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Quantum optical neural networks

Gregory R. Steinbrecher, Jonathan P. Olson, Dirk Englund, Jacques Carolan

arXiv:1808.10047v2quant-ph

TL;DR

The paper addresses how neural-network capabilities can be mapped onto near-term quantum optical hardware. It introduces and numerically evaluates QONNs across quantum processing, simulation, compression, and reinforcement-learning tasks, finding generalization from limited training data and improved training reliability with sufficient depth.

  • Problem

    Practical quantum machine-learning protocols for specific near-term hardware remain an open question, motivating architectures tailored to quantum optical systems.

  • Method

    The paper introduces QONNs that map optical mode mixing, nonlinearities, photonic state encoding, and detector readout into a trainable neural-network architecture.

  • Results

    Across quantum gates, Hamiltonian simulation, state compression, and reinforcement learning, QONNs perform the studied tasks and generalize from limited training data to unseen inputs.

  • Takeaways & Limitations

    QONNs are presented as design tools for quantum optical systems and as a platform for near-term optical quantum information processing and machine learning.

Abstract

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Physically motivated quantum algorithms for specific near-term quantum hardware will likely be the next frontier in quantum information science. Here, we show how many of the features of neural networks for machine learning can naturally be mapped into the quantum optical domain by introducing the quantum optical neural network (QONN). Through numerical simulation and analysis we train the QONN to perform a range of quantum information processing tasks, including newly developed protocols for quantum optical state compression, reinforcement learning, and black-box quantum simulation. We consistently demonstrate our system can generalize from only a small set of training data onto states for which it has not been trained. Our results indicate QONNs are a powerful design tool for quantum optical systems and, leveraging advances in integrated quantum photonics, a promising architecture for next generation quantum processors.

I. INTRODUCTION

The QONN maps neural-network components onto quantum optics, using optical mode mixing and nonlinearities to process photonic quantum states. Numerical studies apply it across quantum information protocols and test generalization from limited training data.

  • I. INTRODUCTION: The QONN is introduced as a neural-network architecture tailored to quantum optical systems.It maps mode mixing and optical nonlinearity onto neural-network features.
  • I. INTRODUCTION: QONNs are motivated as potential design tools for near-term optical quantum information processing and future integrated quantum processors.The proposed route leverages advances in photonic quantum computing and CMOS-compatible microelectronics.
  • I. INTRODUCTION: The architecture is applied to quantum gates, Ising and Bose-Hubbard simulation, quantum optical compression, and inverted-pendulum reinforcement learning.The simulations also test generalization from limited input/output state pairs to previously unseen inputs.
  • I. INTRODUCTION: QONN inputs can be dual-rail photonic states encoding qubits, with optical processing followed by photon-number detection.A photon in the top or bottom rail represents |0⟩ or |1⟩, respectively.
  • I. INTRODUCTION: Integrated quantum photonics can implement key neural-network components, including linear transformations across optical modes and optical nonlinearities.Beam-splitter arrays and programmable phase shifts realize mode transformations, while Kerr effects and other mechanisms provide nonlinearities.

II. ARCHITECTURE

The QONN encodes photonic quantum states, alternates programmable linear optical unitaries with single-mode nonlinearities, and trains its parameters against measured outputs. Training can occur directly on hardware or through simulated measurements.

  • II. ARCHITECTURE: QONN inputs are encoded as dual-rail qubits or more general Fock states across optical modes.Dual-rail encoding uses two modes per photon, while Fock states specify photon occupations in modes.
  • II. ARCHITECTURE: Each QONN layer combines an m-mode linear optical unitary with single-mode χ(3) interactions that phase-shift additional photons.The linear unitary is parameterized by phase shifts, and the nonlinear strength is typically fixed at φ = π.
  • II. ARCHITECTURE: Single-photon detectors measure photon numbers at the output, and the resulting measurements are optimized against K desired input/output pairs.The parameter vector is variationally minimized using the measurement results and training data.
  • II. ARCHITECTURE: The in situ training approach directly optimizes the quantum optical processor using measurements from detectors at the circuit output.The approach targets figures of merit that can be estimated with measurement counts scaling polynomially with photon number.
  • II. ARCHITECTURE: Benchmarking evaluates training success using an error threshold of less than 10^-4 across CNOT, Bell-state, and GHZ-state tasks.The benchmark figure reports success percentage as a function of QONN layer count.

III. BENCHMARKING

The QONN is benchmarked on representative quantum state and gate tasks to test whether its architecture can express and reliably discover desired operations. Increasing layer depth improves optimization reliability across the studied tasks.

  • III. BENCHMARKING: Benchmark tasks include Bell-state projection or generation, GHZ-state generation, and CNOT-gate implementation.The training sets represent the full basis sets for the corresponding quantum operations.
  • III. BENCHMARKING: QONNs are trained with layer depths from N = 2 →10 and nonlinear strength φ = π.Shorter networks frequently terminate at non-optimal local minima across all studied tasks.
  • III. BENCHMARKING: Increasing layer depth makes it consistently easier to find local minima close to the global minimum.The paper reports that a single layer may implement a CNOT gate, while deeper networks improve reliable discovery of the correct operation.

IV. HAMILTONIAN SIMULATION

The QONN learns quantum-system dynamics from limited input/output examples and generalizes to unseen states across Ising and Bose-Hubbard Hamiltonians. Deeper networks reduce simulation error, while larger systems remain more difficult to train.

  • A QONN trained on limited input/output state pairs can simulate quantum-system evolution and generalize to previously unseen inputs.The approach is demonstrated for both Ising and Bose-Hubbard Hamiltonians.
  • For n = 2 Ising spins, a three-layer QONN reliably converges across J/B ∈ [−5, 5] and predicts evolution from an untrained |↑↑⟩ initialization state.Training used 20 random two-photon states and testing used 50 different states.
  • 10.1% average test error is reached for the n = 3 Ising case, motivating more advanced methods for efficiently training deeper QONNs.The reported approaches include backpropagation and layer-wise training.
  • 42% mean test error for a single-layer Bose-Hubbard system falls to 0.1% at seven layers for the strongly interacting (2, 4) model.The benchmark uses U/thop = 20 and t = 1 with square-lattice connectivity.
  • Using five layers as a tradeoff between error and computational tractability, the Bose-Hubbard experiments achieve 2.9±1.3% mean test error across U/thop ∈ [−20, 20].The uncertainty is the standard deviation over 22 experiments.
  • The same input/output-mimicking strategy may support learning quantum-system representations when circuit decompositions are unknown or compiling known circuits.

V. QUANTUM OPTICAL AUTOENCODER

The quantum optical autoencoder compresses families of quantum states by disentangling qubits into a latent space and fixed reference states. For molecular-hydrogen ground states, structured training outperforms global-unstructured training in the reported simulations.

  • Quantum autoencoders compress n-qubit state families into a lower-dimensional k-qubit latent space and generalize beyond the training set.They disentangle n−k qubits and set them to a fixed reference state.
  • The molecular-hydrogen test family contains STO-3G ground states mapped to four logical qubits, represented as four photons in eight optical modes.The states vary with H2 bond length and use a dual-rail encoding.
  • The autoencoder objective is to map training states so that selected qubits occupy a fixed reference state, whose fidelity proxies the decoded-state fidelity.
  • Local-structured training sequentially disentangles one qubit at a time before a final global refinement, while global strategies optimize the full architecture.The global-unstructured strategy trains a six-layer system acting on all four qubits.
  • 92.0% fidelity is achieved by both structured optimizations, whereas global-unstructured optimization reaches 57.9% fidelity on the H2 training states.The training set uses bond lengths of 0.5, 1.0, 1.5, and 2.0 angstroms.
  • The iterative approach may become more efficient with stricter convergence criteria, but its asymptotic scaling or accuracy relative to global optimization remains unclear.

VI. QUANTUM REINFORCEMENT LEARNING

The QONN learns a reinforcement-learning policy for balancing an inverted pendulum, encoding four observations onto four qubits and improving fitness across training generations. Its performance was comparable to an equally sized classical network, although the authors caution against direct comparison.

  • Task: The task balances an inverted pendulum on a bounded one-dimensional track by choosing force in the +x or −x direction from four observed state values.A run ends when position, pole angle, or t reaches a boundary, and fitness is the number of time steps before failure.
  • QONN encoding: Four values—x, ẋ, θ, and θ̇—are encoded onto four qubits, and the QONN’s first two output modes select the applied action.Each input value is compressed into γ ∈ [0, π/2] and encoded as cos(γ)|0⟩+sin(γ)|1⟩.
  • Training results: Fitness increases with training generation across five training cycles using a 6-layer QONN, indicating longer pole-balancing runs on new instances.Each generation averages fitness over 80 distinct runs, with batch size 100 for approximate-gradient estimation.
  • Classical comparison: After 1000 generations, mean fitness was 66.4 for the QONN versus 37.1 for the classical network.The classical comparison used 4-neuron, 6-layer networks; the authors state both systems could likely be optimized and caution against direct comparison.

VII. DISCUSSION

The discussion presents QONNs as near-term quantum-optical architectures that map neural-network features onto photonic systems. It positions them as an intermediate route toward larger-scale photonic quantum technologies.

  • Discussion: QONNs map many features of classical neural networks onto near-term quantum optical systems and support a broad range of quantum information-processing tasks.The paper includes quantum optical state compression and black-box quantum simulation among these applications.
  • Technology outlook: Integrated photonics, nanofabrication, and single-photon readout advances provide a feasible route toward large-scale implementations.The architecture is not limited to systems with strong single-photon nonlinearities.
  • Technology outlook: QONNs may learn practical quantum operations with weak or noisy nonlinearities that are unsuitable for fault-tolerant quantum computing.The authors describe this as an intermediate regime for photonic quantum technologies.

Appendix A: Computational Techniques

The simulations use optimized classical code and gradient-free optimization to model QONNs. Computing the multiphoton transformation is the dominant computational step, while gradients are difficult to obtain efficiently.

  • Computational techniques: The multiphoton unitary transform U(θ_i), derived from the single-photon unitary, is the most computationally intensive simulation step.Its calculation involves the permanent of matrices with dimension n × n.
  • Computational techniques: The simulations use custom Python code with performance-sensitive sections translated to Cython and some large operations GPU-accelerated with Numba.The computations ran on a workstation with a 12-core Intel Core i7-5820K, 64GB RAM, and an Nvidia Tesla K40 GPU.
  • Optimization: The training relies on gradient-free optimization because computing and backpropagating gradients likely requires knowledge of the internal quantum state.The authors note that this may be acceptable for small simulated systems but limits scalability to larger systems.

Appendix B: Benchmarking Training

The benchmark suite tests whether QONNs can learn representative optical quantum-information operations from prescribed input-output mappings. It includes Bell-state projection and generation, CNOT implementation, and GHZ-state generation.

  • Benchmark tasks: Bell-state projection uses the full set of four Bell states as dual-rail input states and maps them to detector-distinguishable binary-encoded output states.The reverse mapping can generate Bell states from the corresponding input Fock states.
  • Benchmark tasks: The CNOT benchmark uses a full input-output basis set with binary-encoded output states.The output configurations are specified explicitly for the four basis inputs.
  • Benchmark tasks: The GHZ generator is trained on a single selected input-output configuration.This contrasts with the full basis-set training used for the Bell-state and CNOT benchmarks.

Appendix C: Simulated Hamiltonians

The paper introduces the Ising and Bose-Hubbard Hamiltonians used for quantum simulation, defining their physical interaction parameters and operator meanings.

  • The Ising model is specified by a Hamiltonian describing spin interactions in a magnetic field.
  • B represents each spin’s interaction with a magnetic field in the x direction, while J is the interaction strength between spins in an orthogonal direction.
  • The Bose-Hubbard model is introduced through its Hamiltonian as the second simulated quantum system.
  • The operators b_i† and b_i represent creation and annihilation in mode i, while n_i is the number operator.
  • ω, t_hop, and U denote the on-site potential, hopping amplitude, and on-site interaction strength, respectively.
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