General phase spaces: from discrete variables to rotor and continuum limits

Abstract

We provide a basic introduction to discrete-variable, rotor, and continuous-variable quantum phase spaces, explaining how the latter two can be understood as limiting cases of the first. We extend the limit-taking procedures used to travel between phase spaces to a general class of Hamiltonians (including many local stabilizer codes) and provide six examples: the Harper equation, the Baxter parafermionic spin chain, the Rabi model, the Kitaev toric code, the Haah cubic code (which we generalize to qudits), and the Kitaev honeycomb model. We obtain continuous-variable generalizations of all models, some of which are novel. The Baxter model is mapped to a chain of coupled oscillators and the Rabi model to the optomechanical radiation pressure Hamiltonian. The procedures also yield rotor versions of all models, five of which are novel many-body extensions of the almost Mathieu equation. The toric and cubic codes are mapped to lattice models of rotors, with the toric code case related to U(1) lattice gauge theory.

22 pages, 3 figures; part of special issue on Rabi model; v2 minor changes

References in corpus (85)

• Fault-tolerant quantum computation by anyons
• Anyons in an exactly solved model and beyond
• Quantum Spin Hall Effect
• Encoding a qubit in an oscillator
• On the Integrability of the Rabi Model
• On mutually unbiased bases
• Local stabilizer codes in three dimensions without string logical operators
• On a Lorentz-Invariant Interpretation of Noncommutative Space-Time and Its Implications on Noncommutative QFT
• Fracton Topological Order, Generalized Lattice Gauge Theory and Duality
• A quantum memory for orbital angular momentum photonic qubits
• Quantum orders in an exact soluble model
• On integrability of the Yang-Baxter $\si$-model
• Strong magnetic coupling between an electronic spin qubit and a mechanical resonator
• Discrete phase space based on finite fields
• Foundations of the AdS_5 x S^5 Superstring. Part I
• An integrable deformation of the AdS_5 x S^5 superstring action
• Integrability for the Full Spectrum of Planar AdS/CFT II
• Integrability for the Full Spectrum of Planar AdS/CFT
• Yang-Baxter $\sigma$-models and dS/AdS T-duality
• Parafermionic edge zero modes in Z_n-invariant spin chains
• Quantum spectral curve for AdS_5/CFT_4
• Integrable interpolations: From exact CFTs to non-Abelian T-duals
• Topological Equivalence between the Fibonacci Quasicrystal and the Harper Model
• Quantum many particle systems in ring-shaped optical lattices
• Mechanical spin control of nitrogen-vacancy centers in diamond
• Superstring Theory on AdS_2 x S^2 as a Coset Supermanifold
• Scale invariance of the eta-deformed AdS5 x S5 superstring, T-duality and modified type II equations
• Jordanian deformations of the AdS_5xS^5 superstring
• Quasi-probability representations of quantum theory with applications to quantum information science
• On integrable deformations of superstring sigma models related to AdS_n x S^n supercosets
• Target space supergeometry of $\eta$ and $\lambda$-deformed strings
• An Integrable Deformation of the AdS5 x S5 Superstring
• Thermodynamic Bethe Ansatz for planar AdS/CFT: a proposal
• Yang-Baxter deformations, AdS/CFT, and twist-noncommutative gauge theory
• Wigner Functions for Arbitrary Quantum Systems
• Free parafermions
• Coherent states for a quantum particle on a circle
• Quantum phase transition in the Dicke model with critical and non-critical entanglement
• Supergravity backgrounds for deformations of AdS_n x S^n supercoset string models
• The ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ mirror model as a string
• Discrete Abelian Gauge Theories for Quantum Simulations of QED
• Qudit surface codes and gauge theory with finite cyclic groups
• Supergravity background of the lambda-deformed AdS_3 x S^3 supercoset
• The integrable harmonic map problem versus Ricci flow
• Aharonov-Bohm effect in the tunnelling of a quantum rotor in a linear Paul trap
• Double Wick rotating Green-Schwarz strings
• An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group
• Uncertainty Relation for the Discrete Fourier Transform
• Quasiprobability distribution functions for periodic phase-spaces: I. Theoretical Aspects
• Quantum Rabi model for N-state atoms
• Quantum deformations of the flat space superstring
• The Quantum Deformed Mirror TBA II
• Achieving translational symmetry in trapped cold ion rings
• Generalized Kitaev Models and Slave Genons
• Deformed Neumann model from spinning strings on (AdS_5 x S^5)_η
• Supersymmetry in quantum optics and in spin-orbit coupled systems
• Parafermionic clock models and quantum resonance
• Assembling a ring-shaped crystal in a microfabricated surface ion trap
• S-Matrices and Quantum Group Symmetry of k-Deformed Sigma Models
• Quantum mechanics in finite dimensional Hilbert space
• Extended Cahill and Glauber formalism for finite dimensional spaces II. Applications in quantum tomography and quantum teleportation
• From Large N Matrices to the Noncommutative Torus
• The calculation of longitude and latitude from geodesic measurements
• Anyonic statistics with continuous variables
• The S-matrix algebra of the AdS2 X S2 superstring
• Yang-Baxter invariance of the Nappi-Witten model
• Encoding many qubits in a rotor
• Integrability of the eta-deformed Neumann-Rosochatius model
• Discrete squeezed states for finite-dimensional spaces
• Generalized Toric Codes Coupled to Thermal Baths
• Detecting Topological Entanglement Entropy in a Lattice of Quantum Harmonic Oscillators
• On discrete structures in finite Hilbert spaces
• Entanglement vs. the quantum-to-classical transition
• Schwinger, Pegg and Barnett approaches and a relationship between angular and Cartesian quantum descriptions II: Phase Spaces
• Energy spectrum and critical exponents of the free parafermion $Z_N$ spin chain
• Phase space methods for particles on a circle
• $Z_N$ symmetric chiral Rabi model: a new $N$-level system
• Properties of finite Gaussians and the discrete-continuous transition
• $Z_3$ generalization of the Kitaev's spin-1/2 model
• Schwinger, Pegg and Barnett and a relationship between angular and Cartesian quantum descriptions
• Orbit method quantization of the AdS${}_2$ superparticle
• Sylvester-t' Hooft generators of sl(n) and sl(n|n), and relations between them
• Travelling to exotic places with ultracold atoms
• Berezinskii-Kosterlitz-Thouless transition in Josephson junction arrays
• UV-IR transmutation for hybrid realizations of Z_k parafermion systems

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• Fractons
• Fracton-Elasticity Duality
• Robust encoding of a qubit in a molecule
• Quantum Error Correction with the Toric-GKP Code
• Emergent Phases of Fractonic Matter
• Twisted Fracton Models in Three Dimensions
• Continuous symmetries and approximate quantum error correction
• Cage-Net Fracton Models
• Pinch Point Singularities of Tensor Spin Liquids
• Crystal-to-Fracton Tensor Gauge Theory Dualities
• Symmetric Tensor Gauge Theories on Curved Spaces
• Fractonic line excitations : an inroad from 3d elasticity theory
• Higher Rank Deconfined Quantum Criticality at the Lifshitz Transition and the Exciton Bose Condensate
• Subsystem analysis of continuous-variable resource states
• Homological Quantum Rotor Codes: Logical Qubits from Torsion
• Modeling Position and Momentum in Finite-Dimensional Hilbert Spaces via Generalized Pauli Operators
• On the discrete Wigner function for SU(N)
• $SU(2)$-particle sigma model: Momentum-space quantization of a particle on the sphere $S^3$
• Clifford operations and homological codes for rotors and oscillators
• Viewing protected superconducting qubits through the lens of the cat qubit
• Spin chains, defects, and quantum wires for the quantum-double edge
• Topological stabilizer models on continuous variables
• Utilizing discrete variable representations for decoherence-accurate numerical simulation of superconducting circuits
• Generalized Conditional Displacement
• Enhancement in Breaking of Time-reversal Invariance in the Quantum Kicked Rotor