Diagonal-unitary 2-designs and their implementations by quantum circuits

TL;DR

This paper introduces efficient quantum circuits for generating diagonal-unitary 2-designs, enabling the creation of random quantum states with applications in quantum information processing.

Contribution

The paper presents novel quantum circuits that implement diagonal-unitary 2-designs exactly or approximately, improving efficiency and randomness in quantum state generation.

Findings

Exact 2-design achieved with $N(N-1)/2$ gates using controlled-phase gates.

Approximate 2-design achieved with $O(N^2(N+ ext{log}1/ ext{epsilon}))$ gates using randomized pairs.

Application demonstrated in generating random quantum states with classical randomness.

Abstract

We study efficient generations of random diagonal-unitary matrices, an ensemble of unitary matrices diagonal in a given basis with randomly distributed phases for their eigenvalues. Despite the simple algebraic structure, they cannot be achieved by quantum circuits composed of a few-qubit diagonal gates. We introduce diagonal-unitary $t$-designs and present two quantum circuits that implement diagonal-unitary $2$-designs with the computational basis in $N$-qubit systems. One is composed of single-qubit diagonal gates and controlled-phase gates with randomized phases, which achieves an exact diagonal-unitary $2$-design after applying the gates on all pairs of qubits. The number of required gates is $N(N-1)/2$. If the controlled-Z gates are used instead of the controlled-phase gates, the circuit cannot achieve an exact $2$-design, but achieves an $\epsilon$-approximate $2$-design by applying gates on randomly selected pairs of qubits. Due to the random choice of pairs, the circuit obtains extra randomness and the required number of gates is at most $O(N^2(N+\log1/\epsilon))$. We also provide an application of the circuits, a protocol of generating an exact $2$-design of random states by combining the circuits with a simple classical procedure requiring $O(N)$ random classical bits.