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Lovász Meets Lieb-Schultz-Mattis: Complexity in Approximate Quantum Error Correction

Jinmin Yi, Ruizhi Liu, Zhi Li

TL;DR

This work tackles the tension between approximate quantum error correction (AQEC) performance and the circuit complexity of preparing code states. By applying the Lovász local lemma to the local indistinguishability of short-range entangled states, the authors derive a distinguishability–complexity trade-off that links the subsystem variance $\varepsilon(d)$ to the light-cone function $f(t)$, establishing a bound of the form $\varepsilon(f(t)) > 1/(e f(4t))$ when two orthogonal code states have depth at most $t$. This leads to concrete consequences: covariant codes with transversal gates incur nontrivial code-space depth bounds, the W state requires extensive preparation resources under local and all-to-all connectivity, and a streamlined AQEC-based proof of Lieb–Schultz–Mattis type constraints emerges. Collectively, the framework provides a unifying lens to understand complexity in quantum information and many-body physics, with promising directions for discrete anomalous symmetries and holographic models.

Abstract

Approximate quantum error correction (AQEC) provides a versatile framework for both quantum information processing and probing many-body entanglement. We reveal a fundamental tension between the error-correcting power of an AQEC and the hardness of code state preparation. More precisely, through a novel application of the Lovász local lemma, we establish a fundamental trade-off between local indistinguishability and circuit complexity, showing that orthogonal short-range entangled states must be distinguishable via a local operator. These results offer a powerful tool for exploring quantum circuit complexity across diverse settings. As applications, we derive stronger constraints on the complexity of AQEC codes with transversal logical gates and establish strong complexity lower bounds for W state preparation. Our framework also provides a novel perspective for systems with Lieb-Schultz-Mattis type constraints.

Lovász Meets Lieb-Schultz-Mattis: Complexity in Approximate Quantum Error Correction

TL;DR

This work tackles the tension between approximate quantum error correction (AQEC) performance and the circuit complexity of preparing code states. By applying the Lovász local lemma to the local indistinguishability of short-range entangled states, the authors derive a distinguishability–complexity trade-off that links the subsystem variance to the light-cone function , establishing a bound of the form when two orthogonal code states have depth at most . This leads to concrete consequences: covariant codes with transversal gates incur nontrivial code-space depth bounds, the W state requires extensive preparation resources under local and all-to-all connectivity, and a streamlined AQEC-based proof of Lieb–Schultz–Mattis type constraints emerges. Collectively, the framework provides a unifying lens to understand complexity in quantum information and many-body physics, with promising directions for discrete anomalous symmetries and holographic models.

Abstract

Approximate quantum error correction (AQEC) provides a versatile framework for both quantum information processing and probing many-body entanglement. We reveal a fundamental tension between the error-correcting power of an AQEC and the hardness of code state preparation. More precisely, through a novel application of the Lovász local lemma, we establish a fundamental trade-off between local indistinguishability and circuit complexity, showing that orthogonal short-range entangled states must be distinguishable via a local operator. These results offer a powerful tool for exploring quantum circuit complexity across diverse settings. As applications, we derive stronger constraints on the complexity of AQEC codes with transversal logical gates and establish strong complexity lower bounds for W state preparation. Our framework also provides a novel perspective for systems with Lieb-Schultz-Mattis type constraints.

Paper Structure

This paper contains 11 sections, 7 theorems, 18 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Distinguishability-Complexity trade-off
  4. Distinguishing SRE states
  5. Complexity of AQEC code states
  6. Applications
  7. Applications
  8. Complexity of covariant codes
  9. W state preparation
  10. Lieb-Schultz-Mattis theorem
  11. Discussion and outlook

Key Result

Theorem 1

Let $|\psi_{1,2}\rangle$ be two $n$-qubit states that are almost orthogonal, i.e., $|\langle\psi_1|\psi_2\rangle|<\delta$. Suppose $\mathscr{C}(|\psi_{1,2}\rangle)\leq t$, then there exists a distinguishing operator ${\hat{O}}$, with $\left\| {\hat{O}} \right\|= 1$ and $|\mathrm{supp}({\hat{O}})| \l

Figures (1)

  • Figure 1: The trade-off between AQEC capability and code state complexity. In the "LRE" region, the whole code subspace has a complexity lower bound. If the code subspace contains two nearly orthogonal SRE states, the capability will fall into the "SREs" region. In the marginal region, there can be at most one SRE state in any orthogonal basis.

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Theorem \ref{thm:local-distinguishability}$'$
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Corollary 4: $U(1)\times T$ LSM
  • Corollary 5: Nonzero momentum implies LRE