Efficient quantum pseudorandomness under conservation laws

TL;DR

This paper constructs explicit local symmetric quantum circuits that efficiently generate symmetric unitary 2-designs, resolving a major open problem and enabling near-optimal covariant quantum error correction.

Contribution

It provides the first explicit polynomial-time constructions of local symmetric circuits for generating symmetric 2-designs under U(1) and SU(d) symmetries.

Findings

Explicit symmetric circuits converge to 2-designs in polynomial time

Construction applies to U(1) and SU(d) symmetries

Enables efficient covariant quantum error correction

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)].