Strategic Code: A Unified Spatio-Temporal Framework for Quantum Error-Correction

TL;DR

This paper introduces a comprehensive spatio-temporal framework for quantum error correction called the 'strategic code', unifying static and dynamical QECCs and providing conditions for fault-tolerance considering correlated errors.

Contribution

It proposes a unified theoretical framework for dynamical and static QECCs using the quantum combs formalism, including error-correction conditions and an optimization approach.

Findings

Unified spatio-temporal QECC framework covering all known codes.

Necessary and sufficient error-correction conditions for correlated errors.

An optimization method for adaptive approximate strategic codes.

Abstract

Quantum error-correcting code (QECC) is the central ingredient in fault-tolerant quantum information processing. An emerging paradigm of dynamical QECC shows that one can robustly encode logical quantum information both temporally and spatially in a more resource-efficient manner than traditional QECCs. Nevertheless, an overarching theory of how dynamical QECCs achieve fault-tolerance is lacking. In this work, we bridge this gap by proposing a unified spatio-temporal QECC framework called the ``strategic code'' built around an ``interrogator'' device which sequentially measures and evolves the spatial QECC in an adaptive manner based on the ``quantum combs'' formalism, a generalization of the channel-state duality. The strategic code covers all existing dynamical and static QECC, as well as all physically plausible QECCs to be discovered in the future, including those that involve adaptivity in its operational dynamics. Within this framework, we show an algebraic and an information-theoretic necessary and sufficient error-correction conditions for a strategic code, which consider spatially and temporally correlated errors. These conditions include the analogous known static QECC conditions as a special case. Lastly, we also propose an optimization-theoretic approach to obtain an approximate strategic code adapting to a correlated error.