On the Importance of Error Mitigation for Quantum Computation

TL;DR

The paper clarifies the role of quantum error mitigation (EM) by distinguishing asymptotic and finite quantum advantages, arguing that EM cannot deliver exponential asymptotic QA but can unlock substantial finite quantum advantages well before full fault-tolerant error correction (EC) is feasible. It introduces the circuit volume boost (CVB) as a concrete metric to quantify EM’s benefit and develops precise estimates for bare and EM-enabled circuit volumes in expectation value estimation (EVE). The work further shows that EM remains valuable alongside EC and, when integrated as logical EM (LEM) with modern EC codes, can yield orders-of-magnitude larger circuit volumes, potentially enabling industry-relevant applications. Collectively, the results position EM as a critical, enduring component of the quantum roadmap, driving near-term finite QAs and enhancing the capabilities of fault-tolerant architectures in the long run.

Abstract

Quantum error mitigation (EM) is a family of hybrid quantum-classical methods for eliminating or reducing the effect of noise and decoherence on quantum algorithms run on quantum hardware, without applying quantum error correction (EC). While EM has many benefits compared to EC, specifically that it requires no (or little) qubit overhead, this benefit comes with a painful price: EM seems to necessitate an overhead in quantum run time which grows as a (mild) exponent. Accordingly, recent results show that EM alone cannot enable exponential quantum advantages (QAs), for an average variant of the expectation value estimation problem. These works raised concerns regarding the role of EM in the road map towards QAs. We aim to demystify the discussion and provide a clear picture of the role of EM in achieving QAs, both in the near and long term. We first propose a clear distinction between finite QA and asymptotic QA, which is crucial to the understanding of the question, and present the notion of circuit volume boost, which we claim is an adequate way to quantify the benefits of EM. Using these notions, we can argue straightforwardly that EM is expected to have a significant role in achieving QAs. Specifically, that EM is likely to be the first error reduction method for useful finite QAs, before EC; that the first such QAs are expected to be achieved using EM in the very near future; and that EM is expected to maintain its important role in quantum computation even when EC will be routinely used - for as long as high-quality qubits remain a scarce resource.

Figures (3)

• Figure 1: Circuit volume boost (CVB) due to unbiased error mitigation with blow up rate $\lambda=2$. As the allowed inaccuracy $\epsilon\rightarrow 0$, the maximal volume possible without EM vanishes, while the maximal volume possible with unbiased EM is independent of $\epsilon$, given the same allowed shot overhead $R$ for both circuit execution methods.
• Figure 2: Predictions for finite QA with EM. Coloured lines indicate the QPU time as a function of active circuit volume, when using QESEM, an unbiased EM method introduced in Ref. QESEMpaper, on QPUs based on superconducting-qubits and trapped-ions, for different 2-qubit gate fidelities $F_{2q}=1-\gamma$, and requiring 95% output accuracy. Differences between the left and right panels stem from the different time scales in the two types of QPUs. The dashed line estimates the run time of state-of-the-art classical simulation algorithms on HPC hardware, using Eq. \ref{['eq:Tc-of-V']}, with dimension $d=2$ and operator spreading velocity $v=0.1$, as an example. Note that since we measure the active volume in native gates and work with $d=2$, the all-to-all connectivity advantage of trapped-ions is not explicitly indicated. Finite QA may be achieved when QESEM’s QPU time is significantly below the dashed line. The purple background shows the velocity $v$ (for $d=2$) as a function of the active volume $V$ and classical run time $T_c$, demonstrating how the dashed line would have looked like for with different choices of the velocity $v$.
• Figure 3: An example for circuit volume boosts (CVBs) with different error reduction strategies, with and without EC. The different curves indicate the CVBs relative to bare circuit execution, due to EC, EM, an ‘external’ mitigation of logical errors (ExtLEM), the combination of EC and post-selection (EC+PS), and a syndrome-aware mitigation of logical errors (SALEM) SALEMpaper. All methods are given a very mild allowed shot overhead $R=2$. The data presented was obtained from numerical simulations of repeated EC cycles (a logical memory circuit), with the well-known Steane code (7-qubit color code), based on the fault-tolerant construction of Ref. Reichardt_2021. The results hold qualitatively for any EC code, but will be quantitatively different, e.g. the blue line will be higher with better EC schemes (cf. Tab. \ref{['Tab: tab']}). For ease of reference, focus on 95% output accuracy. We compare the volume boosts for different methods. For an infidelity of $5\times 10^{-4}$, slightly below the Steane code's threshold, logical errors are slightly smaller than physical errors, and EC provides a small CVB (blue) compared to bare execution. This CVB does not improve with required accuracy, because EC suffers a significant bias due to logical errors (similarly to the bias in the bare computation, discussed in Sec. \ref{['Sec:CVB']}). Unbiased EM (yellow), applied directly to physical qubits, is better than EC for the Steane code and chosen infidelity of $5\times 10^{-4}$. ExtLEM (green), where EM is naively applied to logical gates, enjoys the benefits of both EC and EM, and provides a larger CVB than EM. The remaining methods, EC+PS and SALEM, are discussed in the main text.

Theorems & Definitions (14)

• Definition 2.1: Asymptotic QA
• Definition 2.2: Finite QA
• Remark 1: The size of $I_n$
• Remark 2: Does Asymptotic QA imply finite QA, or vice versa?
• Definition 3.1: Expectation Value Estimation ($\mathsf{EVE}$)
• Remark 3
• Definition 3.2: Active circuit volume
• Remark 4
• Remark 5
• Corollary 1