Entropy Computing, A Paradigm for Optimization in Open Photonic Systems

TL;DR

The paper introduces entropy computing as a hardware paradigm for NP-hard optimization in open photonic systems and demonstrates a hybrid photonic-electronic implementation that encodes variables in time-bin qudits and uses measurement-feedback to realize a loss-based imaginary-time search toward the ground state of a target Hamiltonian. The Dirac-3 machine implements a polynomial objective up to fifth order with non-negativity and a fixed sum constraint, enabling continuous and discrete problems with up to 949 variables; results show advantages over gradient descent on non-convex and over SDP relaxations on Potts/max-cut problems, with solution quality tunable by mean photon number and shot noise. The approach combines TCSPC, EOM, PPLN SFG, SPD, FPGA, and VOAs to embed the optimization into per-time-bin losses, enabling flexible encoding and scalable higher-order interactions. This work positions entropy computing as a scalable, energy-efficient platform for tackling a broad class of NP-hard problems, with potential extensions toward all-optical and quantum variants and applications in grid optimization, resource allocation, and machine learning tasks. Key mathematical elements include the polynomial objective $E = \sum_i C_i v_i + \sum_{i<j} J_{ij} v_i v_j + \sum_{i,j,k} T_{ijk} v_i v_j v_k + \sum_{i,j,k,l} Q_{ijkl} v_i v_j v_k v_l + \sum_{i,j,k,l,m} P_{ijklm} v_i v_j v_k v_l v_m$ under $v_i \ge 0$ and $\sum_i v_i = R$, enabling a rich landscape beyond binary Ising models.

Abstract

Finding better solutions to combinatorial optimization problems could have a large positive impact on many real-world application areas, such as logistics. For this reason, significant efforts have been made to design novel optimisation paradigms. Here we show an early instance of such paradigm in an optical setting, the entropy computing paradigm. Specifically, we experimentally demonstrate the feasibility of entropy computing by building a hybrid photonic-electronic computer that uses optical measurement and feedback to solve non-convex optimization problems. The system functions by using temporal photonic modes to create qudits in order to encode probability amplitudes in the time-frequency degree of freedom of a photon. This scheme, when coupled with with electronic interconnects, allows us to encode an arbitrary Hamiltonian into the system and solve non-convex continuous variables and combinatorial optimization problems. We show that the proposed entropy computing paradigm can act as a scalable and versatile platform for tackling a large range of NP-hard optimization problems.

Figures (5)

• Figure 1: An emulation system for entropy computing using time-energy modes and a measurement-feedback scheme. (a) The optical signal is generated by a continuous-wave laser followed by a set of variable optical attenuators (VOA). It passes through the electo-optic modulator (EOM), the periodic-poled lithium niobate (PPLN) for nonlinear process, and is detected by a single photon detector (SPD). The time correlated single photon counting (TCSPC) results are fed to a field-programmable gate array (FPGA) board, where the radio-wave input to the EOM is calculated and generated in a digital-to-analog converter (DAC) for each time bin. Here, the nonlinear circuit (in this case sum frequency generation via PPLN), photon detector, and the FPGA function together as a mixer/encoder. (b) The quantum states are encoded into a train of time-bin states of light in the photon-number Hilbert space. The linear loss in the Hamiltonian is mapped into probability amplitudes of a wave function. Each variable of the objective function is assigned into each time bin, creating "qudit" equivalent that is widely used in high-dimensional temporal encoding in quantum random number generation, quantum key distribution, and single-photon sensing Rehain21Nguyen18Mower_2013. (c) The single photon collapsed states are collected one by one and accumulate to create the next feedback to tailor new wave functions in each loop. When the wave function evolutions reach a stable distribution, the number of photons collected in each bin are translated directly as a state vector multiplied with a normalization factor.
• Figure 2: Solving a two-variable non-convex quadratic optimization problem. A two-variable non-convex polynomial optimization problem is considered. (a) A visualization of the energy landscape that involves three local minima and a global minimum at $(\textup{x},\textup{y})=(0,0)$. (b) The iterative evolution of the cost function of the proposed hybrid entropy computing solver over 20 runs. (c) Three exemplary evolutions of the optimization variables involved, including the slack variable ($x_3$), over iterations of the entropy computing solver. (d-k) Eight exemplary trajectories of the optimization variables in the two-dimensional $(x,y)$ plane as the solver evolves toward equilibrium while starting from different initial conditions. In these figures, the solid lines show the trajectory of the entropy-computing solver, while the dashed lines depict the trajectory of a gradient-descent solver. More details on the gradient descent method can be found in supplementary note 3.
• Figure 3: Solving a non-convex continuous optimization problem. A non-convex quadratic optimization problem with 50 variables is considered (QPLIB$\_$0018). (a) Energy distribution over 500 runs of Dirac-3 (blue) and gradient descent algorithm (red). (c) Energy evolution versus the number of iterations on Dirac-3 (blue) and gradient descent (red). The inset shows the evolution of the solution versus iterations. (b) Relationship between the mean photon number ($\mu$), quantum fluctuation coefficient,and key performance metrics - average returned energy. The transitioning from left to right, represents an increase in mean photon number ($\mu$), showing the probability that Dirac-3 is operating in single-photon regime. Quantum fluctuation or shot noise is used as $\frac{1}{\sqrt{N}}$ where N is the number of photon count accumulated in each time-bin. From bottom to top of the vertical axis, this coefficient is gradually decreased. (d), (e) Average returned energy and number of ground states found after 500 runs are collected when mean photon number is decreased further up to $0.002\%$, close to the level of dark count of single photon detector.
• Figure 4: The results for solving the max-k-cut problem. The optimization results for solving combinatorial problems of max-cut, max-3-cut, and max-4-cut on a 30-node unweighted graph that is generated randomly with $p=0.5$ probability of connectivity between each two nodes. (a-c) The visualizations of exemplary graph cuts with different numbers of partitions. (d-f) The distribution of the cut sizes over 100 runs of the max-2-cut (traditional max-cut), max-3-cut, and max-4-cut problems on Dirac-3 using relaxation schedule 1. (g-i) Similar distributions when using the schedule 4. (j-l) The distribution of the cut sizes over 100 runs of the same problems using semi-definite programming (SDP).
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