Derandomizing Quantum Circuits with Measurement-Based Unitary Designs

Abstract

Entangled multipartite states are resources for universal quantum computation, but they can also give rise to ensembles of unitary transformations, a topic usually studied in the context of random quantum circuits. Using several graph state techniques, we show that these resources can 'derandomize' circuit results by sampling the same kinds of ensembles quantum mechanically, analogously to a quantum random number generator. Furthermore, we find simple examples that give rise to new ensembles whose statistical moments exactly match those of the uniformly random distribution over all unitaries up to order t, while foregoing adaptive feed-forward entirely. Such ensembles -known as t-designs -often cannot be distinguished from the 'truly' random ensemble, and so they find use in many applications that require this implied notion of pseudorandomness.Introduction -Randomness is an important resource in both classical and quantum information theory, underpinning cryptography, characterisation, and simulation. Random unitary transformations are often considered in the form of random quantum circuits, with wide-ranging applications in, for example, estimating noise[1], private channels[2], modelling thermalisation[3], photonics [4], and even black hole physics [5]. Uniform randomness -sampling from the 'flat' measure on a continuous set -is however very resource intensive. A natural definition of a less costly pseudorandom ensemble is one whose statistical moments are equal to those of the uniform ensemble up to some finite order t -this is the defining property of a t-design. Analogous to combinatorial designs that arise in many areas [6], in the quantum community the concept was first applied to states [7], and later to processes [8], the latter being the topic of much recent work (e.g. [9]) and are our concern here.

Multiqubit randomized benchmarking using few samples

“…Finding the optimal way to perform this fitting procedure is still an open problem [23]. Finally, a major theoretical open problem is the extension of the present bounds to non-qubit systems, different varieties of randomized benchmarking [35,36], and to different 2designs [35,37,38] or even orthogonal 2-designs [39,40]. If these 2-designs are assumed to be groups, similar techniques from representation theory might be used [41] but how this would be done is currently unknown.…”

Efficient quantum pseudorandomness with simple graph states

“…In standard MBQC the correction strategy is given by the gflow [21]-a partial order over the graph as well as a function, which give a time order for the measurements and the dependency respectively. In this work, following [14], we do not adapt the measurement angles so that for each measurement result a potentially different unitary is performed.…”

“…with equal probabilty for m = 0 and m = 1. As in [14], the same idea applied to larger graphs, with more inputs and outputs is the source of the random unitary ensembles we will study in this work.…”

“…(14) All that remains now is to bound the RHS of Equation (14). Using Equation (12) one directly obtains :…”

“…Following [14] our approach is to harness the quantum randomness arising from applying a measurement based (MB) scheme to produce approximate designs. In MB computation [15], unitary determinsitic computation is achieved by making sequential, adaptive measurements on an entangled multipartite state, known as a graph state [16].…”