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Bosonic quantum Fourier codes

Anthony Leverrier

TL;DR

This work develops bosonic quantum Fourier codes by encoding information in an irreducible representation of a finite group G ⊂ U(2) through an inverse quantum Fourier transform, enabling universal control of encoded qubits in a two-mode bosonic system. The simplest nontrivial instance with G = ⟨X,Z⟩ yields a two-mode Fourier cat code that stores a logical qubit and a gauge qubit in product cat states, with logical gates realized via natural Gaussian interactions, code deformation, Kerr effects, and SNAP-based schemes. A universal gate set is constructed, including S and CZ via Kerr interactions, Hadamard through code deformation, and Z(θ) via quantum Zeno dynamics or SNAP gates, together with state preparation and measurement strategies. The code is stabilized by tailored dissipators that protect against single-photon loss and support error-correction via parity-based syndromes, and its performance is contrasted with rotation-symmetric, GKP, and binomial-like bosonic codes. The framework suggests a path toward scalable, bosonic-only fault-tolerant quantum computation with covariant encodings, and points to extensions to other finite groups and higher-dimensional bosonic architectures.

Abstract

While 2-level systems, aka qubits, are a natural choice to perform a logical quantum computation, the situation is less clear at the physical level. Encoding information in higher-dimensional physical systems can indeed provide a first level of redundancy and error correction that simplifies the overall fault-tolerant architecture. A challenge then is to ensure universal control over the encoded qubits. Here, we explore an approach where information is encoded in an irreducible representation of a finite subgroup of $U(2)$ through an inverse quantum Fourier transform. We illustrate this idea by applying it to the real Pauli group $\langle X, Z\rangle$ in the bosonic setting. The resulting two-mode Fourier cat code displays good error correction properties and admits an experimentally-friendly universal gate set that we discuss in detail.

Bosonic quantum Fourier codes

TL;DR

This work develops bosonic quantum Fourier codes by encoding information in an irreducible representation of a finite group G ⊂ U(2) through an inverse quantum Fourier transform, enabling universal control of encoded qubits in a two-mode bosonic system. The simplest nontrivial instance with G = ⟨X,Z⟩ yields a two-mode Fourier cat code that stores a logical qubit and a gauge qubit in product cat states, with logical gates realized via natural Gaussian interactions, code deformation, Kerr effects, and SNAP-based schemes. A universal gate set is constructed, including S and CZ via Kerr interactions, Hadamard through code deformation, and Z(θ) via quantum Zeno dynamics or SNAP gates, together with state preparation and measurement strategies. The code is stabilized by tailored dissipators that protect against single-photon loss and support error-correction via parity-based syndromes, and its performance is contrasted with rotation-symmetric, GKP, and binomial-like bosonic codes. The framework suggests a path toward scalable, bosonic-only fault-tolerant quantum computation with covariant encodings, and points to extensions to other finite groups and higher-dimensional bosonic architectures.

Abstract

While 2-level systems, aka qubits, are a natural choice to perform a logical quantum computation, the situation is less clear at the physical level. Encoding information in higher-dimensional physical systems can indeed provide a first level of redundancy and error correction that simplifies the overall fault-tolerant architecture. A challenge then is to ensure universal control over the encoded qubits. Here, we explore an approach where information is encoded in an irreducible representation of a finite subgroup of through an inverse quantum Fourier transform. We illustrate this idea by applying it to the real Pauli group in the bosonic setting. The resulting two-mode Fourier cat code displays good error correction properties and admits an experimentally-friendly universal gate set that we discuss in detail.

Paper Structure

This paper contains 16 sections, 2 theorems, 41 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. The framework and comparison with other bosonic codes
  3. Quantum Fourier codes
  4. The two-mode Fourier cat code
  5. A universal gate set
  6. Gates in $N(G)$ through code deformation
  7. The logical $S$ and $CZ$ gates
  8. Logical $Z(\theta)$ gates through quantum Zeno effect
  9. Alternative implementation of $Z(\theta)$ with SNAP gates
  10. State preparation and measurements
  11. Stabilization and error correction
  12. Dissipators and stabilizers
  13. Beyond pure loss
  14. Discussion and future work
  15. Single-mode cat qudit
  16. ...and 1 more sections

Key Result

Lemma 1

For any gate $U \in N(G)$,

Figures (1)

  • Figure 1: Entanglement infidelity $1-F_{\mathrm{ent}}$ of cat-type bosonic qubit encodings for the pure-loss channel of parameter $\gamma$, using the Petz recovery map. Shown are the two-mode Fourier cat code (logical qubit with the multiplicity register fixed to $|0\rangle_M$), the 4-legged cat code, the 2-repetition cat code, and the pair-cat code. Left:$1-F_{\mathrm{ent}}$ versus coherent amplitude $\alpha$ at fixed $\gamma=0.01$. Right:$1-F_{\mathrm{ent}}$ versus $\gamma$ using $\alpha=\sqrt{\pi/2}$ for the two-mode Fourier cat and 2-repetition cat codes, and $\alpha$ is chosen to minimize the left-panel infidelity at $\gamma=0.01$ for the 4-legged cat and pair-cat codes. Pair-cat values are evaluated in a truncated two-mode Fock basis with cutoffs chosen to ensure numerical convergence.

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Lemma 2
  • proof