Efficient quantum pseudorandomness under conservation laws

2024-11-07

arXiv.org

AI Summary

This work resolves the long-standing open problem of efficiently constructing local unitary 2-designs under symmetry constraints: conventional pseudorandom circuits converge slowly under continuous symmetries—specifically U(1) charge conservation and SU(d) global symmetry. We explicitly construct a polynomial-depth local symmetric quantum circuit and rigorously prove it converges to a symmetric unitary 2-design in polynomial time in system size. Our approach integrates representation theory, graph theory, and Markov chain analysis to ensure all gate sequences respect the conserved charges. This breakthrough overcomes the fundamental bottleneck imposed by continuous symmetries on design construction. Moreover, it yields the first provably efficient and explicit generation scheme for covariant quantum error-correcting codes, thereby validating and advancing the central conjecture from PRX Quantum (2022) that covariant designs admit efficient implementations.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

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)].

Problem

• Quantum Local Unitary Design
• Quantum Randomness
• Quantum Error Correction

Innovation

• Quantum Pseudo-Randomness
• Quantum Error Correction Codes
• Local Unitary Design