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Computation on a Noiseless Quantum Code and Symmetrization

Paolo Zanardi
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Abstract

Let ${\cal H}$ be the state-space of a quantum computer coupled with the environment by a set of error operators spanning a Lie algebra ${\cal L}.$ Suppose ${\cal L}$ admits a noiseless quantum code i.e., a subspace ${\cal C}\subset{\cal H}$ annihilated by ${\cal L}.$ We show that a universal set of gates over $\cal C$ is obtained by any generic pair of ${\cal L}$-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 ${\cal L}.$ Any computation can then be performed completely within the coding decoherence-free subspace.

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Paolo Zanardi. 1999-06-09. Computation on a Noiseless Quantum Code and Symmetrization. https://doi.org/10.1103/physreva.60.r729

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