We find the ground-state energy of the Ising model using the Cascaded Variational Quantum Eigensolver (CVQE) algorithm with the Guided-Sampling Ansatz (GSA) using up to 63 qubits on a quantum computer. We study a heavy-hex lattice to match the qubit architecture, allowing us to perform calculations in the quantum utility regime. We study both a homogeneous and random-coupling model. We locate the boundary of acceptable quantum errors as a function of the number of qubits and coupling strength. An entropic analysis is performed giving insights into the quantum computing performance. A subspace analysis is performed that suggests that the Ising model is especially suited for near-term quantum computing.
We find the ground-state energy of the Ising model using the Cascaded Variational Quantum Eigensolver (CVQE) algorithm with the Guided-Sampling Ansatz (GSA) using up to 63 qubits on a quantum computer. We study a heavy-hex lattice to match the qubit architecture, allowing us to perform calculations in the quantum utility regime. We study both a homogeneous and random-coupling model. We locate the boundary of acceptable quantum errors as a function of the number of qubits and coupling strength. An entropic analysis is performed giving insights into the quantum computing performance. A subspace analysis is performed that suggests that the Ising model is especially suited for near-term quantum computing.
We present and analyze large-scale simulation results of a hybrid quantum-classical variational method to calculate the ground state energy of the anti-ferromagnetic Heisenberg model. Using a massively parallel universal quantum computer simulator, we observe that a low-depth-circuit ansatz advantageously exploits the efficiently preparable Néel initial state, avoids potential barren plateaus, and works for both one- and two-dimensional lattices. The analysis reflects the decisive ingredients required for a simulation by comparing different ansätze, initial parameters, and gradient-based versus gradient-free optimizers. Extrapolation to the thermodynamic limit accurately yields the analytical value for the ground state energy, given by the Bethe ansatz. We predict that a fully functional quantum computer with 100 qubits can calculate the ground state energy with a relatively small error.
Quantum computers and simulators may offer significant advantages over their classical counterparts, providing insights into quantum many-body systems and possibly improving performance for solving exponentially hard problems, such as optimization and satisfiability. Here, we report the implementation of a low-depth Quantum Approximate Optimization Algorithm (QAOA) using an analog quantum simulator. We estimate the ground-state energy of the Transverse Field Ising Model with long-range interactions with tunable range, and we optimize the corresponding combinatorial classical problem by sampling the QAOA output with high-fidelity, single-shot, individual qubit measurements. We execute the algorithm with both an exhaustive search and closed-loop optimization of the variational parameters, approximating the ground-state energy with up to 40 trapped-ion qubits. We benchmark the experiment with bootstrapping heuristic methods scaling polynomially with the system size. We observe, in agreement with numerics, that the QAOA performance does not degrade significantly as we scale up the system size and that the runtime is approximately independent from the number of qubits. We finally give a comprehensive analysis of the errors occurring in our system, a crucial step in the path forward toward the application of the QAOA to more general problem instances.
In the study of solving the ground state of quantum systems by variational principle, variational quantum algorithm is widely adopted to solve the ground state. As one kind of quantum classical hybrid algorithm, variational quantum eigensolver plays an important role. In this paper, we calculate ground state of quantum Rabi model in the ultra-strong coupling regime by invoking the variational quantum eigensolver in the open-source quantum computing simulator Qiskit. In the 3-qubit computing basis, we use the standard binary encoding method to encode the operators and quantum states in the quantum Rabi model, and measure the minimum average value (ground state energy) on the quantum circuit constructed by the Hamiltonian variational ansatz. Finally, comparing the calculation results of the variational quantum eigensolver with the exact values obtained by classical numerical simulations, we discuss the dependence of computational accuracy of the variational quantum eigensolver on the consumption of the number of the qubits and the coupling strength in quantum Rabi model.
Estimating the ground-state energy of Hamiltonians is a fundamental task for which it is believed that quantum computers can be helpful. Several approaches have been proposed toward this goal, including algorithms based on quantum phase estimation and hybrid quantum-classical optimizers involving parameterized quantum circuits, the latter falling under the umbrella of the variational quantum eigensolver. Here, we analyze the performance of quantum Boltzmann machines for this task, which is a less explored ansatz based on parameterized thermal states and which is not known to suffer from the barren-plateau problem. We delineate a hybrid quantum-classical algorithm for this task and rigorously prove that it converges to an ε-approximate stationary point of the energy function optimized over parameter space, while using a number of parameterized-thermal-state samples that is polynomial in ε−1, the number of parameters, and the norm of the Hamiltonian being optimized. Our algorithm estimates the gradient of the energy function efficiently by means of a quantum circuit construction that combines classical random sampling, Hamiltonian simulation, and the Hadamard test. Additionally, supporting our main claims are calculations of the gradient and Hessian of the energy function, as well as an upper bound on the matrix elements of the latter that is used in the convergence analysis.
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.
The variational quantum eigensolver (VQE) is a hybrid quantum classical algorithm designed for current and near-term quantum devices. Despite its initial success, there is a lack of understanding involving several of its key aspects. There are problems with VQE that forbid a favorable scaling towards quantum advantage. In order to alleviate the problems, we propose and extensively test a quantum annealing inspired heuristic that supplements VQE. The improved VQE enables an efficient initial state-preparation mechanism, in a recursive manner, for a quasidynamical unitary evolution. We conduct an in-depth scaling analysis of finding the ground-state energies with increasing lattice sizes of the Heisenberg model, employing simulations of up to 40 qubits that manipulate the complete state vector. In addition to systematically finding the ground-state energy, we observe that it avoids barren plateaus, escapes local minima, and works with low-depth circuits. For the current devices, we further propose a benchmarking toolkit using a mean-field model and test it on IBM Q devices. Realistic gate execution times estimate a longer computational time to complete the same computation on a fully functional error-free quantum computer than on a quantum computer emulator implemented on a classical computer. However, our proposal can be expected to help accurate estimations of the ground-state energies beyond 50 qubits when the complete state vector can no longer be stored on a classical computer, thus enabling quantum advantage.