Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Efficient quantum pseudorandomness under conservation laws

Zimu Li, Han Zheng, Zi-Wen Liu

TL;DR

The paper addresses the problem of efficiently generating symmetric quantum pseudorandomness under continuous conservation laws, focusing on unitary $2$-designs in the presence of $U(1)$ and $SU(d)$ symmetries. It introduces Convolutional Quantum Alternating (CQA) ensembles and analyzes their second-moment operators using representation theory (Schur–Weyl duality, Young diagrams, YJM elements), Wick projections, and Markov-chain methods, proving polynomial-depth convergence to symmetric $2$-designs. The results provide explicit depth bounds across various interaction topologies (1D nearest-neighbor, star, complete graphs) and symmetry settings, establishing a rigorous, symmetry-preserving route to $2$-designs with wide potential applications, including covariant quantum error-correcting codes. This work resolves a long-standing open problem and supplies a practical framework for studying symmetric random circuits in quantum information and many-body physics.

Abstract

The efficiency of locally generating unitary designs, which capture statistical notions of quantum pseudorandomness, lies at the heart of wide-ranging areas in physics and quantum information technologies. While there are extensive potent methods and results for this problem, the evidently important setting where continuous symmetries or conservation laws (most notably U(1) and SU(d)) are involved is known to present fundamental difficulties. In particular, even the basic question of whether any local symmetric circuit can generate 2-designs efficiently (in time that grows at most polynomially in the system size) remains open with no circuit constructions provably known to do so, despite intensive efforts. In this work, we resolve this long-standing open problem for both U(1) and SU(d) symmetries by explicitly constructing local symmetric quantum circuits which we prove to converge to symmetric unitary 2-designs in polynomial time using a combination of representation theory, graph theory, and Markov chain methods. As a direct application, our constructions can be used to efficiently generate near-optimal covariant quantum error-correcting codes, confirming a conjecture in [PRX Quantum 3, 020314 (2022)].

Efficient quantum pseudorandomness under conservation laws

TL;DR

The paper addresses the problem of efficiently generating symmetric quantum pseudorandomness under continuous conservation laws, focusing on unitary -designs in the presence of and symmetries. It introduces Convolutional Quantum Alternating (CQA) ensembles and analyzes their second-moment operators using representation theory (Schur–Weyl duality, Young diagrams, YJM elements), Wick projections, and Markov-chain methods, proving polynomial-depth convergence to symmetric -designs. The results provide explicit depth bounds across various interaction topologies (1D nearest-neighbor, star, complete graphs) and symmetry settings, establishing a rigorous, symmetry-preserving route to -designs with wide potential applications, including covariant quantum error-correcting codes. This work resolves a long-standing open problem and supplies a practical framework for studying symmetric random circuits in quantum information and many-body physics.

Abstract

The efficiency of locally generating unitary designs, which capture statistical notions of quantum pseudorandomness, lies at the heart of wide-ranging areas in physics and quantum information technologies. While there are extensive potent methods and results for this problem, the evidently important setting where continuous symmetries or conservation laws (most notably U(1) and SU(d)) are involved is known to present fundamental difficulties. In particular, even the basic question of whether any local symmetric circuit can generate 2-designs efficiently (in time that grows at most polynomially in the system size) remains open with no circuit constructions provably known to do so, despite intensive efforts. In this work, we resolve this long-standing open problem for both U(1) and SU(d) symmetries by explicitly constructing local symmetric quantum circuits which we prove to converge to symmetric unitary 2-designs in polynomial time using a combination of representation theory, graph theory, and Markov chain methods. As a direct application, our constructions can be used to efficiently generate near-optimal covariant quantum error-correcting codes, confirming a conjecture in [PRX Quantum 3, 020314 (2022)].

Paper Structure

This paper contains 26 sections, 30 theorems, 241 equations, 1 table.

Table of Contents

  1. Introduction and background
  2. Summary of main results
  3. Unitary designs and symmetries
  4. Symmetric CQA ensembles and convergence time
  5. Discussion
  6. Summary of notations
  7. Preliminaries
  8. Miscellaneous facts about $S_n$ representation theory
  9. Spectral gap of $S_n$ Cayley graphs and Aldous' conjecture
  10. Schur--Weyl duality, space decomposition under $\mathop{\mathrm{SU}}\nolimits(d)$ symmetry and CQA architecture
  11. Permutation modules and space decomposition under $\mathop{\mathrm{U}}\nolimits(1)$ symmetry
  12. Unitary $k$-designs under continuous symmetry and invariant subspaces of moment operators
  13. Induced Markov chain and comparison theorems
  14. $\mathop{\mathrm{U}}\nolimits(1)$-symmetric universality and unitary $k$-designs from CQA
  15. Universality under $\mathop{\mathrm{U}}\nolimits(1)$ symmetry
  16. ...and 11 more sections

Key Result

Theorem 1

The local circuit ensembles defined by $\mathcal{E}_{\mathrm{CQA},\times}$ can converge to symmetric $2$-designs within a polynomial depth. Specifically, for $n$ qubits under $\mathop{\mathrm{U}}\nolimits(1)$ symmetry, Under $\mathop{\mathrm{SU}}\nolimits(d)$ symmetry with adjacent SWAPs, the convergence time is $O(n^3(4n \log d+ \log (1/\epsilon) ))$.

Theorems & Definitions (77)

  • Theorem 1: Informal
  • Definition B.1
  • Definition B.2
  • Definition B.3
  • Definition B.4
  • Proposition B.5
  • Theorem B.6
  • Proposition B.7
  • Definition B.8
  • Lemma B.9
  • ...and 67 more