Quantum pseudorandomness from cluster-state quantum computation

Abstract

We show how to efficiently generate pseudo-random states suitable for quantum information processing via cluster-state quantum computation. By reformulating pseudo-random algorithms in the cluster-state picture, we identify a strategy for optimizing pseudo-random circuits by properly choosing single-qubit rotations. A Markov chain analysis provides the tool for analyzing convergence rates to the Haar measure and finding the optimal single-qubit gate distribution. Our results may be viewed as an alternative construction of approximate unitary 2-designs.PACS numbers: 03.67. Mn, 03.67.Lx, 03.67.Bg, Methods for characterizing and efficiently generating random quantum states and unitary operators have broad conceptual and practical significance across quantum physics. From a fundamental standpoint, a main motivation stems from the challenge of modeling complex quantum behavior, including quantum chaos [1] and typical entanglement in many-body systems [2,3,4,5,6,7,8]. Within quantum information science, states and unitaries sampled from the appropriate uniform (Haar) distribution provide the enabling resource in a growing number of algorithms and protocols. Remarkably, random pure states saturate the classical communication capacity of a noisy quantum channel [9], and allow superdense coding of arbitrary quantum states [10]. Random unitaries find applications in tasks ranging from approximate encryption and remote state preparation [11] to unbiased noise estimation [12,13] and selective process tomography [14].However, implementing exact randomization on a quantum computer is inefficient, as the number of required elementary gates grows exponentially with the number of qubits. Still, it has been shown [5,12,15] that one can generate pseudo-random (PR) quantum states and unitary operators which satisfy certain practical tests of randomness using only a polynomial number of gates. In particular, a framework for quantifying to what extent pseudo-randomness may simulate the Haar distribution for an intended randomization task is offered by the notion of a t-design [16]. In its essence, a state (unitary) tdesign is a probability distribution over pure states (unitaries) whose statistical moments up to order t equal those from the Haar distribution. While efficient exact unitary 2-designs are known [16], constructions of approximate 2-designs as well as alternative schemes tackling higher-order moments are actively investigated [5,6,7]. So far, existing studies have focused only on the circuit model of quantum computation (QC).In this work, we construct an efficient algorithm for PR state generation in the cluster-state paradigm of QC [17]. This is crucial from an implementation perspective, in that many of the above-mentioned applications of random states originate in quantum communication protocols, for which photonic entanglement and clusterstate QC provide a leading approach [18]. Furthermore, we find that reformulating PR algorithms in a clusterstate picture suggests a path to optimize existing circuit constructions. I...

Anticoncentration theorems for schemes showing a quantum speedup

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