Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exploring quantum weight enumerators from the $n$-qubit parallelized SWAP test

Fei Shi, Kaiyi Guo, Xiande Zhang, Qi Zhao

TL;DR

The paper builds a bridge between quantum weight enumerators and the $n$-qubit parallelized SWAP test, showing that shadow enumerators correspond to SWAP-test probabilities and that Shor-Laflamme and Rains unitary enumerators can be derived from these probabilities. It provides an operational meaning for shadow enumerators and a simple, non-negativity-based proof of shadow inequalities via the SWAP-test circuit. Leveraging the quantum MacWilliams identities, the authors show how to compute key code parameters, including distances of QECCs and the $k$-uniformity of pure states, as well as quantify multipartite entanglement measures. This framework enables efficient estimation of quantum weight enumerators on quantum hardware, with practical implications for validating QECC performance and characterizing entanglement in complex quantum systems.

Abstract

Quantum weight enumerators are fundamental tools for analyzing quantum error-correcting codes and multipartite entanglement, offering insights into the existence of quantum error-correcting codes and $k$-uniform states. In this work, we establish a connection between quantum weight enumerators and the $n$-qubit parallelized SWAP test. We demonstrate that each shadow enumerator corresponds to a probability derived from this test, providing a physical interpretation for the shadow enumerators. Leveraging the non-negativity of these probabilities, we present an elegant proof for the shadow inequalities. Additionally, we show that the Shor-Laflamme weight enumerators and the Rains unitary enumerators can be calculated using the $n$-qubit parallelized SWAP test. For applications, we utilize this test to compute the distances of quantum error-correcting codes, determine the $k$-uniformity of pure states, and evaluate multipartite entanglement measures. Our results indicate that quantum weight enumerators can be efficiently estimated on quantum computers, opening a path to calculate and verify the distances of quantum error-correcting codes.

Exploring quantum weight enumerators from the $n$-qubit parallelized SWAP test

TL;DR

The paper builds a bridge between quantum weight enumerators and the -qubit parallelized SWAP test, showing that shadow enumerators correspond to SWAP-test probabilities and that Shor-Laflamme and Rains unitary enumerators can be derived from these probabilities. It provides an operational meaning for shadow enumerators and a simple, non-negativity-based proof of shadow inequalities via the SWAP-test circuit. Leveraging the quantum MacWilliams identities, the authors show how to compute key code parameters, including distances of QECCs and the -uniformity of pure states, as well as quantify multipartite entanglement measures. This framework enables efficient estimation of quantum weight enumerators on quantum hardware, with practical implications for validating QECC performance and characterizing entanglement in complex quantum systems.

Abstract

Quantum weight enumerators are fundamental tools for analyzing quantum error-correcting codes and multipartite entanglement, offering insights into the existence of quantum error-correcting codes and -uniform states. In this work, we establish a connection between quantum weight enumerators and the -qubit parallelized SWAP test. We demonstrate that each shadow enumerator corresponds to a probability derived from this test, providing a physical interpretation for the shadow enumerators. Leveraging the non-negativity of these probabilities, we present an elegant proof for the shadow inequalities. Additionally, we show that the Shor-Laflamme weight enumerators and the Rains unitary enumerators can be calculated using the -qubit parallelized SWAP test. For applications, we utilize this test to compute the distances of quantum error-correcting codes, determine the -uniformity of pure states, and evaluate multipartite entanglement measures. Our results indicate that quantum weight enumerators can be efficiently estimated on quantum computers, opening a path to calculate and verify the distances of quantum error-correcting codes.
Paper Structure (21 sections, 11 theorems, 61 equations, 2 figures, 2 tables)

This paper contains 21 sections, 11 theorems, 61 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. QECCs and $k$-uniform states
  4. Shor-Laflamme weight enumerators and Rains unitary enumerators
  5. Shadow enumerators
  6. The $n$-qubit parallelized SWAP test
  7. Connection between quantum weight enumerators and the $n$-qubit parallelized SWAP test
  8. applications
  9. Calculating the distances of QECCs
  10. Determining the $k$-uniformity of pure states
  11. Evaluating multipartite entanglement measures
  12. Conclusion
  13. Acknowledgement
  14. Two equivalent definitions of shadow enumerators.
  15. Monogamy inequalities from the shadow inequalities
  16. ...and 6 more sections

Key Result

Lemma 4

A subspace ${\cal Q}$ of ${\cal H}_{[n]}$ with dimension $K$ is a pure $((n,K,\delta))_d$ QECC if and only if $|\psi\rangle$ is a $(\delta-1)$-uniform state for all $|\psi\rangle\in {\cal Q}$.

Figures (2)

  • Figure 1: Quantum circuit for the $n$-qubit parallelized SWAP test of two $n$-partite states $\rho$ and $\sigma$. Each $A_i$ is an ancilla qubit, each $H$ is a Hadamard gate, and each controlled-SWAP gate is performed on $A_i$, $B_i$, and $C_i$.
  • Figure 2: The Venn diagram for the proof of Lemma \ref{['lemma:equivalent']}.

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 4: scott2004multipartitehuber2020quantum
  • Lemma 5: rains1998quantumscott2004multipartite
  • Proposition 6
  • Proposition 7
  • Theorem 8
  • Theorem 9
  • Proposition 10
  • ...and 4 more