resulting two-qubit unitary against the one derived from the reference solution.

Run your code to see the quantum state.

This $4\times 4$ unitary is one layer of a width-2 model circuit — the smallest building block of a quantum-volume experiment.