Quantum machine learning with adaptive linear optics Ulysse Chabaud1,2,3, Damian Markham3,4, and Adel Sohbi5 1Department of Computing and Mathematical Sciences, California Institute of Technology 2Université de Paris, IRIF, CNRS, France 3Laboratoire d’Informatique de Paris 6, CNRS, Sorbonne Université, 4 place Jussieu, 75005 Paris, France 4JFLI, CNRS, National Institute of Informatics, University of Tokyo, Tokyo, Japan 5School of Computational Sciences, Korea Institute for Advanced Study, Seoul 02455, Korea We study supervised learning algorithms duction of various subuniversal models—models in which a quantum device is used to per- that are believed to have an intermediate com- form a computational subroutine—either putational power between classical and univer- for prediction via probability estimation, sal quantum computing—such as Boson Sam- or to compute a kernel via estimation of pling [4] or IQP circuits [5], recently culminating quantum states overlap. We design imple- with the experimental demonstration of random mentations of these quantum subroutines circuit sampling [6] and Gaussian Boson Sam- using Boson Sampling architectures in lin- pling [7]. ear optics, supplemented by adaptive mea- Finding practical applications for these subuni- surements. We then challenge these quan- versal models, other than the demonstration of tum algorithms by deriving classical sim- quantum speedup, is a timely issue, as it may en- ulation algorithms for the tasks of output able interesting quantum advantages in the era of probability estimation and overlap estima- Noisy Intermediate-Scale Quantum devices [3]. tion. We obtain different classical simula- Recently, there has been an increased interest bility regimes for these two computational on the possibility of enhancing classical machine tasks in terms of the number of adaptive learning algorithms using quantum computers [8], measurements and input photons. In both which includes the development of quantum neu- cases, our results set explicit limits to the ral networks [9–18] and the development of quan- range of parameters for which a quantum tum kernel methods [19–24]. advantage can be envisaged with adaptive In particular, recent proposals have been driven linear optics compared to classical machine by subuniversal models such as Gaussian Boson learning algorithms: we show that the Sampling [25, 26] or IQP circuits [5, 19]. In the number of input photons and the number latter, the authors considered supervised learning of adaptive measurements cannot be si- algorithms in which some computational subrou- multaneously small compared to the num- tines are executed in a quantum way, namely the ber of modes. Interestingly, our analysis estimation of the output probabilities of quantum leaves open the possibility of a near-term circuits, or the estimation of the overlap of the quantum advantage with a single adaptive output states of quantum circuits. They showed measurement. that IQP circuits alone could not provide a quan- arXiv:2102.04579v2 [quant-ph] 2 Jul 2021 tum advantage for these subroutines and there- 1 Introduction fore considered minimal extensions of these cir- cuits, in terms of circuit depth. Quantum computers promise dramatic advan- Hereafter, we study the use of Boson Sampling tages over their classical counterparts [1,2], but a linear optical interferometers [4], with input pho- fault-tolerant universal quantum computer is still tons, for similar quantum machine learning tasks. far from being available [3]. The quest for near- Instead of extending the depth which was shown term quantum speedup has thus led to the intro- to improve machine learning model performances [19] while making the training phase challenging Ulysse Chabaud: [email protected] [27–29], we allow for adaptive measurements— Adel Sohbi: [email protected] intermediate measurements that drive the rest of Accepted in Quantum 2021-07-02, click title to verify. Published under CC-BY 4.0. 1 the computation—which provide a natural anal- optics which we consider. We give a prescription ogy with the circuit depth in the linear optics pic- for performing probability estimation and overlap ture [30]: by encoding qubits into single-photons estimation with instances of this model, and de- and using a sufficient number of adaptive mea- tail how to use these as subroutines for machine surements, one can perform universal quantum learning problems. In section4, we derive two computing [31]. classical simulation algorithms, one for each of We give a detailed prescription for performing these two tasks, and we analyse the running time quantum machine learning with classical data, us- of these algorithms. We conclude in section5. ing adaptive linear optics for computational sub- routines such as probability estimation and over- 2 Background lap estimation. We also examine the classical simulability of 2.1 Kernel methods for quantum machine these quantum subroutines. More precisely, we learning give classical simulation algorithms whose run- times are explicitly dependent on: (i) the number 2.1.1 Encoding classical data with quantum states of modes m, (ii) the number of adaptive measure- We consider a typical machine learning problem, ments k, (iii) the number of input photons n and such as a classification problem, where a classical (iv) the number of photons r detected during the dataset D = {~x1, . , ~x|D|} from an input set X adaptive measurements. This effectively sets a is given. One way to use quantum computers to limit on the range of parameters for which adap- solve such problems is to encode classical data tive linear optics may provide an advantage for onto quantum states such that there exists a so- machine learning over classical computers using called feature map ~xl 7→ |φ(~xl)i which can be our methods, thus identifying the regimes where processed by a quantum computer. a quantum advantage can be envisaged. Boson Sampling instances [4] correspond to the case Definition 1 (Feature map [20]). Let F be a with no adaptive measurement, while the Knill– Hilbert space, called feature Hilbert space, X a Laflamme–Milburn scheme for universal quantum non empty set, called input set, and ~x ∈ X .A computing [31] corresponds to the case where the feature map is a map φ : X → F from inputs to number of adaptive measurements scales linearly vectors in the Hilbert space. with the size of the computation. For probability estimation, we show that the Many machine learning algorithms perform well classical simulation is efficient whenever the num- in linear cases such as the support vector ma- ber of adaptive measurements or the number of chines which will be used hereafter. However, input photons is constant. Moreover, the num- many real world problems require non-linearity ber of input photons and the number of adaptive to make successful predictions. By using kernel measurements cannot be simultaneously small methods one can introduce non-linearity and use compared to the number of modes (see Table2). estimation methods that are linear in terms of the For overlap estimation, we show a similar be- kernel evaluations. haviour, although in this case our results do not Definition 2 (Kernel [20]). Let X be an input rule out the possibility of a quantum advantage set. A function κ : X × X → C is called a kernel with a single adaptive measurement (see Tables3 if for any finite subset D = {~x1, . , x|D|} with and4). Our main technical contribution is an ex- M ≥ 2 the Gram matrix K with entries Kl,l0 = pression for the inner product of the output states κ(~xl, ~xl0 ) is positive semidefinite. of two adaptive unitary interferometers which is essentially independent of the number of adaptive The kernel corresponds to a dot product in a fea- measurements. ture space (here in a high-dimensional Hilbert The rest of the paper is organised as follows. In space). In [32], it is shown that the notions of section2, we provide a background on quantum feature map in Hilbert space and kernel can be machine learning with classical data and classi- connected. A straightforward way is to define a cal simulation of quantum computations. In sec- (quantum) kernel K from a feature map φ as fol- tion3, we introduce the model of adaptive linear lows: Accepted in Quantum 2021-07-02, click title to verify. Published under CC-BY 4.0. 2 that estimating the overlap in Eq. (3) is hard (for 2 κ(~xl, ~xl0 ) = |hφ(~xl)|φ(~xl0 )iF | , (1) classical computers). We give more formal defi- nitions for these classical simulation tasks in the where xm ∈ X , xm0 ∈ X and h·|·iF is the inner following section. product over the Hilbert space F. 2.2 Classical simulation of quantum computa- 2.1.2 Using Feature Hilbert Spaces for Machine tions Learning Depending on the approach used for simulat- There are two main ways to use feature Hilbert ing classically the functioning of quantum de- spaces: vices, several notions of simulability are com- monly used. One could ask the classical simula- • The quantum variational classification [19] tion algorithm to mimic the output of the quan- or explicit approach [20]: the entire model tum computation [34, 35]: informally, a quantum computation is preformed on a quantum computation is weakly simulable if there exists a device trained by a hybrid variational classical algorithm which outputs samples from quantum-classical or classical algorithm [14, its output probability distribution in time poly- 15, 33]. In this case, the probability distri- nomial in the size of the quantum computation. bution over the possible outcomes is used for