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 -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 -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 non-diagonal two- and one-qubit gates. Our results provide a concrete application of diagonal quantum circuits in quantum informational tasks.
Highlights
Diagonal quantum circuits in the computational basis have recently been attracting much attention [1,2,3,4,5]
This means that the protocol in proposition 1 generates an η (N, t)-approximate ( ) state t-design for t ⩽ 3 by a quantum circuit composed of O N2 two- and one-qubit diagonal gates that has no temporal structure
We investigated protocols of generating a state t-design in an N-qubit system by using a diagonal-unitary t-design in the computational basis applied on a fixed separable state
Summary
Diagonal quantum circuits in the computational basis have recently been attracting much attention [1,2,3,4,5]. One of the applications of diagonal circuits, proposed by two of the authors [7], is related to random states, which are an ensemble of pure states uniformly distributed in a Hilbert space with respect to the unitarily invariant measure They have many utilities in a wide range of applications, e.g., in quantum communicational tasks [8], for efficient measurements [9], for an algorithmic use [10, 11], and for estimation of gate fidelities [12]. From a theoretical point of view, it shows that a diagonal quantum circuit can generate an ensemble of states whose distribution in a Hilbert space is difficult to distinguish from the uniform one when looking at lower order statistical moments This may aid in an intuitive understanding of a strong computational power of diagonal quantum circuits.
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