Simple factorization of unitary transformations

Guise, Hubert de; Matteo, Olivia Di; Sánchez-Soto, Luis L.

doi:10.1103/physreva.97.022328

Cited by 65 publications

(81 citation statements)

References 49 publications

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“…If the global phase of the emitted light is inconsequential, we can assume that the matrix U is in SU(N ), the group of special unitary matrices. Existing architectures for implementing a given SU(N ) matrix U via linear optics rely on first systematically decomposing U into U(2) matrices, which are then identified as beamsplitter and phase transformations acting on different modes of light [19][20][21].…”

Section: Unitary Transformation Decompositionsmentioning

confidence: 99%

Hybrid spatiotemporal architectures for universal linear optics

Dhand

Helt

et al. 2019

Phys. Rev. A

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We present two hybrid linear-optical architectures that simultaneously exploit spatial and temporal degrees of freedom of light to effect arbitrary discrete unitary transformations. Our architectures combine the benefits of spatial implementations of linear optics, namely low loss and parallel operation, with those of temporal implementations, namely modest resource requirements and access to transformations of potentially unbounded size. We arrive at our architectures by devising and employing decompositions of large discrete unitary transformations into smaller ones, decompositions we expect to have broad utility beyond spatio-temporal linear optics. We show that hybrid architectures promise important advantages over both spatial-only and temporal-only architectures. arXiv:1812.07939v2 [quant-ph]

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Section: Unitary Transformation Decompositionsmentioning

confidence: 99%

Hybrid spatiotemporal architectures for universal linear optics

Dhand

Helt

et al. 2019

Phys. Rev. A

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“…Properly assessing uncertainty is important for foundational quantum mechanics [10][11][12] and for quantum information and communication [13][14][15][16]; variance is closer than entropy for practical quantum mechanics, a driving motivation behind research into sum-uncertainty relations (SURs), which deliver state-independent lower bounds [17][18][19][20][21][22][23][24][25]. Here we discuss SURs by showing connections with the algebras of observables, with examples of the Weyl-Heisenberg wh, special unitary su(n) and su(1, 1) and generally semi-simple compact algebras thereby extending the range to applications of SURs in areas such as [26,27] and quantum information [28] where u(n) or su(n) symmetries are prevalent.Indeed, single-photon multi-path quantum optical interferometry provides a convenient way to realize SU (n) symmetry [29][30][31][32] with the experimental signature obtained via sampling photodetection of the photon emerging from each of the n output ports, both by direct detection and by adding special post-processing interferometers at the output followed by photodetection. Photodetection sampling statistics obtained in these ways yield uncertainties from estimates second-order cumulants for the distributions and, through this process, our uncer-(a) (b) FIG.…”

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confidence: 99%

State-independent uncertainty relations

et al. 2018

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The standard state-dependent Heisenberg-Robertson uncertainly-relation lower bound fails to capture the quintessential incompatibility of observables as the bound can be zero for some states. To remedy this problem, we establish a class of tight (i.e., inequalities are saturated) variance-based sum-uncertainty relations derived from the Lie algebraic properties of observables and show that our lower bounds depend only on the irreducible representation assumed carried by the Hilbert space of state of the system. We illustrate our result for the cases of the Weyl-Heisenberg algebra, special unitary algebras up to rank 4, and any semisimple compact algebra. We also prove the usefulness of our results by extending a known variance-based entanglement detection criterion.For ∆w 2 signifying the variance of measurement outcomes for the observable w, Heisenberg's uncertainty relation for position x and momentum p iswhere [x, p] = i1, and 1 is the identity operator. Eq. (1) fortuitously has a constant lower bound due to the appealing algebraic properties of the commutator of x and p. Robertson's generalization to ∆A 2 ∆B 2 ≥ | [A, B] | 2 /4 for arbitrary observables A and B more generally has a state-dependent lower bound [1], and so fails to capture the intrinsic incompatibility of noncommuting observables [2,3]. This cannot be amended as the underlying product of uncertainties is null whenever one of the uncertainties is null, an observation that provided impetus for the emergence of uncertainty relations [4-9] that eschew variance in favor of entropy. Properly assessing uncertainty is important for foundational quantum mechanics [10][11][12] and for quantum information and communication [13][14][15][16]; variance is closer than entropy for practical quantum mechanics, a driving motivation behind research into sum-uncertainty relations (SURs), which deliver state-independent lower bounds [17][18][19][20][21][22][23][24][25]. Here we discuss SURs by showing connections with the algebras of observables, with examples of the Weyl-Heisenberg wh, special unitary su(n) and su(1, 1) and generally semi-simple compact algebras thereby extending the range to applications of SURs in areas such as [26,27] and quantum information [28] where u(n) or su(n) symmetries are prevalent.Indeed, single-photon multi-path quantum optical interferometry provides a convenient way to realize SU (n) symmetry [29][30][31][32] with the experimental signature obtained via sampling photodetection of the photon emerging from each of the n output ports, both by direct detection and by adding special post-processing interferometers at the output followed by photodetection. Photodetection sampling statistics obtained in these ways yield uncertainties from estimates second-order cumulants for the distributions and, through this process, our uncer-(a) (b) FIG. 1: Sum of variances is a measure of total uncertainty. Given a (green) box with the uncertainties as edges, the sum of variances is the squared length of the (red) diagonal. [Here 30000 (blue) points ...

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“…GBS is a model of photonic quantum computing where a Gaussian state is measured in the Fock basis. A general pure Gaussian state can be prepared by using single-mode squeezing and displacement operations in combination with linear-optical interferometry [30,[32][33][34]. It was shown in Ref.…”

Section: A Gaussian Boson Samplingmentioning

confidence: 99%

Exact simulation of Gaussian boson sampling in polynomial space and exponential time

Quesada

Arrazola

2020

Phys. Rev. Research

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We introduce an exact classical algorithm for simulating Gaussian boson sampling (GBS). The complexity of the algorithm is exponential in the number of photons detected, which is itself a random variable. For a fixed number of modes, the complexity is in fact equivalent to that of calculating output probabilities, up to constant prefactors. The simulation algorithm can be extended to other models such as GBS with threshold detectors, GBS with displacements, and sampling linear combinations of Gaussian states. In the specific case of encoding non-negative matrices into a GBS device, our method leads to an approximate sampling algorithm with polynomial runtime. We implement the algorithm, making the code publicly available as part of Xanadu's The Walrus library and benchmark its performance on GBS with random Haar interferometers and with encoded Erdős-Renyi graphs.

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Simple factorization of unitary transformations

Cited by 65 publications

References 49 publications

Hybrid spatiotemporal architectures for universal linear optics

Hybrid spatiotemporal architectures for universal linear optics

State-independent uncertainty relations

Exact simulation of Gaussian boson sampling in polynomial space and exponential time

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