Algebraic Geometry of Quantum Graphical Models

• URL: http://arxiv.org/abs/2308.11538v1
• Date: Tue, 22 Aug 2023 16:07:07 GMT
• Title: Algebraic Geometry of Quantum Graphical Models
• Authors: Eliana Duarte, Dmitrii Pavlov, Maximilian Wiesmann
• Abstract summary: We introduce the foundations to carry out a similar study of quantum counterparts. quantum graphical models are families of quantum states satisfying certain locality or correlation conditions encoded by a graph.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Algebro-geometric methods have proven to be very successful in the study of graphical models in statistics. In this paper we introduce the foundations to carry out a similar study of their quantum counterparts. These quantum graphical models are families of quantum states satisfying certain locality or correlation conditions encoded by a graph. We lay out several ways to associate an algebraic variety to a quantum graphical model. The classical graphical models can be recovered from most of these varieties by restricting to quantum states represented by diagonal matrices. We study fundamental properties of these varieties and provide algorithms to compute their defining equations. Moreover, we study quantum information projections to quantum exponential families defined by graphs and prove a quantum analogue of Birch's Theorem.

Related papers

• Gaussian Entanglement Measure: Applications to Multipartite Entanglement of Graph States and Bosonic Field Theory [50.24983453990065]
  An entanglement measure based on the Fubini-Study metric has been recently introduced by Cocchiarella and co-workers. We present the Gaussian Entanglement Measure (GEM), a generalization of geometric entanglement measure for multimode Gaussian states. By providing a computable multipartite entanglement measure for systems with a large number of degrees of freedom, we show that our definition can be used to obtain insights into a free bosonic field theory.
  arXiv  Detail & Related papers  (2024-01-31T15:50:50Z)
• Quantum Tutte Embeddings [2.8981045877033993]
  This paper describes how to create a graph-drawing quantum circuit from a given graph. We show how a Tutte embedding can be calculated as a quantum state in this circuit that can then be sampled to extract the embedding.
  arXiv  Detail & Related papers  (2023-07-17T21:23:28Z)
• Quantum data learning for quantum simulations in high-energy physics [55.41644538483948]
  We explore the applicability of quantum-data learning to practical problems in high-energy physics. We make use of ansatz based on quantum convolutional neural networks and numerically show that it is capable of recognizing quantum phases of ground states. The observation of non-trivial learning properties demonstrated in these benchmarks will motivate further exploration of the quantum-data learning architecture in high-energy physics.
  arXiv  Detail & Related papers  (2023-06-29T18:00:01Z)
• On Quantum Circuits for Discrete Graphical Models [1.0965065178451106]
  We provide the first method that allows one to provably generate unbiased and independent samples from general discrete factor models. Our method is compatible with multi-body interactions and its success probability does not depend on the number of variables. Experiments with quantum simulation as well as actual quantum hardware show that our method can carry out sampling and parameter learning on quantum computers.
  arXiv  Detail & Related papers  (2022-06-01T11:03:51Z)
• Theory of Quantum Generative Learning Models with Maximum Mean Discrepancy [67.02951777522547]
  We study learnability of quantum circuit Born machines (QCBMs) and quantum generative adversarial networks (QGANs) We first analyze the generalization ability of QCBMs and identify their superiorities when the quantum devices can directly access the target distribution. Next, we prove how the generalization error bound of QGANs depends on the employed Ansatz, the number of qudits, and input states.
  arXiv  Detail & Related papers  (2022-05-10T08:05:59Z)
• From Quantum Graph Computing to Quantum Graph Learning: A Survey [86.8206129053725]
  We first elaborate the correlations between quantum mechanics and graph theory to show that quantum computers are able to generate useful solutions. For its practicability and wide-applicability, we give a brief review of typical graph learning techniques. We give a snapshot of quantum graph learning where expectations serve as a catalyst for subsequent research.
  arXiv  Detail & Related papers  (2022-02-19T02:56:47Z)
• Equivariant Quantum Graph Circuits [10.312968200748116]
  We propose equivariant quantum graph circuits (EQGCs) as a class of parameterized quantum circuits with strong inductive bias for learning over graph-structured data. Our theoretical perspective on quantum graph machine learning methods opens many directions for further work, and could lead to models with capabilities beyond those of classical approaches.
  arXiv  Detail & Related papers  (2021-12-10T00:14:12Z)
• Spectral bounds for the quantum chromatic number of quantum graphs [0.0]
  We obtain lower bounds for the classical and quantum number of a quantum graph using eigenvalues of the quantum adjacency matrix. We generalize all the spectral bounds given by Elphick and Wocjan to the quantum graph setting. Our results are achieved using techniques from linear algebra and a complete definition of quantum graph coloring.
  arXiv  Detail & Related papers  (2021-12-03T05:36:21Z)
• Benchmarking Small-Scale Quantum Devices on Computing Graph Edit Distance [52.77024349608834]
  Graph Edit Distance (GED) measures the degree of (dis)similarity between two graphs in terms of the operations needed to make them identical. In this paper we present a comparative study of two quantum approaches to computing GED.
  arXiv  Detail & Related papers  (2021-11-19T12:35:26Z)
• Geometric properties of evolutionary graph states and their detection on a quantum computer [0.0]
  Geometric characteristics of graph states corresponding to a chain, a triangle, and a square are detected on the basis of calculations on IBM's quantum computer ibmq-manila.
  arXiv  Detail & Related papers  (2021-08-29T20:31:37Z)
• State preparation and measurement in a quantum simulation of the O(3) sigma model [65.01359242860215]
  We show that fixed points of the non-linear O(3) sigma model can be reproduced near a quantum phase transition of a spin model with just two qubits per lattice site. We apply Trotter methods to obtain results for the complexity of adiabatic ground state preparation in both the weak-coupling and quantum-critical regimes. We present and analyze a quantum algorithm based on non-unitary randomized simulation methods.
  arXiv  Detail & Related papers  (2020-06-28T23:44:12Z)

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.