Quantum error mitigation

For quantum computers to successfully solve real-world problems, it is necessary to tackle the challenge of noise: the errors that occur in elementary physical components due to unwanted or imperfect interactions. The theory of quantum fault tolerance can provide an answer in the long term, but in the coming era of noisy intermediate-scale quantum machines one must seek to mitigate errors rather than completely eliminate them. This review surveys the diverse methods that have been proposed for quantum error mitigation, assesses their in-principle efficacy, and describes the hardware demonstrations achieved to date. Commonalities and limitations among the methods are identified, while mention is made of how mitigation methods can be chosen according to the primary type of noise present, including algorithmic errors. Open problems in the field are identified, and the prospects for realizing mitigation-based devices that can deliver a quantum advantage with an impact on science and business are discussed.

Figure 1

Probability density distributions of the unmitigated estimator and the error-mitigated estimator. We see a decrease of bias and an increase of variance after performing error mitigation.

Figure 2

Error-mitigated estimate obtained using zero-noise extrapolation. We perform extrapolation using three noisy expectation values at the circuit fault rate of $\lambda =0.5$, 1, and 1.5, where 0.5 is the lowest circuit fault rate that we can achieve and the other two values are obtained by boosting the noise in the device.

Figure 3

Diagrammatic proof of Eq. (30) for three copies ($M=3$) with the observable acting on the first copy ($m=1$). The proof uses tensor network notations ([22]) and can be easily extended to more copies and/or with $O$ acting on the $m$th copy rather than the first.

Figure 4

Decomposition of the copy-swap operator $S_2$ into transversal qubit-swap operators ${\tilde{S}}_2$. A similar decomposition also applies to other cyclic copy-permutation operators $S_M$ with $M>2$.

Figure 5

The process of learning-based quantum error mitigation. For cases in which we need to construct variants of the input circuit for the error-mitigation function as in Eq. (49), the entire process is similar, but instead of running only the input circuits on the quantum hardware, we need to run all the different variants of the input circuits.

Figure 6

Hadamard test circuit for performing Pauli $O$ measurements. If $O$ is a general unitary, then the Hadamard test circuit in which we measure $X\otimes I$ will output the expectation value $\mathrm{Re}(\mathrm{Tr}[O\rho ])$, while measuring $Y\otimes I$ will output the expectation value $\mathrm{Im}(\mathrm{Tr}[O\rho ])$.

Figure 7

Circuits for measuring $O{S}_{+}=(OS+SO)/2$, in which $O$ is Pauli and $S$ is Hermitian. In the left circuit, we first perform a nondestructive measurement of $O$ and then measure $S$ and take the product of the measurement results. In the right circuit, the nondestructive measurement of $O$ is carried out using Hadamard tests, and the expectation value of $O{S}_{+}$ is obtained by simply measuring $X\otimes S$ at the end.