Character Randomized Benchmarking for Non-Multiplicity-Free Groups With Applications to Subspace, Leakage, and Matchgate Randomized Benchmarking

Randomized benchmarking (RB) is a powerful method for determining the error rate of experimental quantum gates. Traditional RB, however, is restricted to gatesets, such as the Clifford group, that form a unitary 2-design. The recently introduced character RB can benchmark more general gates using techniques from representation theory; up to now, however, this method has only been applied to “multiplicity-free” groups, a mathematical restriction on these groups. In this paper, we extend the original character RB derivation to explicitly treat non-multiplicity-free groups, and derive several applications. First, we derive a rigorous version of the recently introduced subspace RB, which seeks to characterize a set of one- and two-qubit gates that are symmetric under swap. Second, we develop a new leakage RB protocol that applies to more general groups of gates. Finally, we derive a scalable RB protocol for the matchgate group, a group that like the Clifford group is nonuniversal but becomes universal with the addition of one additional gate. This example provides one of the few examples of a scalable non-Clifford RB protocol. In all three cases, compared to existing theories, our method requires similar resources, but either provides a more accurate estimate of gate fidelity, or applies to a more general group of gates. In conclusion, we discuss the potential, and challenges, of using non-multiplicity-free character RB to develop new classes of scalable RB protocols and methods of characterizing specific gates.

Popular Summary

Advances in accurate and scalable methods for characterizing the performance of quantum gates are critical for the realization of large-scale reliable quantum computers. However, when testing quantum processors, it is difficult to determine whether the error comes from a gate’s implementation or from another source. A common approach is randomized benchmarking (RB), which uses variable-length sequences of quantum gates to separate out the effect of gate errors in a large-scale computation. In our work, we extend randomized benchmarking to apply to a larger class of logical operators, in order to determine gate errors for new gates that could not be previously characterized.

We derive a generalization of an existing RB procedure, character RB, to apply to a larger class of gates. Our method, non-multiplicity-free character RB, applies to any gates that form a group, and requires knowledge of the group’s representations. Our method provides a broad framework for benchmarking quantum computers by forming elementary gates into groups.

As initial examples, we demonstrate three concrete applications of our theory. First, we develop a method to benchmark the two-qubit gate currently being investigated in a trapped ion system, in a way that allows for rigorous estimates of its fidelity. Second, we demonstrate how our procedure can determine how often qubits “leak” out of their computational space, in more generality than existing leakage RB proposals. Finally, we demonstrate how to benchmark circuits built out of matchgates, a set of quantum gates that can be efficiently simulated on a one-dimensional line of qubits but becomes universal in two dimensions. Our matchgate RB is scalable in the size of the quantum computer, and represents one of the few examples of a scalable RB group. These applications are highly relevant to cutting-edge technologies, including characterizing the two-qubit gate currently used at Honeywell Quantum Solutions. We anticipate our method will have many additional applications, such as characterizing new elementary gates and discovering further scalable RB groups.

Figure 1

The elements of the benchmarking group $G$ are constructed by composing elementary gates as shown above to implement elements of $G_T$ on the triplet subspace. Each group element contains exactly four entangling gates.

Figure 2

The predicted and measured character-weighted survival probability for a random error channel. The exact decay (green) is an exponential decay given by Eq. (11). We estimate $S_i(N)$ by applying random gates and measuring the final state (blue points). The data is fit to an appropriate function (orange) from which we estimate the fidelity.

Figure 3

The exact and estimated fidelity for a selection of randomly generated error channels. Each estimate is based on data taken over 15 different lengths $N$. Each estimate is arrived at by applying a total of 150 000 benchmarking group elements. This is the same number of elements applied in the experiment described in Ref. [23]. The diagonal line denotes the points where the exact and estimated fidelities are equal. The data agree with the line with a reduced ${\chi }^2$ value of $.9$, indicating good agreement. Note that the error bars are derived from statistical uncertainty in the data, and vanish in the limit of an infinite number of data points.

Figure 4

(a) The extended subfidelity ${\tilde{F}}_{\mathrm{\Lambda }}$ of Ref. [23] versus the exact fidelity $F_{\mathrm{\Lambda }}$ that we can estimate in our paper, for a selection of error channels of varying strengths: intensity errors, which correspond to an overrotation $e^{-iϵZZ}$; optical pumping errors, which cause amplitude damping on each qubit; inhomogeneous fields, which cause phase damping on each qubit; and swap errors, which interchange the qubits. This plot corresponds to the exact value of both $F_{\mathrm{\Lambda }}$ and ${\tilde{F}}_{\mathrm{\Lambda }}$ that one estimates in an experiment. Note that while the ${\tilde{F}}_{\mathrm{\Lambda }}$ agrees with $F_{\mathrm{\Lambda }}$ in the limit $F_{\mathrm{\Lambda }}\to 1$, in general the two do not agree, and there exists worst-case errors such as swap that ${\tilde{F}}_{\mathrm{\Lambda }}$ cannot detect. (b),(c) Simulation of an experiment that estimates $F_{\mathrm{\Lambda }}$ versus ${\tilde{F}}_{\mathrm{\Lambda }}$ for a total of $300000$ unitaries, in the case of (b) intensity and (c) swap errors of varying strengths. These plots correspond to experiments that estimate the exact values shown in (a). We see that the difference between $F_{\mathrm{\Lambda }}$ and ${\tilde{F}}_{\mathrm{\Lambda }}$ can be discerned in a realistic experiment.

Figure 5

The predicted and measured $S_0(N)$ for a single randomly generated error channel. The actual decay (green) is an exponential decay given by Eq. (18). We estimate $S_0(N)$ by applying random gates and measuring the final state (blue points). The data is fit to a function of the form of Eq. (18), from which we estimate $L$ and $S$.

Figure 6

The exact and estimated leakage and seepage for a selection of randomly generated error channels. Each estimate is based on data taken over 15 different lengths $N$. Each estimate is arrived at by applying a total of 300 000 unitary group elements. The diagonal line denotes the points where the exact and estimated fidelities are equal. The data agree with this line with a reduced ${\chi }^2$ value of $1.3$, indicating good agreement.

Figure 7

The predicted and measured character-weighted survival probability for a random error channel. The exact decay (green) is an exponential decay given by one of Eq. (23). We estimate $S_i(N)$ by applying random gates and measuring the final state (blue points). The data is fit to an appropriate function (orange) from which we estimate the fidelity.

Figure 8

The exact and estimated fidelity for a selection of random errors. Each estimate is based on data taken over 15 different lengths $N$. Each estimate is arrived at by applying a total of 300 000 unitary group elements. The diagonal line denotes the points where the exact and estimated fidelities are equal. The data agree with the line with a reduced ${\chi }^2$ value of $1.0$, indicating good agreement.