Gelfand-Tsetlin basis for partially transposed permutations, with applications to quantum information

• URL: http://arxiv.org/abs/2310.02252v1
• Date: Tue, 3 Oct 2023 17:55:10 GMT
• Title: Gelfand-Tsetlin basis for partially transposed permutations, with applications to quantum information
• Authors: Dmitry Grinko, Adam Burchardt, Maris Ozols
• Abstract summary: 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.
• Score: 0.9208007322096533
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study representation theory of the partially transposed permutation matrix algebra, a matrix representation of the diagrammatic walled Brauer algebra. This algebra plays a prominent role in mixed Schur-Weyl duality that appears in various contexts in quantum information. Our main technical result is an explicit formula for the action of the walled Brauer algebra generators in the Gelfand-Tsetlin basis. It generalizes the well-known Gelfand-Tsetlin basis for the symmetric group (also known as Young's orthogonal form or Young-Yamanouchi basis). We provide two applications of our result to quantum information. First, we show how to simplify semidefinite optimization problems over unitary-equivariant quantum channels by performing a symmetry reduction. Second, we derive an efficient quantum circuit for implementing the optimal port-based quantum teleportation protocol, exponentially improving the known trivial construction. As a consequence, this also exponentially improves the known lower bound for the amount of entanglement needed to implement unitaries non-locally. Both applications require a generalization of quantum Schur transform to tensors of mixed unitary symmetry. We develop an efficient quantum circuit for this mixed quantum Schur transform and provide a matrix product state representation of its basis vectors. For constant local dimension, this yields an efficient classical algorithm for computing any entry of the mixed quantum Schur transform unitary.

Related papers

• Geometric Quantum Machine Learning with Horizontal Quantum Gates [41.912613724593875]
  We propose an alternative paradigm for the symmetry-informed construction of variational quantum circuits. We achieve this by introducing horizontal quantum gates, which only transform the state with respect to the directions to those of the symmetry. For a particular subclass of horizontal gates based on symmetric spaces, we can obtain efficient circuit decompositions for our gates through the KAK theorem.
  arXiv  Detail & Related papers  (2024-06-06T18:04:39Z)
• Optimized synthesis of circuits for diagonal unitary matrices with reflection symmetry [0.0]
  It is important to optimize the quantum circuits in circuit depth and gate count, especially entanglement gates, including the CNOT gate. We show that the quantum circuit by our proposed algorithm achieves nearly half the reduction in both the gate count and circuit depth.
  arXiv  Detail & Related papers  (2023-10-10T14:52:17Z)
• The mixed Schur transform: efficient quantum circuit and applications [0.0]
  The Schur transform is an important primitive in quantum information and theoretical physics. We give a generalization of its quantum circuit implementation due to Bacon, Chuang, and Harrow (SODA 2007) We show how the mixed Schur transform enables efficient implementation of unitary-equivariant channels in various settings.
  arXiv  Detail & Related papers  (2023-10-02T20:03:56Z)
• A Bottom-up Approach to Constructing Symmetric Variational Quantum Circuits [0.0]
  We show how to construct symmetric quantum circuits using representation theory. We show how to derive the particle-conserving exchange gates, which are commonly used in constructing hardware-efficient quantum circuits.
  arXiv  Detail & Related papers  (2023-08-17T10:57:15Z)
• Monte Carlo Graph Search for Quantum Circuit Optimization [26.114550071165628]
  This work proposes a quantum architecture search algorithm based on a Monte Carlo graph search and measures of importance sampling. It is applicable to the optimization of gate order, both for discrete gates, as well as gates containing continuous variables.
  arXiv  Detail & Related papers  (2023-07-14T14:01:25Z)
• Vectorization of the density matrix and quantum simulation of the von Neumann equation of time-dependent Hamiltonians [65.268245109828]
  We develop a general framework to linearize the von-Neumann equation rendering it in a suitable form for quantum simulations. We show that one of these linearizations of the von-Neumann equation corresponds to the standard case in which the state vector becomes the column stacked elements of the density matrix. A quantum algorithm to simulate the dynamics of the density matrix is proposed.
  arXiv  Detail & Related papers  (2023-06-14T23:08:51Z)
• Quantum Circuit Completeness: Extensions and Simplifications [44.99833362998488]
  The first complete equational theory for quantum circuits has only recently been introduced. We simplify the equational theory by proving that several rules can be derived from the remaining ones. The complete equational theory can be extended to quantum circuits with ancillae or qubit discarding.
  arXiv  Detail & Related papers  (2023-03-06T13:31:27Z)
• Expanding the reach of quantum optimization with fermionic embeddings [2.378735224874938]
  In this work, we establish a natural embedding for this class of LNCG problems onto a fermionic Hamiltonian. We show that our quantum representation requires only a linear number of qubits. We provide evidence that this rounded quantum relaxation can produce high-quality approximations.
  arXiv  Detail & Related papers  (2023-01-04T19:00:01Z)
• Quantum Worst-Case to Average-Case Reductions for All Linear Problems [66.65497337069792]
  We study the problem of designing worst-case to average-case reductions for quantum algorithms. We provide an explicit and efficient transformation of quantum algorithms that are only correct on a small fraction of their inputs into ones that are correct on all inputs.
  arXiv  Detail & Related papers  (2022-12-06T22:01:49Z)
• Quantum algorithms for grid-based variational time evolution [36.136619420474766]
  We propose a variational quantum algorithm for performing quantum dynamics in first quantization. Our simulations exhibit the previously observed numerical instabilities of variational time propagation approaches.
  arXiv  Detail & Related papers  (2022-03-04T19:00:45Z)
• Quantum Circuits in Additive Hilbert Space [0.0]
  We show how conventional circuits can be expressed in the additive space and how they can be recovered. In particular in our formalism we are able to synthesize high-level multi-controlled primitives from low-level circuit decompositions. Our formulation also accepts a circuit-like diagrammatic representation and proposes a novel and simple interpretation of quantum computation.
  arXiv  Detail & Related papers  (2021-11-01T19:05:41Z)

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.