Bounding Entanglement Entropy with Contracted Graphs

• URL: http://arxiv.org/abs/2310.19874v1
• Date: Mon, 30 Oct 2023 18:00:01 GMT
• Title: Bounding Entanglement Entropy with Contracted Graphs
• Authors: Cynthia Keeler, William Munizzi, Jason Pollack
• Abstract summary: We study contracted graphs for stabilizer states, W states and Dicke states. We derive an upper bound on the number of entropy vectors that can be generated using any $n$-qubit Clifford circuit. We speculate on the holographic implications for the relative proximity of gravitational duals of states within the same Clifford orbit.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Bit by Bit: Gravity Through the Lens of Quantum Information [0.0]
  dissertation reviews recent advances at the intersection of quantum information and holography. In holography, properties of quantum systems admit a gravitational interpretation via the AdS/CFT correspondence. Magic and entanglement play complementary roles when describing emergent phenomena in AdS/CFT.
  arXiv  Detail & Related papers  (2024-06-03T18:00:17Z)
• CKGConv: General Graph Convolution with Continuous Kernels [24.58050212186722]
  We propose a novel and general graph convolution framework by parameterizing the kernels as continuous functions of pseudo-coordinates derived via graph positional encoding. We name this Continuous Kernel Graph Convolution (CKGConv) We show that CKGConv-based Networks outperform existing graph convolutional networks and perform comparably to the best graph transformers across a variety of graph datasets.
  arXiv  Detail & Related papers  (2024-04-21T10:26:13Z)
• Entanglement entropy in conformal quantum mechanics [68.8204255655161]
  We consider sets of states in conformal quantum mechanics associated to generators of time evolution whose orbits cover different regions of the time domain. States labelled by a continuous global time variable define the two-point correlation functions of the theory seen as a one-dimensional conformal field theory.
  arXiv  Detail & Related papers  (2023-06-21T14:21:23Z)
• Clifford Orbits from Cayley Graph Quotients [0.0]
  We describe the structure of the $n$-qubit Clifford group $mathcalC_n$ via Cayley graphs. In order to obtain the action of Clifford gates on a given quantum state, we introduce a quotient procedure. We extend our study to non-stabilizer states, including the W and Dicke states.
  arXiv  Detail & Related papers  (2023-06-01T18:00:02Z)
• Tensor Rank and Other Multipartite Entanglement Measures of Graph States [5.8087716340417765]
  Graph states play an important role in quantum information theory through their connection to measurement-based computing and error correction. We show that several multipartite extensions of bipartite entanglement measures are dichotomous for graph states based on the connectivity of the corresponding graph.
  arXiv  Detail & Related papers  (2022-09-13T21:59:31Z)
• An Entropic Lens on Stabilizer States [0.0]
  We show how the two subgraphs already present at two qubits are embedded into more complicated subgraphs at three and four qubits. We argue that no additional types of subgraph appear beyond four qubits, but that the entropic structures within the subgraphs can grow progressively more complicated as the qubit number increases.
  arXiv  Detail & Related papers  (2022-04-15T18:00:12Z)
• Extending the Graph Formalism to Higher-Order Gates [0.0]
  We show how a $mathcalC_3$ gate acting on a stabilizer state splits it into two stabilizer states. We discuss applications of our algorithm to circuit identities and finding low stabilizer rank representations of magic states.
  arXiv  Detail & Related papers  (2021-08-05T15:39:31Z)
• A Hyperbolic-to-Hyperbolic Graph Convolutional Network [46.80564170208473]
  We propose a hyperbolic-to-hyperbolic graph convolutional network (H2H-GCN) that directly works on hyperbolic manifold. The H2H-GCN achieves substantial improvements on the link prediction, node classification, and graph classification tasks.
  arXiv  Detail & Related papers  (2021-04-14T16:09:27Z)
• A Thorough View of Exact Inference in Graphs from the Degree-4 Sum-of-Squares Hierarchy [37.34153902687548]
  We tackle the problem of exactly recovering an unknown ground-truth binary labeling of the nodes from a single corrupted observation of each edge. We apply a hierarchy of relaxations known as the sum-of-squares hierarchy, to the problem. We show that the solution of the dual of the relaxed problem is related to finding edge weights of the Johnson and Kneser graphs.
  arXiv  Detail & Related papers  (2021-02-16T08:36:19Z)
• Hamiltonian systems, Toda lattices, Solitons, Lax Pairs on weighted Z-graded graphs [62.997667081978825]
  We identify conditions which allow one to lift one dimensional solutions to solutions on graphs. We show that even for a simple example of a topologically interesting graph the corresponding non-trivial Lax pairs and associated unitary transformations do not lift to a Lax pair on the Z-graded graph.
  arXiv  Detail & Related papers  (2020-08-11T17:58:13Z)
• Convergence and Stability of Graph Convolutional Networks on Large Random Graphs [22.387735135790706]
  We study properties of Graph Convolutional Networks (GCNs) by analyzing their behavior on standard models of random graphs. We first study the convergence of GCNs to their continuous counterpart as the number of nodes grows. We then analyze the stability of GCNs to small deformations of the random graph model.
  arXiv  Detail & Related papers  (2020-06-02T18:36:19Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.