Recent studies to solve nuclear structure problems using quantum computers rely on a quantum algorithm known as Variational Quantum Eigensolver (VQE). In this study, we calculate the correlation energy in Helium-6 using VQE, with a \textit{full-term} unitary-paired-coupled-cluster-doubles (UpCCD) ansatz on a quantum computer simulator and implement a set of custom termination criteria to shorten the optimization time. Using this setup, we test out noisy quantum computer simulators of various coherence times and quantum errors to find the required specification for such calculations. We also look into the contribution of errors from the quantum computers and optimization process. We find that the minimal specification of 5~ms coherence times and $10^{-4}$ quantum errors is required to reliably reproduce state-vector results within 8\% discrepancy. Our study indicates the possibility of performing VQE calculations using a full-term UpCCD ansatz on a slightly noisy quantum computer, without implementing quantum error correction.
Recent studies to solve nuclear structure problems using quantum computers rely on a quantum algorithm known as Variational Quantum Eigensolver (VQE). In this study, we calculate the correlation energy in Helium-6 using VQE, with a \textit{full-term} unitary-paired-coupled-cluster-doubles (UpCCD) ansatz on a quantum computer simulator and implement a set of custom termination criteria to shorten the optimization time. Using this setup, we test out noisy quantum computer simulators of various coherence times and quantum errors to find the required specification for such calculations. We also look into the contribution of errors from the quantum computers and optimization process. We find that the minimal specification of 5~ms coherence times and 10−4 quantum errors is required to reliably reproduce state-vector results within 8% discrepancy. Our study indicates the possibility of performing VQE calculations using a full-term UpCCD ansatz on a slightly noisy quantum computer, without implementing quantum error correction.
Exponential scaling presents a significant challenge in electronic structure calculations performed on classical computers. This paper explores how quantum computer algorithms can accurately represent quantum systems. Variational Quantum Eigensolver (VQE) algorithm is used to compute the ground state energy of hydrogen and helium sequences by implementing variational principle and quantum gates as trial wavefunction. This technique combines classical optimization with quantum computing calculations to simulate quantum systems on noisy and resource-limited computers. The resulting calculated energy is highly consistent to the corresponding exact values and Hartree-Fock calculations with a trend of when the number of atoms increases the calculated energy becomes more negative, leading to a decrease in the percentage error. Moreover, the convergence of the ground state energy of hydrogen and helium atoms was effectively optimized. The desired energy was reached, proven by adjusting the expectation value, and gradually achieving unity in state overlap. These findings demonstrate the VQE method's accuracy in calculating simple quantum systems and its scalability for larger atomic and molecular system, such as those in quantum chemistry and material science. However, challenges in quantum computer simulations, such as limited in qubit numbers and the presence of noise, require further advancements. Therefore, implementing a larger basis sets, advanced qubit mapping, specific chemistry ansatz, and flexible optimization techniques is one way to improve overall calculation.
Traditional quantum algorithms with guaranteed performance generally require fully coherent quantum computers to operate, making error-correction a necessity. In contrast, variational quantum algorithms seek to remove this requirement by formulating the computation as the approximate optimization of a functional. This optimization is carried out in the space of heuristic functions defined by a parameterized (variational) quantum circuit. Taking advantage of the flexibility in the definition of these heuristics, we can expand the operability of variational algorithms within the regime of coherence times of noisy intermediate-scale quantum devices, avoiding the necessity of error-correction. In this dissertation, we present the application of the variational quantum computing approach to problems in quantum simulation, quantum state preparation, quantum error-correction, and generative modeling.\nIn the first part, we investigate the use of the variational quantum eigensolver (VQE) for simulating the ground state of fermionic systems. We start by studying the implementation of VQE using the unitary coupled cluster (UCC) ansatz and propose strategies to reduce its cost. We use these insights to carry out some of the first experimental quantum computations of molecular energies using simplified versions of this ansatz. We also propose an extension of UCC to study problems in condensed matter physics, along with a new low-depth circuit ansatz for preparing non-gaussian fermionic states on quantum computers. We show the potential of our approach to describe strongly correlated fermionic systems.\nIn the second part, we develop three new variational quantum algorithms for problems in quantum computing and machine learning, namely: 1) the quantum autoencoder, to compress ensembles of quantum states, 2) the variational quantum error corrector, to find device-tailored quantum encoding and recovery circuits for quantum error correction, and 3) the variational quantum generator, to generate classical probability distributions. These techniques offer efficient ways to design new quantum circuits for state preparation, to find more effective error-correcting codes and to perform generative modeling with quantum computers. Our work provides insights into the design of variational quantum algorithms and establishes practical guidelines to implement these methods on near-term quantum computers, as well as some future research directions for this field.
Variational quantum eigensolver (VQE) is demonstrated to be the promising methodology for quantum chemistry based on near-term quantum devices. However, many problems are yet to be investigated for this methodology, such as the influences of optimization algorithm and basis size on the accuracy and efficiency for quantum computing. To address these issues, five molecules (H2, LiH, HF, N2 and F2) are studied in this work based on the VQE method using unitary coupled cluster (UCC) ansatz. The performance of the gradient optimization L-BFGS-B is compared with that of the direct search method COBYLA. The former converges more quickly, but the accuracy of energy surface is a little lower. The basis set shows a vital influence on the accuracy and efficiency. A large basis set generally provides an accurate energy surface, but induces a significant increase in computing time. The 631g basis is generally required from the energy surface of the simplest H2 molecule. For practical applications of VQE, complete active space (CAS) is suggested based on limited quantum resources. With the same number of qubits, more occupied orbitals included in CAS gives a better accuracy for the energy surface and a smaller evaluation number in the VQE optimization. Additionally, the electronic structure, such as filling fraction of orbitals, the bond strength of a molecule and the maximum nuclear charge also influences the performance of optimization, where half occupation of orbitals generally requires a large computation cost.
Quantum computing of the 6Li nucleus via ordered unitary coupled cluster
The variational quantum eigensolver (VQE) is an algorithm to compute ground and excited state energy of quantum many-body systems. A key component of the algorithm and an active research area is the construction of a parametrized trial wave function—a so-called variational ansatz. The wave function parametrization should be expressive enough, i.e., represent the true eigenstate of a quantum system for some choice of parameter values. On the other hand, it should be trainable, i.e., the number of parameters should not grow exponentially with the size of the system. Here, we apply VQE to the problem of finding ground and excited state energies of the odd-odd nucleus Li6 . We study the effects of ordering fermionic excitation operators in the unitary coupled clusters ansatz on the VQE algorithm convergence by using only operators preserving the Jz quantum number. The accuracy is improved by two orders of magnitude in the case of descending order. We first compute optimal ansatz parameter values using a classical state-vector simulator with arbitrary measurement accuracy and then use those values to evaluate energy eigenstates of Li6 on a superconducting quantum chip from IBM. We post-process the results by using error mitigation techniques and are able to reproduce the exact energy with an error of 3.8% and 0.1% for the ground state and for the first excited state of Li6 , respectively.
Abstract Variational quantum eigensolver (VQE) for electronic structure calculations is believed to be one major potential application of near term quantum computing. Among all proposed VQE algorithms, the unitary coupled cluster singles and doubles excitations (UCCSD) VQE ansatz has achieved high accuracy and received a lot of research interest. However, the UCCSD VQE based on fermionic excitations needs extra terms for the parity when using Jordan–Wigner transformation. Here we introduce a new VQE ansatz based on the particle preserving exchange gate to achieve qubit excitations. The proposed VQE ansatz has gate complexity up-bounded to O ( n 4 ) for all-to-all connectivity where n is the number of qubits of the Hamiltonian. Numerical results of simple molecular systems such as BeH 2 , H 2 O, N 2 , H 4 and H 6 using the proposed VQE ansatz gives very accurate results within errors about 10 −3 Hartree.
The variational quantum eigensolver (VQE) is an algorithm to compute ground and excited state energy of quantum many-body systems. A key component of the algorithm and an active research area is the construction of a parametrized trial wave function---a so-called variational ansatz. The wave function parametrization should be expressive enough, i.e., represent the true eigenstate of a quantum system for some choice of parameter values. On the other hand, it should be trainable, i.e., the number of parameters should not grow exponentially with the size of the system. Here, we apply VQE to the problem of finding ground and excited state energies of the odd-odd nucleus 6Li. We study the effects of ordering fermionic excitation operators in the unitary coupled clusters ansatz on the VQE algorithm convergence by using only operators preserving the Jz quantum number. The accuracy is improved by two orders of magnitude in the case of descending order. We first compute optimal ansatz parameter values using a classical state-vector simulator with arbitrary measurement accuracy and then use those values to evaluate energy eigenstates of 6Li on a superconducting quantum chip from IBM. We post-process the results by using error mitigation techniques and are able to reproduce the exact energy with an error of 3.8 and 0.1 for the ground state and for the first excited state of 6Li, respectively.