Yoshifumi Nakata, Mio Murao
We consider random diagonal-unitary matrices, an ensemble of unitary matrices diagonal in a given basis and with randomly distributed phases of their eigenvalues. They describe typical dynamics in a closed system governed by the Schr\"{o}dinger equation. We investigate efficient constructions of random diagonal-unitary matrices and introduce diagonal-unitary t-designs, which are ensembles of diagonal unitary matrices simulating random diagonal-unitary matrices up to the t-th order moments. We present two efficient constructions of diagonal-unitary 2-designs for N-qubit systems using quantum circuits composed of two-qubit diagonal gates. One of the constructions uses a set of gates where each of gates consists of a tensor product of single-qubit diagonal gates and a controlled-Z gate, and it achieves an approximate diagonal-unitary 2-design by applying randomly chosen $O(N^3)$ gates on randomly chosen pairs of qubits. Another construction uses a set of gates where each of gates consists of a tensor product of single-qubit diagonal gates and a controlled-phase gates with a randomly chosen phase. In this case, an exact diagonal-unitary 2-design is achieved by applying randomly chosen $O(N^2)$ gates, which shows clear enhancement of the phase-randomization power by using the controlled-phase gates.
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http://arxiv.org/abs/1206.4451
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