Vibronic Boson Sampling: Generalized Gaussian Boson Sampling for Molecular Vibronic Spectra at Finite Temperature

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.


Results
Boson Sampling and the Gaussian version. In general, the probability P Π of a projective measurement Π for sampling Bosons, with a problem-dependent scattering operator Ô and an input state in ρˆ, is given by, is the (real-valued) squeezing parameter matrix, with "diag" labeling a diagonal matrix. The thermal state ρˆt h is a product of individual thermal states with potentially different frequencies ω i and temperatures ( k T 1/ Here ˆ⁎ D z is the displacement operator with the displacement vector z * . Accordingly, X and Y in Eq. 3 are identified as U U sinh( ) R † , respectively 20,21 , via the singular value decomposition (SVD). The detailed forms and the actions of the quantum optical operators can be found in Methods.
In the following, to make a distinction from Gaussian Boson Sampling, we define Vibronic Boson Sampling as the class of sampling problems utilizing the most general Gaussian states as the input. The name is motivated by the problem of sampling molecular vibrational transitions at finite temperatures (see Methods). The connection between Boson Sampling and molecular transitions was first made in Huh et al. 16 , where the mode correlations are absorbed into local operations. However, such a procedure is possible for Gaussian states associated with the vacuum state (i.e., zero temperature) only.

Vibronic Transition and Franck-Condon Profile.
In the following, we are going to connect the problem of molecular spectroscopy with the vibronic extension of Boson Sampling. Such a connection was first pointed out in ref. 16 but details are missing. For completeness, here we summarize a self-contained description and extend the result for initial thermal states.
Under the Born-Oppenheimer approximation, the total molecular wavefunction of the nuclear ( = ...ˆR R R ( , , ) 1 2 ) and electronic ( = ...ˆr r r ( , , ) 1 2 ) degrees of freedom are separated, i.e., ˆˆψ φ r R R ( , ) ( ), and the electronic wavefunction ψˆr R ( , ) depends parametrically the nuclear coordinates R . As a result, for transitions involving two electronic levels, |g〉 and |e〉, the molecular Hamiltonian mol  can be approximated as follows:  , we can then focus on the matrix element involving only on the phonon modes, , imposes the energy conservation condition. The initial distribution of the phonon modes is denoted by P in (n), which will be taken to be a thermal distribution. Our goal in this work is to explain how a quantum (optical) simulator can be constructed to efficiently sample the FCP at finite temperature. See, for example, Dierksen and Grimme 26 for the computational difficulties in the evaluation of the FCP.
Scattershot sampling for thermal states. We now address the problem of thermal state preparation in Gaussian Boson Sampling, which is relevant in the thermal extension of the sampling problem 16 for vibronic transitions in molecular spectroscopy. Boson Sampling with thermal input states, as an instance of Gaussian Boson Sampling, has been considered 15 , where it is shown that the distribution can be simulated by a classical computer efficiently. As a result, such a problem belongs to the complexity class BPP NP , which is believed to be less complex than the counting problems in the complexity class #P.
Instead of sampling the thermal distribution, our approach starts with a purification of the mixed initial states, which is a standard method for studying thermo-field dynamics 27 , to prepare and identify the thermally excited Fock states given in the Boltzmann distribution. Specifically, we extend the idea of Scattershot Boson Sampling 6 for the problem of sampling thermalized Bosons. The key idea of Scattershot Boson Sampling is to send half of entangled photons through the optical network, followed by a post-selection for projecting out the single-photon states at the end, The main purpose is to overcome the experimental difficulty of preparing single-photon states required in Boson Sampling.
In order to extend the idea of Scattershot Boson Sampling for thermal initial states, we first consider the purification of every thermal state with ancillary modes, i.e., β ρ = ∑ ⊗ = ∞ 0 n n n n ( ) n 0 th B , where the ancillary Hilbert space 'B' has been introduced. Note that the original thermal state th ρˆ can be obtained after tracing away the ancillary modes, i.e., ˆβ are the annihilation and creation operators of the ancillary modes, and Tr and Tr B trace over the original and ancillary Hilbert spaces, respectively. Here = − is the mean quantum number of the k-th mode. Consequently, the sampling problem involving initial sampling of Fock states can be transformed into a problem involving post-selection only, i.e., There are two major differences between Scattershot Boson Sampling 6 and our approach. The ancillary modes in Scattershot Boson Sampling are not sent to an optical network and only the measurement results involving single-photon detection are relevant. On the other hand, in our case, all the ancillary modes are involved in the optical network in general, and all the measurement outcomes are relevant for the sampling problem. Furthermore, the randomized input Fock states are generated with M two-mode squeezed vacuum state that Eq. (9) is reduced to the Scattershot Boson Sampling by Lund and coworkers 6 when the Gaussian operator includes only the rotation operator, i.e.
= O R U Ĝˆ. In this sense, the Scattershot Boson Sampling is a special instance of Vibronic Boson Sampling. In the following, we shall show that the mode correlation created by the two-mode squeezing operation (V ) can be eliminated through a Bogoliubov transformation.
Gaussian decorrelation. To get started, let us define a new operator,ˆβ G for the Scattershot-fashion probability distribution in Eq. 9. The action of β U( ) of the general Gaussian operator is defined for the collective Boson creation operator vector which is obtained by applying the products of two-mode squeezing operators V( ) β to ˆ † a′ in Eq. 3 and to b † . The resulting parameters are and the hyperbolic matrices are defined as = Using the 2M-dimensional Bogoliubov relation for ˆ † c′ , one can convert the Gaussian Boson Sampling with thermal states into the Gaussian Boson Sampling with squeezed coherent (γ ≠ 0) or vacuum states (γ = 0) as the input states to the linear photon network (Fig. 1). We can achieve this goal by means of the SVD of the matrices 20 is a diagonal matrix with real values, which correspond to the squeezing parameters.
As a result, the unitary operator in the extended Hilbert space, which is going to be projected on a vacuum state, is decomposed asˆˆˆˆˆˆˆˆ⁎ † † The displacement parameter vector after moving the displacement operator from left end to the right end in Eq. 13, γ ′ γ γ = ′ + ′ i R I , can be calculated by, R I R R I I . This linear relation between the displacement parameter vectors for the second equality in Eq. 13 can be found by applying the two set of sequential operators in Eq. 13 and comparing the resulting parameter vectors. See Methods for the derivation.
Finally, the operator (the second equality) in Eq. 13 can be implemented in quantum optical device via preparing the 2M-dimensional single-mode squeezed coherent states and passing the squeezed coherent states, ) is replaced with the 2M-dimensional single-mode squeezed coherent states. In Fig. 1, the quantum optical unraveling of the thermal state is depicted. This can be applied to any thermal state involved problem, e.g. thermal state Boson Sampling and molecular vibronic spectroscopy at finite temperature 28,29 . The connection to the molecular problem is given in Methods.   is defined with the following Bogoliubov matrices and displacement vector, where δ is a molecular displacement vector, and J is defined as follows with the Duschinsky unitary rotation matrix U 29,31 . The FCP at finite temperature can be implemented with linear optical network as depicted in Fig. 1c with the sing-mode squeezed coherent (vacuum) states in Eq. 15.
Numerical Example. We present in Fig. 2, the photoelectron spectrum of sulfur dioxide anion (SO 2 − → SO 2 ) 32 at finite temperature (650 K) 32, 33 as an example for the optical setup in Fig. 1. The FCP at finite temperature in Eq. 16 is computed with a classical computer for this two-dimensional example and presented as sticks in the figure. The corresponding molecular spectroscopic curve from ref. 33 is overlaid in red. The molecular specific parameters are given in Methods. Unlike the FCP at zero temperature, the FCP at finite temperature has peaks in the negative frequency domain due to the thermal excitation of the molecule. By using Eq. 13, the quantum optical apparatus can be constructed for the quantum simulation as depicted in Fig. 1. In Fig. 3, the squeezing parameters in dB of the optical modes including the ancillary modes at varying temperature are shown as solid and dashed lines, respectively. The magnitudes of the squeezing parameters start from below 1 dB at 0 K, and then they increase as temperature increases because the two-mode squeezing parameter θ k increases. The squeezing parameters at 650 K corresponding to the spectrum in Fig. 2 are below 6.5 dB as indicated in Fig. 3.  In closing, we studied the problem of generalizing Gaussian Boson Sampling with initial correlation of input Bosons. This problem is relevant to the molecular spectroscopy problem at finite temperature; and we show this with a specific example of the photoelectron process of SO 2 − at 650 K. We employed a multidimensional Bogoliubov transformation together with an extended Hilbert space to de-correlate the Gaussian input state. Furthermore, we present a hierarchy for clarifying the relationships between various types of Gaussian Boson Samplings. Finally, our results imply an explicit scattershot approach for quantum-optical realization of sampling thermal Bosons without the need of an explicit preparation procedure for the Boltzmann distribution.

Methods
Quantum optical operators. A shorthand notation for the Boson operator vector x has been used, i.e.
in the paper. The displacement, squeezing and rotation operators are defined as follow, respectively, Derivation of Eq. 14. We rewrite Û as = The action of U 0 to ĉ † can be found easily with the identities in Eqs 26,27 and 28,as 20,21 . Now we work out ˆ † † U U c in the two different ways, i.e. for U D U 0ˆ⁎ = γ and ˆˆÛ U D 0 = γ′ . The resulting linear transforms are as follow,   β ω = − − . The probability P(m) is given in an integral form with the Husimi  function, accordingly, is the Glauber-Sudarshan  function of the number state |m〉〈m| 34,35 . As a special case, when z = 0 and = U I R , the generalized Gaussian state ρˆG is reduced to the Gaussian state considered in ref. 15. Such that, the corresponding Husimi  function is obtained from Eq. 40, i.e. . Again, this is a special instance of Eq. 46 at zero temperature (μ = (1, 1, …) t ).