Neural Network-Guided Extrapolation Technique for Quantum Variational Algorithms

Summary

We propose a neural network-driven zero noise extrapolation method to increase the accuracy of VQE in a noisy quantum environment.

Abstract

In the noisy intermediate-scale quantum (NISQ) era, the Variational Quantum Eigensolver (VQE) comes out as an effective algorithm for addressing complex quantum challenges. However, the presence of noise in quantum devices frequently reduces the accuracy and reliability of VQE outcomes. This paper presents an innovative method to address this issue by using a neural network-based extrapolation technique in VQE calculation. By utilizing the Qiskit framework, we designed parameterized quantum circuits using the RY-RZ ansatz, and their performance was analyzed under different levels of depolarizing noise with bit-flip errors, phase-flip errors, and amplitude damping errors. Our inquiry involved analyzing the expected outcomes of a Hamiltonian across various levels of noise intensity with the objective of deducing the ground state energy (GSE). To connect the observed noisy outcomes with the ideal noise-free condition, a Feedforward Neural Network (FFNN) was trained using the error probabilities and their corresponding expectation values. This model accurately predicted the VQE results in an ideal noise-free scenario. Comparison of the result of simulation and real quantum hardware executions revealed noise-induced inconsistencies, highlighting the effectiveness of this neural network-based extrapolation approach in correcting them. This comprehensive method improves the accuracy of VQE calculation on NISQ devices and highlights the significant potential of blending quantum and classical methods to address the threats imposed by quantum noise. The comparison of results between FFNN, convolutional neural network (CNN), and long short-term memory (LSTM) network reveals that FFNN predicts outcomes with more accuracy but in less time.

Introduction

Quantum computing is like a mix of different subjects, blending ideas from quantum mechanics and computer theory. It could change the way we handle information completely. It can offer computational capabilities far beyond the reach of classical computations¹. Although quantum computing holds great promise, it faces significant hurdles. Quantum systems are fragile and easily influenced by noise and errors from different sources. These disturbances can greatly impact the accuracy of computations^2,3,4,5,6.

The Variational Quantum Eigensolver (VQE) is among the most potential algorithms developed to leverage the ability of near-term quantum devices⁷. VQE is a unique approach that combines elements of quantum mechanics with classical optimization methods. Its objective is to pinpoint the lowest energy state of a given Hamiltonian. This blend of quantum and classical techniques makes VQE especially well-suited for the abilities of current noisy intermediate scale quantum (NISQ) devices⁸. Nevertheless, despite the potential offered by tools like VQE, the persistent obstacle that is the presence of noise remains in quantum devices. Quantum noise arises from the communications between quantum systems and their surrounding environment. These interactions lead to errors that can distort results, rendering them unreliable or even completely inaccurate². Quantum devices encounter various types of noise, including depolarizing noise, phase damping, and amplitude damping. The task at hand is to develop strategies that can either reduce the impact of this noise or rectify it after the fact.

In this study, we propose a neural network-based extrapolation technique to enhance the accuracy of the VQE under noisy conditions typical of NISQ-era devices. The core idea is to train neural networks on expectation values computed at various noise levels and then extrapolate the result to a zero-noise limit. This method effectively learns the functional dependence between noise levels and measured outcomes, allowing prediction of the ground state energy in the absence of noise. We specifically explore the performance of Feedforward Neural Networks (FFNN), Convolutional Neural Networks (CNN), and Long Short-Term Memory (LSTM) models. To the best of our knowledge, this is the first study that evaluates and compares the prediction accuracy of multiple neural network architectures for estimating noise-free expectation values in VQE. The results demonstrate that FFNN achieves superior accuracy with lower prediction time, making it well-suited for error mitigation in practical quantum applications.

Unlike Quantum Error Correction (QEC), Quantum Error Mitigation (QEM) enhances the accuracy of the outcome without encoding the quantum state into an entangled one. No extra resource is required in QEM, and even though the error rate is quite high, it can improve quantum computation. Thus, for increasing the performance of NISQ computing, QEM techniques are highly applicable and essential^7,9,10,11,12. For mitigating quantum errors, classical ML models, neural networks, reinforcement learning, variational quantum algorithms, and hybrid models have significant contribution.

Extrapolation^13,14,15, probabilistic error cancellation^14,15, quantum subspace expansion^16,17, and symmetry verification^18,19 are different available QEM methods. Strikis et al.²⁰ describes a machine learning approach that utilizes a Clifford circuit for training. In Czarnik etal.²¹, data regression is utilized using the Clifford circuit. For handling readout errors, some studies illustrate error mitigation techniques^22,23.

Variational quantum algorithms use classical optimization techniques for minimizing errors. By adjusting circuit parameters optimal solution can be found. VQE is well-suited for NISQ devices as classical and quantum computing. Czarnik et al.²¹ introduced variational strategies for removing errors from quantum solutions. Koczor proposed a variational error mitigation technique for optimizing the fidelity of quantum states²⁴. For mitigating errors, hybrid classical quantum models can be used. Classical ML techniques preprocess the data, and then the quantum circuit does further processing. From McArdle's research, a hybrid approach is obtained where quantum errors are corrected in variational quantum algorithms using classical ML models²⁵. Thus, they showed that hybrid models could mitigate quantum errors. Czarnik et al.²⁶ integrated classical ML with quantum circuits. ML-QEM is benchmarked using various machine learning models -- linear regression, random forests, multi-layer perceptron, and graph neural networks on diverse quantum circuits and noise profiles, with results validated through simulations and experiments²⁷.

Protocol

All experiments were conducted using the Qiskit qasm_simulator for classical simulations and on the IBM quantum device ibm_kyoto, which was selected as the least busy backend available at the time of execution using Qiskit's least_busy() function; no human or animal subjects were involved, and thus no ethical approval was required. All software and hardware resources were used in accordance with institutional guidelines. The coding files are provided as Supplementary Coding File 1 and Supplementary Coding File 2.

Setup
The experiments were implemented in a Python-based quantum-computing and machine-learning stack. Quantum circuits were constructed and simulated with an open-source quantum framework (e.g., Qiskit)²⁸, and the neural network was implemented in a standard machine-learning library. Computations ran on a workstation equipped with a modern CPU and GPU. Quantum-noise models and hardware access were provided by publicly available IBM Quantum resources.

Quantum-circuit construction
RY-RZ ansatz is more hardware-efficient and practical to implement. A parameterized two-qubit RY-RZ ansatz is prepared for the VQE. This entails the utilization of rotation gates that are adjustable along both the Y-axis (RY gates) and the Z-axis (RZ gates). The basic idea of an RY gate is a rotation around the Y-axis. The general form of an RY gate is as follows in equation 1.:

     (1)

Analogously, the RZ gate is defined as in equation 2:

      (2)

Each qubit starts in the computational basis state 0, described in Dirac notation as |0 state, followed by a rotation about the Y axis of angle θ, and a rotation about the Z axis of angle θ, and then integration of CNOT gates between successive qubits is implemented to foster entanglement. Figure 1 displays the RY-RZ ansatz for a 12-qubit system, where the same basic ansatz structure is applied across all 12 qubits. The ansatz is defined per 2 qubits, e.g., each qubit undergoes an RY and RZ rotation. In a 12-qubit system, the same basic ansatz structure is applied across all 12 qubits, resulting in a full circuit composed of 6 ansatz circuits.

Figure 1: RY-RZ Ansatz. The figure illustrates the ansatz circuit. The ansatz is defined per 2 qubits where each qubit undergoes an RY and RZ rotation. Please click here to view a larger version of this figure.

The state change of the circuit on n qubits with the quantum state , can be expressed as equation 3:

    (3)

Where indicates the tensor product.

The Hamiltonian for two-qubit system is defined as equation 4

      (4)

Where:
n represents the number of qubits in the system.
C_i represents the local magnetic field coefficients.
σ_i^z represents the Pauli-Z operator acting on qubit i.
J_ij represents the interaction strengths between qubits I and j.

The first sum represents terms involving local magnetic fields along the Z-axis, and the second sum represents qubit-qubit interactions along the Z-axis. The eigenvalues of this operator present the potential results upon measuring the related quantum state, and the eigenvector associated with the lowest eigenvalue signifies the ground state of the system. Our goal is to measure the ground state of the Hamiltonian. So, we do not directly implement the Hamiltonian as a circuit. Instead, we prepare the circuit using a parameterized RY-RZ ansatz and measure the expectation value of the Hamiltonian on that circuit.

Noise model simulation
In this research, we utilize Qiskit's KrausError and NoiseModel classes to integrate these noise channels directly into gate operations²⁸. The noise is applied to the state vector without manually transforming the quantum state into a density matrix. Having the depolarizing noise model available with error probability λ, we implement it onto the quantum circuits constructed using the RY-RZ ansatz. When a density matrix ρ of a qubit state crosses the depolarizing channel, the state transforms as in equation 5

    (5)

A bit-flip error with probability λ can be represented as a quantum channel in equation 6

    (6)

A phase-flip error with probability λ can be represented as a quantum channel in equation 7

    (7)

Where, ρ is the density matrix of the qubit.

The amplitude damping process can be described using Kraus operators in equations 8 and 9

    (8)

    (9)

Where γ is the damping probability

Neural network training
A feed-forward neural network (input: λ, hidden layer: 50 ReLU neurons; output: one linear neuron) maps error probability to the noise-free expectation ₀. The difference, or the error, is measured using a loss function. The mean squared error (MSE) loss function is expressed in equation 10

    (10)

Where,
y_n is the true value for the nth sample.
n is the predicted value by the network for the n^th sample.

The model is trained on a range of error probabilities and their corresponding quantum circuit expectation values. This enables the neural network to capture the complex relationship between noise levels and quantum circuit performance. The steady decline in MSE across epochs demonstrates the model's effective learning process^29,30. CNN consists of 2 convolutional layers with 32 and 64 filters, respectively, each using a kernel size of 3. A ReLU activation function is used after each convolutional layer, followed by a max pooling layer. The output is flattened and passed through a dense layer with 64 units. The LSTM network includes one LSTM layer with 50 units, followed by a fully connected dense layer with 64 units and a final output layer. ReLU activation is used in the dense layer. Adam optimizer (learning rate 10^-3 over 500 epochs) is used. During training, the noisy data λ, are supplied; after convergence, the network extrapolates to λ = 0, predicting 0 for unseen circuits. Network prediction matches the ideal simulation more closely than raw noisy data, validating the zero-noise-extrapolation approach.

Execution on quantum hardware
An available two-qubit IBM Quantum backend with the lowest queue length is selected. For each parameter set θ, prepare the RY-RZ circuit and append measurements on both qubits, transpile to native gates and submit a job with 8192 shots, and retrieve counts for the quantum states |00, |01, |10, |11, and compute

_device=P₀₀+P₁₁-P₀₁-P₁₀

Where Pxy is the probability of outcome |xy.

Device results deviate from the ideal values owing to real-hardware noise; however, neural-network predictions after zero-noise extrapolation nearly coincide with the ideal simulations, confirming the effectiveness of the method.

RRIDs.
Key software resources include the Qiskit suite (RRID:SCR_021282) and the Python language (RRID: SCR_008394). A full Table of Materials accompanies the manuscript.

Results

Quantum simulations under noise
We vary the noise probabilities from 0.01 to 0.05 and measure the expectation value . As noise increases, the observed expectation values deviate from their ideal (noise-free) results, reflecting the detrimental effect of decoherence and errors.

Neural network predictions
A Feed-Forward Neural Network (FFNN) is trained to predict ideal, noise-free expectation values given a noisy input. During training:
Inpu...

Discussion

To find out the effectiveness of the neural network model described here, we started by assessing its capability to get noise-free performance depending on noisy quantum circuit outcomes. Adam optimizer is used and MSE is used as the loss function. The steady decrease in MSE values suggests that the model successfully captured the relationship between error probabilities and the associated outcomes of the quantum circuit.

The comparison is done between real quantum device result, simulation re...

Materials

List of materials used in this article

Name	Company	Catalog Number	Comments
Qiskit =0.39.0	IBM Quantum	https://www.ibm.com/quantum/qiskit	Primary quantum computing framework used for circuit design and simulation ((RRID:SCR_021282)
Python 3.10	Python Software Foundation	https://www.python.org/	Programming language used to implement algorithms and data analysis (RRID: SCR_008394)
Quantum Platform	IBM Quantum	https://quantum.cloud.ibm.com/computers	Quantum computing framework (RRID:SCR_021282)