Unitary 2-designs from random X- and Z-diagonal unitaries

Abstract

Unitary 2-designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries, often referred to as Haar random unitaries. They are used in a wide variety of theoretical and practical quantum information protocols, and also have been used to model the dynamics in complex quantum many-body systems. Here, we show that unitary 2-designs can be approximately implemented by alternately repeating random unitaries diagonal in the Pauli-Z basis and that in the Pauli-X basis. We also provide a converse about the number of repetitions needed to achieve unitary 2-designs. These results imply that the process after repetitions achieves a Θ(d − )approximate unitary 2-design. Based on the construction, we further provide quantum circuits that efficiently implement approximate unitary 2-designs. Although a more efficient implementation of unitary 2-designs is known, our quantum circuit has its own merit that it is divided into a constant number of commuting parts, which enables us to apply all commuting gates simultaneously and leads to a possible reduction of an actual execution time. We finally interpret the result in terms of the dynamics generated by time-dependent Hamiltonians and provide for the first time a random disordered time-dependent Hamiltonian that generates a unitary 2-design after switching interactions only a few times.

Efficient Quantum Pseudorandomness with Nearly Time-Independent Hamiltonian Dynamics

“…As highlighted in Ref. [43,49], the quantum circuits repeating RDC(I 2 ) or RDC (t) disc (I 2 ) and the Hadamard transformation are divided into a constant number of commuting parts. Indeed, only non-commuting parts are the Hadamard parts.…”

“…However, when a system consists of a large number of particles, it is highly inefficient to implement Haar random unitaries, implying that they rarely appear in natural systems composed of many particles especially when the interactions are local. This fact has lead to the research area on finite-degree approximations of Haar random unitaries, so-called unitary designs [32][33][34], and their efficient implementations [35][36][37][38][39][40][41][42][43][44][45]. A unitary t-design is called exact when it simulates all the first t moments of Haar random unitaries and approximate when the simulations are with errors.…”

“…In this paper, we provide new constructions of unitary t-designs and propose a design Hamiltonian, a random Hamiltonian of which dynamics forms a unitary design at any time after a threshold time. The constructions are based on the scheme of repeating random unitaries diagonal in mutually unbiased bases [43,[47][48][49]. We first show that the process on one qudit achieves unitary t-designs after O(t) repetitions if a pair of the two bases satisfies a certain condition, which is met by a pair of any basis and its Fourier basis and that of the Pauli-X and -Z bases.…”

One-Shot Randomized and Nonrandomized Partial Decoupling

“…Note that, when a quantum channel T A→E achieves partial decoupling for a state Ψ AR within a small error, it follows from the decomposition of Ψ av (see (38)) that…”

“…Since the decoupling method makes use of the second statistical moments of the Haar measure, we could use the unitary 2-designs instead of the Haar measure for our tasks. Although a number of efficient implementations of unitary 2-designs have been discovered [30][31][32][33][34][35][36][37][38], and it is also shown that decoupling can be achieved using unitaries less random than unitary 2-designs [39,40], we here need unitary designs in a given DSP form, which we refer to as the DSP unitary designs. Thus, we cannot directly use the existing constructions, posing a new problem about efficient implementations of DSP unitary designs.…”