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arXiv:quant-ph/0109021 · Physical Review Letters

Reducing Constraints on Quantum Computer Design by Encoded Selective Recoupling

Daniel A. Lidar, Lian-Ao Wu

Published Dec 19, 2001• 110 citations
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Abstract

The requirement of performing both single-qubit and two-qubit operations in the implementation of universal quantum logic often leads to very demanding constraints on quantum computer design. We show here how to eliminate the need for single-qubit operations in a large subset of quantum computer proposals: those governed by isotropic and XXZ, XY-type anisotropic exchange interactions. Our method employs an encoding of one logical qubit into two physical qubits, while logic operations are performed using an analogue of the NMR selective recoupling method.

DOI
10.1103/physrevlett.88.017905

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The requirement of performing both single-qubit and two-qubit operations in the implementation of universal quantum logic often leads to very demanding constraints on quantum computer design. We show here how to eliminate the need for single-qubit operations in a large subset of quantum computer proposals: those governed by isotropic and XXZ, XY-type anisotropic exchange interactions. Our method employs an encoding of one logical qubit into two physical qubits, while logic operations are performed using an analogue of the NMR selective recoupling method.
We study the quantum computational power of a generic class of anisotropic solid-state Hamiltonians. A universal set of encoded logic operations are found, which do away with difficult-to-implement single-qubit gates in a number of quantum-computer proposals, e.g., quantum dots and donor atom spins with anisotropic exchange coupling, quantum Hall systems, and electrons floating on helium. We show how to make the corresponding Hamiltonians universal by encoding one qubit into two physical qubits, and by controlling nearest-neighbor interactions.
Proposals for scalable quantum computing devices suffer not only from decoherence due to the interaction with their environment, but also from severe engineering constraints. Here we introduce a practical solution to these major concerns, addressing solid-state proposals in particular. Decoherence is first reduced by encoding a logical qubit into two qubits, then completely eliminated by an efficient set of decoupling pulse sequences. The same encoding removes the need for single-qubit operations, which pose a difficult design constraint. We further show how the dominant decoherence processes can be identified empirically, in order to optimize the decoupling pulses.
In a previous publication [J. Kempe et al., Quantum Computation and Information (Rinton Press, Princeton, NJ, 2001), Vol. 1, special issue, p. 33] we showed that it is possible to implement universal quantum computation with the anisotropic XY-Heisenberg exchange acting as a single interaction. To achieve this we used encodings of the states of the computation into a larger Hilbert space. This proof is nonconstructive, however, and did not explicitly give the trade-offs in time that are required to implement encoded single-qubit operations and encoded two-qubit gates. Here we explicitly give the gate sequences needed to simulate these operations on encoded qubits and qutrits (three-level systems) and analyze the trade-offs involved. We also propose a possible layout for the qubits in a triangular arrangement.
Proposals for scalable quantum computing devices suffer not only from decoherence due to their interaction with the environment, but also from severe engineering constraints. For example, our ability to implement quantum gates is determined, in part, by the experimentally available interactions with which quantum information may be processed. Here we review a practical solution to some of the major concerns, control and error prevention, addressing solid state proposals for quantum computing devices. Some noise is eliminated by encoding a logical qubit into two qubits, other noise is reduced by an efficient set of decoupling pulse sequences. The same encoding removes the need for single-qubit operations which pose a difficult design constraint. We also discuss several generalizations which follow from this work.
Quantum computation can be performed by encoding logical qubits into the states of two or more physical qubits, and control of effective exchange interactions and possibly a global magnetic field. This "encoded universality" paradigm offers potential simplifications in quantum computer design since it does away with the need to control physical qubits individually. Here we show how encoded universality schemes can be combined with fault-tolerant quantum error correction, thus establishing the scalability of such schemes.
We derive an encoded universality representation for a generalized anisotropic exchange Hamiltonian that contains cross-product terms in addition to the usual two-particle exchange terms. The recently developed algebraic approach is used to show that the minimal universality-generating encodings of one logical qubit are based on three physical qubits. We show how to generate both single- and two-qubit operations on the logical qubits, using suitably timed conjugating operations derived from analysis of the commutator algebra. The timing of the operations is seen to be crucial in allowing simplification of the gate sequences for the generalized Hamiltonian to forms similar to that derived previously for the symmetric (XY)(\mathrm{XY})(XY) anisotropic exchange Hamiltonian. The total number of operations needed for a controlled-ZZZ gate up to local transformations is five. A scalable architecture is proposed.