Quantum computing based on complex Clifford algebras

• URL: http://arxiv.org/abs/2201.02246v2
• Date: Thu, 3 Mar 2022 18:43:00 GMT
• Title: Quantum computing based on complex Clifford algebras
• Authors: Jaroslav Hrdina, Ales Navrat, Petr Vasik
• Abstract summary: We propose to represent both $n$--qubits and quantum gates acting on them as elements in the complex Clifford algebra defined on a complex vector space of $2n.$ We demonstrate its functionality by performing quantum computations with several well known examples of quantum gates.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We propose to represent both $n$--qubits and quantum gates acting on them as elements in the complex Clifford algebra defined on a complex vector space of dimension $2n.$ In this framework, the Dirac formalism can be realized in straightforward way. We demonstrate its functionality by performing quantum computations with several well known examples of quantum gates. We also compare our approach with representations that use real geometric algebras.

Related papers

• Absolute dimensionality of quantum ensembles [41.94295877935867]
  The dimension of a quantum state is traditionally seen as the number of superposed distinguishable states in a given basis. We propose an absolute, i.e.basis-independent, notion of dimensionality for ensembles of quantum states.
  arXiv  Detail & Related papers  (2024-09-03T09:54:15Z)
• Efficient Learning for Linear Properties of Bounded-Gate Quantum Circuits [63.733312560668274]
  Given a quantum circuit containing d tunable RZ gates and G-d Clifford gates, can a learner perform purely classical inference to efficiently predict its linear properties? We prove that the sample complexity scaling linearly in d is necessary and sufficient to achieve a small prediction error, while the corresponding computational complexity may scale exponentially in d. We devise a kernel-based learning model capable of trading off prediction error and computational complexity, transitioning from exponential to scaling in many practical settings.
  arXiv  Detail & Related papers  (2024-08-22T08:21:28Z)
• Quantum channels, complex Stiefel manifolds, and optimization [45.9982965995401]
  We establish a continuity relation between the topological space of quantum channels and the quotient of the complex Stiefel manifold. The established relation can be applied to various quantum optimization problems.
  arXiv  Detail & Related papers  (2024-08-19T09:15:54Z)
• From Entanglement to Universality: A Multiparticle Spacetime Algebra Approach to Quantum Computational Gates Revisited [0.0]
  We focus on testing the usefulness of geometric algebras (GAs) techniques in two applications to quantum computing. First, we offer an explicit algebraic characterization of one- and two-qubit quantum states together with a MSTA description of one- and two-qubit quantum computational gates. In this first application, we devote special attention to the concept of entanglement, focusing on entangled quantum states and two-qubit entangling quantum gates.
  arXiv  Detail & Related papers  (2024-05-13T19:51:26Z)
• Gelfand-Tsetlin basis for partially transposed permutations, with applications to quantum information [0.9208007322096533]
  We study representation theory of the partially transposed permutation matrix algebra. We show how to simplify semidefinite optimization problems over unitary-equivariant quantum channels. We derive an efficient quantum circuit for implementing the optimal port-based quantum teleportation protocol.
  arXiv  Detail & Related papers  (2023-10-03T17:55:10Z)
• Quantum entanglement and contextuality with complexifications of $E_8$ root system [91.3755431537592]
  The Witting configuration with 40 complex rays was suggested as a possible reformulation of Penrose model with two spin-3/2 systems based on geometry of dodecahedron. An analysis of properties of suggested configuration of quantum states is provided using many analogies with properties of Witting configuration.
  arXiv  Detail & Related papers  (2022-10-27T11:23:12Z)
• Clifford Algebras, Quantum Neural Networks and Generalized Quantum Fourier Transform [0.0]
  We propose models of quantum neural networks through Clifford algebras. The Clifford algebras are the natural framework for multidimensional data analysis in a quantum setting.
  arXiv  Detail & Related papers  (2022-06-03T20:29:59Z)
• No-signalling constrains quantum computation with indefinite causal structure [45.279573215172285]
  We develop a formalism for quantum computation with indefinite causal structures. We characterize the computational structure of higher order quantum maps. We prove that these rules, which have a computational and information-theoretic nature, are determined by the more physical notion of the signalling relations between the quantum systems.
  arXiv  Detail & Related papers  (2022-02-21T13:43:50Z)
• A Graphical Calculus for Quantum Computing with Multiple Qudits using Generalized Clifford Algebras [0.0]
  We show that it is feasible to envision implementing the braid operators for quantum computation, by showing that they are 2-local operators. We derive several new identities for the braid elements, which are key to our proofs. In terms of quantum computation, we show that it is feasible to envision implementing the braid operators for quantum computation.
  arXiv  Detail & Related papers  (2021-03-30T05:19:49Z)
• Clifford algebras, algebraic spinors, quantum information and applications [0.0]
  We use extensively the algebras $Cl_3,0$ and $Cl_1,3$ as well as tensor products of Clifford algebras for quantum information.
  arXiv  Detail & Related papers  (2020-05-08T16:27:16Z)
• 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.