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arXiv:quant-ph/9705044 · Physical Review Letters

Noiseless Quantum Codes

Paolo Zanardi A, Mario Rasetti

Published Oct 27, 1997• 1,315 citations
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Abstract

In this paper we study a model quantum register $R$ made of $N$ replicas (cells) of a given finite-dimensional quantum system S. Assuming that all cells are coupled with a common environment with equal strength we show that, for $N$ large enough, in the Hilbert space of $R$ there exists a linear subspace ${C}_{N}$ which is dynamically decoupled from the environment. The states in ${C}_{N}$ evolve unitarily and are therefore decoherence-dissipation free. The space ${C}_{N}$ realizes a noiseless quantum code in which information can be stored, in principle, for an arbitrarily long time without being affected by errors.

DOI
10.1103/physrevlett.79.3306

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In this paper we study a model quantum register RRR made of NNN replicas (cells) of a given finite-dimensional quantum system S. Assuming that all cells are coupled with a common environment with equal strength we show that, for NNN large enough, in the Hilbert space of RRR there exists a linear subspace CN{C}_{N}CN​ which is dynamically decoupled from the environment. The states in CN{C}_{N}CN​ evolve unitarily and are therefore decoherence-dissipation free. The space CN{C}_{N}CN​ realizes a noiseless quantum code in which information can be stored, in principle, for an arbitrarily long time without being affected by errors.
A model for a quantum register R\mathcal{R}R made of NNN replicas of a ddd-dimensional quantum system (cell) coupled with the environment is studied by means of a Born-Markov master equation (ME). Dissipation and decoherence are discussed in various cases in which a subdecoherent encoding can be rigorously found. For the quantum-bit case (d=2)(d=2)(d=2) we have solved, for small N,N,N, the ME by numerical direct integration and studied, as a function of the coherence length ${\ensuremath{\xi}}_{c}$ of the bath, fidelity and decoherence rates of states of the register. For large enough ${\ensuremath{\xi}}_{c}$ the singlet states of the global su(2) pseudo spin algebra of the register (noiseless at ${\ensuremath{\xi}}_{c}=\ensuremath{\infty})$ are shown to have a much smaller decoherence rates than the rest of the Hilbert space.
The dynamical evolution of a quantum register of arbitrary length coupled to an environment of arbitrary coherence length is predicted within a relevant model of decoherence. The results are reported for quantum bits (qubits) coupling individually to different environments (``independent decoherence'') and qubits interacting collectively with the same reservoir (``collective decoherence''). In both cases, explicit decoherence functions are derived for any number of qubits. The decay of the coherences of the register is shown to strongly depend on the input states: We show that this sensitivity is a characteristic of both types of coupling (collective and independent) and not only of the collective coupling, as has been reported previously. A nontrivial behavior (``recoherence'') is found in the decay of the off-diagonal elements of the reduced density matrix in the specific situation of independent decoherence. Our results lead to the identification of decoherence-free states in the collective decoherence limit. These states belong to subspaces of the system's Hilbert space that do not get entangled with the environment, making them ideal elements for the engineering of ``noiseless'' quantum codes. We also discuss the relations between decoherence of the quantum register and computational complexity based on the dynamical results obtained for the register density matrix.
In idealized models of a quantum register and its environment, quantum information can be stored indefinitely by encoding it into a decoherence-free subspace (DFS). Nevertheless, perturbations to the idealized register-environment coupling will cause decoherence in any realistic setting. Expanding a measure for state preservation, the dynamical fidelity, in powers of the strength of the perturbations, we prove stability to linear order is a generic property of quantum state evolution. The effect of noise perturbation is quantified by a concise expression for the strength of the quadratic leading order, which we define as the dynamical fidelity susceptibility of DFSs. Under the physical restriction that noise acts on the register kkk-locally, this susceptibility is bounded from above by a polynomial in the system size. These general results are illustrated by two physically relevant examples. Knowledge of the susceptibility can be used to increase coherence times of future quantum computers.
A model for a quantum register dissipatively coupled with a bosonic thermal bath is studied. The register consists of NNN quantum bits (qubits) the bath is described by Nb{N}_{b}Nb​ bosonic modes. The register-bath coupling is chosen in such a way that the total number of excitations is conserved. The Hilbert space splits, allowing the study of the dynamics separately in each sector. Assuming that the coupling with the bath is the same for all qubits, the excitation sectors have a further decomposition according the irreducible representations of the su(2) spin algebra. The stability against environment-generated noise of the information encoded in a quantum state of the register depends on its SU(2) symmetry content. At zero temperature we find that states belonging to the vacuum symmetry sector have, for long time, vanishing fidelity, whereas each lowest spin vector is decoupled from the bath and therefore is decoherence free. Numerical results are shown in the one-excitation space in the case of qubit-dependent bath-system coupling.