Validating quantum computers using randomized model circuits

We introduce a single-number metric, quantum volume, that can be measured using a concrete protocol on near-term quantum computers of modest size ($n\lesssim 50$), and measure it on several state-of-the-art transmon devices, finding values as high as 16. The quantum volume is linked to system error rates, and is empirically reduced by uncontrolled interactions within the system. It quantifies the largest random circuit of equal width and depth that the computer successfully implements. Quantum computing systems with high-fidelity operations, high connectivity, large calibrated gate sets, and circuit rewriting toolchains are expected to have higher quantum volumes. The quantum volume is a pragmatic way to measure and compare progress toward improved system-wide gate error rates for near-term quantum computation and error-correction experiments.

Figure 1

Model circuit. A model circuit consists of $d$ layers of random permutations of the qubit labels, followed by random two-qubit gates. When the circuit width $m$ is odd, one of the qubits is idle in each layer. A final permutation can be applied to the labels of the measurement outcomes.

Figure 2

Experimental data for square (width equals depth) quantum volume circuits using the IBM Q 20-qubit device Tokyo. The ideal simulation results are green plus signs. The noisy simulations, using a depolarizing noise model with average error rates from the qubits used on the device, are red circles. The experiments using 200 circuits are blue squares. The dotted line is the threshold of $\frac{2}{3}$ for heavy output generation and the dashed green line is the asymptotic ideal heavy output probability of $\frac{1+ln2}{2}$ [19], which the ideal simulations quickly approach. In order to set a high confidence level that $h_d$ surpasses the threshold, the point at $m=d=3$ was repeated with 5000 circuits (cyan diamond). This number of shots corresponds to a stricter threshold of 0.68, indicated by the solid line at the experimental points for $m=3$.

Figure 3

Experimental data for square (width equals depth) quantum volume circuits using the IBM Q Johannesburg 20-qubit device. As in Fig. 2, the ideal simulation results are green plus signs, the noisy simulations are red circles, and the experiments are blue squares. Again, the dotted line is the threshold of $\frac{2}{3}$ for heavy output generation and the dashed green line is the asymptotic ideal heavy output probability. The additional point at $m=d=4$ (magenta triangle) is not only repetition with more circuits but experimental results using optimized circuits with the $KAK$ approximation, assuming 1% error gates. The experiments with optimized circuits were run with 1000 circuits. The threshold for this number of circuits is 0.695 and is indicated by the solid line at $m=4$.

Figure 4

The quantum volume increases as a function of inverse gate error $1/\epsilon$. This plot shows numerical simulation results from the top half of Table 3 together with estimates using the expression in Eq. (8) for grid and loop connectivities.

Figure 5

Average gate fidelity for random target gates in the Haar measure, for approximations using zero, one, or two applications of a two-qubit supercontrolled basis gate, with and without freedom to mirror. These approximations are optimal for the case that the basis gate is equivalent to cnot.

Figure 6

Number of basis gate applications used for approximating randomly chosen target gates in the Haar measure, choosing the approximation according to Eq. (B17) as a function of the basis gate fidelity $F_b$ (a) without mirroring, as in Eq. (B17a), and (b) with mirroring, as in Eq. (B17b). The fraction of cases with each number of applications is shown by shading (left axis) and the mean number of basis applications is shown by the dashed line (right axis).

Figure 7

Effective infidelity ratio as a function of basis gate infidelity, with and without freedom to mirror.

Figure 8

Device diagrams used for the experimental data in Table 1: (a) Tenerife, (b) Melbourne, (c) Tokyo, and (d) Johannesburg. The highlighted qubits are those selected for the experiments discussed here. cx gates are available between pairs of qubits connected by a highlighted line.