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Journal of Optics B Quantum and Semiclassical Optics

A quantum code in decoherence-free and interaction-free subspaces

Yong Zhang, Zheng-Wei Zhou, Bo Hao Yu, Guang‐Can Guo

Published May 30, 2003• 2 citations
Publisher

Abstract

A quantum code based on decoherence-free subspace is considered. We investigate a special interaction—exchange interaction in the quantum system—and show that quantum states encoded by the quantum code are free of the exchange interaction. On the basis of this quantum code, universal quantum computation can be implemented by switching 'off and on' exchange interaction.

DOI
10.1088/1464-4266/5/3/317

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A quantum code based on decoherence-free subspace is considered. We investigate a special interaction—exchange interaction in the quantum system—and show that quantum states encoded by the quantum code are free of the exchange interaction. On the basis of this quantum code, universal quantum computation can be implemented by switching 'off and on' exchange interaction.
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.
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.
The exchange interaction between identical qubits in a quantum-information processor gives rise to unitary two-qubit errors. It is shown here that decoherence-free subspaces (DFSs) for collective decoherence undergo Pauli errors under exchange, which, however, do not take the decoherence-free states outside of the DFS. In order to protect DFSs against these errors it is sufficient to employ a recently proposed concatenated DFS quantum-error-correcting code scheme [D. A. Lidar, D. Bacon, and K.B. Whaley, Phys. Rev. Lett. 82, 4556 (1999)].
Most quantum computer realizations require the ability to apply local fields and tune the couplings between qubits, in order to realize single bit and two bit gates which are necessary for universal quantum computation. We present a scheme to remove the necessity of switching the couplings between qubits for two bit gates, which are more costly in many cases. Our strategy is to compute with encoded qubits in and out of carefully designed interaction free subspaces analogous to decoherence free subspaces. We give two examples to show how universal quantum computation is realized in our scheme with local manipulations to physical qubits only, for both diagonal and off diagonal interactions.
A scheme to implement a quantum computer subjected to decoherence and governed by un-tunable qubit-qubit interaction is presented. By concatenating bang-bang (BB) control and decoherence-free subspaces (DFSs) encoding, we protect the quantum computer from environment-induced docoherence that result in quantum information dissipating into the environment. For the un-tunable qubit-qubit interaction in the quantum system, BB control plus DFSs encoding will stabilize encoded quantum states. We show how this quantum system to implement universal quantum computation.
In this paper, I propose new models of quantum information processing using the exchange interaction in physical systems. The partial SWAP operator that can be realized using the exchange interaction is used as the underlying resource for defining models of quantum computation, quantum communication, quantum memory and decoherence-free subspaces. Given the non-commutativity of these operators (for adjacent operators operating on a common qubit), a number of quantum states and entanglement patters can be obtained. This zoo of states can be classified, due to the parity constraints and permutation symmetry of the states, into invariant subspaces that are used for the definition of some of the applications in this paper.