Faster quantum computation with permutations and resonant couplings

Recently, there has been increasing interest in designing schemes for quantum computations that are robust against errors. Although considerable research has been devoted to quantum error correction schemes, much less attention has been paid to optimizing the speed it takes to perform a quantum computation and developing computation models that act on decoherence-free subspaces. Speeding up a quantum computation is important, because fewer errors are likely to result. Encoding quantum information in a decoherence-free subspace is also important, because errors would be inherently suppressed. In this paper, we consider quantum computation in a decoherence-free subspace and also optimize its speed. To achieve this, we perform certain single-qubit quantum computations by simply permuting the underlying qubits. In this paper, we make progress in understanding the extent in which quantum computation can be performed by permuting qubits. Namely, we provide two different subgroups of the single qubit Clifford group that can be computed solely by permutations. To make the quantum computation universal for one of the schemes, we rely on resonant couplings motivated from physics. Our first scheme potentially improves the speed in which a quantum computation can be done. We also reduce the problem of finding the existence of permutational Clifford gates to that finding an upper bound for the rank of certain matrices.