Generating a state $t$-design by diagonal quantum circuits

Abstract

We investigate protocols for generating a state $t$-design by using a fixed separable initial state and a diagonal-unitary $t$-design in the computational basis, which is a $t$-design of an ensemble of diagonal unitary matrices with random phases as their eigenvalues. We first show that a diagonal-unitary $t$-design generates a $O(1/2^N)$-approximate state $t$-design, where $N$ is the number of qubits. We then discuss a way of improving the degree of approximation by exploiting non-diagonal gates after applying a diagonal-unitary $t$-design. We also show that it is necessary and sufficient to use $O(\log_2 t)$-qubit gates with random phases to generate a diagonal-unitary $t$-design by diagonal quantum circuits, and that each multi-qubit diagonal gate can be replaced by a sequence of multi-qubit controlled-phase-type gates with discrete-valued random phases. Finally, we analyze the number of gates for implementing a diagonal-unitary $t$-design by {\it non-diagonal} two- and one-qubit gates. Our results provide a concrete application of diagonal quantum circuits in quantum informational tasks.

ver. 1: 15 pages, 1 figures. ver.2: 16 pages, 2 figures, major changes, we corrected a mistake, which slightly changes a main conclusion, added a new result, and improved a presentation. ver.3: 11 pages, 2 figures, published version