Hypergraph Simplification: Linking the Path-sum Approach to the ZH-calculus

• URL: http://arxiv.org/abs/2003.13564v2
• Date: Mon, 6 Sep 2021 00:59:17 GMT
• Title: Hypergraph Simplification: Linking the Path-sum Approach to the ZH-calculus
• Authors: Louis Lemonnier (ENS Paris-Saclay, Universit\'e Paris-Saclay), John van de Wetering (Radboud Universiteit Nijmegen), Aleks Kissinger (Oxford University)
• Abstract summary: We establish a correspondence between the ZH-calculus and the path-sum formalism. We introduce and prove several new simplification rules for the ZH-calculus. The relatively opaque path-sum rules are shown to arise naturally from two powerful families of rewrite rules.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The ZH-calculus is a complete graphical calculus for linear maps between qubits that admits a straightforward encoding of hypergraph states and circuits arising from the Toffoli+Hadamard gate set. In this paper, we establish a correspondence between the ZH-calculus and the path-sum formalism, a technique recently introduced by Amy to verify quantum circuits. In particular, we find a bijection between certain canonical forms of ZH-diagrams and path-sum expressions. We then introduce and prove several new simplification rules for the ZH-calculus, which are in direct correspondence to the simplification rules of the path-sum formalism. The relatively opaque path-sum rules are shown to arise naturally from two powerful families of rewrite rules in the ZH-calculus. The first is the extension of the familiar graph-theoretic simplifications based on local complementation and pivoting to their hypergraph-theoretic analogues: hyper-local complementation and hyper-pivoting. The second is the graphical Fourier transform introduced by Kuijpers et al., which enables effective simplification of ZH-diagrams encoding multi-linear phase polynomials with arbitrary real coefficients.

Related papers

• Tempered Calculus for ML: Application to Hyperbolic Model Embedding [70.61101116794549]
  Most mathematical distortions used in ML are fundamentally integral in nature. In this paper, we unveil a grounded theory and tools which can help improve these distortions to better cope with ML requirements. We show how to apply it to a problem that has recently gained traction in ML: hyperbolic embeddings with a "cheap" and accurate encoding along the hyperbolic vsean scale.
  arXiv  Detail & Related papers  (2024-02-06T17:21:06Z)
• 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)
• Rewriting and Completeness of Sum-Over-Paths in Dyadic Fragments of Quantum Computing [0.0]
  We give here a new set of rewrite rules for the formalism "Sum-Over-Paths" We show that it is complete for "Toffoli-Hadamard", the simplest approximately universal fragment of quantum mechanics. We also show how to perform sums and concatenation of arbitrary terms.
  arXiv  Detail & Related papers  (2023-07-26T14:40:21Z)
• The ZX-calculus as a Language for Topological Quantum Computation [0.0]
  Unitary fusion categories formalise the theory of topological quantum computation. We represent generators for the Fibonacci and Ising models. We give derivations of the single-qubit braid equations for Fibonacci anyons and the single- and two-qubit braid equations for Ising anyons.
  arXiv  Detail & Related papers  (2022-11-07T20:45:33Z)
• Completeness of Sum-Over-Paths for Toffoli-Hadamard and the Dyadic Fragments of Quantum Computation [0.0]
  "Sum-Over-Paths" formalism is a way to symbolically manipulate linear maps that describe quantum systems. We show that it is complete for "Toffoli-Hadamard", the simplest approximately universal fragment of quantum mechanics.
  arXiv  Detail & Related papers  (2022-05-05T12:29:29Z)
• LOv-Calculus: A Graphical Language for Linear Optical Quantum Circuits [58.720142291102135]
  We introduce the LOv-calculus, a graphical language for reasoning about linear optical quantum circuits. Two LOv-circuits represent the same quantum process if and only if one can be transformed into the other with the rules of the LOv-calculus.
  arXiv  Detail & Related papers  (2022-04-25T16:59:26Z)
• Simplification Strategies for the Qutrit ZX-Calculus [0.0]
  The ZX-calculus is a graphical language for suitably represented tensor networks, called ZX-diagrams. The ZX-calculus has found applications in reasoning about quantum circuits, condensed matter systems, quantum algorithms, quantum error codes, and counting problems.
  arXiv  Detail & Related papers  (2021-03-11T19:17:28Z)
• AKLT-states as ZX-diagrams: diagrammatic reasoning for quantum states [1.1470070927586016]
  We introduce the ZXH-calculus, a graphical language that we use to represent and reason about many-body states entirely graphically. We show how we recover the AKLT matrix-product state representation, the existence of topologically protected edge states, and the non-vanishing of a string order parameter. We also provide an alternative proof that the 2D AKLT state on a hexagonal lattice can be reduced to a graph state, demonstrating that it is a universal quantum computing resource.
  arXiv  Detail & Related papers  (2020-12-02T14:03:27Z)
• Local optimization on pure Gaussian state manifolds [63.76263875368856]
  We exploit insights into the geometry of bosonic and fermionic Gaussian states to develop an efficient local optimization algorithm. The method is based on notions of descent gradient attuned to the local geometry. We use the presented methods to collect numerical and analytical evidence for the conjecture that Gaussian purifications are sufficient to compute the entanglement of purification of arbitrary mixed Gaussian states.
  arXiv  Detail & Related papers  (2020-09-24T18:00:36Z)
• 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)
• A refinement of Reznick's Positivstellensatz with applications to quantum information theory [72.8349503901712]
  In Hilbert's 17th problem Artin showed that any positive definite in several variables can be written as the quotient of two sums of squares. Reznick showed that the denominator in Artin's result can always be chosen as an $N$-th power of the squared norm of the variables.
  arXiv  Detail & Related papers  (2019-09-04T11:46:26Z)

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.