Extending quantum probabilistic error cancellation by noise scaling

We propose a general framework for quantum error mitigation that combines and generalizes two techniques: probabilistic error cancellation (PEC) and zero-noise extrapolation (ZNE). Similar to PEC, the proposed method represents ideal operations as linear combinations of noisy operations that are implementable on hardware. However, instead of assuming a fixed level of hardware noise, we extend the set of implementable operations by noise scaling. By construction, this method encompasses both PEC and ZNE as particular cases and allows us to investigate a larger set of hybrid techniques. For example, gate extrapolation can be used to implement PEC without requiring knowledge of the device's noise model, e.g., avoiding gate-set tomography. Alternatively, probabilistic error reduction can be used to estimate expectation values at intermediate virtual noise strengths (below the hardware level), leading to partially mitigated results at a lower sampling cost. Moreover, multiple results obtained with different noise-reduction factors can be further postprocessed with ZNE to better approximate the zero-noise limit.

Figure 1

Pictorial representation of different error-mitigation techniques. (a) In probabilistic error cancellation (PEC), an ideal expectation value is estimated from different circuits executed at the same hardware noise $\lambda =1$. (b) In zero-noise extrapolation (ZNE), an ideal expectation value is estimated from the same circuit executed at different noise levels. (c) In noise-extended probabilistic error cancellation (NEPEC)—the error-mitigation framework introduced in this work—an ideal expectation value is estimated from a general combination of different circuits evaluated at different noise levels. In fact, the two-dimensional diagram used in this figure is a simplified representation of the more general linear combination involved in the NEPEC method. Indeed, according to Eq. (25), the two axes of the diagram can be multi-index arrays ($\vec{\lambda }$ and $\vec{\alpha }$) since the noise of each gate of a circuit could be scaled differently.

Figure 2

The expectation value of the observable $A=|0\rangle \langle 0|$ is estimated with different techniques for a single-qubit randomized benchmarking circuit of depth 14 such that ${\langle A\rangle }_{\mathrm{ideal}}=1$. The PEC results are based on quasiprobability representations built assuming a fixed estimated noise level of $p=0.01$. When this assumption matches the actual noise level of the hardware, PEC achieves an optimal cancellation of the noise. On the other hand, when the actual noise of the system is different from the assumed value (noise characterization error), PEC results are more biased than NEPEC results. The NEPEC results are obtained using the noise-agnostic representations defined according to Eq. (30) with scale factors $S=\{{\lambda }_1,{\lambda }_2\}=\{1,51\}$ and coefficients given by Eq. (31). Unitary folding is used for digitally scaling the noise as described in Eq. (23) without assuming a specific noise model or noise level. All the noisy expectation values are directly evaluated from simulated density matrices, and therefore, shot noise is absent in this figure (corresponding to the limit of infinite shots). The random fluctuations of the PEC and NEPEC points are due to the finite number (5000) of Monte Carlo samples. For the numerical simulation of this example, we used the error-mitigation software package mitiq [26].

Figure 3

(a) Probabilistic error reduction by virtual noise scaling. A single-qubit randomized-benchmarking circuit composed of 46 gates is simulated assuming a depolarizing noise model with $p=0.015$. The expectation value of $A=|0\rangle \langle 0|$ is estimated at different values of the noise scale factor $\lambda$ via PER according to Eq. (33) and, more specifically, (35). For increasing values of $\lambda$, the results continuously interpolate between PEC ($\lambda =0$) and no mitigation ($\lambda =1$), while the associated statistical variance gets reduced. (b) Error mitigation by linear zero-noise extrapolation with virtual scale factors $S=\{{\lambda }_1,{\lambda }_2\}=\{0.2,1\}$, compared to standard PEC and to PER (with $\lambda =0.2$). We observe that all error-mitigation methods improve the unmitigated result. As expected, virtual ZNE and PER are, on average, more biased than PEC; however, their statistical uncertainty can be significantly smaller. For each noise scale factor, $5\times {10}^4$ measurements (shots) have been used. For each shot, a different circuit was sampled according to PER. To estimate the statistical distribution, the samples have been divided into 25 batches of $2\times {10}^3$ elements each. The violin plots show the statistical distributions with respect to independent batches. The central horizontal segment in each “violin” corresponds to the mean of all the samples, which is equal to the mean of the batches. For the numerical simulation of this example, we used the error-mitigation software package mitiq [26].