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arXiv:quant-ph/9901047 · Physical Review A

Computation on an error-avoiding quantum code and symmetrization

Paolo Zanardi A

Published Aug 1, 1999• 44 citations
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Abstract

Let $\mathcal{H}$ be the state space of a quantum computer coupled with the environment by a set of error operators spanning a Lie algebra $\mathcal{L}.$ Suppose $\mathcal{L}$ admits an error-avoiding quantum code, i.e., a subspace $\mathcal{C}\ensuremath{\subset}\mathcal{H}$ annihilated by $\mathcal{L}.$ We show that a universal set of gates over $\mathcal{C}$ is obtained by any generic pair of $\mathcal{L}$-invariant gates. Such gates---if not available from the outset---can be obtained by resorting to a symmetrization with respect to the group generated by $\mathcal{L}.$ Any computation can then be performed completely within the coding decoherence-free subspace.

DOI
10.1103/physreva.60.r729

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Let H\mathcal{H}H be the state space of a quantum computer coupled with the environment by a set of error operators spanning a Lie algebra L.\mathcal{L}.L. Suppose L\mathcal{L}L admits an error-avoiding quantum code, i.e., a subspace C\ensuremath⊂H\mathcal{C}\ensuremath{\subset}\mathcal{H}C\ensuremath⊂H annihilated by L.\mathcal{L}.L. We show that a universal set of gates over C\mathcal{C}C is obtained by any generic pair of L\mathcal{L}L-invariant gates. Such gates---if not available from the outset---can be obtained by resorting to a symmetrization with respect to the group generated by L.\mathcal{L}.L. Any computation can then be performed completely within the coding decoherence-free subspace.
Decoherence in quantum computers is formulated within the semigroup approach. The error generators are identified with the generators of a Lie algebra. This allows for a comprehensive description which includes as a special case the frequently assumed spin-boson model. A generic condition is presented for errorless quantum computation: decoherence-free subspaces are spanned by those states which are annihilated by all the generators. It is shown that these subspaces are stable to perturbations and, moreover, that universal quantum computation is possible within them.
The notion of symmetry is shown to be at the heart of all error correction/avoidance strategies for preserving quantum coherence of an open quantum system S e.g., a quantum computer. The existence of a non-trivial group of symmetries of the dynamical algebra of S provides state-space sectors immune to decoherence. Such noiseless sectors, that can be viewed as a noncommutative version of the pointer basis, are shown to support universal quantum computation and to be robust against perturbations. When the required symmetry is not present one can generate it artificially resorting to active symmetrization procedures.
Decoherence-free subspaces (DFSs) shield quantum information from errors induced by the interaction with an uncontrollable environment. Here we study a model of correlated errors forming an Abelian subgroup (stabilizer) of the Pauli group (the group of tensor products of Pauli matrices). Unlike previous studies of DFSs, this type of error does not involve any spatial symmetry assumptions on the system-environment interaction. We solve the problem of universal, fault-tolerant quantum computation on the associated class of DFSs. We do so by introducing a hybrid DFS quantum error-correcting-code approach, where errors that arise due to departure of the codewords from the DFS are corrected actively.
Quantum error correcting codes play an essential role in protecting quantum information from the noise and the decoherence. Most quantum codes have been constructed based on the Pauli basis indexed by a finite field. With a newly introduced algebraic class called a nice ring, it is possible to construct the quantum codes such that their alphabet sizes are not restricted to powers of a prime.\n\nSubsystem codes are quantum error correcting schemes unifying stabilizer codes, decoherence free subspaces and noiseless subsystems. We show a generalization of subsystem codes over nice rings. Furthermore, we prove that free subsystem codes over a finite chain ring cannot outperform those over a finite field. We also generalize entanglement-assisted quantum error correcting codes to nice rings. With the help of the entanglement, any classical code can be used to derive the corresponding quantum codes, even if such codes are not self-orthogonal. We prove that an R-module with antisymmetric bicharacter can be decomposed as an orthogonal direct sum of hyperbolic pairs using symplectic geometry over rings. So, we can find hyperbolic pairs and commuting generators generating the check matrix of the entanglement-assisted quantum code.\n\nFault-tolerant quantum computation has been also studied over a finite field. Transversal operations are the simplest way to implement fault-tolerant quantum gates. We derive transversal Clifford operations for CSS codes over nice rings, including Fourier transforms, SUM gates, and phase gates. Since transversal operations alone cannot provide a computationally universal set of gates, we add fault-tolerant implementations of doubly-controlled Z gates for triorthogonal stabilizer codes over nice rings.\n\nFinally, we investigate optimal key exchange protocols for unconditionally secure key distribution schemes. We prove how many rounds are needed for the key exchange between any pair of the group on star networks, linear-chain networks, and general networks.
Proposals for quantum computing devices are many and varied. They each have unique noise processes that make none of them fully reliable at this time. There are several error correction/avoidance techniques which are valuable for reducing or eliminating errors, but not one, alone, will serve as a panacea. One must therefore take advantage of the strength of each of these techniques so that we may extend the coherence times of the quantum systems and create more reliable computing devices. To this end we give a general strategy for using dynamical decoupling operations on encoded subspaces. These encodings may be of any form; of particular importance are decoherence-free subspaces and quantum error correction codes. We then give means for empirically determining an appropriate set of dynamical decoupling operations for a given experiment. Using these techniques, we then propose a comprehensive encoding solution to many of the problems of quantum computing proposals which use exchange-type interactions. This uses a decoherence-free subspace and an efficient set of dynamical decoupling operations. It also addresses the problem of controllability in solid-state quantum dot devices.