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

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 $O(N^2(N+\log 1/\epsilon ))$. 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.

Revised, 22 pages + Appendix, 3 figures; major revision from v2; presentation is improved in v4; v5 is a published version

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Cited by in corpus (12)

• Efficient unitary designs with nearly time-independent Hamiltonian dynamics
• Quantum Commuting Circuits and Complexity of Ising Partition Functions
• Entanglement Typicality
• Unitary $2$-designs from random $X$- and $Z$-diagonal unitaries
• Generating a state $t$-design by diagonal quantum circuits
• Diagonal quantum circuits: their computational power and applications
• Markovianization with approximate unitary designs
• Decoupling with random diagonal unitaries
• A graphical calculus for integration over random diagonal unitary matrices
• Thermal states of random quantum many-body systems
• Quantum-data-driven dynamical transition in quantum learning
• Simple Diagonal State Designs with Reconfigurable Real-Time Circuits