Multiqubit randomized benchmarking using few samples

Randomized benchmarking (RB) is an efficient and robust method to characterize gate errors in quantum circuits. Averaging over random sequences of gates leads to estimates of gate errors in terms of the average fidelity. These estimates are isolated from the state preparation and measurement errors that plague other methods such as channel tomography and direct fidelity estimation. A decisive factor in the feasibility of randomized benchmarking is the number of sampled sequences required to obtain rigorous confidence intervals. Previous bounds were either prohibitively loose or required the number of sampled sequences to scale exponentially with the number of qubits in order to obtain a fixed confidence interval at a fixed error rate. Here, we show that, with a small adaptation to the randomized benchmarking procedure, the number of sampled sequences required for a fixed confidence interval is dramatically smaller than could previously be justified. In particular, we show that the number of sampled sequences required is essentially independent of the number of qubits and scales favorably with the average error rate of the system under investigation. We also investigate the fitting procedure inherent to randomized benchmarking in light of our results and find that standard methods such as ordinary least squares optimization can give misleading results. We therefore recommend moving to more sophisticated fitting methods such as iteratively reweighted least squares optimization. Our results bring rigorous randomized benchmarking on systems with many qubits into the realm of experimental feasibility.

Figure 1

Randomized benchmarking protocol. We perform randomized benchmarking using the Clifford group $\mathsf{\text{C}}$; i.e., all gates that can be constructed by successive application of cnot gates, Hadamard gates, and $\pi /4$ phase gates. We assume the input states $\rho ,\left(\widehat{\rho }\right)$ to be noisy implementations of the states $2^{-q}(I+\mathbf{P}),\left[2^{-q}(I-\mathbf{P})\right]$, and $Q$ is a noisy implementation of the observable $\mathbf{P}$ where $\mathbf{P}$ is a Pauli operator. We denote the length of an RB sequence by $m$, the amount of random sequences for a given $m$ by $N$, and the amount of times a single sequence is repeated by $L$. The goal of this paper is to provide confidence intervals around the empirical average $k_{m,N}$ assuming that individual realizations $k_m\left(\vec{G}\right)$ are estimated to very high precision (corresponding to the case $L\to \infty$). In experimental implementations, running the same sequence many times $\left(L\right)$ is typically easy, but running many different sequences $\left(N\right)$ is hard [25], meaning that the quantity that we want to minimize is $N$. See Sec. 4 for a detailed discussion of the construction of confidence regions around the empirical average $k_{m,N}$.

Figure 2

Improvements in dimensional and sequence length scaling. The number of sequences needed (on a log scale) to obtain a $99\%$ confidence interval around $p_{m,N}$ with $\epsilon ={10}^{-2}$ for a prior infidelity $r={10}^{-3}$ as a function of (a) the sequence length $m$ for a single qubit $(q=1)$ from Eq. (9) (full line red) compared to the single-qubit bound from Ref. [24], Eq. (6)] (dashed green) and a trivial bound that arises from noting that the distribution sampled from is bounded on the interval [0,1] and hence has a variance at most $1/4$ (dot-dashed blue) and (b) the number of qubits from Eq. (11) (full line) for sequence length $m=100$ compared to the multiqubit bound from from Ref. [24], Eq. (4)] (dashed green). In both cases, our bounds are asymptotically constant while the bounds from Ref. [24] diverge. Our bounds are also substantially smaller than the trivial bound. For multiple qubits, we set the SPAM contribution to $\eta =0.05$ while for a single qubit we set the SPAM contribution to $\eta =0$ in both bounds. We also assumed the unitarity to be $u=(1+f^2)/2$, where $f$ is the depolarizing parameter, corresponding to somewhat but not fully coherent noise.

Figure 3

(a) Number of sequences needed for a $99\%$ confidence interval of size $\epsilon =5r$ for various infidelities $r$ (ranging from $r=5\times {10}^{-3}$ to $r={10}^{-4}$), number of qubits $q\in [1,10]$, and sequence length $m=100$ using Eq. (10) under the assumption of negligible SPAM. (similar plots can be made without this assumption). The number of sequences needed increases with decreasing infidelity, reflecting the generic statistical rule that higher precision requires more samples. Note that even in the case of infidelity $r=2\times {10}^{-4}$ the number of sequences required is within experimental limits. (b) Variance, as given by Eq. (11), vs infidelity $r$ (taking $d=16$ and $m=100$ for illustration) for various levels of SPAM $\eta \in \{0,0.01,0.05,0.1,0.5\}$. Note that the size of the SPAM term has a strong influence on the variance and hence the number of sequences required, especially in the small-$r$ limit. As indicated by the visual aids, this is due to the transition from a variance scaling quadratically in infidelity $r$ (small $\eta$) to a variance scaling linearly in the infidelity $r$ (large $\eta$).

Figure 4

(a) Number of sequences needed for a $99\%$ confidence interval of size $\epsilon =0.01$ around $k_{m,N}$ for various values of the unitarity [given by a linear interpolation between $f^2$ and 1 where $\kappa =1$ corresponds to $u=1$ (unitary noise) and $\kappa =0$ corresponds to $u=f^2$ (depolarizing noise)] for fixed infidelity $r=0.01$ and sequence length in the interval $m\in [1,10000]$ (log scale) using the variance Eq. (9). We also assume $d=16$ (four qubits) and ideal SPAM ($\eta =0$). Note that the number of sequences differs radically for $u=1$ (unitary noise). In the case of $u<1$, the number of sequences needed rises with increasing sequence length $m$, peaks, and then decays to zero but for $u=1$ the number of sequences keeps rising with increasing sequence length $m$ until it converges to a nonzero constant (which will be independent of $r$). In Sec. 4h, we argue that this is expected behavior for randomized benchmarking with unitary noise. [(b), (c)] Contour plot of the variance bound with infidelity on the $y$ axis ($r\in [0.01,0.1]$) and sequence length $m$ on the $y$ axis ($m\in [1,100]$). For panel (b), we have set the unitarity to $u=(1+f^2)/2$ corresponding to relatively incoherent noise, and for panel (c), we have set the unitarity $u=1$ corresponding to coherent noise. Note again the radical difference in behavior. For $u=1$, the variance rises monotonically in the sequence length $m$ to a constant independent of the infidelity $r$. Moreover, the variance is monotonically increasing in infidelity $r$. However, for incoherent noise, the variance will peak strongly around $mr\approx 1$ and then decay to zero with increasing sequence length $m$. This means that both upper and lower bounds on the infidelity are required to make full use of the bound in Eq. (9). The looser bound of Eq. (10) does not share this property and can be used with only an upper bound on the infidelity $r$.