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Efficient Approximation of Quantum Channel Fidelity Exploiting Symmetry

Yeow Meng Chee, Hoang Ta, Van Khu Vu

TL;DR

This work tackles the challenge of computing the quantum channel fidelity $\mathrm{F}(\mathcal{N},M)$ by leveraging symmetry in the SDP hierarchy $\mathrm{SDP}_n(\mathcal{N},M)$. By confining the search to the invariant subspace $\mathrm{End}^{\mathfrak{S}_n}(\mathcal{H}^{\otimes n})$ and mapping it to a block-diagonal form, the authors convert the exponentially large SDP into a polynomial-size problem in the level $n$ and input dimension. Consequently, $\mathrm{F}(\mathcal{N},M)$ can be approximated to within $\epsilon$ in time polynomial in $1/\epsilon$ and the input dimension, with the number of blocks and block sizes bounded by polynomials in $n$ and fixed dimensions. The construction rests on a robust representation-theoretic framework for symmetry reduction, including a canonical invariant basis and a PSD-preserving bijection, enabling practical polynomial-time computation for fixed output dimensions and encodings.

Abstract

Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of $ε$ in $\mathrm{poly}(1/ε, \text{input dimension})$ time.

Efficient Approximation of Quantum Channel Fidelity Exploiting Symmetry

TL;DR

This work tackles the challenge of computing the quantum channel fidelity by leveraging symmetry in the SDP hierarchy . By confining the search to the invariant subspace and mapping it to a block-diagonal form, the authors convert the exponentially large SDP into a polynomial-size problem in the level and input dimension. Consequently, can be approximated to within in time polynomial in and the input dimension, with the number of blocks and block sizes bounded by polynomials in and fixed dimensions. The construction rests on a robust representation-theoretic framework for symmetry reduction, including a canonical invariant basis and a PSD-preserving bijection, enabling practical polynomial-time computation for fixed output dimensions and encodings.

Abstract

Determining the optimal fidelity for the transmission of quantum information over noisy quantum channels is one of the central problems in quantum information theory. Recently, [Berta-Borderi-Fawzi-Scholz, Mathematical Programming, 2021] introduced an asymptotically converging semidefinite programming hierarchy of outer bounds for this quantity. However, the size of the semidefinite programs (SDPs) grows exponentially with respect to the level of the hierarchy, thus making their computation unscalable. In this work, by exploiting the symmetries in the SDP, we show that, for a fixed output dimension of the quantum channel, we can compute the SDP in time polynomial with respect to the level of the hierarchy and input dimension. As a direct consequence of our result, the optimal fidelity can be approximated with an accuracy of in time.
Paper Structure (7 sections, 11 theorems, 58 equations, 1 figure, 1 table)

This paper contains 7 sections, 11 theorems, 58 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Approximate quantum error correction
  4. Elementary Representation Theory
  5. Efficient approximation of channel fidelity
  6. A basis for the invariant subspace.
  7. Proof of Lemma \ref{['lem:invariant_matrices']}

Key Result

Theorem 1

For $M, n \in \mathbb{N}_{\geq 1}$, let $\mathcal{N}_{\bar{A}\to B}$ be a quantum channel, and let $d_{\bar{A}}$ denote the input dimension of $\mathcal{N}_{\bar{A}\to B}$. The optimal value of the program $\mathrm{SDP}_{n}(\mathcal{N},M)$ can be computed in $\mathrm{poly}(n,d_{\bar{A}})$ time, for

Figures (1)

  • Figure :

Theorems & Definitions (19)

  • Theorem
  • Theorem 2.1: berta2021semidefinite
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4: Proposition 2.4.4, Polak_thesis
  • Lemma 2.5
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • ...and 9 more