Abstract
Molecular vibroic spectroscopy, where the transitions involve non-trivial Bosonic correlation due to the Duschinsky Rotation, is strongly believed to be in a similar complexity class as Boson Sampling. At finite temperature, the problem is represented as a Boson Sampling experiment with correlated Gaussian input states. This molecular problem with temperature effect is intimately related to the various versions of Boson Sampling sharing the similar computational complexity. Here we provide a full description to this relation in the context of Gaussian Boson Sampling. We find a hierarchical structure, which illustrates the relationship among various Boson Sampling schemes. Specifically, we show that every instance of Gaussian Boson Sampling with an initial correlation can be simulated by an instance of Gaussian Boson Sampling without initial correlation, with only a polynomial overhead. Since every Gaussian state is associated with a thermal state, our result implies that every sampling problem in molecular vibronic transitions, at any temperature, can be simulated by Gaussian Boson Sampling associated with a product of vacuum modes. We refer such a generalized Gaussian Boson Sampling motivated by the molecular sampling problem as Vibronic Boson Sampling.
Highlights
A quantum simulation protocol called “Boson Sampling”, proposed by Aaronson and Arkhipov[1], represents a serious challenge to the extended Church-Turing thesis
We present a hierarchical framework that can reduce all problems in Boson Sampling[1], Scattershot Boson Sampling[6, 7], and Guassian Boson Sampling[15] at any temperature, as an instance of Vibronic Boson Sampling at zero temperature, which is in turn equivalent to Gaussian Boson Sampling at zero temperature using the result in ref
In Gaussian Boson Sampling[15], a product of Gaussian modes are employed as the input, which can be represented by the ing operators following squeezed Sσk, i.e., SΣ = ⊗kM=1 thermal Sσk, and state, ρin = SΣ ρth SΣ†, where SΣ is a product of M single-mode Σ = diag(σ1, ..., σM) is the squeezing parameter squeezmatrix, wtwdieaiittflihhlnyt“edhddieifabfageyc”rtetlihnaoebtnefBRrlieoUnqs,gouwnaehndoicipciaheegsroraωentsoiauarllnsmtsdsaaitntterismtixhfy.peiTenfrhogale[ltouathwkr,eeisranml(g†β]adkl=is=sttrδai1ktble/.ukρTtBtihhToekins)s,atia.peter.,oiρsdthuthc=et noef−seiHnn/dtTitvrhi(dreo−uuHagl)htwhaeitrlhimnHeaal =rstoa∑ptetksMic=wa1βlitnkh epωtwokatoek†rnak-k, P(n) = Tr [RUρinRU† | n〉〈n |]
Summary
A quantum simulation protocol called “Boson Sampling”, proposed by Aaronson and Arkhipov[1], represents a serious challenge to the extended Church-Turing thesis. The Scattershot Boson Sampling[6, 7] using two-mode squeezed states was suggested to overcome the difficulty in preparing the initial Fock states. Non-classical quantum optical states like displaced Fock states, photon-added or subtracted squeezed vacuum states, and cat states have been considered[8,9,10]. Apart from photons, non-optical Boson Sampling, including trapped-ion[13] and superconducting circuits[14] have been proposed theoretically, which aims to overcome the state-preparation problem, and can be readily extended to Gaussian input states[15], including squeezed vacuum and squeezed coherent states for molecular applications[16]. Similar to the original version of Boson Sampling, simulating Gaussian Boson Sampling involving squeezed states is still a hard problem for classical devices.
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