Quantum phase estimation is one of the most fundamental and useful quantum computing algorithms available to researchers, but traditional methods for estimating the phase of a unitary operator acting on its eigenvalue leave information about the phase contained in all of the output from a quantum computer unused. Mean squared error (MSE) for such a scheme is 0.03 for simulated data with and without noise. Machine learning with classical methods, like linear regression (MSE=0.011 for noisy data and 0.0043 for noise-free data) and random forests (MSE=0.01 for noisy data and 0.006 for noise-free data) increase the accuracy of the estimation drastically without requiring additional qubits like the traditional approach. However, neural networks reduce this number further (MSE=0.0077) in the case of noisy data and with noise-free data (MSE=0.003). Examining the performance of neural networks with different modifications also yields insight into their performance on quantum computing output that could be generalized to output from other algorithms and on quantum computers with different numbers of qubits.
Quantum phase estimation is one of the most fundamental and useful quantum computing algorithms available to researchers, but traditional methods for estimating the phase of a unitary operator acting on its eigenvalue leave information about the phase contained in all of the output from a quantum computer unused. Mean squared error (MSE) for such a scheme is 0.03 for simulated data with and without noise. Machine learning with classical methods, like linear regression (MSE=0.011 for noisy data and 0.0043 for noise-free data) and random forests (MSE=0.01 for noisy data and 0.006 for noise-free data) increase the accuracy of the estimation drastically without requiring additional qubits like the traditional approach. However, neural networks reduce this number further (MSE=0.0077) in the case of noisy data and with noise-free data (MSE=0.003). Examining the performance of neural networks with different modifications also yields insight into their performance on quantum computing output that could be generalized to output from other algorithms and on quantum computers with different numbers of qubits.
Quantum computing has the potential to solve problems that are currently intractable to classical computers with algorithms like Quantum Phase Estimation (QPE); however, noise significantly hinders the performance of today’s quantum computers. Machine learning has the potential to improve the performance of QPE algorithms, especially in the presence of noise. In this work, QPE circuits were simulated with varying levels of depolarizing noise to generate datasets of QPE output. In each case, the phase being estimated was generated with a phase gate, and each circuit modeled was defined by a randomly selected phase. The model accuracy, prediction speed, overfitting level and variation in accuracy with noise level was determined for 5 machine learning algorithms. These attributes were compared to the traditional method of post-processing and a 6x–36 improvement in model performance was noted, depending on the dataset. No algorithm was a clear winner when considering these 4 criteria, as the lowest-error model (neural network) was also the slowest predictor; the algorithm with the lowest overfitting and fastest prediction time (linear regression) had the highest error level and a high degree of variation of error with noise. The XGBoost ensemble algorithm was judged to be the best tradeoff between these criteria due to its error level, prediction time and low variation of error with noise. For the first time, a machine learning model was validated using a 2-qubit datapoint obtained from an IBMQ quantum computer. The best 2-qubit model predicted within 2% of the actual phase, while the traditional method possessed a 25% error.
In this thesis we have implemented three many-body methods in quantum mechanics on quantum computers, namely the Quantum Phase Estimation (QPE) algorithm, the Variational Quantum Eigensolver (VQE) algorithm and the Quantum Adiabatic Time Evolution (QATE) algorithm.We utilized these methods to approximate the ground state energy of the pairing model Hamiltonian.We provided a comparison between the ideal simulation of a quantum computer and a simulation including the noise model from some of IBM's actual quantum devices.For the VQE algortihm, we also run the method on the IBM Q London five-qubit device.As the time complexity (circuit depth) of a quantum algorithm (quantum circuit) is often the bottleneck on current quantum computers, we also introduced a circuit optimization scheme, called Recursive Circuit Optimization, which can be utilized to approximate the action of a quantum circuit by learning the parameters of a parametrized unitary operation and only evaluating small parts of the circuit at a time.We tested out the method by trying to optimize a random circuit, the QATE algorithm and the unitary coupled cluster doubles (UCCD) ansatz.Finally, we implemented a quantum circuit that allows us to encode a real data set into the amplitudes of a quantum state, and we utilized this circuit to implement several realizations of dense and recurrent neural networks on quantum computers.We proposed neural network layers that are consisting of hardware-and parameter-efficient parametrized quantum circuits, and we have made a Python framework to construct and train deep quantum neural networks.We then tested the neural networks by trying to learn a simple non-linear function as well as finding the smallest eigenvalue of the pairing Hamiltonian Full Configuration Interaction (FCI) matrix.We found that all the quantum many-body methods showed promise in providing an upper bound on the ground state energy for the ideal simulation of a quantum computer.When performing the simulation with noise and the execution on the real quantum device however, we found that only the VQE algorithm was able to yield reasonable results due to short circuit depth.Still, the success of the method was found to be dependent on the interaction strength g of the quantum system, as the approximation of the ground state energy started to deviate from the actual value as we increased g.For the recursive circuit optimization of a random circuit, we found that we were able to approximate the state given by the random circuit, and we learned that we should expect some error in the approximation due to finite measurements and the flexibility of the parametrized unitary.When utilizing the recursive optimization scheme to approximate the QATE algorithm, we found that although we were not able to exactly reproduce the action of the operator, we did provide an upper bound to the ground state energy close to the one given by the actual QATE algorithm.In the same time, the circuit depth of the algorithm was drastically decreased.When utilizing the scheme for the VQE algorithm, we got promising results while reducing the circuit depth of the UCCD ansatz.Over to the neural networks, we managed to approximate a non-linear function utilizing a hardware-efficient three layered network, with only three qubits in the intermediate layers.We showed that our proposed neural network are suitable for near-term quantum computers by successfully approximating the lowest eigenvalue of the FCI matrix when running a simulation with the noise model from the IBM Q London five-qubit computer.i I would first like to thank my supervisor, Morten Hjorth-Jensen, for being helpful and motivational throughout my time at the University of Oslo.Your door has always been open when help is needed, which has been a great relief in stressful times.I would also like to thank my friends and family for believing in me and providing emotional support.Your continuous interest in my studies and well-being has meant a lot to me.My next thanks goes to Heidi K. J.I highly appreciate your never-ending support.Finally I would like to thank the computational physics study group.You have been invaluable to me in both a social and educational manner.
Quantum computing is revolutionizing artificial intelligence (AI) by offering exponential speed-ups for complex computations. Quantum computing differs from classical computing, which uses bits leverages qubits that enable parallelism and superposition. This advancement enhances AI activities including pattern detection, optimization, and deep learning. By integrating quantum computing into AI, researchers aim to achieve superior computational power, driving breakthroughs in fields like healthcare, finance, and cryptography. In AI applications, machine learning techniques are essential., and their efficiency can be significantly improved using quantum computing. Linear Regression predicts continuous values, Random Forest Regression enhances prediction robustness by averaging multiple decision trees, and Support Vector Machines classify data by finding optimal hyperplanes. Quantum computing accelerates these algorithms by reducing computational complexity and improving model accuracy. Key input parameters for evaluating quantum AI performance include Quantum Bit Processing Speed (GHz), which determines data processing rate, Qubit Coherence Time (µs), which affects stability, and Quantum Gate Fidelity (%), which measures the accuracy of quantum operations. These parameters directly influence AI model efficiency and reliability. The AI Performance Score serves as a comprehensive evaluation metric, assessing computational efficiency, prediction accuracy, and overall robustness of quantum AI models. Keywords: Quantum Computing, Machine Learning, Qubit Coherence Time, AI Performance Score
The quantum phase estimation (QPE) is one of the fundamental algorithms based on the quantum Fourier transform. It has applications in order-finding, factoring, and finding the eigenvalues of unitary operators. The major challenge in running QPE and other quantum algorithms is the noise in quantum computers. In the present work, we study the impact of incoherent noise on QPE, modeled as trace-preserving and completely positive quantum channels. Different noise models such as depolarizing, phase flip, bit flip, and bit-phase flip are taken to understand the performance of the QPE in the presence of noise. The simulation results indicate that the standard deviation of the eigenvalue of the unitary operator has strong exponential dependence upon the error probability of individual qubits. However, the standard deviation increases only linearly with the number of qubits for fixed error probability when that error probability is small.
Quantum Phase Estimation(QPE) is a fundamental quantum algorithm that is used for the estimation of eigenphases of unitary operators. Its main goal is to determine the phase associated with each eigenstate. Usually, it take steps such as prepare quantum states, apply controlled unitaries, inverse quantum fourier transformation, and measurement. This study uses the OpenAI Gym framework to build a customized QPE environment. Here, the phase of a randomly generated target unitary operator is estimated using a quantum circuit. Through interaction with this environment, the DQN agent learns the best course of action to increase phase estimation accuracy. It exhibits more flexibility in noisy environments and reduces estimating mistakes. With its insights and approaches for further study in this area, this effort represents a significant advancement in the use of Deep Reinforcement Learning in quantum computing. A Comparative analysis between IBM Quantum(ibm kyiv) and the Aer Simulator on the OpenAI Gym environment using RL agents has been done.
Major obstacles remain to the implementation of macroscopic quantum computing: hardware problems of noise, decoherence, and scaling; software problems of error correction; and, most important, algorithm construction. Finding truly quantum algorithms is quite difficult, and many of these genuine quantum algorithms, like Shor's prime factoring or phase estimation, require extremely long circuit depth for any practical application, which necessitates error correction. In contrast, we show that machine learning can be used as a systematic method to construct algorithms, that is, to non-algorithmically “program” quantum computers. Quantum machine learning enables us to perform computations without breaking down an algorithm into its gate “building blocks”, eliminating that difficult step and potentially increasing efficiency by simplifying and reducing unnecessary complexity. In addition, our non-algorithmic machine learning approach is robust to both noise and to decoherence, which is ideal for running on inherently noisy NISQ devices which are limited in the number of qubits available for error correction. We demonstrate this using a fundamentally nonclassical calculation: experimentally estimating the entanglement of an unknown quantum state. Results from this have been successfully ported to the IBM hardware and trained using a hybrid reinforcement learning method.