Characterization of Noise using variants of Unitarity Randomized Benchmarking

TL;DR

This paper implements the URB protocol in a quantum simulator with all the parameters and noise model are used from a real quantum device, and alters the m-URB protocol, namely, native gate URB or Ng-URB protocol, to study the noise in the native gates into which the quantum circuits are compiled in a quantum computer.

Abstract

Benchmarking of noise that is induced during the implementation of quantum gates is the main concern for practical quantum computers. Several protocols have been proposed that empirically calculate various metrics that quantify the error rates of the quantum gates chosen from a preferred gate set. Unitarity randomized benchmarking (URB) protocol is a method to estimate the coherence of noise induced by the quantum gates which is measured by the metric \textit{unitarity}. In this paper, we for the first time, implement the URB protocol in a quantum simulator with all the parameters and noise model are used from a real quantum device. The direct implementation of the URB protocol in a quantum device is not possible using current technologies, as it requires the preparation of mixed states. To overcome this challenge, we propose a modification of the URB protocol, namely the m-URB protocol, that enables us to practically implement it on any quantum device. We validate our m-URB protocol using two single-qubit noise channels -- (a) depolarising channel and (b) bit-flip channel. We further alter the m-URB protocol, namely, native gate URB or Ng-URB protocol, to study the noise in the native gates into which the quantum circuits are compiled in a quantum computer. Using our Ng-URB protocol, we can also detect the presence of cross-talk errors which are correlated errors caused due to non-local and entangling gates such as CNOT gate. For illustration, we simulate the noise of the native gates taking the noise parameter from two real IBM-Q processors.

Figures (4)

• Figure 1: URB curves showing the relationship between average shifted purities and sequence lengths (in Clifford unitaries) as a result of simulations of URB single copy protocol for completely depolarising noise with different values of parameter $p$, applied to single qubit register. (a) Depolarising parameter $p=0.9$, estimated unitarity $u=0.81015$. (b) Depolarising parameter $p=0.8$, estimated unitarity $u=0.64081$. (c) Depolarising parameter $p=0.7$, estimated unitarity $u=0.49238$. (d) Depolarising parameter $p=0.6$, estimated unitarity $u=0.36072$. For each simulation, the sequence lengths were taken to be $m \in \mathbb{M}$$= \left\lbrace 1,2, \dots ,10 \right\rbrace$, number of iterations were 15 and for each iteration, the sample size was 5. Overall variance in the estimated unitarities, in the experiments, are $4.495 \times 10^{-6}$, $6.953 \times 10^{-6}$, $1.272 \times 10^{-5}$ and $1.408 \times 10^{-5}$ respectively.
• Figure 2: URB curves showing the relationship between average shifted purities and sequence lengths (in Clifford unitaries) as a result of simulations of URB single copy protocol for the bit-flip channel with different values for probability $p$, applied to single qubit register. (a) For $p=0.975$, the estimated unitarity was found to be $u=0.935424$. (b) For $p=0.95$, the estimated unitarity was found to be $u=0.876434$. (c) For $p=0.9$, the estimated unitarity was found to be $u=0.772098$. (d) For $p=0.8$, the estimated unitarity was found to be $u=0.624513$. For each simulation, the sequence lengths were taken from $\mathbb{M}$, the number of iterations was 15 and for each iteration, the sample size was 5. Overall variance in the estimated unitarities, in the experiments, are $9 \times 10^{-7}$, $1.775 \times 10^{-5}$, $7.21 \times 10^{-5}$ and $1.81 \times 10^{-3}$ respectively.
• Figure 3: Results of experimental implementation of a variant of Standard URB procedure, to benchmark unitarity of native gates of Burlington processor (display_name: ibmq_burlington). URB curves associated with experiments done with native gates chosen as Identity (id), $U2$ (u2), $U3$ (u3), and CNOT (cx) are shown in Fig (a),(b),(c), and (d) respectively. The depths used for each benchmarking experiment were for identity and CNOT gates, $m \in \mathbb{M}$$= \left\lbrace 1,2, \dots ,10 \right\rbrace$ and for $U2$ and $U3$ gates, $m \in \mathcal{M} = \{5k \mid k \in \{1,2,3...10\}\}$. Number of iterations for each benchmarking experiment $= 15$ and number of samples $= 5$. Unitarity of the approximate noise map of id, u2, u3, and cx was found to be 0.998763, 0.998683, 0.997480, and 0.971898 respectively. Overall variance in the estimated unitarities, in the experiments, are $3.229 \times 10^{-9}$, $1.413 \times 10^{-7}$, $4.985 \times 10^{-8}$, and $6.894 \times 10^{-9}$ respectively over 15 iterations of repeating the entire experiment for id, u2 and u3 gates whereas 10 iterations were made for cx gate.
• Figure 4: Results of experimental implementation of Ng-URB procedure, to benchmark unitarity of native gates of Melbourne processor (display_name: ibmq_16_melbourne). URB curves associated with experiments done with native gates chosen as Identity (id), $U2$ (u2), $U3$ (u3), and CNOT (cx) are shown in Fig (a),(b),(c), and (d) respectively. The depths used for each benchmarking experiment are, for identity and CNOT gates $m \in \mathcal{M} = \{5k \mid k \in \{1,2,3...10\}\}$ and for $U2$ and $U3$ gates, $m \in \{ 1,2,3,4 \}$. Number of iterations for each benchmarking experiment $= 15$ and number of samples $= 5$. Unitarity of the approximate noise map of id, u2, u3, and cx was found to be 0.997995, 0.997861, 0.995872, and 0.930389 respectively. Overall variance in the estimated unitarities in the experiments are $2.778 \times 10^{-9}$, $2.928 \times 10^{-8}$, $2.755 \times 10^{-8}$, and $1.314 \times 10^{-7}$ respectively over 15 iterations of repeating the entire experiment for id, u2 and u3 gates whereas 10 iterations were made for cx gate.