Measurement Based Variational Quantum Circuits

mVQE is a Julia library designed to facilitate the implementation of the algorithms introduced in Learning Feedback Mechanisms for Measurement-Based Variational Quantum State Preparation. It leverages the ITensors.jl framework for tensor network computations and integrates seamlessly with Flux.jl for machine learning components.

Table of Contents

1. Features
2. Examples
3. API Breakdown
   • ITensorsExtension
   • StateFactory
   • Gates
   • Layers
   • GirvinProtocol
   • FluxExtension
   • Circuits
   • Optimization
   • ITensorsMeasurement

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Features

• Efficient one-qubit gate application: A custom runcircuit function enables fast forward and backward passes when applying one-qubit gates to a state (MPS).
• Entropy & mutual information: Functions for computing various measures of entanglement (von Neumann entropy, mutual information, etc.) on both pure and mixed states.
• MPS construction: A suite of functions to initialize commonly used matrix product states.
• Circuit construction: Variational circuit components and layers, both standard and measurement-based (with mid-circuit measurements).
• Hamiltonian expectation: An efficient expect function for computing expectation values w.r.t. a Hamiltonian, optimized for backpropagation.
• Neural network feedback: An optional Flux-based approach for feedback and adaptive parameter updates in the mVQE algorithm.
• Girvin protocol: An implementation of the measurement-based protocol from Smith et al., PRX Quantum 4, 020315.

Examples

vmodels = [
        mVQE.Circuits.VariationalCircuitRy(N, depth),
        mVQE.Circuits.VariationalOneQubitM(
            N_state; gate_type="U", sites=state_indices, nr_params=3
        )
    ]
feedback_model = mVQE.FluxExtensions.TabularModel

model = VariationalMeasurementMCFeedback(vmodels, [feedback_model], ancilla_indices)

ψM = model(ψ)

The resulting model can be optimized in a variety of ways. To see full examples you can check the zenodo repository.

API Breakdown

ITensorsExtension

• File: src/ITensorsExtension/apply.jl
• Key Function: runcircuit(psi, circuit; onequbit_gates=true)
  • Efficiently applies one-qubit unitary gates to an MPS state.
  • Includes a faster backpropagation pathway than the default ITensors methods.
• Additional Utilities:
  • Entropy and mutual information calculations for both pure and mixed states.

StateFactory

• File: src/StateFactory.jl
• Contains helper functions to create or initialize various matrix product states (e.g., product states, random states, GHZ states, etc.).

Gates

• File: src/Gates.jl
• Provides definitions of different quantum gates.
• These gate definitions can be used to build custom circuits with measurement-based or standard gate-based approaches.

Layers

• File: src/Layers.jl
• Includes definitions of “layers” for building deeper variational circuits easily.

GirvinProtocol

• File: src/GirvinProtocol.jl
• Implements the measurement-based protocol described by Smith et al. (PRX Quantum 4, 020315).

FluxExtension

• Folder: src/FluxExtension/
• Implements Flux-based recurrent neural network that can be used in the feedback steps of the mVQE algorithm.
• Directly integrates with Circuits to enable gradient-based updates and advanced optimization routines.

Circuits

• Folder: src/Circuits/
• Houses the definitions of variational circuits used by the mVQE algorithm.
• MeasurementCircuits sub-module: Provides circuits with mid-circuit measurement and classical feedback, allowing measurement-based VQE protocols.

Optimization

• File: src/Optimization.jl
• Contains the expect function for Hamiltonian expectation value computations.
• Optimized to replace ITensors’ standard inner(psi', H, psi) call, offering better performance when backpropagating fot the calulcations of gradients.

ITensorsMeasurement

• Folder: src/ITensorsMeasurement/
• Implements projective measurements on both pure and mixed states.
• Handles backpropagation through measurement steps, keeping the workflow end-to-end differentiable.