Single-Photon Image Classification

1 INTRODUCTION . Both quantum mechanics and machine learning play a major role in modern technology , and the emerging field of AI applications of quantum computing may well enable major breakthroughs across many scientific disciplines . Yet , as the majority of current machine learning practitioners do not have a thorough understanding of quantum mechanics , while the majority of quantum physicists only have an equally limited understanding of machine learning , it is interesting to look for “ Rosetta Stone ” problems where simple and widely understood machine learning ideas meet simple and widely understood quantum mechanics ideas . It is the intent of this article to present a setting in which textbook quantum mechanics sheds a new light on a textbook machine learning problem , and vice versa , conceptually somewhat along the lines of Google ’ s TensorFlow Playground ( Smilkov et al . ( 2017 ) , ) which was introduced as a teaching device to illustrate key concepts from Deep Learning to a wider audience . Specifically , we want to consider the question what the maximal achievable accuracy on common one-out-of-many image classification tasks is if one must make a decision after the detection of the very first quantum of light ( i.e . photon ) that passed a filter showing an example image from the test set . In this setting , we do not have a one-to-one correspondence between example images from the training ( respectively test ) set and classification problems . Instead , every example image defines a probability distribution for the ( x , y ) detector pixel location on which the first photon passing an image filter lands , the per-pixel probability being the pixel ’ s brightness relative to the accumulated ( across all pixels ) image brightness . So , from every ( 28× 28 pixels ) example image , we can sample arbitrarily many photon-detection-event classifier examples , where the features are a pair of integer pixel coordinates , and the label is the digit class . On the MNIST handwritten digit dataset ( LeCun and Cortes ( 2010 ) ) , any machine learning system that only gets to see a single such “ photon detected at coordinates ( x , y ) ” event as its input features , of the pixel that flashed up are the only input features , is limited in accuracy by the maximum likelihood estimate , since we have : P ( Image class C|Photon detected at ( x , y ) ) =∑ E P ( Image class C|Example E ) P ( Example E|Photon detected at ( x , y ) ) . On photon detection events generated each by first randomly picking an example image , and then randomly picking a brightness-weighted pixel from that , we can not do any better than predicting the most likely digit class given these input features – the two pixel coordinates . As performance is measured on the test set , no classifier could possibly ever outperform one that is built to achieve maximal performance on the test set . This is obtained by determining , for each pixel , what the most likely class is , where examples from the test set are weighted by the fraction of total example-image brightness that comes from the pixel in question . Figure 2 ( b ) shows the most likely image-class per pixel . ( For MNIST , some pixels are dark in every test set example . ) No classifier can outperform one that simply looks up the pixel-coordinates at which a photon was detected in Figure 2 ( b ) and returns the corresponding class , and this optimal classifier ’ s accuracy is 22.96 % for the the MNIST dataset – substantially higher than random guessing ( 10 % ) . Appendix A.2 provides a detailed ( but mostly straightforward ) optimality proof of this accuracy threshold . We can not , for example , outperform it by redistributing light intensity between pixels , since any such redistribution could only destroy some of the available useful information , not magically create extra useful information . An entirely different situation arises when we allow quantum mechanics to enter the stage : For a single photon passing through a coherently illuminated image filter , with all pixels at the same optical phase on the incoming wave , we can imagine putting some precision optical device between the image filter and the detector array that redistributes not the probabilities ( which correspond to light intensity when aggregating over many photons ) , but the amplitudes that make up the spatial part of the photon wave-function . Illuminating such a set-up with many photons would show a hologram-like interference pattern on the detector array . This transformation of the ( single- ) photon wave function by linear optical elements then has tuneable parameters which we can adjust to improve classifier accuracy . Quantum mechanics tells us that every ( lossless ) linear optical device can be represented by a linear unitary transform on the photon state : The action of any complex optical device consisting of ( potentially very many ) components which transforms a N -component photon state ( in our case , N = 282 amplitudes in the spatial part of the photon wavefunction ) can be described by an element of the N2-dimensional unitary matrix Lie group U ( N ) . Vice versa , Reck et al . ( 1994 ) describes a constructive algorithm by which any U ( N ) transformation matrix can be translated back to a network of optical beam splitters and phase shifters . 1.1 RELATED WORK . Conceptually , exploiting interference to enhance the probability of a quantum experiment producing the sought outcome is the essential idea underlying all quantum computing . The main difference between this problem and modern quantum computing is that the latter tries to perform calculations by manipulating quantum states of multiple “ entangled ” constituents , typically coupled two-state quantum systems called “ qubits , ” via “ quantum gates ” that are controlled by parts of the total quantum system ’ s quantum state . Building a many-qubit quantum computer hence requires delicate control over the interactions between constituent qubits . This usually requires eliminating thermal noise by going to millikelvin temperatures . For the problem studied here , the quantum state can be transformed with conventional optics at room temperature : the energy of a green photon is 2.5 eV , way above the typical room temperature thermal radiation energy of kT ' 25 meV . The price to pay is that it is challenging to build a device that allows multiple photons to interact in the way needed to build a many-qubit quantum computer . Nevertheless , Knill , Laflamme , and Milburn ( Knill et al . ( 2001 ) ) devised a protocol to make this feasible in principle , avoiding the need for coherency-preserving nonlinear optics ( which may well be impossible to realize experimentally ) by clever exploitation of ancillary photon qubits , boson statistics , and the measurement process . In all such applications , the basic idea is to employ coherent multiphoton quantum states to do computations with multiple qubits . In the problem studied here , there is only a single photon , the only relevant information that gets processed is encoded in the spatial part of its wave function ( i.e . polarization is irrelevant ) , so the current work resembles the “ optical simulation of quantum logic ” proposed by Cerf et al . ( 1998 ) where a N-qubit system is represented by 2N spatial modes of a single photon . Related work studied similar “ optical simulations of quantum computing ” for implementing various algorithms , in particular ( small ) integer factorization ( Clauser and Dowling ( 1996 ) ; Summhammer ( 1997 ) ) , but to the best of the present authors ’ knowledge did not consider machine learning problems . This work can be described as belonging to the category of machine learning methods on quantum non-scalable architectures . Alternatively , one can regard it as a quantum analogue of recent work that demonstrated digital circuit free MNIST digit classification via classical nonlinear optics , for instance via saturable absorbers ( Khoram et al . ( 2019 ) . ) Apart from providing an accessible and commonly understandable toy problem for both quantum and ML research communities , this simplequantum/simple-ML corner also may be of interest for teaching the physics of the measurement process ( “ the collapse of the wave function ” ) in a more accessible setting . Whereas explanations of the measurement process are forced to remain vague where they try to model “ the quantum states of the observer ” ( typically unfathomably many states that one would never hope to be able to model in terms of actual numbers ) , using machine learning as a sort-of cartoon substitute for high level mental processes actually allows us to come up with fully concrete toy models of the measurement process on low-dimensional ( such as : D < 1000 ) Hilbert spaces that nevertheless capture many of the essential aspects – to the extent that “ ML classifies the measurement as showing the image of a shoe ” can be regarded as a crude approximation to “ observer sees a shoe ” . Looking closer at the relation between the present article and Khoram et al . ( 2019 ) , both articles study the general feasibility of realizing Machine Learning classifiers in the form of an analog optical computer at the theoretical level , using numerical optimization to produce a blueprint of a device that can perform inference for a specific problem . In both articles , the primary problem under study is MNIST handwritten digit classification , the input is encoded as spatial dependency of a ( monochromatic ) laser beam ’ s light intensity , and classification happens by using interference to funnel optical energy onto a detector array . In both cases , stochastic gradient descent is used to shape how this funneling of optical energy happens . Indeed , even the loss function used for training ( cross entropy ) is essentially equivalent . The key differences are that Khoram et al . ( 2019 ) only considers the many-photon limit of classical wave optics , which allows one the luxury of using non-linear optical components , specifically saturable absorbers , to implement non-linearities . This has no analog for the single photon case . Also , having many photons available allows identifying the target class that receives most laser light and calling this the prediction of the model . This is clearly not possible when a decision has to be made after seeing only a single photon . If one sent many photons through an interference device as described in this article and picked the target class with the highest photon count , one would observe classification accuracies of about 90 % rather than the claimed about-40 % for a single photon . This is considerably higher than the accuracies of about 80 % presented inKhoram et al . ( 2019 ) as the focus of that article is on manufacturability , running gradient backpropagation directly on a Finite Difference Frequency Domain PDE simulation of Maxwell ’ s equations and taking materials engineering constraints into account , whereas our work focuses on upper and lower bounds for achievable accuracy , exploiting the one-to-one equivalence between linear optical devices and unitary transforms . Our work directly trains the parameters of the unitary transform , which only afterwards get mapped to a blueprint for an experimental device realization . Speculatively , if a device were built experimentally that was designed by the methods inKhoram et al . ( 2019 ) , subject to the extra constraint that no non-linear elements can be used , and then deployed in a low-light-intensity single-photon setting , using a suitable detector such as a SPAD array , it may manage to realize better-than-classically-achievable classifier performance , for reasons explained in the current work .

Summary: This paper focuses on the quantum computing based machine learning and proposes a toy model to illustrate the quantum information processing. On the common used handwritten digit dataset MNIST, more than 40% images can be classified accurately. The proposed method looks interesting and the focused problem (combining quantum computing and machine learning) is of certain significance.