The Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical algorithm for near-term quantum computers. It offers an approach to solving central problems in quantum chemistry, such as estimating molecular ground-state energies by creating a parametrized trial wavefunction or ansatz state and iteratively updating the parameters using classical optimization to minimize the energy expectation value. A critical challenge in VQE implementation arises from the non-commutativity of operators in the ansatz circuit, constructed using first-order Suzuki-Trotter expansions. The underlying operator order significantly impacts the performance of VQE calculations. In this thesis, we systematically investigate the effects of the operator order of the first-order Trotterization of the Unitary Coupled Cluster on VQE performance for different molecules. Using numerical simulations, we demonstrate that arbitrarily chosen orders can lead to substantial variations in convergence behavior and inaccurate ground-state energy estimation. Furthermore, we attempt to evaluate ordering strategies based on the Hamiltonian to test the possibility of using a predecided order strategy. Our studies show that no discernible ordering strategy emerges based on the Hamiltonian and specific random orders perform significantly better than others.
The Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical algorithm for near-term quantum computers. It offers an approach to solving central problems in quantum chemistry, such as estimating molecular ground-state energies by creating a parametrized trial wavefunction or ansatz state and iteratively updating the parameters using classical optimization to minimize the energy expectation value. A critical challenge in VQE implementation arises from the non-commutativity of operators in the ansatz circuit, constructed using first-order Suzuki-Trotter expansions. The underlying operator order significantly impacts the performance of VQE calculations. In this thesis, we systematically investigate the effects of the operator order of the first-order Trotterization of the Unitary Coupled Cluster on VQE performance for different molecules. Using numerical simulations, we demonstrate that arbitrarily chosen orders can lead to substantial variations in convergence behavior and inaccurate ground-state energy estimation. Furthermore, we attempt to evaluate ordering strategies based on the Hamiltonian to test the possibility of using a predecided order strategy. Our studies show that no discernible ordering strategy emerges based on the Hamiltonian and specific random orders perform significantly better than others.
The variational quantum eigensolver (VQE) has emerged as one of the most promising near-term quantum algorithms that can be used to simulate many-body systems such as molecular electronic structures. Serving as an attractive ansatz in the VQE algorithm, unitary coupled cluster (UCC) theory has seen a renewed interest in recent literature. However, unlike the original classical UCC theory, implementation on a quantum computer requires a finite-order Suzuki-Trotter decomposition to separate the exponentials of the large sum of Pauli operators. While previous literature has recognized the nonuniqueness of different orderings of the operators in the Trotterized form of UCC methods, the question of whether or not different orderings matter at the chemical scale has not been addressed. In this Letter, we explore the effect of operator ordering on the Trotterized UCCSD ansatz, as well as the much more compact k -UpCCGSD ansatz recently proposed by Lee et al. [ J. Chem. Theory Comput., 2019, 15, 311. arXiv, 2019, quant-ph:1909.09114. https://arxiv.org/abs/1909.09114 ]. We observe a significant, system-dependent variation in the energies of Trotterizations with different operator orderings. The energy variations occur on a chemical scale, sometimes on the order of hundreds of kcal/mol. This Letter establishes the need to define not only the operators present in the ansatz but also the order in which they appear. This is necessary for adhering to the quantum chemical notion of a “model chemistry”, in addition to the general importance of scientific reproducibility. As a final note, we suggest a useful strategy to select out of the combinatorial number of possibilities, a single well-defined and effective ordering of the operators.
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 The variational quantum eigensolver (VQE) algorithm combines the ability of quantum computers to efficiently compute expectation values with a classical optimization routine in order to approximate ground state energies of quantum systems. In this paper, we study the application of VQE to the simulation of molecular energies using the unitary coupled cluster (UCC) ansatz. We introduce new strategies to reduce the circuit depth for the implementation of UCC and improve the optimization of the wavefunction based on efficient classical approximations of the cluster amplitudes. Additionally, we propose an analytical method to compute the energy gradient that reduces the sampling cost for gradient estimation by several orders of magnitude compared to numerical gradients. We illustrate our methodology with numerical simulations for a system of four hydrogen atoms that exhibit strong correlation and show that the circuit depth of VQE using a UCC ansatz can be reduced without introducing significant loss of accuracy in the final wavefunctions and energies.
This is a case study of the variational quantum eigensolver (VQE) method using numerical simulations to test the influence of noise on the accuracy of the underlying circuit ansatz. We investigate a computational chemistry application of VQE to calculate the electronic ground state and its energy for Sodium Hydride (NaH), a prototypical two-electron problem. Using a one-parameter ansatz derived from unitary coupled cluster (UCC) theory, we simulate the effects of noise on the energy expectation value and variance with respect to the ansatz parameter. These numerical simulations provide insights into the accuracy of the prepared quantum state and the efficiency of the classical optimizer that iteratively refines the ansatz. We conduct a comparative study between analytical results derived for the UCC ansatz in the absence of noise and the noisy numerical simulation results obtained using an isotropic depolarizing noise model for each gate. We also compare the relative increase in noise on logically equivalent UCC ansatz circuits generated by randomized compiling. Notably, we observe that the intrinsic variance in the energy due to the simplicity of the ansatz itself compares with the noise induced by the bare circuit.
Variational quantum eigensolvers (VQEs) are among the most promising quantum algorithms for solving electronic structure problems in quantum chemistry, particularly during the noisy intermediate-scale quantum (NISQ) era. In this study, we investigate the capabilities and limitations of VQE algorithms implemented on current quantum hardware for determining molecular ground-state energies, focusing on the adaptive derivative-assembled pseudo-Trotter ansatz VQE (ADAPT-VQE). To address the significant computational challenges posed by molecular Hamiltonians, we explore various well known strategies to simplify the Hamiltonian, optimize the ansatz, and improve classical parameter optimization through modifications of the COBYLA optimizer. These enhancements are integrated into a tailored quantum computing implementation designed to minimize the circuit depth and computational cost. Using benzene as a benchmark system, we demonstrate the application of these optimizations on an IBM quantum computer. Despite these improvements, our results highlight the limitations imposed by current quantum hardware, particularly the impact of quantum noise on state preparation and energy measurement. The noise levels in today's devices prevent meaningful evaluations of molecular Hamiltonians with sufficient accuracy to produce reliable quantum chemical insights. Finally, we extrapolate the requirements for future quantum hardware to enable practical and scalable quantum chemistry calculations using VQE algorithms. This work provides an assessment of current quantum algorithms for molecular modeling on real quantum hardware, highlighting the impact of noise and hardware limitations on the achievable accuracy.
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.