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arXiv:quant-ph/9710041 · Modern Physics Letters B

Error Avoiding Quantum Codes

Paolo Zanardi A, Mario Rasetti

Published Oct 30, 1997• 134 citations
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Abstract

The existence is proved of a class of open quantum systems that admits a linear subspace [Formula: see text] of the space of states such that the restriction of the dynamical semigroup to the states built over [Formula: see text] is unitary. Such subspace allows for error-avoiding (noiseless) encoding of quantum information.
DOI
10.1142/s0217984997001304

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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.
Quantum error avoiding codes are constructed by exploiting a geometric interpretation of the algebra of measurements of an open quantum system. The notion of a generalized Dirac operator is introduced and used to naturally construct families of decoherence free subspaces for the encoding of quantum information. The members of the family are connected to each other by the discrete Morita equivalences of the algebra of observables, which render possible several choices of noiseless code in which to perform quantum computation. The construction is applied to various examples of discrete and continuous quantum systems.
The essential insight of quantum error correction was that quantum information can be protected by suitably encoding this quantum information across multiple independently erred quantum systems. Recently it was realized that, since the most general method for encoding quantum information is to encode it into a subsystem, there exists a novel form of quantum error correction beyond the traditional quantum error correcting subspace codes. These new quantum error correcting subsystem codes differ from subspace codes in that their quantum correcting routines can be considerably simpler than related subspace codes. Here we present a class of quantum error correcting subsystem codes constructed from two classical linear codes. These codes are the subsystem versions of the quantum error correcting subspace codes which are generalizations of Shor's original quantum error correcting subspace codes. For every Shor-type code, the codes we present give a considerable savings in the number of stabilizer measurements needed in their error recovery routines.
A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (two-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not necessarily independently, the resulting n qubits can be used to faithfully reconstruct the original quantum state of the k encoded qubits. Quantum error-correcting codes are shown to exist with asymptotic rate k/n=1-2H2{\mathit{H}}_{2}H2​(2t/n) where H2{\mathit{H}}_{2}H2​(p) is the binary entropy function -plog2{\mathrm{log}}_{2}log2​p-(1-p)log2{\mathrm{log}}_{2}log2​(1-p). Upper bounds on this asymptotic rate are given. \textcopyright{} 1996 The American Physical Society.
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.