Bounding Entanglement Entropy with Clifford Double Cosets

Abstract

Following on our previous work arXiv:2204.07593 and arXiv:2306.01043 studying the orbits of quantum states under Clifford circuits via `reachability graphs', we introduce `contracted graphs' whose vertices represent classes of quantum states with the same entropy vector. These contracted graphs represent the double cosets of the Clifford group, where the left cosets are built from the stabilizer subgroup of the starting state and the right cosets are built from the entropy-preserving operators. We study contracted graphs for stabilizer states, as well as W states and Dicke states, discussing how the diameter of a state's contracted graph constrains the `entropic diversity' of its $2$-qubit Clifford orbit. We derive an upper bound on the number of entropy vectors that can be generated using any $n$-qubit Clifford circuit, for any quantum state. We speculate on the holographic implications for the relative proximity of gravitational duals of states within the same Clifford orbit. Although we concentrate on how entropy evolves under the Clifford group, our double-coset formalism, and thus the contracted graph picture, is extendable to generic gate sets and generic state properties.

34 pages, 17 figures, 8 tables

References in corpus (18)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• Holographic Derivation of Entanglement Entropy from AdS/CFT
• Variational Quantum Algorithms
• Quantum computational advantage using photons
• Improved Simulation of Stabilizer Circuits
• Universal Quantum Computation with ideal Clifford gates and noisy ancillas
• Quantum corrections to holographic entanglement entropy
• Holographic Mutual Information is Monogamous
• Stabilizer Rényi entropy
• The Holographic Entropy Cone
• The Holographic Entropy Cone for Five Regions
• Constrained Quantum Optimization for Extractive Summarization on a Trapped-ion Quantum Computer
• Primitive Quantum Gates for an SU(2) Discrete Subgroup: BT
• Generators and relations for n-qubit Clifford operators
• The Speed of Quantum Information Spreading in Chaotic Systems
• Stabilizer information inequalities from phase space distributions
• Qudit Dicke state preparation
• An Entropic Lens on Stabilizer States

Cited by in corpus (1)

• Gravitational back-reaction is magical