Scalable Evaluation of Quantum-Circuit Error Loss Using Clifford Sampling

A major challenge in developing quantum computing technologies is to accomplish high precision tasks by utilizing multiplex optimization approaches, on both the physical system and algorithm levels. Loss functions assessing the overall performance of quantum circuits can provide the foundation for many optimization techniques. In this Letter, we use the quadratic error loss and the final-state fidelity loss to characterize quantum circuits. We find that the distribution of computation error is approximately Gaussian, which in turn justifies the quadratic error loss. It is shown that these loss functions can be efficiently evaluated in a scalable way by sampling from Clifford-dominated circuits. We demonstrate the results by numerically simulating 10-qubit noisy quantum circuits with various error models as well as executing 4-qubit circuits with up to ten layers of 2-qubit gates on a superconducting quantum processor. Our results pave the way toward the optimization-based quantum device and algorithm design in the intermediate-scale quantum regime.

Figure 1

(a) An example quantum circuit. The circuit $(\mathit{F},\mathit{R})$ consists of a circuit frame $\mathit{F}$ and single-qubit unitary gates $\mathit{R}$. Single-qubit gates are the green dashed squares, and all other operations form the frame. To evaluate $L_{\mathbb{R}}(\mathit{F})$, the frame is always the same, but we sample random configurations of single-qubit gates. (b) The noise model. The system (Sys.) formed by qubits and the environment (Env.) are initialized in the state ${\rho }_i$. Following the initialization, there is a sequence of completely positive maps applied. The map ${\mathcal{M}}_{\tau }$ describes the evolution applied at the time $\tau$ to realize a layer (column) of quantum gates, which actually acts on the system and environment because of imperfections. Finally, we implement the measurement on each qubit. The operator of observable is $E_f$, which acts on both the system and environment because of imperfections.

Figure 2

Numerical results. (a) The unitary sampling of $\text{error}(\mathit{F},\mathit{R})$. The red curve denotes the Gaussian distribution $\mathcal{N}(0,L_{\mathbb{U}})$. (b) The Clifford sampling of $\text{error}(\mathit{F},\mathit{R})$. A 10-qubit circuit with 100 2-qubit gates is used to compute the mean of a Pauli operator. The error rate per gate is 0.002.

Figure 3

Experiment setup. (a) Diagram of four Xmon qubits coupled to a central bus resonator. (b) A single-qubit gate is realized using an $R_{xy}$ gate followed by an $R_z$ gate. The $R_{xy}$ gate has two parameters ${\theta }_{xy}$ and ${\varphi }_{xy}$, which are controlled by the amplitude and phase of the $XY$ pulse (red). The pulse length is 40 ns. The $R_z$ gate has only one parameter ${\theta }_z$, which is controlled by the amplitude of the $Z$ pulse (blue), whose length is 10 ns. (c) Pulse sequences (square $Z$ pulse in blue and sinusoidal microwave in red for each qubit) for the 2-qubit gate $U_{\text{phase}}$. A short $Z$ pulse is applied on one of the qubits before the long $Z$ pulse to align the $x$ axes of their Bloch spheres. We note that the driving field (${\mathrm{\Omega }}_i$ on the qubit $Q_i$) is different in $U_{\text{phase}}$ gates on different qubits. The length of $U_{\text{phase}}$ is around 300 ns. Phases of driving fields are inverted at the middle of the gate. (d) The quantum circuit used to demonstrate the Clifford sampling.

Figure 4

Experimental results. (a) The unitary sampling of $\text{error}(\mathit{F},\mathit{R})$. The red curve denotes the Gaussian distribution $\mathcal{N}(0,L_{\mathbb{U}})$. (b) The Clifford sampling of $\text{error}(\mathit{F},\mathit{R})$. (c) Moments of two sampling approaches. ${\mu }_n=E[{\text{error}}^n]$ is the $n$th-order moment, and ${\mu }_n^G$ is the moment of $\mathcal{N}(0,L_{\mathbb{U}})$. The quantum circuit used to generate data in (a)–(c) is in Fig. 3. (d) Error losses of unitary sampling versus Clifford sampling for 50 randomly generated frame-operation configurations with one to ten layers of 2-qubit gates. Each layer has one or two $U_{\text{phase}}$ gates. When only one gate is applied, we protect the other two qubits from dephasing by applying dynamically driving fields [76]. The observable is $|0⟩⟨0|$ for one of four qubits. See the Supplemental Material [62] for details. The error bar denotes the standard deviation due to finite sampling.