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Quantum Process Realization of LDPC Code Dualities and Product Constructions

Shuhan Zhang, Deepak Aryal, Yi-Zhuang You

Abstract

We realize a broad class of code constructions, including Kramers-Wannier duality, tensor product, and check product, as quantum processes consisting of ancilla initialization, local unitaries, and projective measurements. Using ZX-calculus, we represent these transformations diagrammatically and provide a systematic algorithm for extracting quantum circuits. Central to our framework is the observation that the physical content of a classical LDPC code is captured by the operator algebra associated with its Tanner graph, and that code transformations correspond to maps between such algebras. Kramers-Wannier duality then admits a natural interpretation as gauging, while tensor and check products correspond to coupled-layer constructions in which interlayer coupling and projection implement a quotient on stacked operator algebras. Together, these results establish a unified framework connecting code transformations, quantum circuits, and mappings between distinct quantum phases of matter.

Quantum Process Realization of LDPC Code Dualities and Product Constructions

Abstract

We realize a broad class of code constructions, including Kramers-Wannier duality, tensor product, and check product, as quantum processes consisting of ancilla initialization, local unitaries, and projective measurements. Using ZX-calculus, we represent these transformations diagrammatically and provide a systematic algorithm for extracting quantum circuits. Central to our framework is the observation that the physical content of a classical LDPC code is captured by the operator algebra associated with its Tanner graph, and that code transformations correspond to maps between such algebras. Kramers-Wannier duality then admits a natural interpretation as gauging, while tensor and check products correspond to coupled-layer constructions in which interlayer coupling and projection implement a quotient on stacked operator algebras. Together, these results establish a unified framework connecting code transformations, quantum circuits, and mappings between distinct quantum phases of matter.
Paper Structure (33 sections, 2 theorems, 87 equations, 13 figures, 3 tables, 1 algorithm)

This paper contains 33 sections, 2 theorems, 87 equations, 13 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction and motivation
  2. Classical and quantum LDPC codes
  3. Definitions
  4. Operator algebra and Hamiltonian for cLDPC codes
  5. Dualities and their quantum process realizations
  6. ZX diagram representation
  7. ZX diagram to quantum process
  8. Physical interpretation as gauging
  9. Defect Condensation
  10. Minimal Coupling
  11. Phase interpretation
  12. Product constructions and coupled layer processes
  13. Tensor product code
  14. ZX realization as quantum process
  15. Phase of matter
  16. ...and 18 more sections

Key Result

Proposition 1

Let $\mathbb{H} \in \mathbb{F}_2^{m \times n}$ be the parity-check matrix of a cLDPC code $\mathcal{C}$, and let ${\mathsf D}$ denote the generalized KW duality map. Then the minimal number of ancilla qubits and projective measurements required to realize ${\mathsf D}$ as a quantum process are $k^\t

Figures (13)

  • Figure S1: (a) Tanner graph representation of a cLDPC code. (b) ZX-diagram representation of KW duality operator ${\mathsf D}$.
  • Figure S2: Basic move in circuit extraction: for row operation $R_2 \mapsto R_1 +R_2$ of the biadjacency matrix of the ZX diagram, we extract a CNOT(control = $q_2$, target = $q_1$) gate.
  • Figure S3: Defect condensation picture: ZX-diagram representation of the quantum process realization with $k^\top$ ancillae and $k$ measurements ${\mathsf D} = {\mathsf P}_\mu \, {\mathsf U}_{\mu,\eta} \, {\mathsf P}_\eta$.
  • Figure S4: Minimal coupling picture: ZX-diagram representation of the quantum process realization with $m$ ancilla qubits and $n$ measurements $\mathsf{D} = {\mathsf P}_\gamma \, {\mathsf U}'_{\gamma,\kappa} \, {\mathsf P}_\kappa$.
  • Figure S5: ZX diagram and quantum process realization of the tensor product construction.
  • ...and 8 more figures

Theorems & Definitions (2)

  • Proposition 1: Minimal resources for quantum process realizing ${\mathsf D}$
  • Proposition 2: Unitary ZX diagrams