Recongurable Quantum Photonic Convolutional Neural Network Layer Utilizing Recongurable Photonic Gate and Teleportation Mechanism

This article, proposes a reconﬁgurable quantum photonic convolutional layer (QPCL) based on the reconﬁgurable photonic gates. The QPCL is used in the classical photonic CNN, where, an array of reconﬁgurable photonic gates (RPG) are arranged in a systematic way. The designed reconﬁgurable photonic gate serves as a unit cell for quantum photonic operations such as beam splitting, rotation, displacement, squeezing, and cubic- phase shifting. The designed RPG provides the features namely broadband operation, low insertion loss and compact layout. The entangled states are created based on the normalized pixel value of the input image. The conﬁguration of reconﬁgurable photonic gate is accomplished using electro-optic P-i-N carrier injection mechanism. As compared to Mach-Zehnder interferometer (MZI) based realization, the proposed silicon reconﬁgurable photonic gate provides scalable operation and compact footprint. The reconﬁgurable photonic gate is modeled using 2D ﬁnite element beam propagation method (FE-BPM). Finally, a compact numerical model is developed which performs Gaussian based continuous-variable (CV) quantum photonic operations and are veriﬁed with Xanadu’s strawberryﬁelds quantum photonic simulator and PennyLane deep learning framework. The optimized accuracy (loss) is obtained with the utilization of QPCL layer and the values are 0.7627 (0.9595), this optimum result is obtained using a single QPCL layer with an epoch number of 30. Finally, a comparative analysis is made between quantum CNN and classical photonic CNN, where the quantum CNN resulted in 6.553% high accuracy and 6.988% low loss compared to the classical photonic CNN.


Introduction
The application specific photonic ICs perform dedicated functions, where the function can not be altered after manufacturing. Reconfigurable photonic ICs are capable of performing multiple functionality in the same photonic IC after manufacturing, which reduces the development time, cost and hardware flexibility [1]. The reconfigurable photonic IC can be implemented with different structures such as square, rectangular [2], triangular and hexagonal with array of either 2×2 MZI or 2×2 directional couplers. With the suitable configuration of each element, the required photonic applications can be mapped. The MZI based photonic array resulted more insertion loss and larger layout dimensions, which limits the denser integration of configurable Mach-Zhender interferometer (MZI) for large-scale applications.
The silicon photonics draws a great attention nowadays since it make use of the design tools, approach and fabrication procedures adopted in silicon based electronic IC industry. The main requirements for the quantum photonic applications are low bending and propagation losses, minimum cross-talk and photoluminence [3][4][5]. For silicon based MZI switches, modulation and phase-shifting operation are realized using free carrier plasma dispersion effect by following any one of the mechanism namely, carrier injection (forward biased P-i-N junction), carrier depletion (reverse biased P-N junction), and carrier accumulation (MOS capacitor structure). In P-i-N configuration, the carrier injection in the intrinsic silicon waveguide core changes the total carrier concentration, which in turn changes the refractive index and absorption coefficient [6,7]. Due to the change in refractive index, the propagation constant can also be changed, which makes the light to slow-down. The carrier-injection produces insertion loss due to the free-carrier absorption and addition of phase shifter increases the insertion loss. The intrinsic problems with the directional coupler are the crosstalk and attenuation due to the undesirable coupling and fabrication variations in the coupling factor. These inherent problems does not allow the directional coupler to completely transfer the power/intensity to adjacent waveguide and leads to ≈100% transfer. Due to the advanced fabrication technology, these inherent problems are in control, hence we focus on the reconfigurable photonic structures with directional couplers [8][9][10][11].
Artificial neural network (ANNs) are emerging due to its proven ability to solve complex real world problems. The features such as faster and power efficient operation of photonic circuit has made it suitable for ANNs. The inherent wavelength multiplexing capability make the photonic circuit to handle multiple light signals in parallel, which ensures faster operation. Due to this reduced footprint of the directional coupler, it consumes lesser power compared to digital electronic counterpart. The high performance quantum photonic applications require an alternative approach of using a generic reconfigurable photonic circuits. The impact of quantum machine learning becomes significant over past few years, which are evidenced by the publications [?,2,5,6,16,34,35]. Quantum machine learning approach provides probabilistic results which leads to exponential speed improvement due to the superposition of qumodes, and serves as the powerful components in the machine learning applications. Our research primarily focuses on utilizing reconfigurable photonic gate to perform different quantum photonic operations namely arbitrary beam splitting, displacement, rotation, squeezing, and cubic phase operation. By extending the convolutional neural network to the quantum context, a quantum feed-forward neural network [12] is developed for the character recognition.
In this work, we developed a quantum photonic neural network that performs the quantum photonic operations, shows an efficient approach for achieving high performance in deep learning applications. In our current approach, a feed-forward convolutional quantum photonic architecture is adopted. Using the reconfigurable photonic gates, we designed a quantum photonic convolutional layer that performs the sequence of complex quantum photonic operations such as sequence operations such as quantum interferometric operations, squeezing of entangled states, displacement and cubic-phase non-Gaussian operations. The degree of freedom in arbitrary power coupling and the phase shifts are realized using the P-i-N configuration.

Reconfigurable quantum photonic gate
We adopted the silicon on insulator (SoI) photonic platform to design the reconfigurable photonic gate for the analysis of large scale reconfigurable quantum photonic circuit [5,13]. In SoI platform, a silicon device layer is bonded to a SiO 2 layer deposited on the silicon substrate. The dimensions of the silicon device layer is 500×220 nm with slab height of 90 nm. The upper and lower cladding of SiO 2 with a thickness of 2 µm is considered and the transverse-electric (TE) polarization is employed in the simulation [8]. In the proposed SoI platform, silicon device layer (n 1 =3.54 at 1550 nm) is covered by SiO 2 cladding layer (n 2 =1.54 at 1550 nm), this structure results in a 40% high index contrast ratio. The Fig.1 shows the schematic representation of the proposed 2×2 reconfigurable photonic gate, also known as reversible photonic gate suitable for quantum photonic applications [14,15]. The power coupling coefficient in the directional coupler controls the amount of light intensity to be given as an output. The phase shifter have a control over the output light which serves two purposes. One is to introduce the required phase shift for the output light and the other one is to provide required rotation for the linear unitary operations. This 2×2 reconfigurable photonic gate has an advantage of behaving like an arbitrary reconfigurable modulator [16] . A new type of photonic component is developed using mode coupling mechanism with a phase shifter to provide the desired rotation for the entangled incoming photons [17,18].
The characteristic relations of the directional coupler is expressed using transfer matrix method as follows.
where, κ represents the field-coupling factor of the directional coupler. The phase shifter (PS) [15] in the top and bottom ports of the output arm is given using the following relation.
where, the φ 1 and φ 2 are the phase-shifts introduced by the phase shifters in the top and bottom phase shifters respectively. The final linear transformation matrix of the proposed reconfigurable photonic gate is obtained using (1) and (2).
Finally, the input and output electric fields of the light with the transfer matrix T RP G are described as follows.
where, the E i1 , and E i2 are the electric fields of the input lights. The parameters E o1 , and E o2 represents the output electric fields of the light. The voltage controlled field-coupling factor κ(V dc ) is described by means of plasma dispersion effect as follows.
where, L dc is the length of the coupling region. The voltage controlled phase-shift in the top arm is given by the following relation.
Similarly, the voltage controlled phase-shift at the bottom arm is given by, where, the ∆n ef f is the change in effective refractive index, λ is the free-space wavelength, and L π is the length of the phase shifter. Hence, the voltage controlled input-output relationship of the reconfigurable photonic gate is given by, where, the parameters κ(V dc ), φ 1 (V 1 ), and φ 2 (V 2 ) are the voltage controlled fieldcoupling coefficient, top and bottom phase-shift values respectively. The basic 2×2 reconfigurable photonic gate is designed using FE-BPM and an quasi-rigorous compact numerical model is developed using StrawberryFields (SF) quantum photonic simulator [19] to utilize it in PennyLane, a quantum machine learning framework. The design of reconfigurable photonic gate using different simulation methods ensure its accuracy and provides a more accurate and low loss in the analysis of large-scale quantum photonic circuits for image processing applications. The electro-optic modulation in silicon photonics is achieved with the help of plasma dispersion effect adopting carrier injection and carrier depletion mechanisms. The change in refractive index (∆n) due to the change in carrier density at 1550 nm is described as follows [20,21]. However, the change in carrier density also leads to the change in absorption (∆α) of the waveguide, which can be described as follows: where ∆N and ∆P are the carrier densities of electrons and holes [cm −3 ] respectively. The voltage-power coupling coefficient relation is shown in Fig.2, the relationship is less stable beyond 1.1 V, since the increase in voltage leads to abrupt change in power coupling factor. Hence, it is desirable to use the voltage range from 0 to 1.15 V to get various power coupling values.

Continuous variable quantum photonic gates
The continuous nature of the physical quantity can be viewed as the infinitesimal quanta of its discrete nature. Classical digital computers are best known for processing discrete quantity of information compared to continuous information.
In continuous variable approach, the quantum information is represented with the superposition of continuous real values, whereas in qubit approach (a discrete version of CV approach) [22,23], superposition of discrete values are used. The quadrature fields of lights such as amplitude and phase are used to represent the quantum information to utilize the continuous degrees of freedom of light. In CV approach, the quantum information is described as follows [24][25][26][27][28].
where, |ψ is the eigenstate ofx and ψ(x) is the encoding function where the quantum information is to be encoded. The Gaussian gates are the key elements in CV approach in which unitary transformations [29] involve Hamiltonian of linear or quadraticx andp. The Gaussian gates such as displacement gate, phaseshift/rotation gate, interferometer gate, beam splitter gate, and non-Gaussian cubic-phase gates are the basic gates used in the CV quantum information processing applications [2,19,30] , which are implemented using the proposed RPGs.
The main advantage of CV approach over qubit approach in implementing quantum teleportation is that the former provides deterministic results and the later provides probabilistic results [6,17,[31][32][33]. Based on the teleportation mechanism various CV quantum photonic gates are realized as follows [34]. The displacement gate utilizing the reconfigurable photonic gate is depicted in Fig.3. The displacement gate includes three RPGs, where the RPG#1 and RPG#2 are the two 50:50 beam splitters and the RPG#3 acts as a typical electro-optic modulator (EOM). The squeezing gate utilizing the RPG is depicted in Fig.4, which includes two RPGs, where the RPG#1 is a typical 50:50 beamsplitter and the RPG#2 acts as a typical EOM. The cubic-phase gate employing the RPGs is depicted in Fig.5, which includes three RPGs, where the RPG#1 and RPG#2 are typical 50:50 beam splitters. The phase-shifting operation is accomplished by the phase shifter of RPG#2 and the RPG#3 acts as a typical EOM.

Quantum Photonic Convolutional Layer
For image processing applications convolutional neural network is the standard machine learning approach [35,36]. In this approach, instead of handling the fulldepth of input data, a local convolution processing is carried out. In classical approach, the input image is segmented in standard sizes such as 2×2, 3×3, 5×5, and 9×9. The segmented portions are processed by a single kernel with respective size sequentially. Utilizing photonic processing feature and considering the quantum nature of light, it is efficient to process the image using classical photonic components. In this work a convolutional neural network approach is extended to quantum photonics. The considered 2×2 image portion is handled by quantum photonic functionalities such as displacement gate, squeezing gate, and rotation gate. Here the reconfigurable photonic gate serves all the above said quantum operations. A simple rotation operation is accomplished with the RPG with the help of phase shifters present at the outputs. Such rotations forms the input layer of the proposed QPCL as shown in Fig.6. The hidden layers are configured to perform different unitary operations such as interferometers, displacement gate, squeezing gates, and rotation gates. The final output layer includes the nonlinear photonic gates such as cubic-phase gate. The quantum information from the output layer is measured with homodyne detectors [19], the result further post-processed by electro-optic modulators to get the classical expectation values. The singe quantum photonic convolutional layer (QPCL) is composed of unitary transformation meshes made up of the proposed reconfigurable photonic gate [41]. It involves the following sequence of gate operations. The input image I x is handled by the first 4×4 interferometric section (U 1 ). The results are represented as follows.
After the interferometric operations, the resultant entangled quantum information is squeezed using RPG based squeezing gates (S n ), which is represented as follows.
After the squeezing section, the quantum information is given to the second 4×4 interferometric section (U 2 ), represented by The position displacement of quantum information is done using RPG based displacement gates (D), given by, At the end of QPCL layer, a non-Gaussian operation called cubic-phase gate (CP) operation is performed to introduce the quantum non-linearity in the considered feed-forward quantum neural network. The cubic-phase section is represented by, All the operations stated above are collectively carried in quantum convolution layer and is represented by Q x = QP CL(CP, D, U 2 , S, U 1 )).

Character recognition using quantum photonic convolutional neural network
In order to adopt the multi-photon CV approach, we initialized the PennyLane and strawberryfields framework with fock states [19], which supports Gaussian operations works on the basis of CV. The character recognition setup using quantum photonic processing is shown in Fig. 7. The quantum circuit operations are Fig. 7: The quantum photonic convolutional neural network for character recognition.
developed by creating a dedicated quantum circuit node that performs the rotation operations based on RPG gate, with a single hidden layers [37]. The quantum information from the output layer is measured by means of homodyne detector that represent the classical expectation values. We also developed the multi-photon classical photonic neural network that process the image without quantum circuit.
In order to reduce the image details, a method called preprocessing is accomplished using the developed CV quantum circuit. This preprocessed image is further handled by the quantum circuit and the classical photonic circuit [39,40] . The result of quantum photonic convolution layer is further processed by the classical photonic circuit. The objective of the proposed work is to classify the 10 different digits of the MNIST dataset with high accuracy and low loss. The classical photonic circuit that uses 10 output nodes with softmax activation function, stochastic-gradientdescent optimizer and a cross-entropy loss function. The classical photonic circuit is initialized, trained and validated with the dataset that has been already preprocessed by quantum circuit. The accuracy and loss characteristics of the CV quantum photonic neural network is analyzed as follows. Here we considered three different cases with epoch number of 28, 30 and 45. For the case-1 (epoch=28), Fig.8 shows the accuracy and loss characteristics. In this case, an accuracy of 0.7338 and a log-loss of 1.0494 is obtained with the utilization of a single QPCL layer. Correspondingly, an accuracy of 0.6715 and a log-loss of 1.0704 is obtained  model, the accuracy and the loss characteristics [38] for different epoch number varies from 1 to 100 is depicted in Fig.11. Different accuracy and loss values are obtained within the span of epoch value of 1 to 100. But, the results are considered as an optimum if both accuracy and loss levels are within the expected value with a minimum epoch value.

Conclusion
In this work, we designed a reconfigurable quantum photonic convolutional neural network for MNIST dataset character recognition. A 2×2 reconfigurable photonic gate (RPG) using silicon platform with with P-i-N carrier injection mechanism for power coupling and phase shift. The designed RPG, provides faster operation and reduced power consumption compared to the electronic counterpart. Due to the reduced foot-print, the proposed silicon reconfigurable quantum photonic tensor core is suitable for large-scaled reconfigurable photonic integrated circuit applications. The proposed RPG, acts as the base element in the reconfigurable quantum photonic circuits. With the design of quantum photonic convolutional layer (QPCL), an optimized accuracy(loss) values are obtained for the epoch number of 30 and the values are 0.76271(0.95953), and 0.69718(1.02941) with and without QPCL layer respectively. This optimum accuracy(loss) is achieved with the help of QPCL layer, hence we conclude that the inclusion of quantum photonic layer in photonic neural network resulted in 6.553% high accuracy and 6.988% low loss compared to the classical photonic convolutional neural network.