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Codeword Stabilized Codes from m-Uniform Graph States

Sowrabh Sudevan, Sourin Das, Thamadathil Aswanth, Nupur Patanker, Navin Kashyap

TL;DR

This work presents a principled method to construct pure quantum error-correcting codes with fixed minimum distance from m-uniform graph states via the codeword stabilized (CWS) framework, linking graph-state uniformity, classical codes, and symplectic weight conditions. The core result (Theorem 1) characterizes when a CWS code built from a graph state and a binary linear code is a pure [[n,k,m+1]]_2 QECC, and shows that for m-regular graphs, any [n,k,d] code with d > m(m+1) suffices to yield such a code. The authors instantiate this with 1D and 2D cluster states to produce explicit infinite families: [[2^{2r}-1, 2^{2r}-2r-3, 3]]_2 and [[(2^{4r}-1)^2, (2^{4r}-1)^2-32r-7, 5]]_2 codes, including a refined construction using 2D 2-dimensional cyclic codes that improves the dimension for fixed distance 5. Additionally, measurement-based encoding and recovery protocols are developed for additive CWS codes, with the tent-peg framework providing a practical encoding structure. These results advance principled code construction from highly entangled graph resources and open avenues for MBQC-inspired QECC implementations and higher-dimensional generalizations.

Abstract

An m-uniform quantum state on n qubits is an entangled state in which every m-qubit subsystem is maximally mixed. Starting with an m-uniform state realized as the graph state associated with an m-regular graph, and a classical [n,k,d \ge m+1] binary linear code with certain additional properties, we show that pure [[n,k,m+1]]_2 quantum error-correcting codes (QECCs) can be constructed within the codeword stabilized (CWS) code framework. As illustrations, we construct pure [[2^{2r}-1,2^{2r}-2r-3,3]]_2 and [[(2^{4r}-1)^2, (2^{4r}-1)^2 - 32r-7, 5]]_2 QECCs. We also give measurement-based protocols for encoding into code states and for recovery of logical qubits from code states.

Codeword Stabilized Codes from m-Uniform Graph States

TL;DR

This work presents a principled method to construct pure quantum error-correcting codes with fixed minimum distance from m-uniform graph states via the codeword stabilized (CWS) framework, linking graph-state uniformity, classical codes, and symplectic weight conditions. The core result (Theorem 1) characterizes when a CWS code built from a graph state and a binary linear code is a pure [[n,k,m+1]]_2 QECC, and shows that for m-regular graphs, any [n,k,d] code with d > m(m+1) suffices to yield such a code. The authors instantiate this with 1D and 2D cluster states to produce explicit infinite families: [[2^{2r}-1, 2^{2r}-2r-3, 3]]_2 and [[(2^{4r}-1)^2, (2^{4r}-1)^2-32r-7, 5]]_2 codes, including a refined construction using 2D 2-dimensional cyclic codes that improves the dimension for fixed distance 5. Additionally, measurement-based encoding and recovery protocols are developed for additive CWS codes, with the tent-peg framework providing a practical encoding structure. These results advance principled code construction from highly entangled graph resources and open avenues for MBQC-inspired QECC implementations and higher-dimensional generalizations.

Abstract

An m-uniform quantum state on n qubits is an entangled state in which every m-qubit subsystem is maximally mixed. Starting with an m-uniform state realized as the graph state associated with an m-regular graph, and a classical [n,k,d \ge m+1] binary linear code with certain additional properties, we show that pure [[n,k,m+1]]_2 quantum error-correcting codes (QECCs) can be constructed within the codeword stabilized (CWS) code framework. As illustrations, we construct pure [[2^{2r}-1,2^{2r}-2r-3,3]]_2 and [[(2^{4r}-1)^2, (2^{4r}-1)^2 - 32r-7, 5]]_2 QECCs. We also give measurement-based protocols for encoding into code states and for recovery of logical qubits from code states.
Paper Structure (22 sections, 15 theorems, 63 equations, 3 figures, 1 algorithm)

This paper contains 22 sections, 15 theorems, 63 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Prior Work on QECCs from $m$-Uniform States
  4. Organization
  5. Preliminaries
  6. Graph states and $m$-uniformity
  7. CWS codes from graph states
  8. Main Results
  9. Pure $[[n,k,3]]$ QECCs from $1$-dimensional cluster states
  10. Pure $[[n^2,k,5]]$ QECCs from $2$-dimensional cluster states
  11. Measurement-Based Encoding and Recovery Protocols
  12. Encoding
  13. Sequential encoding
  14. Recovering the logical qubits
  15. Partial recovery of logical qubits
  16. ...and 7 more sections

Key Result

Proposition 1

A graph state $\ket{\mathrm{G}}$ associated with a graph $G$ is $m$-uniform iff all the stabilizers in $\mathcal{S} = \langle S_1,S_2,\ldots,S_n \rangle$, other than the identity $I$, have weight at least $m+1$. In particular, if $\ket{\mathrm{G}}$ is $m$-uniform, then every vertex of $G$ has degree

Figures (3)

  • Figure 1: $\Lambda_{3,3}$: An example of $2$-dimensional rectangular lattice with periodic boundary conditions. The periodic boundaries are shown with dotted lines.
  • Figure 2: Encoding one logical qubit: the input qubit contains the information to be encoded, and the bottom array of $n$ qubits is initially in a graph state. Each solid line connecting two qubits represents a controlled-$Z$ gate.
  • Figure 3: Tent peg protocol: The top horizontal array of dots represents $k$ input qubits holding logical information. The bottom horizontal array of dots represents the $n$ target qubits, initially in a graph state. Each solid line between a pair of qubits represents a controlled-$Z$ gate between the two qubits.

Theorems & Definitions (49)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3
  • Example 1
  • Proposition 4
  • Example 2
  • Theorem 1
  • ...and 39 more