Quantum Volume

A single number that measures a quantum computer's usable size — it folds qubit count, connectivity, gate fidelity, and crosstalk together into one figure. Adding more qubits does not move the number if they cannot work together across a deep enough circuit.

The question it tries to answer

"How many qubits do you have" is a dangerously misleading metric. A machine advertised at 50 qubits but plagued by high gate error and sparse connectivity may be far less useful than a 27-qubit machine with high fidelity and full connectivity. Quantum Volume (QV) exists precisely to answer a more honest question with one number: how large a circuit can this machine run reliably? Here "large" means both width (qubit count) and depth (number of gate layers), because real algorithms need both.

The crucial design choice is that QV tests square circuits — running circuits of depth n on n qubits. Tying width to depth means any single weakness drags the whole score down; no one can game the metric by simply piling on qubits.

Intuition — why random circuits

The test circuits are generated randomly: each layer randomly pairs the qubits two by two and applies a random SU(4) two-qubit gate to every pair. There are two reasons for this. First, random circuits smear out errors, making it hard for any single error source to be optimised away — and hard to "teach to the test." Second, the ideal output distribution of a random circuit is highly non-uniform: some bitstrings are naturally more probable (the "heavy outputs") and some less so.

On real hardware, each random SU(4) gate must be compiled into a sequence of native single- and two-qubit gates, so QV tests both the quality of the hardware and the capability of the compiler.

An ideal, noise-free machine samples those heavy-output bitstrings at a noticeably elevated rate. Noise does the opposite: it pushes the output distribution toward uniform, flattening the peaks and valleys. So "how often can I still sample a heavy output" becomes a sensitive ruler for noise.

The criterion — heavy-output probability and the 2/3 threshold

For a given random circuit, you first classically simulate the ideal statevector and use it to decide which output bitstrings count as "heavy" (their ideal probability exceeds the median over all outputs). Then you sample the real machine many times and measure how often the results land in the heavy-output set — the heavy-output probability (HOP).

For a sufficiently random circuit of large enough size, the ideal heavy-output probability is expected to be about (1+ln 2)/2 ≈ 0.846; under full decoherence (output tending to uniform) it approaches 0.5. The criterion is: with statistical significance, the machine's average heavy-output probability over a family of random circuits must exceed 2/3. The largest square size n that reliably clears the 2/3 threshold gives

QV = 2n

Note the exponential form: going from n to n+1 doubles QV. This means the QV number itself grows fast, while the n that actually changed underneath it only increments by one — always convert back to n in your head when reading it.

What it measures — and what it doesn't

• It is a holistic single number. QV will not tell you the fidelity of one particular gate or the connectivity of two specific qubits on its own; it folds all of that, together with crosstalk and compilation quality, into one figure. The upside is that it is hard to cheat; the downside is that it is hard to use for locating a bottleneck.
• It does not measure speed. Two machines with the same QV can differ greatly in how many circuits they complete per second. To address this, the field added throughput metrics such as CLOPS (circuit layer operations per second).
• Its square constraint saturates. When the qubit count far exceeds the depth QV can reach, more qubits cannot raise QV, because depth has become the bottleneck. For machines with thousands of qubits, QV alone is no longer enough to characterise their capability.
• It depends on classical simulability. Deciding heavy outputs requires classically computing the ideal distribution, so QV can only be measured at scales that are still classically simulable — which is exactly what caps the largest size it can certify.

The measurement procedure, step by step

1. Choose a square size n and randomly generate a batch of depth-n model circuits.
2. For each circuit, classically compute the ideal statevector and determine its heavy-output set.
3. Sample each circuit many times on the real machine, counting how often results fall in the heavy-output set.
4. Average the heavy-output probability over the whole batch and run a statistical significance test.
5. Take the largest n that reliably clears the 2/3 threshold, giving QV = 2ⁿ.

Generating a model circuit

Read the kernel below against the principles above: it applies, layer by layer on n qubits, random SU(4) gates acting on randomly paired qubits — exactly the random square circuits the QV protocol prescribes. get_state retrieves the ideal statevector used to determine the heavy-output set; sample simulates sampling the real machine, and its frequencies are then compared against that ideal distribution to compute the heavy-output probability.

import qalgora

@qalgora.kernel
def qv_circuit(n: int, perms: list[int], angles: list[float]):
    q = qalgora.qvector(n)
    # layers of random SU(4) gates on permuted qubit pairs ...
    mz(q)

# heavy-output probability is compared to the ideal statevector
ideal = qalgora.get_state(qv_circuit, n, perms, angles)
counts = qalgora.sample(qv_circuit, n, perms, angles, shots_count=10000)

References

• A. W. Cross, L. S. Bishop, S. Sheldon, P. D. Nation, J. M. Gambetta, "Validating quantum computers using randomized model circuits," Phys. Rev. A 100, 032328 (2019). arXiv:1811.12926