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Training Gaussian boson sampling by quantum machine learning

Abstract

We use neural networks to represent the characteristic function of many-body Gaussian states in the quantum phase space. By a pullback mechanism, we model transformations due to unitary operators as linear layers that can be cascaded to simulate complex multi-particle processes. We use the layered neural networks for non-classical light propagation in random interferometers, and compute boson pattern probabilities by automatic differentiation. This is a viable strategy for training Gaussian boson sampling. We demonstrate that multi-particle events in Gaussian boson sampling can be optimized by a proper design and training of the neural network weights. The results are potentially useful to the creation of new sources and complex circuits for quantum technologies.

Introduction

The development of new models and tools for machine learning (ML) is surprisingly affecting the study of many-body quantum systems and quantum optics (Huang et al. 2021). Neural networks (NN) enable representations of high-dimensional systems and furnish a universal ansatz for many purposes, like finding the ground state of many-body Hamiltonians (Carleo et al. 2019), including dissipative systems (Vicentini et al. 2019; Mangini et al. 2021).

Unsupervised and supervised learning endow new designs for quantum circuits (Marquardt 2021), metrology and cryptography (Lumino et al. 2018; Fratalocchi et al. 2021), multilevel gates (Marcucci et al. 2020), and Bell tests (Melnikov et al. 2020). NN are also triggering new fundamental investigations in quantum neuromorphic and wave computing (Marcucci et al. 2020; Hughes et al. 2019; Ballarini et al. 2020; Nokkala et al. 2020; Marković and Grollier 2020; Silva et al. 2021), quantum thermodynamics (Sgroi et al. 2021), and topological photonics (Pilozzi et al. 2021).

The impact of ML in quantum optics and many-body physics is related to the versatile representation that the NN models furnish for functions of an arbitrary number of variables. Also, the powerful application programming interfaces (APIs), as TensorFlow, enable many new features and tools to compute and design many-body Hamiltonians or large-scale quantum gates (Broughton et al. 2020).

Here, we show that NN models are also useful when considering representations in the phase space, as the characteristic functions χ or the Q-representation (Barnett and Radmore 1997). Unitary operators, as squeezers or displacers, act on the phase-space as variable transformations that correspond to layers in the NN model. Hence, a multilayer NN may encode phase-space representations of complex many-body states. This encoding has two main advantages: on the one hand, one can quickly build complex quantum states by combining NN layers; on the other hand, one can use the automatic graph building and API differentiation technology to compute observables. Also, graphical and tensor processing units (GPU and TPU) may speed up the computation.

In the following, we show how to compute the probability of multi-particle patterns when Gaussian states propagate in a system made of squeezers and interferometers. This problem corresponds to the renowned Gaussian Boson sampling (Hamilton et al. 2017; Quesada et al. 2018), which recently demonstrated the quantum advantage at an impressing scale (Zhong et al. 2020), following earlier realizations (Tillmann et al. 2012; Broome et al. 2013; Spring et al. 2013; Spagnolo et al. 2014; Carolan et al. 2014; Wang et al. 2019) of the original proposal by Aharanson and Arkhipov (Aaronson and Arkhipov 2013). The theory of Gaussian Boson sampling (GBS) heavily relies on phase-space methods (Kruse et al. 2019), making it an exciting NN test-bed supported by recently reported trainable hardware (Arrazola et al. 2021; Hoch et al. 2021; Zhong et al. 2021).

A notable outcome of adopting NN models in the phase space is the possibility of training multi-particle statistics (Banchi et al. 2020) and other features as the degree of entanglement. Indeed, most of the reported investigations in quantum ML focus either on using NN models as a variational ansatz or tailoring the input/output response of a quantum gate. On the contrary, ML in the phase space permits optimizing many-particle features, for example, to increase the probability of multi-photon events. NN may open new strategies to generate non-classical light or enhance the probability of observing large-scale entanglement with relevance in many applications. Here, we derive the NN representing the characteristic function of the Gaussian boson sampling setup. Proper NN training increases the photon-pair probability by orders of magnitude.

Figure 1 shows the general workflow of the proposed methodology, the different steps enable to define a trainable model for optmizing Gaussian boson sampling. In Section 2, we introduce the way we adopt a neural network to compute the characteristic function. In Section 3, we detail how to compute the observable as derivatives of the characteristic function neural network. In Section 4, we show how to compute the Gaussian boson sampling patterns. In Section 5, we introduce the loss function and describe the training of the model to optimize specific patterns. Conclusions are drawn in Section 6.

Fig. 1
figure1

Workflow of the proposed methodology to train Boson sampling by representing the characteristic function as a neural network

Characteristic function as a neural network

In the phase space, we represent a n-body state by the complex characteristic function χ(x) = χR(x) + ıχI(x) of a real vector x (Gardiner and Zoller 2004; Barnett and Radmore 1997). x has dimension 1 × N with N = 2n. For Gaussian states (Wang et al. 2007)

$$ \chi(\textbf{x})= e^{-\frac{1}{4}\textbf{x}\textbf{g}\textbf{x}^{\top}+\imath \textbf{x}\textbf{d}}. $$
(1)

with g the real covariance N × N matrix, and d the real displacement N × 1 vector. In our notation, we omit the symbols of the dot product such that xd and xgx are scalars. One has (\(j,k=0,1,2\dots ,N-1\))

$$ \langle \hat{R}_{j}\rangle =d_{j}= \left.\frac{\partial \chi}{\partial x_{j}}\right\vert_{\textbf{x} =0}\text{,} $$
(2)

and

$$ g_{jk}=2\langle(\hat{R}_{j}-d_{j})(\hat{R}_{k}-d_{k})\rangle-\imath J_{jk}, $$
(3)

being \(\textbf {J}=\bigoplus _{j=0}^{n-1} \textbf {J}_{1}\), \(\textbf {J}_{1}=\big (\begin {smallmatrix} 0 &1 \\ -1& 0\end {smallmatrix}\big )\) (Wang et al. 2007). In Eq. 2, the canonical variables, \(\hat {q}_{j}=\hat {R}_{2j}\) and \(\hat {p}_{j}=\hat {R}_{2j+1}\), with \(j=0,1,\dots ,n-1\), are organized in the N × 1 operator array \(\hat {\textbf {R}}\). As shown in Fig. 2a, the characteristic function is a NN layer with two real outputs χR and χI. The χ layer has two inputs: x, and a auxiliary bias N × 1 vector a, for later convenience.

Fig. 2
figure2

(a) A neural network model for the characteristic function. Two inputs, a data vector x with shape 1 × N and a bias vector a with shape N × 1 seed the model that compute χ and returns the real and imaginary parts of χ(x)eıxa. (b) A layer representing a linear transformation of the state by a unitary operator represented by a symplectic N × N matrix M and a displacement N × 1 vector \(\mathbf {d}^{\prime }\). With such a definition layers can be cascaded, and one can represent single mode squeezers, interferometers, and other unitary operators. (c) A model representing a state with characteristic function χ, subject to a unitary transformation. This is a pullback of a linear transform from the original state, which produces a new state with characteristic function \(\tilde \chi \) [see Eq. 4]

The vacuum state is a Gaussian state with g = 1 and d = 0. From the vacuum, one can generate specific states by unitary transformations, as displacement or squeezing operators. These transform the canonical variables as \(\hat {\widetilde {\mathbf {R}}}=\textbf {M} \hat {\textbf {R}}+\textbf {d}^{\prime }\), where the symplectic matrix M and the vector \(\textbf {d}^{\prime }\) depend on the specific operator (detailed, e.g., in Wang et al. (2007)). The characteristic function changes as

$$ \tilde\chi(\mathbf{x})=\chi(\mathbf{x}\mathbf{M})e^{\imath \mathbf{x} \mathbf{d}^{\prime}+\imath\mathbf{x} \mathbf{a}}= \chi(\mathbf{x}\mathbf{M})e^{\imath (\mathbf{x}\mathbf{M})\mathbf{M}^{-1}(\mathbf{d}^{\prime}+\mathbf{a})} $$
(4)

We represent the linear transformation as a NN layer with two inputs x and a and two outputs xM and \(\textbf {M}^{-1}(\textbf {d}^{\prime }+\textbf {a})\) (Fig. 2b). By this definition, Eq. 4 is as a two-layer NN.

Figure 2c shows \(\tilde {\chi }\) as the “pullback” of the linear layer from the χ layer. The two layers form a NN that can be implemented with common APIs. Footnote 1Given the vacuum state with characteristic function χ, one can build the NN model of an arbitrary state by multiple pullbacks. Indeed, we defined the linear layers in a way that they can be cascaded. Figure 4a below shows a n-mode squeezed vacuum as a multiple pullback of single mode squezers, each acting on a different mode.

Observables

Observables are computed as derivatives of the NN model. For example, the mean photon number per mode is related to the derivatives of the characteristic function. The mean photon number for mode j, is

$$ \langle \hat{n}_{j}\rangle=-\left.\frac{1}{2}\left( {\nabla_{j}^{2}}+1\right)\chi\right|_{\mathbf{x}=0} $$
(5)

being \({\nabla _{j}^{2}}=\partial ^{2}_{q_{j}}+\partial ^{2}_{p_{j}}\) and qj = x2j and pj = x2j+ 1. The differential photon number of modes j and k is

$$ \langle \left( \hat{n}_{j}-\hat{n}_{k}\right)^{2}\rangle= \left.\left[\frac{1}{4}\left( {\nabla_{j}^{2}}-{\nabla_{k}^{2}}\right)^{2}-\frac{1}{2}\right]\chi\right|_{\mathbf{x}=0}. $$
(6)

Automatic differentiation packages enables an efficient computations of the derivatives of the NN model.

Gaussian boson sampling with the neural network model

In the GBS protocol, one considers a many-body squeezed vacuum state propagating in an Haar inteferometer, which distributes the photons in the output modes. For modelling GBS, we hence need squeezing layers and a layer representing the transmission through random interferometers. The squeezing layers are realized by a proper design of the corresponding symplectic matrices M with d = 0. We implement the Haar matrix operator by QuTiP software (Johansson et al. 2013). Figure 3 shows a pseudo-code to build the neural network model by composing different layers.

Fig. 3
figure3

Pseudo-code for the creation of a neural network representing a Gaussian boson sampling experiment

Figure 4b is a graphical representation of the GBS NN model (Hamilton et al. 2017).

Fig. 4
figure4

(a) A multiple pullback that represents a many-body squeezed vacuum, obtained by a vacuum state χ by cascading n identical single mode squeezers. The order of the squeezers is not relevant as they act of different modes. χR and χI are the real and imaginary part of the resulting characteristic function. (b) GBS setup, a n-body squeezed vacuum enters an Haar inteferometer. Note that the order of the operators, from the vacuum to the interferometer goes from right to left. (c) GBS setup including a trainable random interferometer before entering the Haar interferometer. The multiple squeezers are represented as a single block. The trainable interferometer can optimize the probability of pair generation. The right panel shows the architecture of the TensorFlow model for n = 6

Boson sampling corresponds to computing the probability Pr\((\bar {\mathbf {n}})\) of finding \(\bar {n}_{0}\) photons in mode 0, \(\bar {n}_{1}\) photons in mode 1, and so forth. \(\bar {\mathbf {n}}=(\bar {n}_{0},\bar {n}_{1},\ldots ,\bar {n}_{n-1})\) is a given photon pattern. Letting \(\hat {\rho }\) the density matrix, one has

$$\Pr(\bar{\mathbf{n}})=\text{Tr}[\hat{\rho}|\bar{\mathbf{n}}\rangle\langle\bar{\mathbf{n}}|],$$

with

$$|\bar{\mathbf{n}}\rangle\langle\bar{\mathbf{n}}|=\otimes_{j=0}^{n-1}|\bar{n}_{j}\rangle\langle \bar{n}_{j}|.$$

Correspondingly (Kruse et al. 2019),

$$ \Pr(\bar{\mathbf{n}})=\left.\frac{1}{\bar{\textbf{n}}!} \prod\limits_{j=0}^{n-1} {\left( \frac{\partial^{2}}{\partial \alpha_{j}\partial \alpha_{j}^{*}}\right)}^{\bar{n}_{j}} e^{{\sum}_{j}|\alpha|_{j}^{2}}Q_{\rho}({ \alpha},{ \alpha}^{*})\right|_{{ \alpha}=0} $$
(7)

where \(\bar {\mathbf {n}}!=\bar {n_{0}}!\bar {n}_{1}!\ldots \bar {n}_{n-1}!\) and

$$Q_{\rho}=\pi^{n} \langle { \alpha} | \rho | { \alpha} \rangle$$

is the Q-rapresentation of the density matrix (Gardiner and Zoller 2004; Barnett and Radmore 1997) with \({ \alpha }=\left (\alpha _{0},\alpha _{1},\ldots ,\alpha _{n-1}\right )\) complex displacements.

We introduce the N × 1 real vector k as

$$ k_{2j} = \frac{\alpha_{j}^{*}+\alpha_{j}}{\sqrt{2}}\\ k_{2j+1} = \frac{\alpha_{j}^{*}-\alpha_{j}^{*}}{\sqrt{2}\imath} $$

and we have

$$ \Pr(\bar{\mathbf{n}})=\left.\frac{1}{\bar{\textbf{n}}! 2^{\bar{n}_{T}}} \left( \prod\limits_{j} {\tilde\nabla_{j}}^{2\bar{n}_{j}}\right) e^{\frac{\textbf{k}^{2}}{2}}Q_{\rho}(\textbf{k})\right|_{\textbf{k}=0} $$
(8)

with \({\tilde \nabla _{j}^{2}}=\partial ^{2}/\partial k_{2j}+\partial ^{2}/\partial k_{2j+1}\) and \(\bar {n}_{T}={\sum }_{j=0}^{n-1}\bar {n}_{j}\). Qρ in Eq. 8 can be evaluated explicitly as a multidimensional Gaussian integral:

$$ \Pr(\bar{\mathbf{n}}) = \left. \frac{1}{\bar{\textbf{n}}! 2^{\bar{n}_{T}}} \left( \prod\limits_{j}\tilde\nabla_{j}^{2\bar{n}_{j}}\right) \mathcal{Q}(\textbf{k})\right|_{\textbf{k}=0} $$
(9)

with (p,q = 0,1,..,N − 1)

$$ \mathcal{Q}(\textbf{k})= \frac{1}{\sqrt{2^{n} \det A}} e^{\frac{1}{2}\textbf{k}^{2}} e^{-\frac{1}{2}{\sum}_{pq} A^{-1}_{pq} (k_{p}-d_{q})(k_{p}-d_{q})} $$
(10)

being \(A_{pq}=\frac {1}{2}\left (g_{pq}+\delta _{pq}\right )\). Equations 9 and 10 can be implemented as further layers of the NN, and the probability of a given pattern computed by running the model. Figure 5a shows an example of the pattern probability distribution with n = 6, obtained by using the NN model in Fig. 4b with squeezing parameters rj = 0.88 and ϕj = π/4, such that all the single mode squeezers are identical, each with mean photon number \(\sinh (r_{j})^{2}\simeq 1\). As in Hamilton et al. (2017), we consider patterns with \(\langle \hat {n}_{j}\rangle =\{0,1\}\).

Fig. 5
figure5

(a) Probability distribution of patterns with two photons for n = 6 in the model in Fig. 4c, before training. The insets detail the particle distribution in the patterns. (b) As in (a) after training, the probability of finding a pair in mode 0 and 1 is enhanced by more than one order of magnitude. (c) Mean photon number in mode 0 and 1 during the training epochs (green), and expected differential photon number \(\langle \left (\hat {n}_{0}-\hat {n}_{1}\right )^{2}\rangle \) in the two modes, which vanishes after thousands of epochs. The statistical distribution of pairs changes at a constant photon number per mode. Data generated by the code in https://github.com/nonlinearxwaves/BosonSampling

Training Gaussian boson sampling

Our interest is understanding if we can train the model to maximize the generation of specific patterns, e.g., a photon pair in modes 0 and 1. Using complex media to tailor linear systems is a well renowned technique as, for example, to synthesize specific gates (Leedumrongwatthanakun et al. 2020; Taballione et al. 2019) or taming entanglement (Valencia et al. 2020). Here, we use the NN model in the phase space to optimize multi-particle events.

One could use the squeezing parameters in the model in Fig. 4b as training parameters. However, the degree of squeezing affects the number of particles per mode, and we want to alter the statistical properties of states without changing the average number of particles. We hence consider a GBS setup with an additional trainable interferometer as in Fig. 4c, which is typically realized by amplitude or phase modulators.

In Fig. 4c, n squeezed vacuum modes impinge on a trainable interferometer and then travel through a Haar interferometer. Instead of two distinct interferometers, one could use a single device (i.e., combine the Haar interferometer with the trainable interferometer), but we prefer to distinguish the trainable part from the mode-mixing Haar unitary operator.

Given n modes, our goal is to maximize the probability of patterns that contains a pair of photons in the mode 0 or 1. For example, for n = 6, this means maximizing the probability of \(\bar {\mathbf {n}}=(1,1,0,0,0,0)\) with respect to \(\bar {\mathbf {n}}=(1,0,0,1,0,0)\). We use as loss function

$$ \mathcal{L}=e^{\langle(\hat{n}_{0}-\hat{n}_{1})^{2}\rangle} $$
(11)

which is minimal when the expected differential number of photons in mode 0 and mode 1 vanishes. This is the case when the state has a particle pair in mode 0 and mode 1. We stress the difference in using other cost functions, which involve the expected number of photons per mode as, e.g.,

$$ \mathcal{L}_{0}=e^{{(\langle \hat{n}_{0}\rangle-\langle \hat{n}_{1}\rangle)}^{2}}. $$

The linear interferometer does not affect the average number of photons (which are mixed by the Haar layer). Correspondingly, training using \({\mathscr{L}}_{0}\) is not be effective to generate entangled pairs. On the contrary, \({\mathscr{L}}\) in Eq. 11 contains \(\langle \hat {n}_{0}\hat {n}_{1}\rangle \), which is maximal with a photon pair in modes 0 and 1.

Figure 5a shows the computed probabilities of pairs for the model in Fig. 4c, with a random instance of the Haar and the linear inteferometers. Training strongly alters this statistical distribution, as shown in Fig. 5b.

Figure 5c shows the trend during the training epochs of \(\langle (\hat {n}_{0}^{2}-\hat {n}_{1}^{2})\rangle \), which goes to zero while the mean photon numbers \(\langle \hat {n}_{0}\rangle \) and \(\langle \hat {n}_{1}\rangle \) remain unaltered.

Training also maximizes higher photon events, as in the pattern \(\bar {\mathbf {n}}=(1,1,1,1,0,0)\) with 4 photons and n = 6. Fig. 6a shows the pattern probability with 4 photons. After training with the loss function in Eq. 11, \(\Pr (\bar {\mathbf {n}})\) substantially increases for the patterns with four photons containing 1 pair in modes 0 and 1 (Fig. 6b).

Fig. 6
figure6

(a) Probability distribution of patterns with 4 photons (n = 6) in the model in Fig. 4c before training. The insets detail the particles in each pattern. (b) As in (a) after training; the probability of patterns with two photons in modes 0 and 1 is maximized. Data generated by the code in https://github.com/nonlinearxwaves/BosonSampling

Conclusions

We have shown that a many-body characteristic function may be reformulated as a layered neural network. This approach enables to build complex states for various applications, as gate design or boson sampling.

A common argument in criticizing quantum neural networks is that the linear quantum mechanics does not match with the nonlinearity-eager NN models. However, recent investigations show that nonlinearity may be introduced in quantum neural networks (Zhao and Gao 2021). Our remark is that if we formulate quantum mechanics in the phase space, nonlinearity arises in the characteristic function (or other representation). We analyzed this strategy in the simplest case of Gaussian states. The resulting model is universal and may be trained for different purposes. For this reason, phase space models allow naturally in dealing with non-classical states and computing observables by derivatives. This formulation opens many opportunities. For example, the optimization of multi-particle events can be extended to fermionic fields. As a drawback, computing boson patterns probabilities by NN APIs is not expected to be competitive with highly optimized algorithms running on large-scale clusters (Quesada and Arrazola 2020; Li et al. 2020). Still, it appears to be a versatile and straightforward methodology.

Here, we have shown many-body quantum state design and engineering by TensorFlow. We have demonstrated how to enhance multi-particle generation, with many potential applications in quantum technologies. In addition, the proposed method enables training Boson sampling without explicitly computing derivatives of the Hafnian (Banchi et al. 2020; Broughton et al. 2020), but resorting to automatic computational packages. We have tested the algorithm with a conventional workstation with a single commercial GPU (NVIDIA QUADRO RTX 4000), with a computational time of the order of few minutes with 6 modes.

The method can be generalized to other boson sampling setups, as including Glauber layers and multi-mode squeezers. Also, it readily allows to test different loss functions for tailoring the boson sampling patterns. Extension beyond Gaussian states can be envisaged by using a general machine learning networks with an arbitrary number of layers and different nonlinearity.

Notes

  1. 1.

    A TensorFlow implementation in a Jupyter notebook is available at https://github.com/nonlinearxwaves/BosonSampling

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Funding

Open access funding provided by Università degli Studi di Roma La Sapienza within the CRUI-CARE Agreement. We acknowledge support from Horizon 2020 Framework Programme QuantERA grant QUOMPLEX, by National Research Council (CNR), Grant 731473.

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Correspondence to Claudio Conti.

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Conti, C. Training Gaussian boson sampling by quantum machine learning. Quantum Mach. Intell. 3, 26 (2021). https://doi.org/10.1007/s42484-021-00052-y

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Keywords

  • Machine learning
  • Gaussian Boson sampling