The Zak transform: a framework for quantum computation with the Gottesman-Kitaev-Preskill code

Abstract

The Gottesman-Kitaev-Preskill (GKP) code encodes a qubit into a bosonic mode using periodic wavefunctions. This periodicity makes the GKP code a natural setting for the Zak transform, which is tailor-made to provide a simple description for periodic functions. We review the Zak transform and its connection to a Zak basis of states in Hilbert space, decompose the shift operators that underpin the stabilizers and the correctable errors, and we find that Zak transforms of the position wavefunction appear naturally in GKP error correction. We construct a new bosonic subsystem decomposition (SSD) -- the modular variable SSD -- by dividing a mode's Hilbert space, expressed in the Zak basis, into that of a virtual qubit and a virtual gauge mode. Tracing over the gauge mode gives a logical-qubit state, and preceding the trace with a particular logical-gauge interaction gives a different logical state -- that associated to GKP error correction.

arXiv comment15 pages, 5 figures

References in corpus (32)

• Circuit Quantum Electrodynamics: Coherent Coupling of a Single Photon to a Cooper Pair Box
• Circuit Quantum Electrodynamics
• Encoding a qubit in an oscillator
• A Theory of Quantum Error-Correcting Codes
• Macroscopically distinct quantum superposition states as a bosonic code for amplitude damping
• Encoding a qubit in a trapped-ion mechanical oscillator
• Fault-Tolerant Measurement-Based Quantum Computing with Continuous-Variable Cluster States
• Quantum tensor product structures are observable-induced
• Performance and structure of single-mode bosonic codes
• Bosonic Quantum Codes for Amplitude Damping
• All-Gaussian universality and fault tolerance with the Gottesman-Kitaev-Preskill code
• Robust encoding of a qubit in a molecule
• Generating Grid States From Schrödinger Cat States without Post-Selection
• Encoding a Qubit into a Cavity Mode in Circuit-QED using Phase Estimation
• Error Analysis For Encoding A Qubit In An Oscillator
• Sequential modular position and momentum measurements of a trapped ion mechanical oscillator
• Modular Bosonic Subsystem Codes
• Equivalence of approximate Gottesman-Kitaev-Preskill codes
• Cost-Reduced All-Gaussian Universality with the Gottesman-Kitaev-Preskill Code: Resource-Theoretic Approach to Cost Analysis
• Enhanced noise resilience of the surface-GKP code via designed bias
• Quantum information processing in phase space: A modular variables approach
• Doubly Nonlinear Superconducting Qubit
• Realizing modular quadrature measurements via a tunable photon-pressure coupling in circuit-QED
• Phase-space methods for representing, manipulating, and correcting Gottesman-Kitaev-Preskill qubits
• Subsystem analysis of continuous-variable resource states
• Streamlined quantum computing with macronode cluster states
• Efficient simulation of Gottesman-Kitaev-Preskill states with Gaussian circuits
• Building a bigger Hilbert space for superconducting devices, one Bloch state at a time
• Hidden qubit cluster states
• Universal Continuous Variable Quantum Computation Without Cooling
• Universal quantum computing with thermal state bosonic systems
• Wigner distribution on a double cylinder phase space for studying quantum error correction protocol

Cited by in corpus (4)

• Sufficient condition for universal quantum computation using bosonic circuits
• Logical channel for heralded and pure loss with the Gottesman-Kitaev-Preskill code
• Bosonic Pauli+: Efficient Simulation of Concatenated Gottesman-Kitaev-Preskill Codes
• Achievable rates for concatenated square Gottesman-Kitaev-Preskill codes