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arXiv:quant-ph/9807004 · Physical Review Letters

Decoherence-Free Subspaces for Quantum Computation

Daniel A. Lidar, Isaac L. Chuang, K. Birgitta Whaley

Published Sep 21, 1998• 1,860 citations
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Abstract

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.

DOI
10.1103/physrevlett.81.2594

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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.
It was shown recently [D. A. Lidar et al., Phys. Rev. Lett. 81, 2594 (1998)] that within the framework of the semigroup Markovian master equation, decoherence-free (DF) subspaces exist which are stable to first order in time to a perturbation. Here this result is extended to the non-Markovian regime and generalized. In particular, it is shown that within both the semigroup and the non-Markovian operator sum representation, DF subspaces are stable to all orders in time to a symmetry-breaking perturbation. DF subspaces are thus ideal for quantum memory applications. For quantum computation, however, the stability result does not extend beyond the first order. Thus, to perform robust quantum computation in DF subspaces, they must be supplemented with quantum error correcting codes.
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 $\mathcal{C}\ensuremath{\subset}\mathcal{H}$ 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.
Universal quantum computation on decoherence-free subspaces and subsystems (DFSs) is examined with particular emphasis on using only physically relevant interactions. A necessary and sufficient condition for the existence of decoherence-free (noiseless) subsystems in the Markovian regime is derived here for the first time. A stabilizer formalism for DFSs is then developed which allows for the explicit understanding of these in their dual role as quantum error correcting codes. Conditions for the existence of Hamiltonians whose induced evolution always preserves a DFS are derived within this stabilizer formalism. Two possible collective decoherence mechanisms arising from permutation symmetries of the system-bath coupling are examined within this framework. It is shown that in both cases universal quantum computation which always preserves the DFS (natural fault-tolerant computation) can be performed using only two-body interactions. This is in marked contrast to standard error correcting codes, where all known constructions using one- or two-body interactions must leave the code space during the on-time of the fault-tolerant gates. A further consequence of our universality construction is that a single exchange Hamiltonian can be used to perform universal quantum computation on an encoded space whose asymptotic coding efficiency is unity. The exchange Hamiltonian, which is naturally present in many quantum systems, is thus asymptotically universal.
Coherence in an open quantum system is degraded through its interaction with a bath. This decoherence can be avoided by restricting the dynamics of the system to special decoherence-free subspaces. These subspaces are usually constructed under the assumption of spatially symmetric system-bath coupling. Here we show that decoherence-free subspaces may appear without spatial symmetry. Instead, we consider a model of system-bath interactions in which to first order only multiple-qubit coupling to the bath is present, with single-qubit system-bath coupling absent. We derive necessary and sufficient conditions for the appearance of decoherence-free states in this model, and give a number of examples. In a sequel paper we show how to perform universal and fault tolerant quantum computation on the decoherence-free subspaces considered in this paper.
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.
A general scheme to perform universal, fault-tolerant quantum computation within decoherence-free subspaces (DFSs) is presented. At most two-qubit interactions are required, and the system remains within the DFS throughout the entire implementation of a quantum gate. We show explicitly how to perform universal computation on clusters of the four-qubit DFS encoding one logical qubit each under spatially symmetric (collective) decoherence. Our results have immediate relevance to quantum computer implementations in which quantum logic is implemented through exchange interactions, such as the recently proposed spin-spin coupled quantum dot arrays and donor-atom arrays.