• Hardware

Quantum Volume

A hardware benchmark that measures the largest square random circuit a quantum computer can run with a success probability above 2/3, capturing qubit count, connectivity, and error rates together in a single number.

Quantum Volume (QV) is a single-number benchmark developed by IBM in 2019 to characterize the overall capability of a quantum processor. A machine with quantum volume 2n2^n can successfully run random square circuits of width nn and depth nn. This matters because a processor with 50 low-quality qubits is often less useful than one with 20 high-quality qubits: QV captures that trade-off in a way that raw qubit counts do not.

The details

A quantum volume test works as follows. Random circuits of width dd and depth dd are generated using layers of random two-qubit gates applied to randomly chosen qubit pairs. The circuit is then run on hardware, and the same circuit is simulated classically to obtain the ideal output distribution.

The test uses the heavy output probability: the probability that a shot from the hardware lands on one of the “heavy” outputs (those with probability above the median in the ideal distribution). For an ideal device, the heavy output probability is approximately 0.850.85. The benchmark passes if the device achieves a heavy output probability greater than 2/32/3 with high statistical confidence.

Quantum Volume is defined as:

QV=2nQV = 2^n

where nn is the largest circuit width (and depth) at which the device passes the heavy output probability test. IBM’s earliest systems had QV=4QV = 4 or QV=8QV = 8. By 2023, IBM reported QV=512QV = 512 on Eagle-class processors.

Why QV captures more than qubit count. A machine with many noisy, poorly connected qubits may fail a QV=64QV = 64 test while a smaller machine with better gates and full connectivity passes QV=256QV = 256. Quantinuum’s H-series trapped-ion processors demonstrate this clearly: despite having tens of qubits rather than hundreds, they have achieved among the highest reported quantum volumes because their two-qubit gate fidelities exceed 99.9% and their all-to-all connectivity removes routing overhead.

Measuring heavy output probability in Qiskit:

from qiskit.circuit.random import random_circuit
from qiskit import transpile
from qiskit_aer import AerSimulator
import numpy as np

def heavy_output_probability(n_qubits, shots=8192):
    # Generate a random square circuit
    qc = random_circuit(n_qubits, depth=n_qubits, measure=True)

    # Ideal simulation (statevector)
    ideal_sim = AerSimulator(method="statevector")
    ideal_job = ideal_sim.run(transpile(qc, ideal_sim), shots=shots)
    ideal_counts = ideal_job.result().get_counts()

    # Compute median probability
    total = sum(ideal_counts.values())
    probs = {k: v / total for k, v in ideal_counts.items()}
    median_prob = np.median(list(probs.values()))

    # Heavy outputs are those above the median
    heavy_outputs = {k for k, v in probs.items() if v > median_prob}

    # Compute heavy output probability from noisy device counts
    # (here we reuse the ideal counts as a placeholder)
    heavy_shots = sum(ideal_counts.get(k, 0) for k in heavy_outputs)
    return heavy_shots / total

hop = heavy_output_probability(n_qubits=4)
print(f"Heavy output probability: {hop:.3f}")
print(f"Passes QV=16 test: {hop > 2/3}")

Limitations of Quantum Volume. QV does not measure speed. IBM introduced CLOPS (Circuit Layer Operations Per Second) to capture throughput, since a high-QV machine that runs slowly may still be impractical. QV also does not directly measure performance on application-relevant circuits, which may have structure very different from random circuits. Finally, QV scales exponentially in label but the underlying circuit width only scales linearly, so large QV numbers can be misleading without context.

Why it matters for learners

When evaluating hardware announcements, QV lets you compare machines across vendors on a common scale. A vendor claiming “1,000 qubits” tells you very little if those qubits have gate error rates of 5%. A vendor reporting QV=512QV = 512 tells you something concrete about usable circuit complexity.

Understanding QV also builds intuition for the relationship between qubit count, gate fidelity, and connectivity; all three must improve together for quantum processors to become practically useful.

Common misconceptions

Misconception 1: Higher qubit count means higher Quantum Volume. Qubit count is only one factor. Two-qubit gate fidelity and qubit connectivity are often the binding constraints. A 20-qubit processor with 99.9% two-qubit fidelity can achieve far higher QV than a 1,000-qubit processor with 99% fidelity.

Misconception 2: QV = 512 means you can run circuits on 512 qubits. The exponent in QV=2nQV = 2^n means n=9n = 9. A machine with QV=512QV = 512 can reliably run square circuits on 9 qubits. The number 512 is not the qubit count.

Misconception 3: QV is the definitive benchmark for quantum hardware. QV is one useful benchmark, but researchers and engineers use many others: CLOPS for speed, application benchmarks for problem-specific performance, and randomized benchmarking for gate-level fidelity. No single number fully characterizes a quantum processor.

See also