Here we demonstrate the numerical implementation of neural error mitigation, a novel method that uses neural networks to improve estimates of ground states and ground-state observables obtained using VQE on near-term quantum computers. This method, introduced in "Neural Error Mitigation of Near-term Quantum Simulations" https://arxiv.org/abs/2105.08086, is composed of two main steps:

(A) First, we perform neural quantum state tomography (NQST) to train a neural quantum state (NQS) ansatz to represent the approximate ground state prepared by a noisy quantum device, using experimentally accessible measurements. - NeuralErrorMitigationTrainer uses NQSTomographyTrainer for this step

(B) We then apply the variational Monte Carlo algorithm (VMC) on the same neural quantum state ansatz (which we call the NEM ansatz) to improve the representation of the unknown ground state. - NeuralErrorMitigationTrainer uses VMCTrainer_Regularized for this step

physics_models: Hamiltonians and additional structure for the physical systems studied.

nqs_models: Neural Quantum States

trainers: Training algorithms for neural error mitigation including a neural error mitigation trainer composed of a neural quantum state tomography (NQST) trainer and a variational Monte Carlo (VMC) trainer.

utils: Utilities including a class Complex which represents complex tensors and operations on them by by wrapping around PyTorch tensors. Utilities to construct unitaries implementing basis changes, and for setting up the logging.

exact_solvers: Exact diagonalization solver that finds the exact ground state from the Hamiltonians found in physics_models. The class ExactDiagonalizer, which diagonalizes a Hamiltonian, and GenericExactState, which keeps a (non-trainable) table of amplitudes, and some methods to sample measurement results and query the amplitudes in different bases.

data_utils: Measurement data utilities for neural quantum state tomography.

Contains the variational quantum simulator measurement data for the systems studied in the paper.

H2molecule: Measurement data for the H2 molecule simulated numerically using a hardware-efficient variational quantum eigensolver ansatz with depolarizing noise.

LiHmolecule: Measurement data for the LiH molecule simulated numerically using a hardware-efficient variational quantum eigensolver ansatz with depolarizing noise as well as measurement data obtained from VQE on IBMQ's Rome device.

LatticeSchwinger: Measurement data for the lattice Schwinger model simulated numerically with depolarizing noise.

Tutorials to demonstrate the application of neural error mitigation to the variational simulation of physical systems studied in "Neural Error Mitigation of Near-term Quantum Simulations." More specifically, improving estimates of the ground state and ground state observables for the electronic structure of LiH and H2 molecules as well as the lattice Schwinger model. Additionally, demo files are included to show how the variational simulations are implemented.

In order to reproduce the results of our paper, one would perform neural error mitigation (with hyperparameter values listed in Table S1) from the VQE data saved in data for each parameter value and run.

NeuralErrorMitigation_LiHmolecule_ibmqrome: We demonstrate the application of NEM to the estimation of the LiH molecular ground states prepared experimentally using VQE on IBM’s five-qubit chip, IBMQRome. Here, we input measurements taken on the final optimized VQE result (from saved data in data/LiHmolecule/IBMQRome/ folder) in our neural error mitigation protocol. (Standard runtime ~ 4 minutes)

NeuralErrorMitigation_LiHmolecule_depolarizing_noise: We demonstrate the application of NEM to the estimation of the LiH molecular ground states prepared using classically simulated VQE with a depolarizing noise channel. Here, we input measurements taken on the final optimized VQE result (from saved data in data/LiHmolecule/DepolarizingNoise/ folder) and use those measurements in our neural error mitigation protocol. (Standard runtime ~ 4 minutes)

NeuralErrorMitigation_H2molecule: We demonstrate the application of NEM to the estimation of the H2 molecular ground states prepared using classically simulated VQE with a depolarizing noise channel. Here, we input measurements taken on the final optimized VQE result (from saved data in data/H2molecule/DepolarizingNoise/ folder) in our neural error mitigation protocol. (Standard runtime ~ 0.5 minute)

NeuralErrorMitigation_LatticeSchwinger: We demonstrate the application of NEM to the approximate ground state of the lattice Schwinger model obtained by numerically simulating a VQE algorithm for N = 8 sites. (Standard runtime ~ 3 minutes)

VariationalQuantumSimulation_LatticeSchwinger: Demonstrate how we performed the classically simulated variational quantum simulation with a depolarizing noise chanenl for the lattice Schwinger model with N=8 sites. (Standard runtime ~ 20 seconds)

VariationalQuantumEigensolver_QuantumChemistry Demonstrates the performance of the hardware efficient variational quantum eigensolver classically simulated with a depolarizing noise channel to find ground state of the H2 and LiH molecule. (Standard runtime for H2 ~ 20 seconds, for LiH ~ 4 minutes)

environment.yml is a list of the developer's Python environment used during development. Included are all the necessary packages and versions.

To create an environment from the environment.yml file:

conda env create -f environment.yml

Then activate the NEM environment

conda activate NEM