Error mitigation for short-depth quantum circuits

TL;DR

Mitigating errors in short-depth quantum circuits is essential for near-term quantum simulations. The paper proposes two complementary strategies: zero-noise extrapolation via Richardson's deferred limit and probabilistic error cancellation through quasi-probability representations (QPR). The zero-noise method uses a time/Hamiltonian rescaling to generate multiple effective noise rates and combines estimates to cancel higher-order noise terms, with an explicit bound on the residual error. The probabilistic cancellation method decomposes ideal gates into a weighted, possibly signed, mixture of noisy operations (with overhead gamma), enabling unbiased estimates by Monte Carlo sampling of noisy circuits. Numerical experiments with depolarizing, amplitude-damping, and non-Markovian noise on Clifford+T circuits show substantial error reductions at modest overheads, highlighting the practical potential for running larger short-depth circuits on noisy devices without additional qubits.

Abstract

Two schemes are presented that mitigate the effect of errors and decoherence in short depth quantum circuits. The size of the circuits for which these techniques can be applied is limited by the rate at which the errors in the computation are introduced. Near-term applications of early quantum devices, such as quantum simulations, rely on accurate estimates of expectation values to become relevant. Decoherence and gate errors lead to wrong estimates of the expectation values of observables used to evaluate the noisy circuit. The two schemes we discuss are deliberately simple and don't require additional qubit resources, so to be as practically relevant in current experiments as possible. The first method, extrapolation to the zero noise limit, subsequently cancels powers of the noise perturbations by an application of Richardson's deferred approach to the limit. The second method cancels errors by resampling randomized circuits according to a quasi-probability distribution.

Figures (5)

• Figure 1: (color online) The plots show a random Hamiltonian evolution for $N=4$ system qubits and $d=6$ drift steps, each for time $t = 2$. For all systems plot the error $\Delta E = |E^* - \hat{E}^n_{K}(\lambda)|$ for $n = 0,1,2,3$. Here $\lambda^{1},n=0$ corresponds to the uncorrected error. The noise parameter $\lambda = -1/2\log(1-\epsilon)$ is chosen so that all plots have the same perturbation measured in the depolarizing strength $\epsilon = 10^{-3} \ldots 10^{-2}$. The plot shows the mitigation of (a) Depolarizing noise (b) Amplitude damping / dephasing noise and (c) non-Markovian noise, for $\{c_j\}$ chosen as random partition of in the interval $[1,4]$.
• Figure 2: Simulation precision $\delta(\beta) = |\hat{E}({\boldsymbol{{\beta}}}) - E^*({\boldsymbol{{\beta}}})|$ for $500$ randomly generated ideal Clifford+$T$ circuits on $N=6$ qubits with depth $d=20$. The gates are subject to single- and two-qubit depolarizing noise $\epsilon = 10^{-2}$. The figure shows results for simulations without (a) and with (b) error cancellation. In both cases each ideal circuit was simulated by $M = 4000$ runs of the noisy circuit. For each circuit ${\cal U }_{{\boldsymbol{{\beta}}}}$ we defined the observable $A$ as a projector $\Pi_{out}$ onto the subset of $2^{N-1}$ basis vectors with the largest weight in the final state. The results are consistent with $\gamma_\beta \approx 4.3$ so that $\gamma_\beta M^{-1/2} \approx 0.07$.
• Figure 3: (color online) The figure (a) represents the ideal circuit we want to simulate. It is comprised of single- and two-qubit gates $\{U_{12},\ldots,U_{5} \}$. We assume that a complete set of noisy gates exist $\Omega=\{{\cal O }_{12}^{\alpha_{12}},\ldots,{\cal O }_{5}^{\alpha_{5}}\}$, which serve as an operator basis in which the action of the ideal set can be expanded. It is then sufficient to sample circuits, as given in figure (b), where the gates are drawn from the probability distribution $P_{{\boldsymbol{{\beta}}}}$ in Eq. (\ref{['ppr1']}).
• Figure 4: Distribution of the ideal circuits according to their output probability $E^*({\boldsymbol{{\beta}}})$.
• Figure 5: Simulation precision for $\approx 500$ randomly generated ideal Clifford+$T$ circuits on $n=6$ qubits with depth $d=20$. The left and the right panels show results for simulations with and without error cancellation. In both cases each ideal circuit was simulated by $M=4000$ runs of the noisy circuit.