Error Mitigation for Short-Depth Quantum Circuits

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 do not 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 quasiprobability distribution.

Figure 1

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 $\mathrm{\Delta }E=|E^{*}-{\widehat{E}}_K^n(\lambda )|$ 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 $\epsilon ={10}^{-3}\dots {10}^{-2}$. 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 $\delta (\beta )=|\widehat{E}(\mathit{\beta })-E^{*}(\mathit{\beta })|$ 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 ${\mathcal{U}}_{\mathit{\beta }}$ we defined the observable $A$ as a projector ${\mathrm{\Pi }}_{\mathrm{out}}$ 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 ${\gamma }_{\beta }{M}^{-1/2}\approx 0.07$.