1State Key Laboratory of Optoelectronic Materials and Devices, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China
2College of Materials Science and Opto-Electronic Technology, University of Chinese Academy of Sciences, Beijing 100049, China
3School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, Beijing 100049, China
4State Key Laboratory of Information Photonics and Optical Communications and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
5United Microelectronics Center Co., Ltd., Chongqing 401332, China
6China Academy of Space Technology (Xi’an), Xi’an 710100, China
【AIGC One Sentence Reading】:This paper proposes Ising-model-based intelligent computing for dynamic PIC reconfiguration, enabling high-speed, large-scale analog signal processing with demonstrated efficacy.
【AIGC Short Abstract】:This paper introduces an innovative method using Ising-model intelligent computing to dynamically reconfigure large-scale programmable photonic integrated circuits (PICs). By modeling MZI units as spin qubits, the approach reformulates PIC functionality as a path-planning problem, enabling high-speed, large-scale analog signal processing.
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Abstract
Programmable photonic integrated circuits (PICs) have emerged as a promising platform for analog signal processing. Programmable PICs, as versatile photonic integrated platforms, can realize a wide range of functionalities through software control. However, a significant challenge lies in the efficient management of a large number of programmable units, which is essential for the realization of complex photonic applications. In this paper, we propose an innovative approach using Ising-model-based intelligent computing to enable dynamic reconfiguration of large-scale programmable PICs. In the theoretical framework, we model the Mach–Zehnder interferometer (MZI) fundamental units within programmable PICs as spin qubits with binary decision variables, forming the basis for the Ising model. The function of programmable PIC implementation can be reformulated as a path-planning problem, which is then addressed using the Ising model. The states of MZI units are accordingly determined as the Ising model evolves toward the lowest Ising energy. This method facilitates the simultaneous configuration of a vast number of MZI unit states, unlocking the full potential of programmable PICs for high-speed, large-scale analog signal processing. To demonstrate the efficacy of our approach, we present two distinct photonic systems: a wavelength routing system for balanced transmission of four-channel NRZ/PAM-4 signals and an optical neural network that achieves a recognition accuracy of 96.2%. Additionally, our system demonstrates a reconfiguration speed of 30 ms and scalability to a port network with 2000 MZI units. This work provides a groundbreaking theoretical framework and paves the way for scalable, high-speed analog signal processing in large-scale programmable PICs.
1. INTRODUCTION
Electronic integrated circuits (EICs) serve as the foundational technology of the modern digital information society. However, as scaling laws approach their fundamental physical limits, further advancements in transistor density and power efficiency within EICs are becoming increasingly difficult [1]. At the same time, analog signals play a vital role in real-world applications, especially in scenarios requiring real-time information processing, such as wavelength routing, neural networks, medical diagnostic imaging, robotic control, and remote sensing. These applications demand high reconfigurability, low latency, high bandwidth, and superior energy efficiency. Yet, within the digital framework of EICs [2–5] these tasks are often not performed as efficiently or effectively as they could be.
Programmable photonic integrated circuits (PICs) enable real-time manipulation of light flow on a chip through precise adjustments of their programmable units. This capability allows light to be dynamically redistributed under software control, enabling the execution of diverse functions via signal interference across multiple pathways. As a result, programmable PICs serve as powerful hardware platforms for analog signal processing, aiming to maximize functionality while minimizing cost [6–12]. Reconfigurability is an essential feature of programmable PICs, allowing for dynamic adaptation to a range of analog signal processing scenarios [13], including but not limited to optical communication [14,15], computation [16–19], LiDAR [20–23], sensing [24–26], and cryptography [27]. Current research has demonstrated that programmable PICs composed of tens to hundreds of MZI units have successfully achieved increasingly complex functions, such as routing, linear matrix operations, and filtering, fully validating their programmability, reconfigurability, and scalability [28–32]. In 2022, the team at Xizhi Technology, leveraging 3D optoelectronic integration technology, unveiled a optical matrix photonic computing processor integrating over 10,000 photonic components, further proving the significant advantages of programmable PICs in large-scale scalability [33].
As the scale of programmable PICs increases, a critical challenge has emerged: numerical modeling across hundreds or even thousands of units becomes time-consuming, with computational resource demands escalating exponentially. Suitable algorithmic models are essential, as they directly determine the efficiency and scope of their functional configurations. The genetic algorithm (GA) [34] is utilized for reconfiguration optimization of programmable PICs, solving the single-path-planning task for a chip with 40 MZIs. However, the computation time remains at the second level per single path, highlighting the limitations of traditional algorithms in achieving real-time performance. The Dijkstra algorithm maps a single MZI to a 4-node graph structure and is used to optimize single-path optical routing in programmable PICs with 30 and 81 MZIs [35]. Subsequently, an improved Dijkstra algorithm was employed, modeling each MZI with 8 nodes and targeting multipath planning [36]. These studies reveal a critical challenge: the exponential growth in computation time as both the number of MZIs and the complexity of planning paths increase. The light-processing function optimization strategy, based on recursive methods and dispersionless approximation, addresses single-path () and multipath () planning for programmable PICs with 55 MZIs [37,38]. However, even at the same computational scale, this method faces the challenge of increasing computation time as the number of paths grows. Therefore, these results demonstrate that traditional CPU-based reconfiguration optimization methods all face the challenge of exponential growth in computation times as the chip scale increases. This limitation underscores the need for more efficient optimization strategies in large-scale programmable PICs. Hence, there is an urgent need for a novel intelligent computing model that is specifically tailored to the structure of programmable PICs, and capable of accurately finding global optimum solutions with shorter reconfiguration times.
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The function of programmable PICs is implemented by deploying all programmable units, which can be transformed into the path-planning problem, a type of combinatorial optimization problem (COP). The Ising model, a groundbreaking approach for solving various COPs [39–41], can be ingeniously applied to fields such as battery modeling [42], material growth [43], financial computing [44], and circuit design [45]. It provides an innovative approach for configuring large-scale programmable PICs and handling complex analog signal processing tasks. The Ising model can be solved using simulators on a CPU, optical parametric oscillator Ising machines [46], and microwave photonic Ising machines (MPIMs) [47]. For instance, the MPIM, a specialized hardware platform designed for the Ising model, can significantly improve the efficiency of solving COPs. By leveraging the principle of minimum gain to implement reverse annealing, the MPIM can harness the inherent parallelism of light to accelerate solution discovery and reduce computational complexity. This capability enables the rapid identification of optimal optical paths, dramatically improving the efficiency of optical path planning in programmable PICs [48–56].
In this work, we present a groundbreaking Ising-model-based intelligent computing approach for controlling large-scale programmable PICs, enabling multifunctional analog signal processing. By mapping the fundamental Mach–Zehnder interferometer (MZI) units in programmable PICs to spin qubits with binary decision variables, the complex issue of light wave propagation within programmable PICs is transformed into a path-planning problem that can be solved via the Ising model. This innovative process not only allows for arbitrary programmability but also significantly streamlines the reconfiguration procedure of the PIC. We demonstrate the effectiveness of this approach through two photonic systems for analog signal processing: a wavelength routing system that achieves balanced transmission of four-channel NRZ/PAM-4 signals, and an optical neural network system that achieves handwritten digit recognition with 96.2% accuracy. This analog signal processing framework can be further adapted to programmable PICs with a port count of and 2000 MZI units with a reconfigurable time of approximately 30 ms. This capability showcases unprecedented flexibility and adaptability for optical communication and computation, providing a strong foundation for future advancements in photonic integration technology.
2. METHODOLOGY AND RESULTS
A. Programmable PIC Equivalent Ising Model
The programmable PICs, as depicted in Fig. 1(a), are meticulously crafted from an interconnected cascade of hexagonal structures [6,12]. Within this configuration, six MZIs are strategically interconnected to form a cohesive hexagonal ring. Each MZI, as a primary programmable functional unit, is equipped with a phase shifter in one arm to control the optical phase using the thermo-optic effect [57,58]. The state switching time ranges from 20 to 40 μs [59]. This allows precise control of the light transmission direction for signal processing and opens up a wide range of applications, such as wavelength routing, neural networks, and a multitude of other functionalities. The transmission matrix of MZI, denoted as , represents the transfer vector with an additional phase shift introduced by the phase shifter. The phase is adjusted to or , thereby achieving the desired unit state within the cross (“0” state) or bar (“1” state). Consequently, the light wave exiting from the MZI is accurately represented by the matrix
Figure 1.Programmable PICs and the equivalent Ising model. (a) The structure of programmable PICs consists of a cascading arrangement of hexagonal structures, with each MZI as a programmable basic unit. The MZI is designed with a phase shifter in one arm, enabling precise control over the optical power. (b) The correspondence states between the MZI and the binary decision variable. The MZI’s cross state at corresponds to the node state “0,” whereas the MZI’s bar state at is equated with the node state “1.” (c) The equivalent Ising network model and Ising-model intelligent computation process. A complete Ising matrix is formulated by incorporating the constraints matrix into the Ising model and then finding the optimal solution via an Ising machine to enable dynamic control over the PICs. MZI, Mach–Zehnder interferometer; PS, phase shifter.
The conversion relationship between the optical intensity and the phase shift can be obtained as
To meet diverse input–output combination requirements, the transmission matrices of the MZI units on the PICs are tailored accordingly. The transmission matrix in Eq. (1) is equivalent to the binary variable in the statistical analysis. As depicted in Fig. 1(b), the representation of the MZI can be expressed as follows: where the binary decision variable is introduced to represent the usage status of the th port on the th MZI unit, with a value of 1 or 0 indicating the presence or absence of usage, respectively. The table in Fig. 1(b) shows the unit structure states corresponding to different values, where a value of 1 or 0 indicates whether the th port is used. A value of 0 implies that the path does not pass through this segment, whereas a value of 1 implies that the path passes through this segment. Therefore, a unit structure MZI needs to be determined by four binary decision variables.
Once the modeling of the unit structure is complete, a constraint matrix needs to be established to solve the path-planning problem. In multi-input–output path-planning scenarios, it is crucial to consider that varying path lengths lead to different optical losses, thereby influencing the transmitted signals unevenly. To mitigate this, it is necessary to follow the principle of path loss balancing, ensuring that each path exerts a uniform impact on the transmitted signal.
In the MZI unit structure, two constraints need to be considered. First, the input and output ports must be located on opposite sides of the MZI. This implies that the binary decision variable values cannot be , , , or , , , . Second, each path requires planning the state of the same MZI unit, and the same MZI unit can only have one state across all the planned paths. This means that in multi-input multi-output path planning for programmable PICs, a single set of four binary decision variables is insufficient when dealing with multiple pairs of input and output ports. Therefore, in the k input–output path-planning scenarios, sets of four binary decision variables should be used to represent the states of the th MZI unit. The constraint matrix of MZI unit can be expressed as follows: where represents the number of MZI units of whole programmable PICs. Importantly, each given MZI unit can be used by at most one path at a time. To ensure this, at most one element in the set should be equal to 1, which requires the introduction of constraint functions between paths. Accordingly, the constraint function for a shared MZI unit across different paths can be expressed as where represent the number of MZI unit roads of whole programmable PICs. The entrance-exit matrix exists when the entrance and exit of each path are selected as follows: where sets the binary decision variable values for the input and output ports to 1. To ensure a consistent impact of each path on the transmitted signal, length-balancing constraints are introduced among multiple paths. For a specific case of inputs and outputs, the consistency is evaluated by calculating the variance of the lengths of the selected paths. The constraint function is as follows:
Here refers to the shortest path optimization matrix, which is the sum of all binary decision variables and can be expressed as follows:
The constraint matrix of the programmable PICs can be written as matrix (for our designed programmable PICs, the constraint matrix is detailed in Appendix A):
Among them, represents the weight of each matrix. The functions , , and are the objective functions, and and are the constraint functions. In Eq. (9), the initial weights are confirmed by using the maximum quadratic coefficient (MQC) method after the input and output ports are defined [45,60]. By extracting the maximum coefficient from the constraint matrix using the MQC method, it rapidly initializes the weights and concurrently balances the impacts of both the objective function and the constraint function. This approach effectively prevents the slow convergence challenge with large weights and the constraint failure issue with small weights. After setting the initial weights, the final weight values are refined through multiple iterations, progressively converging toward the global optimum. For the different input/output demands, the weight values need to be recalculated.
To solve the path-planning problem in the Ising machine, the matrix needs to be converted into the Hamiltonian H. Then, the binary decision variable is converted into spin qubit , as follows:
The binary decision variables are linearly mapped onto the spin qubit . Equation (10) is subsequently introduced into Eq. (9), and the matrix can be expressed as the Hamiltonian , as follows (the Ising energy solution process is detailed in Appendix B): where , and the Ising connection matrix represents the interaction strengths between spin qubit and spin qubit , respectively. is the sum of the weights between all the spin qubits. The solution process can also be implemented in the MPIM hardware. In the MPIM system, the microwave pulse in the system minimum energy state is used to represent the solved spin qubit in Eq. (11), where an upper microwave pulse denotes a value of and a lower microwave pulse denotes a value of . A series of spin qubits are transformed back into binary variables. And the binary decision variables are subsequently converted into the control voltage of the MZI (MZI voltage mapping from the binary variables is detailed in Appendix C).
B. PIC Design and Performance
The programmable PIC device is depicted in Fig. 2(a). It is fabricated on a silicon-on-insulator (SOI) wafer using a CMOS-compatible process from Advanced Micro Foundry (AMF, Singapore), with a height of 220 nm and a width of 500 nm. The strip waveguide exhibits propagation losses of 2 dB/cm. The microscope image in Fig. 2(a) shows 46 MZIs arranged into 10 hexagonal rings, forming programmable PICs. The device has a compact footprint of 4.3 mm by 2.8 mm, with 50 electrical pads positioned around its periphery. The grating couplers, located at the edge of the chip, facilitate light coupling both inwardly and outwardly. The MZI, consisting of a pair of directional couplers (DCs) and a phase shifter within a single arm, serves as the core of the programmable PICs. In the MZI, the DC is employed as an alternative to the multimode interference (MMI) device, resulting in a significant reduction in length by 38%, from 74 to 46 μm. The DC-based MZI PIC, which is 62% the size of the MMI-based MZI PIC, provides a more compact and efficient design solution.
Figure 2.Characteristics of programmable PICs. (a) Microscope image of programmable PICs. (b) Packaged programmable PICs. (c) Test results of five cascaded DCs. (d) Transmission performance of an MZI unit. DC, directional coupler.
The packaged programmable PICs are described in Fig. 2(b). A 14-channel fiber array with a pitch of 127 μm is attached to the switch chip using UV light-curable adhesive [61], which boasts a refractive index similar to that of silicon dioxide. The electrical pads are positioned close to the chip edges and are connected to a printed circuit board (PCB) through wire bonding. A thermoelectric cooler (TEC) placed beneath the chip is activated to maintain a stable temperature of 25°C, minimizing thermal crosstalk and ensuring the accuracy of the experimental results.
Figure 2(c) shows the splitting ratio for a series of five cascaded DCs, clearly depicting their attenuation characteristics across various wavelengths. The linear fitting results indicate a splitting ratio of 3.29 dB at 1530 nm, 3.07 dB at 1540 nm, and 2.76 dB at 1550 nm. Notably, the DC splitting ratio deviates by , approaching the 3 dB target within the wavelength range of 1530–1550 nm. This variation is primarily due to the inherent wavelength sensitivity of the DC. Consequently, the designed DC is suitable for operating in the C-band, aligning well with its typical operational parameters. Figure 2(d) shows the working performance of the MZI at various voltages. With increasing voltage, the MZI smoothly switches from the “0” state to the “1” state. Since the fully programmable PICs cannot individually test the state of each MZI, the state variation of a single MZI is used to represent the behavior of all MZI structures on the chip, assuming consistent fabrication processes [35]. Experimental results show that at 1 V, the MZI unit is in the cross state, corresponding to the binary variable’s “0” state; when the voltage rises to 4.3 V, the MZI unit transitions to the bar state, corresponding to the binary variable’s “1” state. The inset shows a spectrum at the “1” state, demonstrating a crosstalk level below across a wavelength range of approximately 20 nm, and additionally, the insertion loss for a single unit structure is approximately 0.4 dB. For further sensitivity analysis, please refer to Appendix D.
Figure 3(a) shows the flowchart for intelligent computation based on the Ising model. The process of Ising-model intelligent computation is mainly divided into four steps. (1) Weight calculation: for our designed programmable PICs, the weights are calculated using the MQC method, as mentioned in Eq. (9). (2) Ising energy calculation: after confirming the weight values, the Ising model computation evolves gradually from a random initial state toward a lower energy state, according to the dynamic equations detailed in Appendix B, and ultimately converges to a stable-state solution. (3) Spin qubits calculation: the minimized Ising energy is obtained within a predefined number of iterations, and the state of the spin qubits is identified in the lowest Ising energy, following Eq. (10). (4) Voltage mapping: the states of the spin qubits are linearly mapped to the state of the binary variables. A lookup table is employed to correlate the values of various binary variables with their respective MZI driving voltages in the programmable PICs (MZI voltage mapping from the binary variables is detailed in Appendix C).
Figure 3.Process of Ising-model intelligent computation. (a) Flow chart of the Ising model-based intelligent computation; (b) minimum Ising energy list for each of the 25 times of iterative calculations. (c) Ising energy evolution process over time of the 7th calculation and the result for four-input four-output intelligent computation. (d) Test results of the stochastic path analog signal processing. (e) Test results of selective path analog signal processing after reconfiguration.
Figure 3(b) illustrates the Ising energy resolution process with the weight values , , , , and , when the input ports are set to 2, 3, 7, and 9, corresponding to the output ports 13, 22, 18, and 20. This resolution process is repeated 25 times through numerical simulation, and the optimal solution is identified as the one with the lowest energy value. On the basis of this optimal solution, the state of the MZI unit can be derived. The energy list from 25 times iterative calculations [Fig. 3(b)] reveals that the 7th and 14th iterations yield the optimal solution, characterized by their lowest energy values within the series. The inset [Fig. 3(b)] depicts the energy evolution process throughout the 7th calculation, with the curve exhibiting a downward trend that reaches a minimum of approximately 0.363T (T is the calculation time of one simulation solution) and then stabilizes. The set of spin qubits that correspond to a stable energy is the solutions derived from the calculation, which are tailored for the PICs. An enlarged view of the first 0.02T of the calculation trend [Fig. 3(c)] shows that the Ising energy rapidly decays, followed by gradual stabilization, reaching the minimum Ising energy as the calculation iterations accumulate. The inset [Fig. 3(c)] shows intelligent computation results corresponding to the minimum Ising energy. According to Ref. [32] on a microwave photonics Ising machine in our lab, the computational time of the 100 bit scale is μ, where μ is the roundtrip time, which corresponds to the calculation time T in the numerical simulation, and is the number of roundtrips.
Powered by Ising-model intelligent computation, Fig. 3(c) illustrates the detailed planning process for complex four-input, four-output analog signal processing within PICs. Strategically, while ports 2, 3, 7, and 9 are designed as input channels, the planned signals are output from 3, 22, 18, and 20 after processing. To address the demand of multipath simultaneous analog signal processing, the Ising model involves an expanded set of parameters and refined boundary conditions to ensure that a comprehensive solution aligns with the Ising energy toward the global optimum. For stochastic path analog signal processing [Fig. 3(d)] using the Ising model, the spin qubit is solved within the Ising-based computational model. It is then inversely mapped to the voltage of each MZI, allowing dynamic adjustments to achieve the desired reconfigurations for analog signal processing [Fig. 3(e)]. These results confirm that programmable PICs can be successfully reconfigured using Ising-model intelligent computation. Therefore, by redefining the input and output ports, the entire chip can be rapidly reconfigured to adapt to different scenarios.
C. Programmable PIC-Based Analog Signal Processing across Various Scenarios
To validate the effectiveness of the Ising-model intelligent computation in facilitating the performance of programmable PICs for analog signal processing across various scenarios, we design two distinct photonic system applications for the PICs: a wavelength routing system for the balanced transmission of four-channel NRZ/PAM-4 signals and an optical neural network system for the handwritten digit recognition.
D. Programmable PICs for Wavelength Routing
Figure 4 shows the programmable PICs for wavelength routing. The schematic diagram in Fig. 4(a) illustrates the complete workflow of the experimental system. It includes the hardware setup for wavelength routing, the Ising-based algorithm for computing the status of PIC nodes, and the control of the PIC hardware state. In accordance with the target requirements, the Ising-based algorithm for computing the status of PIC nodes can be executed within our in-house MPIM. The binary variables, denoted as “0” and “1,” are adeptly mapped to the driving voltages of the MZIs. The manipulation of the programmable PICs is adeptly executed by leveraging the upper computer to govern the voltage source, thereby ensuring precise and effective control over the PIC’s operational states.
Figure 4.The programmable PICs in analog signal processing for wavelength routing. (a) Schematic diagram. LD, laser diode; PC, polarization controller; DRV, driver; AM, amplitude modulator; EDFA, erbium-doped optical fiber amplifier; PD, photodetector. (b) The results of Ising-model-based intelligent computation for wavelength routing path planning. (c) Output spectrum with wavelengths at 1535 nm for Path 1, 1540 nm for Path 2, 1545 nm for Path 3, and 1550 nm for Path 4. (d1)–(d4) Eye diagram of 10 Gbaud NRZ signals. (d5)–(d8) Eye diagram of 10 Gbaud NRZ signals after passing through the programmable PICs. (e1)–(e4) Eye diagram of 10 Gbaud PAM-4 signals. (e5)–(e8) Eye diagram of 10 Gbaud PAM-4 signals after passing through the programmable PICs.
In the hardware setup part, a laser diode sequentially emitted continuous waves at wavelengths of 1535 nm, 1540 nm, 1545 nm, and 1550 nm, which were subsequently fed into the Mach–Zehnder modulator (EOSPACE, AX-0mvs-65-PFA-SHN823). A signal generator (Keysight, M9502A) produced 10 Gbaud non-return-to-zero (NRZ) signals and 10 Gbaud pulse amplitude modulation (PAM-4) electrical signals. These signals were amplified by a radio frequency driver (SHF, S804B) and then applied to the Mach–Zehnder modulator. This integration merged the electrical signal with the optical signal. The optical signal is further amplified by an erbium-doped optical fiber amplifier (EDFA, Keopsys, CEFA-C-HG-PM-50) compensating for any optical loss incurred by previous components and guided into an input port of the programmable PICs. Prior to entering the chip, a PC ensures that the signal is set to transverse-electric (TE) polarization, and then it is coupled into programmable PICs to perform the wavelength routing. Here, we employed a computational platform equipped with an Intel Core i7-9750H CPU (6 cores, 12 threads at 2.6–4.5 GHz) and 16 GB DDR4 RAM, and implemented the Ising intelligent computing process using Python. To control the hardware state of the PICs, an in-house developed multi-channel voltage source adjusts the states of the corresponding MZIs, effectively reconfiguring the programmable PICs according to the path determined by the algorithm. Subsequently, the optical signal emerging from the PICs is converted into the electrical signal at the photodetector (Finisar, XPDV2120RA-VF-FA) and captured by an oscilloscope (Keysight N1092C), allowing for a thorough analysis of the signal quality.
Figure 4(b) illustrates the Ising-model-based intelligent computation process and the result for wavelength routing. In this process, the input ports are set to 1, 3, 5, and 7, corresponding to the output ports 12, 16, 15, and 22, and the computed weight values are , , , , and . The solving process is iteratively computed 25 times using numerical simulation, with the lowest energy solution obtained in the 16th iteration. According to the results, four paths are identified on the basis of the intelligent computation corresponding to the minimum Ising energy. Each path is meticulously planned to ensure a uniform length of 10. This planning avoids any disparity in path lengths, thereby eliminating potential variations in optical signal loss across the four paths.
To establish a baseline for comparison, the eye diagram of the system was first tested in a back-to-back (BtB) transmission setup prior to conducting the actual signal transmission experiment. Figures 4(c)–4(e) show the experimental results of wavelength routing in programmable PICs, harnessing the Ising model-based intelligent computation for multipath planning. Figure 4(c) shows the optical spectrum at the wavelengths of 1535 nm, 1540 nm, 1545 nm, and 1550 nm in the solved four paths. Since the equalization of the length between paths is used as one of the constraints in solving the Ising model, the difference in the insertion loss among the four paths is less than 0.7 dB. The results indicate that the four paths exhibit similar transmission loss across different optical carriers. This further confirms that Ising-model-based intelligent computation is effective for wavelength routing and that the consistency of the unit devices in the programmable PICs meets the requirements for subsequent applications. Figures 4(d) and 4(e) display the performance outcomes for 10 Gbaud NRZ and PAM-4 signals under the BtB and post-transmission through the programmable PICs with the extinction ratio degradation less than 1 dB. It can be concluded that the programmable PICs, powered by Ising-model intelligent computation, are capable of supporting transparent transmission. They also enable non-blocking wavelength routing in wavelength division multiplexing networks, while maintaining multipath power balance.
1. Programmable PICs for Optical Neural Network
Figure 5 shows the programmable PICs-enabled optical neural network (ONN). We utilized matrix multiplication operations constructed with programmable PICs to implement the convolutional neural network. To validate the performance of our programmable-PICs-based ONN, we conducted benchmark testing using the MNIST handwritten digit dataset—a conventional choice in optical computing research—as demonstrated in Fig. 5(a) [62–65]. The only distinguishing thing from Fig. 4(a) is that the field programmable gate array (FPGA, Xilinx, XC7Z100-2FFG900I) is both the signal generation module and the signal reception module. Here, passive programmable PICs, along with external modulators and detectors, are used as the photonic convolutional layer to construct the linear matrix. The modulators and detectors primarily handle the electro-optical and opto-electrical conversion tasks. The non-linear activation function and one fully connected layer are realized on the FPGA. The network model in the FPGA uses the cross-entropy loss function and the stochastic gradient descent optimizer, with the ReLU activation function. The high-speed digital-to-analog converter (DAC, AD5767) module in the FPGA transmits the data required for convolutional computations to the reconfigured programmable PICs, while the high-speed analog-to-digital converter (ADC, 12D1000) collects and digitizes the results from the programmable PICs for further computation. A multi-voltage source control loads voltage signals to configure arbitrary linear matrices on the programmable PICs.
Figure 5.ONN recognition with the programmable PIC-based linear matrix as the convolutional kernel. (a) Schematic diagram of an ONN incorporating programmable PICs for the convolution layer and FPGA for the fully connected layer. (b) The path-planning results of Ising-model-based intelligent computation using the linear matrix X. (c) The comparison between the experimental results and the theoretical values. (d) ONN recognition results for handwritten digits with the programmable PICs as the convolutional kernel.
For a desired linear matrix X, the light transmission path is pre-established by Ising-model intelligent computing, given the specified input ports and output ports, as illustrated in Fig. 5(b). In this solving process, the input ports are set to 7, 8, 9, and 10, corresponding to the output ports 19, 18, 17, and 20, and the computed weight values are , , , , and . According to the results of the 16th iteration, each path is meticulously planned to ensure a uniform length of 10. Subsequently, as depicted in Fig. 5(c), the light wave propagation follows the designated path with the error between the experimental and theoretical results of less than 1%, as does the variation in optical power for each output port of less than 0.9%. These small power errors are caused solely by the process errors in the optical path itself. The programmable PICs enabled by Ising-based intelligent computation not only enable accurate reconfiguration of linear matrices but also avoid differences between paths caused by imbalances in path planning. Through intelligent computation based on the Ising model, different linear matrices can be reconfigured and implemented. Moreover, more complex linear matrices can be achieved by applying multibit quantization to each node in the Ising model.
The linear matrix X reconfigured by the PICs is then integrated into an ONN for MNIST handwritten digit recognition. As shown in Fig. 5(d), the recognition results for the handwritten digits 1, 5, and 9 are consistent with the original images, with an average recognition accuracy of up to 96.2%. The power consumption of the ONN is approximately 400 mW, with an energy efficiency ranging from 40 pJ/MAC to 80 pJ/MAC, while a single MZI consumes around . This represents a substantial improvement compared to the Google TPU’s efficiency of 430 fJ/MAC [66]. It can be concluded that the programmable PICs, powered by Ising-model intelligent computation, can achieve the linear matrices required for ONN computation and perform image recognition.
3. DISCUSSION
A. Performance Comparison of the Solution Algorithms
Table 1 presents the performance comparison of the representative state-of-the-art solution algorithms for programmable PICs. The GA [34], tailored for scale, is utilized to address the 40 MZI-based PICs, with a total computation time of up to 7.44 s. The Dijkstra algorithm (4-core CPU at 2.80 GHz) requires a total computation time of 78.85 s for a scale involving 42 MZIs [36]. The light-processing functions (16-core Intel Xeon E7-4850 CPU at 2.1 GHz) achieve a single calculation time of 65.40 s for a scale involving 55 MZIs [38]. In our study, the Ising model is implemented on a CPU (Intel Core i7-9750H, 6 cores, 12 threads at 2.6–4.5 GHz) for a scale involving 46 MZIs, achieving a total solution time of 71.50 s [67]. When solving the same problem using an MPIM [47], the total solution time was reduced to 0.70 ms, representing a 5-order-of-magnitude improvement over the CPU-based approach.
Performance Comparison of Our Solution Algorithm
Algorithm
MZIs
Solution Scale
Processor
Single Calculation Time
Total Calculation Time
GA [34]
40
/
0.07 s
7.44 s
Dijkstra [36]a
42
4-core CPU at 2.80 GHz
0.78 s
78.85 s
Light-processing functions [38]b
55
16-core Intel Xeon E7-4850 CPU at 2.1 GHz
65.40 s
/
Ising (this work)
46
6-core Intel Core i7-9750H CPU (12 threads at 2.6–4.5 GHz)
1.43 s
71.50 s
MPIM [47]
Expected from this work
NVIDIA A100 GPU
c
MPIM [47]
The state-of-the-art multipath optimized Dijkstra algorithm for programmable PICs [36].
By employing a recursive method under dispersionless approximation, the light-processing function of programmable PICs is derived and subsequently optimized [38].
This time is estimated based on the solution method proposed in the referenced literature [68,69].
When scaling up to programmable PICs involving 2000 MZIs, the large data volume necessitates distributed parallel computing. When using an NVIDIA A100 GPU, the estimated computation time exceeds 1000 h () [68,69], severely limiting the programmability of PICs. In contrast, when employing optimal MPIM-based hardware [47], the estimated total computation time is reduced to 30 ms. This represents a nearly 8-order-of-magnitude improvement over traditional GPU-based hardware solutions, demonstrating the superior efficiency of the MPIM approach. However, the arbitrary and complex reconfiguration process of programmable PICs requires the use of multi-bit Ising spins for characterization and solutions. The detailed mapping theory and methodology are provided in Appendix E, which imposes more stringent requirements on the optical buffer capacity of the MPIM.
B. Programmable PIC Scale Expansion
According to the hexagonal arrangement shown in Fig. 6, based on the current silicon-based process platform and the performance of commonly used devices, a reasonable estimation of the chip scale can be made. On the silicon-based platform, the maximum input optical power for silicon waveguides is 25 dBm, and the power received by the detector is not less than . With coupling losses at 1.5 dB/face, the chip can tolerate a loss of 30 dB. Currently, the loss of a single MZI is approximately 0.4 dB, so a single path can contain up to 75 MZIs. Based on the hexagonal arrangement shown in Fig. 6, the number of MZIs along the diagonal is set to the maximum that a single path can accommodate. In this case, a single path can contain up to 75 MZIs, and the largest achievable hexagonal network scale is estimated to be , comprising approximately 2290 MZIs. Simultaneously, executing 56-path planning requires a minimum of 100,000 spins. Currently, the refined MZI can achieve an insertion loss of 0.2 dB per unit, and an extended programmable PIC scale is expanded to approximately , containing 8921 MZIs. The optimization of 113 paths would necessitate no less than 1 million spin qubits.
Figure 6.Schematic diagram of the hexagonal arrangement of the programmable PIC scale expansion.
Our proposal of Ising-model intelligent computation has successfully sped up the reconfiguration of programmable PICs for diverse analog signal processing applications. By mapping MZI units into the spin qubits with binary decision variables, the light wave propagation challenge in programmable PICs is theoretically transformed into path-planning problems, which can be then solved using the Ising model. This approach has significantly streamlined the reconfiguration process for programmable PICs. Our results have successfully demonstrated a wavelength routing system for balanced transmission of four-channel NRZ/PAM-4 signals and an optical neural network with 96.2% recognition accuracy, both powered by Ising-based computations. Furthermore, the programmable PIC achieves a potential reconfiguration speed of 30 ms for a large-scale network of ports, housing 2000 MZI units. Our approach sets a new benchmark for programmable PICs. The Ising-based algorithm has markedly improved the programmable PIC reconfigurability, processing speed, and intelligence, paving the way for photonic platforms to play a pivotal role in optical communication, computation, LiDAR, sensing, and cryptography.
APPENDIX A: ISING EQUIVALENT MODEL MATRIX
In the process of converting the physical characteristics of the programmable PICs into an Ising model, the ports of the programmable PICs must be numbered first. Based on the scale of our programmable PICs, the port numbering is shown in Fig. 4(a). Here, we define ports 1 to 11 as input ports and ports 12 to 22 as output ports; the constraint matrix for this unit structure can be written as follows: where represents the number of the MZI units, and is the number of the decomposition layers. For a PIC network, we name the four paths as , , , and . is the constraint function used to guarantee that at most one element in the set is equal to 1, and represents the number of MZI unit roads. The designed programmable PICs include 101 roads; the constraint function for the same road between different layers can be written as follows:
Then we specify the input and output ports. Therefore, we have , , , and , , , , and the entering matrix can be written as
Therefore, the balanced matrix can be written as
The shortest path optimization matrix can be written as
APPENDIX B: ISING ENERGY SOLUTION PROCESS
The key to solving the Ising model using dynamical equations lies in mapping the optimization problem to the evolution process of a nonlinear dynamical system [70], with the goal of minimizing the Ising Hamiltonian [Eq. (11)]. By constructing the dynamical equations, the spin variables are mapped to continuous variables (such as amplitude), ensuring that the system’s energy function aligns with the Ising Hamiltonian. The core mechanism involves creating a bistable potential field to achieve phase locking and guiding the system toward the ground state of the Ising model through coupling control. The dynamical equations are given in Ref. [67], where represents the effect of pump gain and loss. represents the amplitude of the dynamical system corresponding to the Ising spin. When the coupling term in the dynamical equation is removed (), the potential function exhibits a double-well structure for . In this case, the stable equilibrium points of the system are , while the saddle point () divides the phase space into two symmetric regions, as shown in Fig. 7. As a result, the amplitude representing the Ising spins is symmetrically distributed in the phase space, forming a double-well potential field. This causes the system to eventually converge to two stable points (corresponding to the states of the Ising spins). Any initial perturbation drives the variable to rapidly converge to , and the corresponding discrete spin state achieves phase locking of the Ising spins (with a phase difference of 0 or ), stabilizing their values at and .
Figure 7.Schematic diagram of the system amplitude representing Ising spins.
The coupling term is a variant of the constraint matrix in the Ising model, where represents the set of weight relationships between these Ising spins. In the absence of the coupling term, the initial distribution of Ising spins is uniform and symmetric. The introduction of the coupling term aims to break this uniformity and symmetry, thereby transforming the path-planning problem into an Ising spin distribution. By guiding this distribution toward the lowest energy state, the system’s energy function becomes the Hamiltonian of the Ising model. Finding the minimum point of the system’s energy corresponds to minimizing the Hamiltonian, which yields the optimal solution to the problem. Specifically, the energy landscape is modulated through the interactions of Ising spins, driving the system to evolve toward a spin configuration with global minimum energy. This process is equivalent to moving along the monotonic descent direction of the gradient of the system’s energy function [71], thereby achieving energy optimization,
Due to non-idealities in physical implementation, the amplitudes of different Ising spins often exhibit significant disparities (i.e., amplitude heterogeneity). This phenomenon arises from the sensitivity of the system’s nonlinearity to initial noise. Initial amplitude perturbations are exponentially amplified by the cubic term , causing nodes with larger amplitudes to saturate to steady states more rapidly. The accumulation of such heterogeneity leads to asynchronous phase transitions, trapping the system in local energy minima and preventing convergence to the global ground state. To suppress amplitude non-uniformity, we propose replacing the linear term in the coupling term with the hyperbolic tangent function . Its saturation property compresses the input amplitudes to the range [, 1], and the modified dynamical equation is
Through the nonlinear filtering effect of , the system mitigates the impact of amplitude disparities on coupling strength while preserving the sign information to maintain spin state regulation. This approach effectively balances the differences in amplitude convergence rates, ensuring phase-synchronized evolution and enabling the system to more stably approach the target ground state.
APPENDIX C: MZI VOLTAGE MAPPING FROM THE BINARY VARIABLES qn,i
We determine the values of node states , , , and their corresponding MZI states through the resulting variable state lookup table (Table 2).
This mapping clearly describes how the values of variables correspond to the path states of the physical system, ultimately determining the functional state of the MZI. This lookup table is a crucial step in mapping the optimization results to physical implementation, used to reconfigure the behavior of MZI in the programmable PICs.
APPENDIX D: SENSITIVITY ANALYSIS OF PROGRAMMABLE PICS TO TEMPERATURE VARIATIONS AND FABRICATION ERRORS
Temperature variations and fabrication errors are the primary sources of state discrepancies in programmable PICs. Since the refractive index of silicon is affected by temperature changes, additional phase shifts are induced in silicon waveguides [72]. Therefore, when designing the structural parameters of the MZI and the overall packaging of the programmable PICs, the impact of temperature variations on the phase-shifting arms of the MZI must be considered. Additionally, fabrication errors can lead to state differences between MZIs, primarily due to the sensitivity of the DC splitting ratio to fabrication precision. The DC splitting ratio is mainly influenced by the spacing and waveguide width. Thus, the influence of fabrication accuracy on DC performance must be evaluated during the design phase.
Figure 8(a) shows the schematic diagram of the multi-physics simulation model setup. When the waveguide spacing of the MZI phase-shifting arm is set to 10 μm and the thermal tuning electrode size is μμ, a voltage of 1–5 V is applied to the thermal tuning electrode. By analyzing the thermal field distribution generated by the thermal tuning, the temperature distributions of Waveguide 1 and Waveguide 2 in the MZI phase-shifting arm are obtained, as shown in Fig. 8(b). As the voltage increases, the temperature of the thermal tuning electrode gradually rises from 302 to 335 K. Due to thermal diffusion, the temperature around the thermal tuning electrode also increases, resulting in a temperature rise of approximately 10 K around Waveguide 2.
Figure 8.Sensitivity analysis of programmable PICs to temperature variations and fabrication errors. (a) Schematic diagram of the multi-physics simulation model setup (not to scale). (b) Temperature distribution around the waveguide as a function of applied voltage. (c) Phase shift of the waveguide as a function of thermal tuning voltage. (d) Splitting ratio variation caused by ±100 nm changes in the waveguide width of the DC. (e) Splitting ratio variation caused by ±100 nm changes in the gap of the DC coupling region.
In Fig. 8(c), the solid and dashed lines represent the phase shifts of Waveguide 1 and Waveguide 2, respectively, as a function of the applied voltage. The curves indicate that when the voltage applied to the thermal tuning electrode is around 4.5 V, Waveguide 1 can achieve a phase shift of , which aligns with the actual test results. For Waveguide 2, when the voltage is around 4.5 V, a temperature increase of 10 K results in a phase shift of . This demonstrates that a 10 K rise in ambient temperature can induce a phase shift in silicon waveguides. It also shows that when the waveguide spacing is greater than or equal to 10 μm, the induced phase shift is less than . To mitigate the impact of temperature variations on the performance of programmable PICs, a TEC module is integrated into the packaging process, with the temperature control precision designed to be 1/1000. The stable operation time is determined by the TEC operating time, ensuring the robustness of the chip’s test results.
Figures 8(d) and 8(e) illustrate the variation in the splitting ratio of the DC at 1550 nm within a fabrication precision range of . When the waveguide width of the DC varies by , the DC splitting ratio changes by 18 dB, whereas a variation in the coupling gap results in a splitting ratio change of up to 16 dB. Here, we select a fabrication platform with an error margin of . Since fabrication errors are unavoidable, for large-scale programmable PICs, it is necessary to achieve a fabrication precision of to ensure that the cumulative errors from dozens or even hundreds of MZIs do not significantly impact the reconfiguration optimization results and test outcomes.
APPENDIX E: MULTI-QUBIT QUANTIZATION
To further enhance the applicability of this method in addressing complex problems, more binary decision variables can be introduced within a single MZI to represent different optical transmission scenarios, as illustrated in Fig. 9.
Figure 9.The correspondence between the unit structure MZI and the Ising equivalent model after multi-qubit quantization.
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