Random Unitary Matrices Associated to a Graph

Kondratiuk, Paweł; Życzkowski, Karol

doi:10.12693/aphyspola.124.1098

Cited by 3 publications

(1 citation statement)

References 26 publications

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“…Products of unitary operators are therefore natural objects to study as they form building blocks for quantum algorithms. Random quantum circuits with random unitary operators providing interaction among qubits have been studied in this context [35][36][37]. They are known to be approximate unitary t-designs that simulate Haar distributed unitaries [38,39].…”

Section: Introductionmentioning

confidence: 99%

Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength

Jonnadula

Mandayam

Życzkowski

et al. 2020

Phys. Rev. Research

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Entanglement properties of bipartite unitary operators are studied via their local invariants, namely the entangling power e p and a complementary quantity, the gate typicality g t. We characterize the boundaries of the set K 2 representing all two-qubit gates projected onto the plane (e p , g t) showing that the fractional powers of the SWAP operator form a parabolic boundary of K 2 , while the other bounds are formed by two straight lines. In this way, a family of gates with extreme properties is identified and analyzed. We also show that the parabolic curve representing powers of SWAP persists in the set K N for gates of higher dimensions (N > 2). Furthermore, we study entanglement of bipartite quantum gates applied sequentially n times, and we analyze the influence of interlacing local unitary operations, which model generic Hamiltonian dynamics. An explicit formula for the entangling power of a gate applied n times averaged over random local unitary dynamics is derived for an arbitrary dimension of each subsystem. This quantity shows an exponential saturation to the value predicted by the random matrix theory, indicating "thermalization" in the entanglement properties of sequentially applied quantum gates that can have arbitrarily small, but nonzero, entanglement to begin with. The thermalization is further characterized by the spectral properties of the reshuffled and partially transposed unitary matrices.

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Section: Introductionmentioning

confidence: 99%

Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength

Jonnadula

Mandayam

Życzkowski

et al. 2020

Phys. Rev. Research

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Direct product of random unitary matrices: Two-point correlations and fluctuations

2021

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Derandomizing Quantum Circuits with Measurement-Based Unitary Designs

Turner

Markham

2016

Phys. Rev. Lett.

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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.

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Random Unitary Matrices Associated to a Graph

Cited by 3 publications

References 26 publications

Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength

Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength

Direct product of random unitary matrices: Two-point correlations and fluctuations

Derandomizing Quantum Circuits with Measurement-Based Unitary Designs

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