[Paper Review] Fundamental limits of quantum error mitigation

Ryuji Takagi, Suguru Endo+2 more|arXiv (Cornell University)|Sep 9, 2021

Quantum Computing Algorithms and Architecture8 citations

TL;DR

This paper establishes fundamental limits on quantum error mitigation by deriving universal bounds on sampling overhead required to reduce computation errors. It proves that for local depolarizing noise in layered circuits, the required sampling overhead grows exponentially with circuit depth, and shows probabilistic error cancellation is optimal for mitigating local dephasing noise across any number of qubits.

ABSTRACT

The inevitable accumulation of errors in near-future quantum devices represents a key obstacle in delivering practical quantum advantages, motivating the development of various quantum error-mitigation methods. Here, we derive fundamental bounds concerning how error-mitigation algorithms can reduce the computation error as a function of their sampling overhead. Our bounds place universal performance limits on a general error-mitigation protocol class. We use them to show (1) that the sampling overhead that ensures a certain computational accuracy for mitigating local depolarizing noise in layered circuits scales exponentially with the circuit depth for general error-mitigation protocols and (2) the optimality of probabilistic error cancellation among a wide class of strategies in mitigating the local dephasing noise on an arbitrary number of qubits. Our results provide a means to identify when a given quantum error-mitigation strategy is optimal and when there is potential room for improvement.

Motivation & Objective

• To establish universal performance bounds for quantum error-mitigation protocols without adaptive operations.
• To identify when a given error-mitigation strategy is optimal or fundamentally limited.
• To provide a benchmark—maximum estimator spread—for evaluating error-mitigation performance across all non-adaptive protocols.
• To clarify whether observed exponential error growth in existing methods is a fundamental limitation or a protocol-specific shortcoming.
• To determine the ultimate potential of error mitigation in near-term quantum devices.

Proposed method

• Proposes a formal framework defining error mitigation as any non-adaptive quantum protocol that classically post-processes repeated device runs.
• Introduces 'maximum estimator spread' as a universal benchmark for error-mitigation performance, quantifying the number of device runs needed to achieve a target accuracy.
• Derives fundamental lower bounds on the maximum estimator spread using sub-fidelity and trace distance as measures of state distinguishability under noise.
• Applies the bounds to two noise models: local depolarizing noise and local dephasing noise, analyzing their impact on variational quantum circuits.
• Uses GHZ states as test states to evaluate distinguishability degradation under noise, enabling analytical derivation of bounds.
• Compares sub-fidelity-based bounds to exact trace distance to validate the tightness of the derived limits.

Experimental results

Research questions

• RQ1 What is the minimum sampling overhead required for any non-adaptive error-mitigation protocol to achieve a given accuracy in the presence of local depolarizing noise?
• RQ2 Is the exponential error growth seen in existing error-mitigation techniques a fundamental limitation or a consequence of suboptimal protocols?
• RQ3 Can probabilistic error cancellation be proven optimal for mitigating local dephasing noise on an arbitrary number of qubits?
• RQ4 How do the performance limits of error mitigation depend on circuit depth and noise model?
• RQ5 To what extent can sub-fidelity provide a tight approximation to the true trace distance in bounding error-mitigation overhead?

Key findings

• The sampling overhead required to mitigate local depolarizing noise in layered circuits scales exponentially with circuit depth for all general error-mitigation protocols.
• Probabilistic error cancellation is optimal for mitigating local dephasing noise on any number of qubits, minimizing the maximum estimator spread.
• The sub-fidelity-based bound closely approximates the true trace distance in the small-error regime, validating its use as a tight performance proxy.
• For global depolarizing noise, the trace distance becomes independent of system size when the number of repetitions 𝑄𝐾=1, but the sub-fidelity bound remains tight for small errors.
• The derived bounds reveal that exponential overhead is unavoidable for local depolarizing noise, confirming that observed exponential scaling in existing methods is not a protocol flaw but a fundamental limit.
• The framework enables universal identification of when a given error-mitigation strategy is already optimal or has room for improvement.