Characterization of randomness in quantum circuits of continuous gate sets

Abstract

In the accompanying paper of arXiv:2408.13472, we have established the method of characterizing the maximal order of asymptotic unitary designs generated by symmetric local random circuits, and have explicitly specified the order in the cases of ${\mathbb{Z}}_2$, U(1), and SU(2) symmetries. Here, we provide full details on the derivation of the main theorems for general symmetry and for concrete symmetries. Furthermore, we consider a general framework where we have access to a finite set of connected compact unitary subgroups, which includes symmetric local unitary gate sets.

49 pages, 2 figures

References in corpus (28)

• Topological Insulators
• Quantum Spin Hall Effect in Graphene
• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Topological superconductors: a review
• "Deconfined" quantum critical points
• Entanglement spectrum of a topological phase in one dimension
• Characterizing Quantum Supremacy in Near-Term Devices
• Predicting Many Properties of a Quantum System from Very Few Measurements
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm
• Exact and Approximate Unitary 2-Designs: Constructions and Applications
• Evidence for deconfined quantum criticality in a two-dimensional Heisenberg model with four-spin interactions
• Scalable Noise Estimation with Random Unitary Operators
• Chaos and complexity by design
• Restrictions on Transversal Encoded Quantum Gate Sets
• Local random quantum circuits are approximate polynomial-designs
• Measurement-induced topological entanglement transitions in symmetric random quantum circuits
• Symmetry enriched phases of quantum circuits
• One-shot decoupling
• Random quantum circuits are approximate unitary $t$-designs in depth $O\left(n{t}^{5+o(1)}\right)$
• Restrictions on realizable unitary operations imposed by symmetry and locality
• Detecting and stabilizing measurement-induced symmetry-protected topological phases in generalized cluster models
• On squares of representations of compact Lie algebras
• Rotationally-Invariant Circuits: Universality with the exchange interaction and two ancilla qubits
• Theory of Quantum Circuits with Abelian Symmetries
• Designs from Local Random Quantum Circuits with SU(d) Symmetry
• Unitary k-designs from random number-conserving quantum circuits
• Unitary Designs of Symmetric Local Random Circuits

Cited by in corpus (7)

• Unitary Designs of Symmetric Local Random Circuits
• Toward Super-polynomial Quantum Speedup of Equivariant Quantum Algorithms with SU($d$) Symmetry
• Non-Haar random circuits form unitary designs as fast as Haar random circuits
• Non-Universality from Conserved Superoperators in Unitary Circuits
• More global randomness from less-random local gates
• Dynamical cluster-based strategy for improving tensor network algorithms in quantum circuit simulations
• Comprehensive Study on Heisenberg-limited Quantum Algorithms for Multiple Observables Estimation