Quantum Generative Adversarial Networks in a Silicon Photonic Chip with Maximum Expressibility

AMZI is an indispensable component of our chip, whose combination leads to the generation of amplitude-adjustable entangled states, which is crucial for our scheme. In this subsection, we describe the transformation that AMZI applies to the quantum state of photon pairs under post-filtering, which results in the measured coincidence rate (CR) varying with the phase changes of AMZI. The relationship between CR and AMZI’s phase angle $\varphi$ is theoretically derived and experimentally verified.

We begin with theoretical derivations. Figure 1a displays a typical AMZI, where signal/idler photon pairs at different wavelengths are injected from port $a$ and split to port $c$ and $d$. The transmission matrix of the AMZI is

where $\beta_{s/i}$ is the transimission constant of the signal/idler photon and $\triangle l$ is the length difference of two arms. Assuming the free spectral range (FSR) of AMZI is designed to be twice the wavelength difference of photon pairs, we have $|\beta_{s}\triangle l-\beta_{i}\triangle l|=\pi$. Therefore, the transmission matrices of the signal and idler photons can be expressed as

The initial state of photon pairs can be written in terms of the creation operator on the vacuum state as $a^{\dagger}_{s}a^{\dagger}_{i}|vac\rangle$. The AMZI performs $U_{s}$ and $U_{i}$ for signal and idler photons, respectively, where the process can be described as

By post-selecting the case $c^{\dagger}_{s}d^{\dagger}_{i}$ with filters, the success probability of coincidence is $sin^{4}(\varphi/2)$. If the photons enter from port $b$, the result becomes $cos^{4}(\varphi/2)=sin^{4}((\varphi+\pi)/2)$. Thus, we always have

where $C_{max}$ is the measured maximum CR, which occurs when signal and idler photons are deterministically separated. This result is also experimentally verified and shown in Figure 1b. The dots are measured CRs by tuning the two AMZIs’ phases $\varphi_{1}$ and $\varphi_{2}$ in Figure 2a, respectively, while the solid lines are fitted results using $asin^{4}(\alpha I^{2}+\beta)+b$.

The quantum GANs are exhibited on a digital silicon quantum photonic chip, as shown schematically in Figure 2a. The chip is pumped with continuous-wave (CW) light at 1550.92 nm, which is amplified to 30 mW by an erbium-doped fiber amplifier (EDFA). After being adjusted by a polarization controller (PC) and filtered by a dense wavelength division multiplexer (DWDM), light is coupled into the chip via a grating coupler with a coupling loss of about 4.5 dB. Photon pairs are created in two 1-cm-long waveguide spiral sources using spontaneous four-wave mixing (SFWM) and routed through single-mode waveguides with 450nm$\times$220nm. Interferemeters, which include multimode interferometers (MMIs), beam-splitters and thermo-optic phase shifters (PSs) driven by a current source, are used to manipulate photonic qubits. The output photons are filtered using DWDMs, which selectively isolate signal photons at 1555.75nm (red) for the control qubit and idler photons at 1546.12nm (blue) for the target qubit. After polarization adjustments, photons are detected by superconducting nanowire single-photon detectors (SNSPDs) (Photec, 50Hz dark counts, 90% efficiency). A TimeTagger (Swabian) is utilized to perform coincidence counting, with a coincidence window of 300 ps. The measured coincidence rate (CR) is approximately 3 kHz. A classical computer is used to automate the quantum GANs, encompassing quantum gate manipulation, data acquisition, instant analysis, and the implementation of classical neural networks (NNs).

When pumped, the two spiral sources have equal power, resulting in the production of two-photon NOON state $(|2_{s,i},0\rangle+|0,2_{s,i}\rangle)/\sqrt{2}$. The photon pairs from two sources are filtered and partially split by two AMZIs, where the FSR of each AMZI is 19.20nm, equaling twice the wavelength difference of the photon pairs. The photons are then switched by a waveguide crossing, yielding the path-entangled state

where $A$ is the normalization coefficient, and $C_{1}(\varphi_{1})$ and $C_{2}(\varphi_{2})$ represent the measured CR of source1 and source2, respectively, which can be calculated using Equation (4). Due to the global phase introduced by AMZI in Equation (1), a relative phase occurs in the entangled state, which can be compensated for by adjusting $\theta_{8}$. The state can be further described as

where the rotation angle $\varphi$ is determined by both the two tunable split ratios of AMZIs, with the relationship $tan(\varphi)=\sqrt{C_{2}(\varphi_{2})/C_{1}(\varphi_{1})}$. As a result, entangled state with arbitrary amplitude is generated.

Next, a $C\hat{U}$ operation is carried out using the approach in Ref. [31]. The space of idler photon is expanded by adding two waveguide modes. The path of the photon introduces the third dimention indicated by $|0_{p}\rangle$ and $|1_{p}\rangle$, choosing which operation the idler qubit experiences — $\hat{V}$ or $\hat{U}$. The path is controlled by the state of the signal photon, resulting in a superposition of $\hat{V}$ and $\hat{U}$ operations

A waveguide crossing and two MMIs are then used to erase the path information. By coincidence, we have the state after $C\hat{U}$

it can be rewritten in the computational basis, which is

where $p_{0}+p_{1}+p_{2}+p_{3}=1$ and the relative phases have been ignored. Obviously, the coefficient before each basis is arbitrary, thus the state can be any two-qubit pure state. This is a deviation from prior works^{[32, 33, 34, 35, 36]}. Finally, additional PSs and MZIs are added as single-qubit gates to perform state tomography or other sophisticated operations.

The equivalent quantum circuit is depicted in Figure 2b, correlating one-to-one with the chip’s PS values, except for $\theta_{10}$ and $\theta_{11}$ because of broken PS $\theta_{11}$. However, this does not affect our experiments because the issue can be addressed by altering $\theta_{8}$ and $\theta_{9}$ (See Appendix A). The parameterized gate $\hat{R}_{y}(\varphi)$ is constructed by the two AMZIs and PS $\theta_{8}$ combined. The remaining $\hat{R}_{z}$ and $\hat{R}_{y}$ gates are built respectively using PSs added to the waveguides and the MZIs. In the quantum circuit context, the ability of our technique to generate arbitrary states is further confirmed by Ref. [37], which states that the combination of $\hat{R}_{y}$ gate on the control qubit and two unitaries controlled by $|0\rangle_{C}$ and $|1\rangle_{C}$ can prepare any two-qubit state. For the sake of simplicity, we will illustrate our chip’s configuration in subsequent experiments using quantum circuit language.

Generating quantum states is vital in quantum computing^[30]. We first exhibited the PQ-GAN, whose architecture is illustrated in Figure 3a. The black box provides quantum real data defined by a density matrix $\sigma$, yet the internal physical structure and quantum process are not exposed. The generator G, which corresponds to the quantum circuit inside the red dashed box in Figure 3b, can generate any quantum state (single-qubit in our case), including pure and mixed states. The mixed state is created by partially tracing the entangled state. D performs quantum measurements ($\mathcal{M}$) on both the real and produced states, intending to differentiate between them based on the statistical features of the measurement results $p_{\rho}=tr(\mathcal{M}\rho)$ and $p_{\sigma}=tr(\mathcal{M}\sigma)$. Figure 3b depicts D’s quantum circuit as the portion within the blue dashed box. The final two gates of the signal qubit are unused, with settings set to perform the identity operation $I$. The optimization mission of PQ-GAN is

where $\bm{\theta_{g}}$ and $\bm{\theta_{d}}$ represent the trainable phases, which are initialized with normal distribution $\mathcal{N}(\mu=0,\sigma=0.2)$. During the training process, G and D are alternatively optimized until they converge. To eliminate the influence of random factors, for pure and mixed state learning, we trained with different and identical beginning parameters, respectively, for 5 rounds, with 200 epochs per round. In each training epoch, D was trained for three steps first, then G for one. Gradient descent (ascent) is used to update the parameters. The detailed experimental parameters can be found in Appendix B.

A pure single-qubit state is learned first. The real state is selected as $\sigma_{p}=(|0\rangle+|1\rangle)(\langle 0|+\langle 1|)/2$, which is prepared by setting $\hat{R}_{y}(\phi)=I$ and $\hat{V}=H$. The density matrix, depicted in Figure 3d, is acquired by quantum state tomography (QST). Figure 3c plots the curve of negative D loss, with the solid line representing the mean value and the shaded area representing the standard deviation (STD) over 5 rounds. The negative D loss keeps decreasing and approaches zero, implying that D is unable to discern the source of the state and that G has successfully replicated the genuine state. To track the resemblance between the produced and real states, we retrieved $\rho_{p}$ by QST and estimated the fidelity with $F(\rho,\sigma)=[Tr(\sqrt{{\sqrt{\rho}\sigma\sqrt{\rho}}})]^{2}$ at each epoch. As illustrated in Figure3c, $F$ approaches 1 after approximately 100 epochs. The final generated state is depicted in Figure 3d, with a fidelity as high as $99.41\pm 0.54$%. Such a high fidelity implies that our chip can easily learn pure states.

Then, the learning of a mixed state $\sigma_{m}=(7|0\rangle\langle 0|+3|1\rangle\langle 1|)/10$ is investigated. The real state $\sigma_{m}$ is created by randomly preparing $\{|0\rangle,|1\rangle\}$ with probabilities $\{0.7,0.3\}$. Figure 3e plots the evolution of negative D loss and fidelities, which oscillate around 0 and 1 after 75 epochs, respectively. This disconvergence of mixed state learning is known as limit cycles, which result from the bilinear nature of the loss function when using gradient descent to generate mixed states in adversarial training^[14]. The occurance of bilinear problem may be due to the mixed state being positioned inside the Bloch sphere, where the nonlinear restrictions from the Bloch boundary are not satisfied^[14]. We displayed $\sigma_{m}$ and the final generated $\rho_{m}$ in Figure 3f, with the fidelity $98.39\pm 0.60$%. The ability to demonstrate mixed-state learning comes from the fact that the chip can produce single-qubit states in any mixedness by tracing out two-qubit pure states.

Quantum GANs can encode classical data into quantum states using a low-depth quantum circuit by training the generator, overcoming the data input bottleneck in QML^{[19, 20]}. Since the distribution is arbitrary, a generator with high expressivity is required. We demonstrate that our chip has the capabilities by performing three distribution loading tasks. To achieve this goal, an HQC-GAN architecture is adopted, and its diagram is depicted in Figure 4a. It is the quantum version of Wassertein GAN with gradient penalty (WGAN-GP)^{[38, 39]}, which employs Wassertein distance (or Earth mover’s distance) as the loss function, alleviating

The HQC-GAN has two components: a quantum G and a classical critic. G is a PQC parameterized with $\bm{\theta_{g}}$, and the generated data is encoded into the occurance probabilities of basis states, which can be acquired by measuring along the computational basis. The measurement operation also incorporates the stochastic aspect required by G. In our experiment, the chip plays the role of G. In Figure 4b, the chip is set to realize a quantum circuit with only three trainable parameters, as the $\hat{R}_{y}$ gate is sufficient to change the probability on each basis. It can create any 2-qubit state with real amplitudes as illustrated in Equation (9), representing any distribution with four data points $\vec{p}=[p_{0},p_{1},p_{2},p_{3}]^{\top}$. The critic, parameterized with $\bm{\theta_{c}}$, is comparable to the discriminator in standard GAN^[42], except that the final sigmoid layer has been removed, resulting in an unbounded output. We construct the critic as a fully-connected NN implemented with Pytorch, which consists of an input layer, two hidden layers and an output layer with a node arrangement of 4-5-3-1. Leaky ReLU functions are used in both the input and hidden layers^[43]. The real data is obtained by sampling from a classical random distribution.

During the training, the critic is fed both real and generated data, and the output is used to alternately train G and the critic. The optimization mission of HQC-GAN is

where $\bm{x}$ is the real data distribution, $p_{\bm{\hat{x}}}(\bm{\hat{x}})$ indicates the distribution of $\bm{\hat{x}}=\bm{x}+\zeta(G(\bm{\theta_{g}})-\bm{x})$ with $\zeta\sim U(0,1)$ and $\lambda$ is the gradient penalty coefficient. The introduction of the gradient penalty aims to meet the 1-Lipschitz continuity criterion required for WGAN^{[40, 41]}. See Appendix B for more detailed experimental settings.

We used our chip to learn three distributions: the normal distribution with $X\sim\mathcal{N}(\mu=1.5,\sigma=1)$, the log-normal distribution with $X\sim\mathcal{LN}(\mu=0.5,\sigma=1)$ and the bimodel distribution being the superposition of $X\sim\mathcal{N}(\mu=0,\sigma=0.5)$ and $X\sim\mathcal{N}(\mu=2,\sigma=0.3)$. The real data, represented by the black solid lines in Figure 4c-e, are obtained by sampling 10,000 points and truncating them into the interval $[0,3]$. We trained the network in five rounds with varied initialization settings, each with 500 epoches. For each epoch, the critic is trained three times first, then G is trained once.

The blue histograms in Figure 4c-e exhibit the average value of learning results, which mimic the target distributions. Figure 4f-h show the evolution of the critic loss and the Kullback-Leibler divergence (KLD), with the solid line representing the mean value over 5 rounds and the shadow part indicating the STD. In traditional GANs, the KLD is commonly used to estimate the similarity of two distributions and is often employed as a convergence indicator. As noted, the critic loss and KLD have a comparable convergence tendency and finally approach 0. The final KLD value for all three distributions is less than 0.05, suggesting that the real distributions are successfully loaded.

As an important application, classical GANs have been utilized to generate realistic images that do not exist in the training dataset. In this application, nonlinearity is critical for capturing complicated patterns and nonlinear correlations in the data. Unfortunately, when a PQC is substituted for a classical G, the quantum G can only conduct linear transformations since quantum evolution is unitary. To introduce nonlinearity, one typical way is to trace the qubits out^{[23, 44, 45]}, i.e., to add auxiliary qubits to the generator, and discard them during the measurement stage. However, this method is not suitable for our experiment since only two qubits are available. To address this challenge, we introduce a new method by placing a classical NN in front of the original quantum G, giving rise to a new hybrid G that performs well in the job of learning compressed MNIST images.

Figure 5a depicts the training framework, which resembles the structure seen in Figure 4a. The significant distinction is the addition of a noise source $Z$ and the use of the hybrid G. The noise vector $\bm{z}$ is obtained by sampling from a uniform distribution within the range [0, 1]. As previously stated, the hybrid G is made up of two parts: one classical, which is a fully-connected NN, and the other quantum, which is a PQC. The incorporation of the classical NN enables straightforward nonlinear data transformations using activation functions. In this experiment, a $2\times 2$ NN activated by Leaky ReLU functions is used. Figure 5b illustrates the quantum part implemented by our chip, which consists of an encoding layer and a training layer $U_{G}(\bm{\theta_{q}})$. The encoding layer occurs because the equation $\hat{R}_{y}(\phi_{1}+\phi_{2})=\hat{R}_{y}(\phi_{1})+\hat{R}_{y}(\phi_{2})$ holds true. To acquire generated data, the noise vector $\bm{z}$ is first input into the classic part of G, and undergoes a nonlinear transformation to become $G(\bm{\theta_{c}};\bm{z})$. The results are then encoded into the angle of $\hat{R}_{y}(z)$ gates and transformed into the output state $|G(\bm{\theta_{c}},\bm{\theta_{q}};\bm{z})\rangle=U_{G}(\bm{\theta_{q}})|G(\bm{\theta_{c}};\bm{z})\rangle$. Finally, measuring along the computational basis yields the generated data $G(\bm{\theta_{c}},\bm{\theta_{q}};\bm{z})$.

The training process is similar to the data loading task, except that in each epoch, a batch of data is sampled separately from the generated and real dataset, rather than a single distribution, to optimize G and D. The hyperparameters used in this experiment can be found in Appendix B. The optimization objective is

where $\bm{x_{i}}$ is a sample from the training dataset, and N is the batch size. The backpropagation approach is ineffective for optimizing $\bm{\theta_{g_{c}}}$ due to the quantum component between the classical NN and the output. Thus, we utilize the finite difference approach to get the approximate gradient, which is

where $\epsilon$ is a small incremental value.

As a demonstration, we use this framework to train the chip to produce MNIST images. The MNIST dataset contains 60,000 handwritten images ranging from 0 to 9 with a size of $k=28\times 28$, exceeding our chip’s capabilities. Inspired by Ref. [46, 47], we preprocess the pictures by compressing them to dimension $k=3$ with the principal component analysis (PCA) algorithm, resulting in the feature data $\vec{x}=[x_{0},x_{1},x_{2}]^{\top}$ for each compressed image. To match the image data with the quantum state represented in Equation (9), we augment $\vec{x}$ to ${x}^{\prime}=[\vec{x}^{\top},0.5]^{\top}$, and utilize the map $p_{i}=x_{i}^{\prime}/\sum_{i=0}^{3}x_{i}^{\prime}$, which assures a one-to-one mapping between $\vec{x}$ and $\vec{p}$, retaining all information.

We then trained the chip to generate all nine digits (0–9) one at a time. The batch size is set to be $N=5$, and 5 rounds with each 200 epochs are conducted. To visualize the quality of the created images, we utilize the inverse PCA (IPCA) method to retrieve the data to the size of $28\times 28$, and display the real and generated images with binarization in Figure 5c. The images are randomly selected from training and generated datasets. As can be seen, we produced images that are comparable in quality to the original. Figure 5d displays the curve of critic loss and KLD of digit 0, which converge to be approximately 0. The convergence and creation of high-quality images demonstrate the efficacy of our strategy.