Error Mitigation of Fault-Tolerant Quantum Circuits with Soft Information

TL;DR

This work reframes quantum error mitigation as a tool that remains valuable in the fault-tolerant era by leveraging decoder soft information produced during quantum error correction. It presents a complete end-to-end protocol for logical-level QEM, including unbiased estimators of logical channels and three practical techniques: soft-information-based post-selection with runtime abort, efficient logical PEC, and soft-information-based ZNE. The methods are demonstrated on surface-code architectures using tensor-network and MWPM decoders, achieving up to $87.4\%$ spacetime overhead reductions relative to GST-based approaches and more than $100\times$ reductions in logical error rates with minimal shot loss. The results emphasize the practicality and efficiency of QEM when integrated with QEC, suggesting substantial performance gains for fault-tolerant quantum computers.

Abstract

Quantum error mitigation (QEM) is typically viewed as a suite of practical techniques for today's noisy intermediate-scale quantum devices, with limited relevance once fault-tolerant quantum computers become available. In this work, we challenge this conventional wisdom by showing that QEM can continue to provide substantial benefits in the era of quantum error correction (QEC), and in an even more efficient manner than it does on current devices. We introduce a framework for logical-level QEM that leverages soft information naturally produced by QEC decoders, requiring no additional data, hardware modifications, or runtime overhead beyond what QEC protocols already provide. Within this framework, we develop and analyze three logical-level QEM techniques: post-selection and runtime abort policies, probabilistic error cancellation, and zero-noise extrapolation. Our techniques reduce logical error rates by more than 100x while discarding fewer than 0.1% of shots; they also provide in situ characterization of logical channels for QEM protocols. As a proof of principle, we benchmark our approach using a surface-code architecture and two state-of-the-art decoders based on tensor-network contraction and minimum-weight perfect matching. We evaluate logical-level QEM on random Clifford circuits and molecular simulation algorithms and find that, compared to previous approaches relying on QEC only or QEC combined with QEM, we can achieve up to 87.4% spacetime overhead savings. Our results demonstrate that logical-level QEM with QEC decoder soft information can reliably improve logical performance, underscoring the efficiency and usefulness of QEM techniques for fault-tolerant quantum computers.

Figures (14)

• Figure 1: Qualitative resource overhead map of the four architectures to implement FTQC algorithms: QEC only, QEM only, GST-based QEC+QEM, and soft information-based QEC+QEM. A larger radial coordinate indicates a higher resource overhead. Our soft information-based QEC+QEM reduces demands in multiple dimensions, leading to a much more feasible architecture for fault tolerance.
• Figure 2: A fault-tolerant gadget for a unitary $U$ that we consider in this paper.
• Figure 3: Boundary condition enforcement for a distance-3 surface code. Top: Decoding with virtual boundaries connected by zero-weight edges. Bottom: Four decoding graphs with different boundary vertex types (defect/normal) corresponding to logical equivalence classes $\bar{I}$, $\bar{X}$, $\bar{Z}$, and $\bar{Y}$. Solid edges represent the MWPM solution for each class.
• Figure 4: Transversal CNOT between two surface code patches of the same size. (a) Physical $X$ errors propagation from control to target qubits and sequential decoding strategy. (b) Error propagation for both $X$ and $Z$ errors on control and target qubits.
• Figure 5: Fault-tolerant logical CNOT via lattice surgery. (a) Circuit diagram and syndrome extraction cycle (dashed box). (b) Physical layout of the three patches involved. (c) Breakdown of the logical error channel.