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Qudit stabiliser codes for $\mathbb{Z}_N$ lattice gauge theories with matter

Luca Spagnoli, Alessandro Roggero, Nathan Wiebe

Abstract

In this work we extend the connection between Quantum Error Correction (QEC) and Lattice Gauge Theories (LGTs) by showing that a $\mathbb{Z}_N$ gauge theory with prime dimension $N$ coupled to dynamical matter can be expressed as a qudit stabilizer code. Using the stabilizer formalism we show how to formulate an exact mapping of the encoded $\mathbb{Z}_N$ gauge theory onto two different bosonic models, uncovering a logical duality generated by error correction itself. From this perspective, quantum error correction provides a unifying language to expose dual descriptions of lattice gauge theories. In addition, we generalize earlier $\mathbb{Z}_2$ constructions on qubits to $\mathbb{Z}_N$ on $N$-level qudits and demonstrate how universal fault-tolerant gates can be implemented via state injection between compatible qudit codes.

Qudit stabiliser codes for $\mathbb{Z}_N$ lattice gauge theories with matter

Abstract

In this work we extend the connection between Quantum Error Correction (QEC) and Lattice Gauge Theories (LGTs) by showing that a gauge theory with prime dimension coupled to dynamical matter can be expressed as a qudit stabilizer code. Using the stabilizer formalism we show how to formulate an exact mapping of the encoded gauge theory onto two different bosonic models, uncovering a logical duality generated by error correction itself. From this perspective, quantum error correction provides a unifying language to expose dual descriptions of lattice gauge theories. In addition, we generalize earlier constructions on qubits to on -level qudits and demonstrate how universal fault-tolerant gates can be implemented via state injection between compatible qudit codes.
Paper Structure (22 sections, 156 equations, 2 figures, 2 tables)

This paper contains 22 sections, 156 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Stabilizer formalism for Qudits
  3. Abelian lattice gauge theories
  4. From free-fermions to gauge theories
  5. Encoding of gauge operators
  6. Fermionic encoding
  7. Hamiltonian in terms of generalized Pauli matrices
  8. Gauss's law embedded in $N$-level systems
  9. Gauss's law error correcting code
  10. Universal gate set
  11. Logical duality
  12. Conclusions
  13. Clifford and non-Clifford gates
  14. State injection
  15. Quantum Fourier Transform
  16. ...and 7 more sections

Figures (2)

  • Figure 1: Quantum circuit to measure error syndromes in the phase-flip code. The first ancilla will measure the stabilizer $S_1 = \mathcal{X}_1 \mathcal{X}_2^{-1}$, while the second ancilla will measure $S_2 = \mathcal{X}_2 \mathcal{X}_3^{-1}$.
  • Figure 2: Example circuit to show the use of flag qudits to identify $\mathcal{Z}$ errors to deduce their propagation through a circuit as explained in Durso-Sabina.