PaperlayerView exact revision on arXiv

Source-linked AI summary

Machine learning method for state preparation and gate synthesis on photonic quantum computers

Juan Miguel Arrazola, Thomas R. Bromley, Josh Izaac, Casey R. Myers, Kamil Brádler, Nathan Killoran

arXiv:1807.10781v1quant-ph

TL;DR

The paper addresses how to automatically construct photonic quantum circuits for preparing target states and synthesizing target transformations. It optimizes a parameterized continuous-variable quantum neural network using machine-learning methods, obtaining high-fidelity results with short-depth circuits across diverse states and gates. The approach is intended to automate circuit discovery for quantum-computing subroutines.

  • Problem

    Automating state preparation and gate synthesis could reduce the large gate overhead and computational expense of conventional decomposition methods.

  • Method

    A parameterized continuous-variable quantum neural network is optimized by gradient-based methods to match desired input-output transformations.

  • Results

    The method synthesizes high-fidelity states and gates with short-depth circuits across the studied examples, including a single photon at 99.998% fidelity.

  • Takeaways & Limitations

    Circuit discovery for several photonic quantum-computing subroutines can be largely automated by specifying a target state or transformation and optimizing the circuit.

Abstract

from arXiv · show

We show how techniques from machine learning and optimization can be used to find circuits of photonic quantum computers that perform a desired transformation between input and output states. In the simplest case of a single input state, our method discovers circuits for preparing a desired quantum state. In the more general case of several input and output relations, our method obtains circuits that reproduce the action of a target unitary transformation. We use a continuous-variable quantum neural network as the circuit architecture. The network is composed of several layers of optical gates with variable parameters that are optimized by applying automatic differentiation using the TensorFlow backend of the Strawberry Fields photonic quantum computer simulator. We demonstrate the power and versatility of our methods by learning how to use short-depth circuits to synthesize single photons, Gottesman-Kitaev-Preskill states, NOON states, cubic phase gates, random unitaries, cross-Kerr interactions, as well as several other states and gates. We routinely obtain high fidelities above 99\% using short-depth circuits, typically consisting of a few hundred gates. The circuits are obtained automatically by simply specifying the target state or gate and running the optimization algorithm.

I. INTRODUCTION

The paper uses machine learning and optimization to automate short-depth photonic quantum circuits for state preparation and gate synthesis. A continuous-variable quantum neural network is optimized through parameterized optical gates and gradient-based methods.

  • I. INTRODUCTION: Machine-learning optimization finds circuits that prepare target states or reproduce target unitary transformations from specified input-output relations.The framework treats state preparation as the single-input case and gate synthesis as its generalization to multiple relations.
  • I. INTRODUCTION: The approach largely automates circuit discovery and achieves high-fidelity states and gates with short-depth circuits across the studied examples.The paper positions short depth as useful for near-term devices and reports broad applicability across state-preparation and gate-synthesis tasks.
  • II. AUTOMATED CIRCUIT DESIGN: The method uses a variational continuous-variable neural network whose layers combine interferometers, squeezing, displacement, and non-Gaussian gates.The architecture is designed to provide substantial representational power while retaining a fixed circuit structure with trainable parameters.
  • II. AUTOMATED CIRCUIT DESIGN: Gradient descent optimizes gate parameters by minimizing a task-specific cost function, with the best observed parameters defining the proposed circuit.Strawberry Fields and automatic differentiation support the numerical optimization of these parameterized circuits.
  • II. AUTOMATED CIRCUIT DESIGN: The simulations use Kerr non-Gaussian gates and truncated Hilbert spaces, requiring sufficiently large cutoffs and attention to active-gate parameter magnitudes.Displacement, squeezing, and Kerr parameters may need upper bounds because of simulation or hardware constraints.

III. STATE PREPARATION

State preparation is formulated as learning a parameterized circuit that maps a fixed reference input to a desired target state. The method fixes a neural-network architecture and optimizes its parameters to obtain short-depth, high-fidelity preparations.

  • III. STATE PREPARATION: The state-preparation task seeks a unitary circuit that maps a canonical input state, fixed here as vacuum, to a target quantum state.Only the unitary's action on the chosen input is constrained; other components remain free.
  • III. STATE PREPARATION: Direct decomposition methods can be computationally expensive and produce circuits with very large gate counts, especially for continuous-variable systems.The paper motivates optimization as an alternative to choosing free unitary parameters and decomposing the result into universal gates.
  • III. STATE PREPARATION: The proposed method selects the network depth and minimizes a cost function measuring how closely the parameterized circuit output matches the target state.The desired condition is U(θ)|0⟩ = |Ψ_t⟩, corresponding to unit fidelity with the target.
  • III. STATE PREPARATION: Experiments on canonical single- and two-mode states show that circuit search can be largely automated while retaining short depth and very high fidelity.The examples are used to demonstrate the approach's effectiveness and versatility for state preparation.

A. Examples

The variational quantum neural network prepares a single photon with very high fidelity, while optimization outcomes vary across runs and network depth trades off fidelity against computational cost.

  • 99.998% fidelity to a perfect single photon is achieved after 5000 gradient-descent steps.The circuit uses an 8-layer single-mode network containing 40 gates.
  • Random initialization and stochastic gradient descent produce different optimization performance across independent runs.The authors recommend multiple independent optimization sessions and selecting the best output.
  • Network depth trades off state-preparation performance against parameter count, simulation resources, and optimization time.The single-photon state can be well approximated using only a few layers.

2. ON state

The method prepares nonclassical resource states with short variational circuits, achieving high fidelity for both an N = 9 ON state and a finite-energy Hex GKP state.

  • 2. ON state: ON states are vacuum–Fock superpositions that can be used through gate teleportation to apply exp(iτ x^N) to first order in τ.The paper selects N = 9 and sets the superposition coefficient a = 1.
  • 2. ON state: 99.93% fidelity is obtained for an N = 9 ON state using a 20-layer, 100-gate network.The target is an equal superposition of vacuum and a nine-photon Fock state, optimized for 5000 steps.
  • 3. GKP state: Hex GKP states are continuous-variable resource states studied here because they are better suited than original GKP states for correcting loss errors.The finite-energy states are modulated by a Gaussian envelope.
  • 3. GKP state: 99.60% fidelity is obtained for a Hex GKP state with µ = 1 and ∆ = 0.3.The more complex target uses 25 layers, 125 gates, a cutoff dimension of 51, and 10,000 optimization steps.

4. Random state

The method prepares unstructured random and two-mode NOON states with high fidelity, extending demonstrations beyond symmetric single-mode targets.

  • 4. Random state: 99.82% fidelity was achieved for a random d = 15 state using a 25-layer, 125-gate network.The simulation used cutoff dimension 20 and 5000 optimization steps.
  • 5. NOON state: The two-mode example is computationally harder because its simulated Hilbert-space dimension is quadratically larger than for a single mode.
  • 5. NOON state: 99.89% fidelity was achieved for a two-mode NOON state with N = 5 using a 20-layer, 100-gate network.The simulation used cutoff dimension 10.

IV. GATE SYNTHESIS

Gate synthesis generalizes state preparation by optimizing a circuit to reproduce a target unitary over multiple basis-state input-output relations in a restricted Fock subspace.

  • IV. GATE SYNTHESIS: Traditional decomposition methods can incur computational overhead, many elementary gates, and errors from Trotterization and commutator approximations.
  • IV. GATE SYNTHESIS: Gate synthesis approximates a target unitary by reproducing its action on the first d Fock-basis states.For infinite-dimensional unitaries, this restricts the learned transformation to a photon-number subspace matching practical energy constraints.
  • IV. GATE SYNTHESIS: The method provides deterministic, fixed-depth realizations while giving users control over the number of elementary gates.Performance is measured by average fidelity over states supported in the input d-dimensional Fock subspace.
  • IV. GATE SYNTHESIS: The gate-synthesis examples use largely automated procedures to construct relatively short-depth approximations of target unitaries.

A. Examples

The examples include cubic-phase-gate synthesis, whose optimized circuit closely matches the target transformation while exhibiting run-to-run optimization variability across subspace dimensions.

  • A. Examples: 99.86% average fidelity was obtained for the cubic phase gate with γ = 0.01 on a 10-dimensional subspace.The circuit used 25 layers, corresponding to 125 gates, and was optimized for 4000 steps.
  • A. Examples: The synthesized and ideal cubic-phase transformations are compared through their real and imaginary matrix components in the Fock basis.
  • A. Examples: Optimization outcomes vary substantially across independent runs, with many runs becoming trapped in local minima far from optimal.For d = 6, fewer runs become trapped in poor local minima than for the larger subspace.
  • A. Examples: Increasing the number of input-output relations makes the optimization landscape more complex and adds computational overhead.

2. Quantum Fourier transform

The study synthesizes increasingly complex single- and two-mode gates with variational photonic circuits, achieving high fidelities while using short-depth architectures.

  • Quantum Fourier transform: 98.89% average gate fidelity was achieved for the d = 8 quantum Fourier transform using 40 layers and 200 gates.The synthesis used 8000 optimization steps and tested the learned transformation on an equal superposition state.
  • Random unitary: 99.5% average fidelity was achieved for a Haar-random unitary on the five-dimensional, at-most-four-photon subspace using 25 layers and 125 gates.The target and learned unitaries were visualized alongside their action on a d = 5 superposition state.
  • Cross-Kerr interaction: 99.994% average fidelity was achieved for a two-mode cross-Kerr interaction with κ = 0.1 using 25 layers and 125 gates.The synthesized circuit used negligible displacement and squeezing, consistent with Kerr gates and beamsplitters being sufficient for this interaction.
  • Cross-Kerr interaction: The cross-Kerr circuit requires 125 elementary operations, compared with approximately 1000 for a decomposition precision of ∼0.1.The comparison is made for κ = 0.1 and highlights the short-depth decomposition obtained by the approach.

V. CONCLUSION

The paper presents machine-learning and optimization methods for automatically discovering circuits that reproduce desired quantum transformations. It reports high-fidelity states and gates with short-depth circuits suited to near-term devices, while identifying simulation scalability and software support as limitations.

  • V. CONCLUSION: The method largely automates discovery of circuits that perform specified subroutines of quantum algorithms.It leverages machine learning and optimization to reproduce desired transformations between input and output states.
  • V. CONCLUSION: High-fidelity states and gates are synthesized with short-depth circuits, making the techniques particularly suited to near-term quantum devices.The conclusion identifies short depth as a key feature of the approach.
  • V. CONCLUSION: Automatic differentiation can transfer to other quantum-computing settings, but Strawberry Fields is identified as the only library natively supporting it.Further work is required to bring native automatic differentiation to other simulation libraries.
  • V. CONCLUSION: The approach requires classical circuit simulation, which becomes intractable as the number of modes increases.Extending the method to complex transformations across several modes may therefore require specialized quantum optimization techniques.

Appendix A: Simulation cutoff size

The simulations represent states in a truncated Fock basis, so the cutoff dimension must be chosen large enough to preserve reliable normalization and tolerate intermediate excursions outside the target subspace.

  • Appendix A: Simulation cutoff size: The simulations replace each state |Ψ⟩ with its projection ΠD |Ψ⟩ onto a fixed D-dimensional truncated Hilbert space.A suitably large cutoff dimension is required for reliable simulations.
  • Appendix A: Simulation cutoff size: For restricted state or gate support, D is set approximately five steps above the restricted dimension to allow intermediate circuit mappings outside that space.For gate synthesis, D also keeps target states close to normalized within the truncated space.
  • Appendix A: Simulation cutoff size: The cutoff is chosen as the smallest D satisfying the stated tolerance condition with ϵ = 0.0001.Increasing D significantly increases computational overhead.

Appendix B: Fidelity for gate synthesis

Gate-synthesis fidelity is evaluated through process fidelity, which uses entangled-state overlaps to estimate average fidelity without sampling the entire state space. A cubic-phase subtlety requires accounting for leakage beyond the restricted subspace.

  • Appendix B: Fidelity for gate synthesis: The average fidelity of unitary operations is defined using an integral over the Haar measure on the state space.The appendix introduces this as the reference fidelity measure before describing process fidelity.
  • Appendix B: Fidelity for gate synthesis: Process fidelity is obtained by applying the target and learned unitaries to half of a maximally entangled state and taking the resulting state overlap.This provides the process-fidelity quantity used in the gate-synthesis evaluation.
  • Appendix B: Fidelity for gate synthesis: Process fidelity avoids sampling the entire state space because it is bounded by average fidelity over two complementary input-state sets.It is related directly to average fidelity through the stated relationship.
  • Appendix B: Fidelity for gate synthesis: For the cubic phase gate, the learned operation can map states from the d-dimensional subspace into the larger D-dimensional cutoff space.The average gate fidelity is therefore numerically approximated using a quantity that accounts for this behavior.
Loading 1807.10781v1…