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 random covariant quantum error-correcting codes, confirming a conjecture in [PRX Quantum 3, 020314 (2022)].

翻译：局部生成捕获量子伪随机性统计特性的酉设计，其效率是物理学和量子信息技术广泛领域的核心问题。尽管针对该问题已有大量有效方法与结果，但涉及连续对称性或守恒定律（最显著的是U(1)和SU(d)）的情形已知存在根本性困难。特别是，即使对于局部对称电路能否高效（在系统规模的多项式时间内）生成2-设计这一基本问题，尽管经过深入研究，目前仍未解决，且尚无被严格证明的电路构造。本工作通过结合表示论、图论与马尔可夫链方法，显式构造了局部对称量子电路，并证明其在多项式时间内收敛于对称酉2-设计，从而解决了U(1)和SU(d)对称性下的这一长期开放问题。作为直接应用，我们的构造可用于高效生成近似最优的随机协变量子纠错码，这证实了[PRX Quantum 3, 020314 (2022)]中的一个猜想。