Complexity and order in approximate quantum error-correcting codes

by 子文 刘 (清华大学丘成桐数学科学中心)

Thursday 27 Mar 2025, 15:10 → 16:10 Asia/Shanghai

B105 (CHEP)

Quantum codes achieving approximate quantum error correction (AQEC) are useful, often fundamentally important, from both practical and physical perspectives but lack a systematic understanding. In this work, we establish general rigorous connections between quantum circuit complexity and approximate quantum error correction (AQEC) properties, covering both all-to-all and geometric scenarios including lattice systems, through a fundamental code parameter that we dub system variance. This theory of AQEC provides a versatile framework for understanding the quantum complexity and order of many-body quantum systems. In addition to showcasing applications to a wide variety of AQEC codes arising from diverse contexts spanning computer science and physics, we demonstrated how our theory offers new physical insights through the lens of both gapped and gapless systems. Specifically, it enables a long-sought rigorous understanding of the gap between strict definitions of gapped topological order and the widely-used long-range entanglement and topological entanglement entropy (TEE) signatures, as well as a discussion of how AQEC codes exhibiting physically significant power-law-error behavior emerge at low energies of conformal field theory (CFT), potentially advancing the understanding of quantum gravity through holography. We observe from various different perspectives that roughly O(1/n) represents a common, physically significant “error scaling threshold” for features associated with nontrivial quantum order, and suggest further studies of "marginal orders" that live close to this threshold.  [Ref: Nature Physics 20, 1798–1803 (2024) & ongoing]

Bio:
Zi-Wen Liu joined the faculty of Yau Mathematical Sciences Center, Tsinghua University in 2023, prior to which he was a 5-Year Senior Postdoctoral Fellow at Perimeter Institute for Theoretical Physics. He obtained his PhD in Physics from MIT. His main recent interests are the interactions between quantum information/computation and physics and mathematics.