An Entropic Lens on Stabilizer States

• URL: http://arxiv.org/abs/2204.07593v3
• Date: Fri, 16 Dec 2022 19:07:29 GMT
• Title: An Entropic Lens on Stabilizer States
• Authors: Cynthia Keeler, William Munizzi, Jason Pollack
• Abstract summary: 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The $n$-qubit stabilizer states are those left invariant by a $2^n$-element subset of the Pauli group. The Clifford group is the group of unitaries which take stabilizer states to stabilizer states; a physically--motivated generating set, the Hadamard, phase, and CNOT gates which comprise the Clifford gates, imposes a graph structure on the set of stabilizers. We explicitly construct these structures, the "reachability graphs," at $n\le5$. When we consider only a subset of the Clifford gates, the reachability graphs separate into multiple, often complicated, connected components. Seeking an understanding of the entropic structure of the stabilizer states, which is ultimately built up by CNOT gate applications on two qubits, we are motivated to consider the restricted subgraphs built from the Hadamard and CNOT gates acting on only two of the $n$ qubits. 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. Starting at four qubits, some of the stabilizer states have entropy vectors which are not allowed by holographic entropy inequalities. We comment on the nature of the transition between holographic and non-holographic states within the stabilizer reachability graphs.

Related papers

• Geometry of degenerate quantum states, configurations of $m$-planes and invariants on complex Grassmannians [55.2480439325792]
  We show how to reduce the geometry of degenerate states to the non-abelian connection $A$. We find independent invariants associated with each triple of subspaces. Some of them generalize the Berry-Pancharatnam phase, and some do not have analogues for 1-dimensional subspaces.
  arXiv  Detail & Related papers  (2024-04-04T06:39:28Z)
• Bounding Entanglement Entropy with Contracted Graphs [0.0]
  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.
  arXiv  Detail & Related papers  (2023-10-30T18:00:01Z)
• Quantum control landscape for generation of $H$ and $T$ gates in an open qubit with both coherent and environmental drive [57.70351255180495]
  An important problem in quantum computation is generation of single-qubit quantum gates such as Hadamard ($H$) and $pi/8$ ($T$) Here we consider the problem of optimal generation of $H$ and $T$ gates using coherent control and the environment as a resource acting on the qubit via incoherent control.
  arXiv  Detail & Related papers  (2023-09-05T09:05:27Z)
• An efficient algebraic representation for graph states for measurement-based quantum computing [0.0]
  Graph states are main computational building blocks of measurement-based computation. We show how to efficiently express a graph state through the generators of the stabilizer group. We provide a framework to manipulate the graph states with a reduced number of stabilizers.
  arXiv  Detail & Related papers  (2022-12-23T01:40:03Z)
• Transformations of Stabilizer States in Quantum Networks [0.0]
  We study party-local Clifford transformations among stabilizer states. Traits arise as a physically motivated extension of local operations in quantum networks. We show that PLC transformations among graph states are equivalent to a generalization of the well-known local complementation.
  arXiv  Detail & Related papers  (2022-03-08T17:04:58Z)
• Quantum phase transition in the one-dimensional Dicke-Hubbard model with coupled qubits [20.002319486166016]
  We study the ground state phase diagram of a one-dimensional two qubits Dicke-Hubbard model with XY qubit-qubit interaction.
  arXiv  Detail & Related papers  (2021-11-05T13:17:49Z)
• Improved Graph Formalism for Quantum Circuit Simulation [77.34726150561087]
  We show how to efficiently simplify stabilizer states to canonical form. We characterize all linearly dependent triplets, revealing symmetries in the inner products. Using our novel controlled-Pauli $Z$ algorithm, we improve runtime for inner product computation from $O(n3)$ to $O(nd2)$ where $d$ is the maximum degree of the graph.
  arXiv  Detail & Related papers  (2021-09-20T05:56:25Z)
• 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 Gap Between the Hypergraph and Stabilizer Entropy Cones [0.20999222360659606]
  We show that the stabilizer and hypergraph entropy cones coincide for four parties, leading to a conjecture of their equivalence at higher party numbers. We improve the characterization of stabilizer entropies and show that all linear rank inequalities at five parties, except for classical monotonicity, form facets of the stabilizer cone.
  arXiv  Detail & Related papers  (2020-06-29T18:08:37Z)
• 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)
• Radiative topological biphoton states in modulated qubit arrays [105.54048699217668]
  We study topological properties of bound pairs of photons in spatially-modulated qubit arrays coupled to a waveguide. For open boundary condition, we find exotic topological bound-pair edge states with radiative losses. By joining two structures with different spatial modulations, we find long-lived interface states which may have applications in storage and quantum information processing.
  arXiv  Detail & Related papers  (2020-02-24T04:44:12Z)

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.