Variational Quantum Eigensolver Algorithm Research Articles

Chemical simulations of quantum systems using quantum computers - review of algorithms and their experimental verification

Computer simulations using ever-increasing computing power and machine learning techniques allow advanced molecular modelling, molecular dynamics simulations and studies of intermolecular interactions. However, due to the complexity of biological systems and chemical processes at the molecular level, their accurate representation using classical computer models and techniques has faced a number of significant limitations for many years. A new and promising direction for the development of computational science and its potential applications in biochemistry is quantum computing and its integration with classical high-performance supercomputing systems. This article responds to the growing interest in the use of available quantum computers in exemplary applications. In this paper, we aim to provide an overview of the basic notions involved in the development of quantum algorithms and simulations related to issues at the interface of quantum chemistry and biochemistry. In addition, the article introduces the basic principles of performing simulations using the state-of-the-art quantum computers in the era of Noisy Intermediate-Scale Quantum (NISQ). Experimental results of the classical-quantum algorithm Variational Quantum Eigensolver (VQE) for example molecules H2 and CH+ are also presented. Despite the many shortcomings of currently available quantum computers, the analysed VQE algorithm proved to be effective in approximating the ground state of molecules using a minimal functional basis.

Simulating chemical reaction dynamics on quantum computer.

The electronic energies of molecules have been successfully evaluated on quantum computers. However, more attention is paid to the dynamics simulation of molecules in practical applications. Based on the variational quantum eigensolver (VQE) algorithm, Fedorov et al. proposed a correlated sampling (CS) method and demonstrated the vibrational dynamics of H2 molecules [J. Chem. Phys. 154, 164103 (2021)]. In this study, we have developed a quantum approach by extending the CS method based on the VQE algorithm (labeled eCS-VQE) for simulating chemical reaction dynamics. First, the CS method is extended to the three-dimensional cases for calculation of first-order energy gradients, and then, it is further generalized to calculate the second-order gradients of energies. By calculating atomic forces and vibrational frequencies for H2, LiH, H+ + H2, and Cl- + CH3Cl systems, we have seen that the approach has achieved the CCSD level of accuracy. Thus, we have simulated dynamics processes for two typical chemical reactions, hydrogen exchange and chlorine substitution, and obtained high-precision reaction dynamics trajectories consistent with the classical methods. Our eCS-VQE approach, as measurement expectations and ground-state wave functions can be reused, is less demanding in quantum computing resources and is, therefore, a feasible means for the dynamics simulation of chemical reactions on the current noisy intermediate-scale quantum-era quantum devices.

Folded Spectrum VQE: A Quantum Computing Method for the Calculation of Molecular Excited States.

The recent developments of quantum computing present novel potential pathways for quantum chemistry as the scaling of the computational power of quantum computers could be harnessed to naturally encode and solve electronic structure problems. Theoretically exact quantum algorithms for chemistry have been proposed (e.g., quantum phase estimation), but the limited capabilities of current noisy intermediate-scale quantum devices motivated the development of less demanding hybrid algorithms. In this context, the variational quantum eigensolver (VQE) algorithm was successfully introduced as an effective method to compute the ground-state energies of small molecules. This study investigates the folded spectrum (FS) method as an extension of the VQE algorithm for the computation of molecular excited states. It provides the possibility of directly computing excited states around a selected target energy using the same quantum circuit as for the ground-state calculation. Inspired by the variance-based methods from the quantum Monte Carlo literature, the FS method minimizes the energy variance, thus, in principle, requiring a computationally expensive squared Hamiltonian to be applied. We alleviate this potentially poor scaling by employing a Pauli grouping procedure to identify sets of commuting Pauli strings that can be evaluated simultaneously. This allows for a significant reduction in the computational cost. We applied the FS-VQE method to small molecules (H2, LiH), obtaining all electronic excited states with chemical accuracy on ideal quantum simulators. Furthermore, we explore the application of quantum error mitigation techniques, demonstrating improved energy accuracy on noisy simulators compared with simulations without mitigation.

Cascaded variational quantum eigensolver algorithm

Cascaded variational quantum eigensolver algorithm

A Simulation of Hydrazine Molecule’s Potential Energy Surface using Variational Quantum Eigensolver Algorithm

Quantum computing is a technology that utilizes the principles of quantum mechanics to perform complex computational processes. In this work, we use Qiskit Module from IBM to do quantum computational calculation using Variational Quantum Eigensolver (VQE) algorithm. VQE is a hybrid quantum-classical method that combines a quantum computer to measure energies and a classical computer to process the measurement results and update the parameters of the quantum computer. The purpose of VQE is to find the ground state energy of a chemical system. In the previous study, many of the VQE calculations have been done on simple molecules. So, in this study, we would like to use Hydrazine molecule as our object of VQE calculation. Furthermore, these results will be compared with the results from the classical calculation (MP2, CCSD(T), QCISD(T), and CASSCF) methods for testing the effectiveness of VQE using Unitary Coupled-Cluster Single and Double excitations (UCCSD) Ansatz. The quantum algorithm based on the UCCSD Ansatz led to a simplification of the algorithm by reducing the circuit depth. Then, the possibility to use active space approximation, can be used to reduce the quantum gates while trying to keep a good level of accuracy. In this study, we chose (2,2) and (4,4) active spaces. Based on the results, as we increase the size of the active space during the evaluation of the single-point energy, the estimated ground states obtained from the VQE algorithm yield nearly identical values. Conversely, in CASSCF calculations, expanding the active space introduces more energy corrections, thus making it more sensitive. Additionally, when examining potential energy surfaces, VQE demonstrates results that gradually align with CCSD(T) and QCISD(T) methods.

Variational quantum eigensolver for closed-shell molecules with non-bosonic corrections.

The realization of quantum advantage with noisy-intermediate-scale quantum (NISQ) machines has become one of the major challenges in computational sciences. Maintaining coherence of a physical system with more than ten qubits is a critical challenge that motivates research on compact system representations to reduce algorithm complexity. Toward this end, the variational quantum eigensolver (VQE) used to perform quantum simulations is considered to be one of the most promising algorithms for quantum chemistry in the NISQ era. We investigate reduced mapping of one spatial orbital to a single qubit to analyze the ground state energy in a way that the Pauli operators of qubits are mapped to the creation/annihilation of singlet pairs of electrons. To include the effect of non-bosonic (or non-paired) excitations, we introduce a simple correction scheme in the electron correlation model approximated by the geometrical mean of the bosonic (or paired) terms. Employing it in a VQE algorithm, we assess ground state energies of H2O, N2, and Li2O in good agreement with full configuration interaction (FCI) models respectively, using only 6, 8, and 12 qubits with quantum gate depths proportional to the squares of the qubit counts. With the adopted seniority-zero approximation that uses only one half of the qubit counts of a conventional VQE algorithm, we find that our non-bosonic correction method reaches reliable quantum chemistry simulations at least for the tested systems.

Unitary coupled-cluster for quantum computation of molecular properties in a strong magnetic field.

In truncated coupled-cluster (CC) theories, non-variational and/or generally complex ground-state energies can occur. This is due to the non-Hermitian nature of the similarity transformed Hamiltonian matrix in combination with CC truncation. For chemical problems that deal with real-valued Hamiltonian matrices, complex CC energies rarely occur. However, for complex-valued Hamiltonian matrices, such as those that arise in the presence of strong magnetic fields, complex CC energies can be regularly observed unless certain symmetry conditions are fulfilled. Therefore, in the presence of magnetic fields, it is desirable to pursue CC methods that are guaranteed to give upper-bound, real-valued energies. In this work, we present the first application of unitary CC to chemical systems in a strong magnetic field. This is achieved utilizing the variational quantum eigensolver algorithm applied to the unitary coupled-cluster singles and doubles (UCCSD) method. We benchmark the method on the H2 molecule in a strong magnetic field and then calculate UCCSD energies for the H4 molecule as a function of both geometry and field angle. We show that while standard CCSD can yield generally complex energies that are not an upper-bound to the true energy, UCCSD always results in variational and real-valued energies. We also show that the imaginary components of the CCSD energy are largest in the strongly correlated region. Last, the UCCSD calculations capture a large percentage of the correlation energy.

Using Differential Evolution to avoid local minima in Variational Quantum Algorithms

Variational Quantum Algorithms (VQAs) are among the most promising NISQ-era algorithms for harnessing quantum computing in diverse fields. However, the underlying optimization processes within these algorithms usually deal with local minima and barren plateau problems, preventing them from scaling efficiently. Our goal in this paper is to study alternative optimization methods that can avoid or reduce the effect of these problems. To this end, we propose to apply the Differential Evolution (DE) algorithm to VQAs optimizations. Our hypothesis is that DE is resilient to vanishing gradients and local minima for two main reasons: (1) it does not depend on gradients, and (2) its mutation and recombination schemes allow DE to continue evolving even in these cases. To demonstrate the performance of our approach, first, we use a robust local minima problem to compare state-of-the-art local optimizers (SLSQP, COBYLA, L-BFGS-B and SPSA) against DE using the Variational Quantum Eigensolver algorithm. Our results show that DE always outperforms local optimizers. In particular, in exact simulations of a 1D Ising chain with 14 qubits, DE achieves the ground state with a 100% success rate, while local optimizers only exhibit around 40%. We also show that combining DE with local optimizers increases the accuracy of the energy estimation once avoiding local minima. Finally, we demonstrate how our results can be extended to more complex problems by studying DE performance in a 1D Hubbard model.

Modular Cluster Circuits for the Variational Quantum Eigensolver.

The variational quantum eigensolver algorithm recently became a popular method to compute the quantum chemical properties of molecules on noisy intermediate scale quantum (NISQ) devices. In order to avoid noise accumulation from the NISQ device in the quantum circuit, it is important to keep the so-called quantum depth of the circuit at a minimum, defined as the minimum number of quantum gates that must be operated sequentially. In the present work, we introduce a modular 2-qubit cluster circuit that allows for the design of a shallow-depth quantum circuit compared to previously proposed architectures without loss of chemical accuracy. Moreover, by virtue of the simplicity of the cluster circuit, it is possible to assign a valence bond chemical interpretation to the cluster circuit. The design was tested on the H2, (H2)2, and LiH molecules, as well as the finite-size transverse-field Ising model, as the latter provides additional insights into the construction of the circuit in a resonating valence bond picture.

Fermionic Correlation Functions from Randomized Measurements in Programmable Atomic Quantum Devices.

We provide an efficient randomized measurement protocol to estimate two- and four-point fermionic correlations in ultracold atom experiments. Our approach is based on combining random atomic beam splitter operations, which can be realized with programmable optical landscapes, with high-resolution imaging systems such as quantum gas microscopes. We illustrate our results in the context of the variational quantum eigensolver algorithm for solving quantum chemistry problems.

Nuclear shell-model simulation in digital quantum computers

The nuclear shell model is one of the prime many-body methods to study the structure of atomic nuclei, but it is hampered by an exponential scaling on the basis size as the number of particles increases. We present a shell-model quantum circuit design strategy to find nuclear ground states by exploiting an adaptive variational quantum eigensolver algorithm. Our circuit implementation is in excellent agreement with classical shell-model simulations for a dozen of light and medium-mass nuclei, including neon and calcium isotopes. We quantify the circuit depth, width and number of gates to encode realistic shell-model wavefunctions. Our strategy also addresses explicitly energy measurements and the required number of circuits to perform them. Our simulated circuits approach the benchmark results exponentially with a polynomial scaling in quantum resources for each nucleus. This work paves the way for quantum computing shell-model studies across the nuclear chart and our quantum resource quantification may be used in configuration-interaction calculations of other fermionic systems.

Quantum Circuit Matrix Product State Ansatz for Large-Scale Simulations of Molecules.

As demonstrated in the density matrix renormalization group (DMRG) method, approximating many-body wave function of electrons using a matrix product state (MPS) is a promising way to solve electronic structure problems. The expressibility of an MPS is determined by the size of the matrices or, in other words, the bond dimension, which unfortunately may be required to be very large in quantum chemistry simulations. In this study, we propose to calculate the ground state energies of molecular systems by variationally optimizing quantum circuit MPS (QCMPS) with a relatively small number of qubits. It is demonstrated that with carefully chosen circuit structure and orbital localization scheme, QCMPS can reach a similar accuracy as that achieved in DMRG with an exponentially large bond dimension. QCMPS simulation of a linear hydrogen molecular chain with 50 orbitals can reach the chemical accuracy using only 6 qubits at a moderate circuit depth. These results suggest that QCMPS is a promising wave function ansatz in the variational quantum eigensolver algorithm for molecular systems.

Efficient Application of the Factorized form of the Unitary Coupled-Cluster Ansatz for the Variational Quantum Eigensolver Algorithm by Using Linear Combination of Unitaries

The variational quantum eigensolver is one of the most promising algorithms for near-term quantum computers. It has the potential to solve quantum chemistry problems involving strongly correlated electrons with relatively low-depth circuits, which are otherwise difficult to solve on classical computers. The variational eigenstate is constructed from a number of factorized unitary coupled-cluster terms applied onto an initial (single-reference) state. Current algorithms for applying one of these operators to a quantum state require a number of operations that scale exponentially with the rank of the operator. We exploit a hidden SU(2) symmetry to allow us to employ the linear combination of unitaries approach, Our Prepare subroutine uses n+2 ancilla qubits for a rank-n operator. Our Select(U^) scheme uses O(n)Cnot gates. This results in a full algorithm that scales like the cube of the rank of the operator n3, a significant reduction in complexity for rank five or higher operators. This approach, when combined with other algorithms for lower-rank operators (when compared to the standard implementation), will make the factorized form of the unitary coupled-cluster approach much more efficient to implement on all types of quantum computers.

Applications and resource reductions in measurement-based variational quantum eigensolvers

We discuss the procedure for obtaining measurement-based implementations of quantum algorithms given by quantum circuit diagrams and how to reduce the required resources needed for a given measurement-based computation. This forms the foundation for quantum computing on photonic systems in the near term. To demonstrate that these ideas are well grounded we present three different problems which are solved by employing a measurement-based implementation of the variational quantum eigensolver algorithm (MBVQE). We show that by utilising native measurement-based gates rather than standard gates, such as the standard controlled not gate (CNOT), measurement-based quantum computations may be obtained that are both shallow and have simple connectivity while simultaneously exhibiting a large expressibility. We conclude that MBVQE has promising prospects for resource states that are not far from what is already available today.

Implementing Jastrow-Gutzwiller operators on a quantum computer using the cascaded variational quantum eigensolver algorithm

A Jastrow-Gutzwiller operator adds many-body correlations to a quantum state. However, the operator is nonunitary, making it difficult to implement directly on a quantum computer. We present an implementation of the Jastrow-Gutzwiller operator using the cascaded variational quantum eigensolver algorithm. We demonstrate the method on IBM Q Lagos for a Hubbard model.

Measurement Optimization Techniques for Excited Electronic States in Near-Term Quantum Computing Algorithms.

The variational quantum eigensolver (VQE) remains one of the most popular near-term quantum algorithms for solving the electronic structure problem. Yet, for its practicality, the main challenge to overcome is improving the quantum measurement efficiency. Numerous quantum measurement techniques have been developed recently, but it is unclear how these state-of-the-art measurement techniques will perform in extensions of VQE for obtaining excited electronic states. Assessing the measurement techniques' performance in the excited state VQE is crucial because the measurement requirements in these extensions are typically much greater than in the ground state VQE, as one must measure the expectation value of multiple observables in addition to that of the electronic Hamiltonian. Here, we adapt various measurement techniques to two widely used excited state VQE algorithms: multistate contraction and quantum subspace expansion. Then, the measurement requirements of each measurement technique are numerically compared. We find that the best methods for multistate contraction are ones utilizing Hamiltonian data and wave function information to minimize the number of measurements. In contrast, randomized measurement techniques are more appropriate for quantum subspace expansion, with many more observables of vastly different energy scales to measure. Nevertheless, when the best possible measurement technique for each excited state VQE algorithm is considered, significantly fewer measurements are required in multistate contraction than in quantum subspace expansion.

Two-qubit quantum photonic processor manufactured by femtosecond laser writing

We present an experimental implementation of a two-qubit photonic quantum processor fabricated using femtosecond laser writing technology. We employ femtosecond laser writing to create a low-loss reconfigurable photonic chip, implementing precise single-qubit and two-qubit operations. We present a careful characterization of the performance of single- and two-qubit gates. An exemplary application of estimating the ground state energy of an H2 molecule using the variational quantum eigensolver algorithm is demonstrated. Our results highlight the potential of the femtosecond laser writing technology to deliver high quality small-scale quantum photonic processors.

Constructing Local Bases for a Deep Variational Quantum Eigensolver for Molecular Systems

Current quantum computers are limited in the number of qubits and coherence time, constraining the algorithms executable with sufficient fidelity. The variational quantum eigensolver (VQE) is an algorithm to find an approximate ground state of a quantum system and is expected to work on even such a device. The deep VQE [K. Fujii, et al., arXiv:2007.10917] is an extension of the original VQE algorithm, which takes a divide-and-conquer approach to relax the hardware requirement. While the deep VQE is successfully applied for spin models and periodic material, its validity on a molecule, where the Hamiltonian is highly nonlocal in the qubit basis, is still unexplored. Here, we discuss the performance of the deep VQE algorithm applied to quantum chemistry problems. Specifically, we examine different subspaceforming methods and compare their accuracy and complexity on a 10 H-atom treelike molecule as well as a 13 H-atom version. Additionally, we examine the performance on the natural occurring molecule retinal. This work also proposes multiple methods to lower the number of qubits required to calculate the ground state of a molecule. We find that the deep VQE can simulate the electron-correlation energy of the ground state to an error of below 1%, thus helping us to reach chemical accuracy in some cases. The accuracy differences and qubits' reduction highlights the basis creation method's impact on the deep VQE.

Quantum Simulation of Molecules in Solution.

Quantum chemical calculations on quantum computers have been focused mostly on simulating molecules in the gas phase. Molecules in liquid solution are, however, most relevant for chemistry. Continuum solvation models represent a good compromise between computational affordability and accuracy in describing solvation effects within a quantum chemical description of solute molecules. In this work, we extend the variational quantum eigensolver to simulate solvated systems using the polarizable continuum model. To account for the state dependent solute-solvent interaction we generalize the variational quantum eigensolver algorithm to treat non-linear molecular Hamiltonians. We show that including solvation effects does not impact the algorithmic efficiency. Numerical results of noiseless simulations for molecular systems with up to 12 spin-orbitals (qubits) are presented. Furthermore, calculations performed on a simulated noisy quantum hardware (IBM Q, Mumbai) yield computed solvation free energies in fair agreement with the classical calculations.

Constrained quantum optimization for extractive summarization on a trapped-ion quantum computer

Realizing the potential of near-term quantum computers to solve industry-relevant constrained-optimization problems is a promising path to quantum advantage. In this work, we consider the extractive summarization constrained-optimization problem and demonstrate the largest-to-date execution of a quantum optimization algorithm that natively preserves constraints on quantum hardware. We report results with the Quantum Alternating Operator Ansatz algorithm with a Hamming-weight-preserving XY mixer (XY-QAOA) on trapped-ion quantum computer. We successfully execute XY-QAOA circuits that restrict the quantum evolution to the in-constraint subspace, using up to 20 qubits and a two-qubit gate depth of up to 159. We demonstrate the necessity of directly encoding the constraints into the quantum circuit by showing the trade-off between the in-constraint probability and the quality of the solution that is implicit if unconstrained quantum optimization methods are used. We show that this trade-off makes choosing good parameters difficult in general. We compare XY-QAOA to the Layer Variational Quantum Eigensolver algorithm, which has a highly expressive constant-depth circuit, and the Quantum Approximate Optimization Algorithm. We discuss the respective trade-offs of the algorithms and implications for their execution on near-term quantum hardware.