Simple qudit ZX and ZH calculi, via integrals

• URL: http://arxiv.org/abs/2304.03310v2
• Date: Fri, 28 Jun 2024 14:11:16 GMT
• Title: Simple qudit ZX and ZH calculi, via integrals
• Authors: Niel de Beaudrap, Richard D. P. East,
• Abstract summary: ZX calculus and ZH calculus use diagrams to denote quantum operations, using rewrite rules' to transform between diagrams which denote the same operator through a functorial semantic map. We describe semantic maps for ZX and ZH diagrams, well-suited to analyse unitary circuits and measurements on qudits of any fixed dimension D>1 as a single ZXH-calculus' We demonstrate rewrite rules for the stabiliser fragment' of the ZX calculus and a multicharacter fragment' of the ZH calculus.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The ZX calculus and ZH calculus use diagrams to denote and compute properties of quantum operations, using `rewrite rules' to transform between diagrams which denote the same operator through a functorial semantic map. Different semantic maps give rise to different rewrite systems, which may prove more convenient for different purposes. Using discrete measures, we describe semantic maps for ZX and ZH diagrams, well-suited to analyse unitary circuits and measurements on qudits of any fixed dimension D>1 as a single `ZXH-calculus'. We demonstrate rewrite rules for the `stabiliser fragment' of the ZX calculus and a `multicharacter fragment' of the ZH calculus.

Related papers

• Disentangled Representation Learning with the Gromov-Monge Gap [65.73194652234848]
  Learning disentangled representations from unlabelled data is a fundamental challenge in machine learning. We introduce a novel approach to disentangled representation learning based on quadratic optimal transport. We demonstrate the effectiveness of our approach for quantifying disentanglement across four standard benchmarks.
  arXiv  Detail & Related papers  (2024-07-10T16:51:32Z)
• Equivalence Classes of Quantum Error-Correcting Codes [49.436750507696225]
  Quantum error-correcting codes (QECC's) are needed to combat the inherent noise affecting quantum processes. We represent QECC's in a form called a ZX diagram, consisting of a tensor network.
  arXiv  Detail & Related papers  (2024-06-17T20:48:43Z)
• 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)
• Geometry of Interaction for ZX-Diagrams [0.0]
  ZX-Calculus is a versatile graphical language for quantum computation equipped with an equational theory. We propose a token-machine-based asynchronous model of both pure ZX-Calculus and its extension to mixed processes.
  arXiv  Detail & Related papers  (2022-06-22T08:52:58Z)
• Constrained mixers for the quantum approximate optimization algorithm [55.41644538483948]
  We present a framework for constructing mixing operators that restrict the evolution to a subspace of the full Hilbert space. We generalize the "XY"-mixer designed to preserve the subspace of "one-hot" states to the general case of subspaces given by a number of computational basis states. Our analysis also leads to valid Trotterizations for "XY"-mixer with fewer CX gates than is known to date.
  arXiv  Detail & Related papers  (2022-03-11T17:19:26Z)
• Addition and Differentiation of ZX-diagrams [0.0]
  We introduce a general, inductive definition of the addition of ZX-diagrams. We provide an inductive differentiation of ZX-diagrams. We also apply our results to deduce a diagram for an Ising Hamiltonian.
  arXiv  Detail & Related papers  (2022-02-23T09:52:26Z)
• 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)
• Completeness of the ZH-calculus [0.0]
  We study the ZH-calculus, an alternative graphical language of string diagrams. We find a set of simple rewrite rules for this calculus and show it is complete with respect to matrices over $mathbbZ[frac12]$. We construct an extended version of the ZH-calculus that is complete with respect to matrices over any ring $R$ where $1+1$ is not a zero-divisor.
  arXiv  Detail & Related papers  (2021-03-11T11:22:51Z)
• 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)
• Well-tempered ZX and ZH Calculi [0.0]
  We present renormalised generators for the ZX calculus, which form a bialgebra precisely. We also present a similar renormalised version of the ZH calculus.
  arXiv  Detail & Related papers  (2020-06-03T22:03:55Z)
• Spatial Pyramid Based Graph Reasoning for Semantic Segmentation [67.47159595239798]
  We apply graph convolution into the semantic segmentation task and propose an improved Laplacian. The graph reasoning is directly performed in the original feature space organized as a spatial pyramid. We achieve comparable performance with advantages in computational and memory overhead.
  arXiv  Detail & Related papers  (2020-03-23T12:28:07Z)

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.