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note
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Any valid scheme for universal quantum computation can be used, in principle, to simulate quantum systems, and the schemes for a UQS presented in this paper are nothing but particular realizations of a quantum computer. In practice, however, experimental limitations drastically interfere with the ideal performance of any of these devices. Imperfect manipulations and finite decoherent times introduce errors at a rate far above the estimated threshold for achieving fault-tolerant controlled quantum evolutions. In this situation, a quantum computer that operates by, say, performing CNOT gates and one-qubit rotations is notoriously less succesful at tracing the dynamics of an array of interacting qubits than a device especially designed for this purpose.
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By a short gate we mean one that does not differ much from the identity operator.
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Multi-qubit systems with n-body interactions, n > 2, d-level systems with d > 2 and more exotic systems such as fermions can also be simulated using our schemes. In this cases we encounter, however, that the time required to perform the simulation does no longer grow linearly in the simulated time, but according to some higher order polynomial.
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note
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Through the paper we use the convention ℏ = 1.
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34
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note
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By symmetrizing the evolution according to the spectral method for dynamical evolution, see numerical recipes, one can easily reduce the control complexity x to scale as √T′/ε instead of T′/ε.
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35
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note
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For simplicity, we describe only a ID optical lattice. For 2D [3D] lattices, one uses four (six) interfering laser beams, one pair for each dimension.
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