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arXiv:2104.11937 · Physical Review A

Noncommutative graphs based on finite-infinite system couplings: Quantum error correction for a qubit coupled to a coherent field

Grigori Gennadievich Amosov, Мокеев Александр Сергеевич, Alexander Nikolaevich Pechen

Published Apr 7, 2021• 7 citations
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Abstract

Quantum error correction plays a key role for quantum information transmission and quantum computing. In this work, we develop and apply the theory of noncommutative operator graphs to study error correction in the case of a finite-dimensional quantum system coupled to an infinite-dimensional system. We consider as an explicit example a qubit coupled via the Jaynes-Cummings (JC) Hamiltonian with a bosonic coherent field. We extend the theory of noncommutative graphs to this situation and construct, using Gazeau-Klauder coherent states, the corresponding noncommutative graph. As the result, we find the quantum anticlique, which is the projector on the error-correcting subspace, and analyze it as a function of the frequencies of the qubit and the bosonic field. The general treatment is also applied to the analysis of the error-correcting subspace for certain experimental values of the parameters of the Jaynes-Cummings Hamiltonian. The proposed scheme can be applied to any system that possess the same decomposition of spectrum of the Hamiltonian into a direct sum as in JC model, where eigenenergies in the two direct summands form strictly increasing sequences.

DOI
10.1103/physreva.103.042407

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Quantum error correction plays a key role for quantum information transmission and quantum computing. In this work, we develop and apply the theory of noncommutative operator graphs to study error correction in the case of a finite-dimensional quantum system coupled to an infinite-dimensional system. We consider as an explicit example a qubit coupled via the Jaynes-Cummings (JC) Hamiltonian with a bosonic coherent field. We extend the theory of noncommutative graphs to this situation and construct, using Gazeau-Klauder coherent states, the corresponding noncommutative graph. As the result, we find the quantum anticlique, which is the projector on the error-correcting subspace, and analyze it as a function of the frequencies of the qubit and the bosonic field. The general treatment is also applied to the analysis of the error-correcting subspace for certain experimental values of the parameters of the Jaynes-Cummings Hamiltonian. The proposed scheme can be applied to any system that possess the same decomposition of spectrum of the Hamiltonian into a direct sum as in JC model, where eigenenergies in the two direct summands form strictly increasing sequences.
Abstract We consider error correction, based on the theory of non-commutative graphs, for a model of a qubit interacting with quantum oscillator. The dynamics of the composite system is governed by the Schrödinger equation which generates positive operator-valued measure (POVM) for the system dynamics. We construct a quantum channel generating the non-commutative graph as a linear envelope of the POVM. The idea is based on applying a generalized version of a quantum channel using the apparatus of von Neumann algebras. The results are analyzes for a non-commutative graph generated by a qubit interacting with quantum oscillator. For this model the quantum anticlique which determines the error correcting subspace has an explicit expression.
Quantum error-correcting codes are constructed that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables. These codes exploit the noncommutative geometry of phase space to protect against errors that shift the values of the canonical variables q and p. In the setting of quantum optics, fault-tolerant universal quantum computation can be executed on the protected code subspace using linear optical operations, squeezing, homodyne detection, and photon counting; however, nonlinear mode coupling is required for the preparation of the encoded states. Finite-dimensional versions of these codes can be constructed that protect encoded quantum information against shifts in the amplitude or phase of a d-state system. Continuous-variable codes can be invoked to establish lower bounds on the quantum capacity of Gaussian quantum channels.
Quantum computers have been shown to be able to perform computations that are thus far thought to be infeasible through classical computation. This is achieved by taking advantage of certain properties and phenomena unique to quantum systems. Unfortunately, engineering particular quantum systems that are useful for computation is difficult and there is still a long road towards building a robust, large-scale quantum computer. A key approach to building quantum computers that are robust to noise is the development of Quantum Error Correcting codes. In recent years, there have been a surge of progress in the development and realisation of bosonic codes, which are quantum error correcting codes that encode quantum information into the infinite-dimensional Hilbert space of an oscillator. Due to its large Hilbert space, bosonic codes are able to protect quantum information using a single, or a few physical systems. This thesis will take two different but complementary approaches to the development of bosonic codes and bosonic code qubits. The first will be a numerical study of errors affecting bosonic codes within a fault-tolerant system. Specifically, we introduce the concatenated Bacon-Shor rotation-symmetric bosonic code and evaluate the entanglement fidelity of a teleportation based error correction scheme against pure loss errors. In the second approach, we study the recently proposed gyrator qubit by Rymarz et al., which realises a bosonic code known as the Gottesman-Kitaev-Preskill (GKP) code in its low energy eigenspace. We support this study with a Wentzel-Kramers-Brillouin (WKB) analysis of the Zak transformed Hamiltonian of the qubit. This allows us to derive the low-energy spectrum and eigenstates of the qubit and derive matrix elements corresponding to transitions between qubit states when the system is weakly coupled to a thermal environment.
Abstract The “noncommutative graphs” which arise in quantum error correction are a special case of the quantum relations introduced in Weaver (Quantum relations. Mem Am Math Soc 215(v–vi):81–140, 2012). We use this perspective to interpret the Knill–Laflamme error-correction conditions (Knill and Laflamme in Theory of quantum error-correcting codes. Phys Rev A 55:900-911, 1997) in terms of graph-theoretic independence, to give intrinsic characterizations of Stahlke’s noncommutative graph homomorphisms (Stahlke in Quantum zero-error source-channel coding and non-commutative graph theory. IEEE Trans Inf Theory 62:554–577, 2016) and Duan, Severini, and Winter’s noncommutative bipartite graphs (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013), and to realize the noncommutative confusability graph associated to a quantum channel (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013) as the pullback of a diagonal relation. Our framework includes as special cases not only purely classical and purely quantum information theory, but also the “mixed” setting which arises in quantum systems obeying superselection rules. Thus we are able to define noncommutative confusability graphs, give error correction conditions, and so on, for such systems. This could have practical value, as superselection constraints on information encoding can be physically realistic.
We theoretically investigate quantum Rabi oscillations in a system consisting of a two-level atom (qubit) strongly coupled to a one-dimensional open waveguide. In contrast to conventional cavity quantum electrodynamics, the qubit interacts with a continuum of propagating modes, which gives rise to fundamentally different dynamical behavior. Within the rotating-wave approximation, we express the multimode Jaynes-Cummings Hamiltonian in terms of collective bosonic operators and show that the system possesses two integrals of motion, enabling an exact diagonalization of the Hamiltonian in the single-excitation subspace. We show that for a strongly interacting qubit-photon system, the dynamics are captured by a reduced two-level model. In this framework, each level is defined as the product of the atomic excited state and a specific field mode, which takes the form of a multiphoton Fock-like state. In this picture, the Rabi oscillations represent a collective phenomenon corresponding to oscillations between multiphoton states differing by a single photon. We then extend our analysis to multiphoton processes in which the initial field is a coherent state with a continuous spectrum. In this case, the Rabi frequency is shown to be sensitive to the spectral profile of the function that generates the coherent state.