Adaptive Quantum Subspace Diagonalization for Transition-State Mapping in Battery Electrolyte Reactions

The Variational Quantum Eigensolver (VQE) is a method of choice to solve the electronic structure problem for molecules on near-term gate-based quantum computers. However, the circuit depth is expected to grow significantly with problem size. Increased depth can both degrade the accuracy of the results and reduce trainability. In this work, we propose a novel approach to reduce ansatz circuit depth. Our approach, called PermVQE, adds an additional optimization loop to VQE that permutes qubits in order to solve for the qubit Hamiltonian that minimizes long-range correlations in the ground state. The choice of permutations is based on mutual information, which is a measure of interaction between electrons in spin-orbitals. Encoding strongly interacting spin-orbitals into proximal qubits on a quantum chip naturally reduces the circuit depth needed to prepare the ground state. For representative molecular systems, LiH, H$_2$, (H$_2$)$_2$, H$_4$, and H$_3^+$, we demonstrate for linear qubit connectivity that placing entangled qubits in close proximity leads to shallower depth circuits required to reach a given eigenvalue-eigenvector accuracy. This approach can be extended to any qubit connectivity and can significantly reduce the depth required to reach a desired accuracy in VQE. Moreover, our approach can be applied to other variational quantum algorithms beyond VQE.

The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm for finding the minimum eigenvalue of a Hamiltonian that involves the optimization of a parameterized quantum circuit. Since the resulting optimization problem is in general nonconvex, the method can converge to suboptimal parameter values that do not yield the minimum eigenvalue. In this work, we address this shortcoming by adopting the concept of variational adiabatic quantum computing (VAQC) as a procedure to improve VQE. In VAQC, the ground state of a continuously parameterized Hamiltonian is approximated via a parameterized quantum circuit. We discuss some basic theory of VAQC to motivate the development of a hybrid quantum-classical homotopy continuation method. The proposed method has parallels with a predictor-corrector method for numerical integration of differential equations. While there are theoretical limitations to the procedure, we see in practice that VAQC can successfully find good initial circuit parameters to initialize VQE. We demonstrate this with two examples from quantum chemistry. Through these examples, we provide empirical evidence that VAQC, combined with other techniques (an adaptive termination criteria for the classical optimizer and a variance-based resampling method for the expectation evaluation), can provide more accurate solutions than “plain” VQE, for the same amount of effort.

Reducing circuit depth is essential for implementing quantum simulations of electronic structure on near-term quantum devices. In this work, we propose a variational quantum eigensolver (VQE) based perturbation theory (PT) algorithm to accurately simulate electron correlation of periodic materials with shallow ansatz circuits, which are generated from adaptive derivative-assembled pseudo-trotter or qubit-excitation-based VQE calculations using a loose convergence criteria. Here, the major part of the electron correlation is described using the VQE ansatz circuit and the remaining correlation energy is described by either multireference or similarity transformation-based PT. Numerical results demonstrate that the new algorithms are able to accurately describe electron correlation of the LiH crystal with only one circuit parameter, in contrast with ~30 parameters required in the adaptive VQE to achieve the same accuracy. Meanwhile, for fixed-depth ansätze, e.g. unitary coupled cluster, we demonstrate that the VQE-based PT provides an appealing scheme to improve their accuracy.

Variational quantum algorithms are suitable for use on noisy quantum systems. One of the most important use-cases is the quantum simulation of materials, using the variational quantum eigensolver (VQE). To optimize VQE performance, a suitable parameterized quantum circuit (ansatz) must be selected. We investigate a class of ansatze that incorporates knowledge of the quantum hardware, namely the hardware efficient ansatze. The performance of hardware efficient ansatze is affected differently by noise, and our goal is to study the effect of noise on evaluating which ansatz gives more accurate results in practice. First, we study the effect of noise on the different hardware efficient ansatze by benchmarking and ranking the performance of each ansatz family (i) on a chemistry application using VQE and (ii) by the recently established metric of "expressibility". The results demonstrate the ranking of optimal circuits does not remain constant in the presence of noise. Second, we evaluate the suitability of the expressibility measure in this context by performing a correlation study between expressibility and the performance of the same circuits on a chemistry application using VQE. Our simulations reveal a weak correlation and therefore demonstrate that expressibility is not an adequate measure to quantify the effectiveness of parameterized quantum circuits for quantum chemistry. Third, we evaluate the effect of different quantum device noise models on the ordering of which ansatz family is best. Interestingly, we see that to decide which ansatz is optimal for use, one needs to consider the specific hardware used even within the same family of quantum hardware.

Quantum chemistry is one of the most promising near-term applications of quantum computers. Quantum algorithms such as variational quantum eigen solver (VQE) and variational quantum deflation (VQD) algorithms have been mainly applied for molecular systems and there is a need to implement such methods for periodic solids. Using Wannier tight-binding Hamiltonian (WTBH) approaches, we demonstrate the application of VQE and VQD to accurately predict both electronic and phonon bandstructure properties of several elemental as well as multi-component solid-state materials. We apply VQE–VQD calculations for 307 spin–orbit coupling based electronic WTBHs and 933 finite-difference based phonon WTBHs. Also, we discuss a workflow for using VQD with lattice Green’s function that can be used for solving dynamical mean-field theory problems. The WTBH model solvers can be used for testing other quantum algorithms and models also.

We explore the use of symmetry-adapted perturbation theory (SAPT) as a simple and efficient means to compute interaction energies between large molecular systems with a hybrid method combining NISQ-era quantum and classical computers. From the one- and two-particle reduced density matrices of the monomer wavefunctions obtained by the variational quantum eigensolver (VQE), we compute SAPT contributions to the interaction energy [SAPT(VQE)]. At first order, this energy yields the electrostatic and exchange contributions for non-covalently bound systems. We empirically find from ideal statevector simulations that the SAPT(VQE) interaction energy components display orders of magnitude lower absolute errors than the corresponding VQE total energies. Therefore, even with coarsely optimized low-depth VQE wavefunctions, we still obtain sub kcal mol−1 accuracy in the SAPT interaction energies. In SAPT(VQE), the quantum requirements, such as qubit count and circuit depth, are lowered by performing computations on the separate molecular systems. Furthermore, active spaces allow for large systems containing thousands of orbitals to be reduced to a small enough orbital set to perform the quantum portions of the computations. We benchmark SAPT(VQE) (with the VQE component simulated by ideal statevector simulators) against a handful of small multi-reference dimer systems and the iron center containing human cancer-relevant protein lysine-specific demethylase 5 (KDM5A).

Abstract: This paper explores hybrid quantum-classical optimization algorithms implemented across multiple quantum software development kits (SDKs): Qiskit and Cirq,. We evaluate the Variational Quantum Eigensolver (VQE) and Quantum Approximate Optimization Algorithm (QAOA) in terms of convergence rate, final cost value, and fidelity, using both simulators and real quantum hardware backends. Our results offer comparative insights into SDK performance and propose a standardized benchmarking approach for future quantum algorithm research. Keywords: Hybrid Quantum Algorithms, Qiskit, Cirq, VQE, QAOA, Quantum SDKs, Quantum Optimization, NISQ

Quantum computers are the next evolution of computing hardware. Quantum devices are being exposed through the same familiar cloud platforms used for classical computers, and enabling seamless execution of hybrid applications that combine quantum and classical components. Quantum devices vary in features, e.g., number of qubits, quantum volume, CLOPS, noise profile, queuing delays and resource cost. So, it may be useful to split hybrid workloads with either large quantum circuits or a large number of quantum circuits, into smaller units. In this paper, we profile two workload splitting techniques on IBM's Quantum Cloud: (1) Circuit parallelization, to split one large circuit into multiple smaller ones, and (2) Data parallelization to split a large number of circuits that run on one device to smaller batches of circuits that run on concurrently on multiple devices. These can improve the utilization of heterogeneous quantum hardware, but involve trade-offs. We evaluate these techniques on two key algorithmic classes: Variational Quantum Eigensolver (VQE) and Quantum Support Vector Machine (QSVM), and measure the impact on quality of the result, circuit execution times, quantum resource usage, and processing overheads, relative to a baseline without such parallelization. The results are obtained on real hardware and complemented by simulations. We see that (1) VQE with circuit cutting is ~ 39% better in ground state estimation than the uncut version, and (2) QSVM that combines data parallelization with a reduced feature set yields upto 3 ×improvement in quantum workload execution time and reduces quantum resource use by 3 ×, while providing comparable accuracy. Error mitigation can improve the accuracy by ≈ 7% and reduce resource footprint by ≈4% compared to the best case among the considered scenarios.

Variational Quantum Eigensolver (VQE) provides a powerful solution for approximating molecular ground state energies by combining quantum circuits and classical computers. However, estimating probabilistic outcomes on quantum hardware requires repeated measurements (shots), incurring significant costs as accuracy increases. Optimizing shot allocation is thus critical for improving the efficiency of VQE. Current strategies rely heavily on hand-crafted heuristics requiring extensive expert knowledge. This paper proposes a reinforcement learning (RL)-based approach that automatically learns shot assignment policies to minimize total measurement shots while achieving convergence to the minimum of the energy expectation in VQE. The RL agent assigns measurement shots across VQE optimization iterations based on the progress of the optimization. This approach reduces VQE's dependence on static heuristics and human expertise. When the RL-enabled VQE is applied to a small molecule, a shot reduction policy is learned. The policy demonstrates transferability across systems and compatibility with other wavefunction Ansätze. In addition to these specific findings, this work highlights the potential of RL for automatically discovering efficient and scalable quantum optimization strategies.

We describe a hybrid quantum-classical approach to treat quantum many-body systems in the thermodynamic limit. This is done by combining numerical linked-cluster expansions (NLCE) with the variational quantum eigensolver (VQE). Here, the VQE algorithm is used as a cluster solver within the NLCE. We test our hybrid quantum-classical algorithm ($\mathrm{NLCE}+\mathrm{VQE}$) for the ferromagnetic transverse-field Ising model on the one-dimensional chain and the two-dimensional square lattice. The calculation of ground-state energies on each open cluster demands a modified Hamiltonian variational ansatz for the VQE. One major finding is convergence of $\mathrm{NLCE}+\mathrm{VQE}$ to the conventional NLCE result in the thermodynamic limit when at least $N/2$ circuit repetitions (known as layers) are used in the VQE ansatz for each cluster with $N$ sites. Our approach demonstrates the fruitful connection of techniques known from correlated quantum many-body systems with hybrid algorithms explored on existing quantum-computing devices.

Materials simulations involving strongly correlated electrons pose fundamental challenges to state‐of‐the‐art electronic structure methods but are hypothesized to be the ideal use case for quantum computing algorithms. To date, no quantum computer has simulated a molecule of a size and complexity relevant to real‐world applications, despite the fact that the variational quantum eigensolver (VQE) algorithm can predict chemically accurate total energies. Nevertheless, because of the many applications of moderately sized, strongly correlated systems, such as molecular catalysts, the successful use of the VQE stands as an important waypoint in the advancement toward useful chemical modeling on near‐term quantum processors. In this paper, we take a significant step in this direction. We lay out the steps, write, and run parallel code for an (emulated) quantum computer to compute the bond dissociation curves of the TiH, LiH, NaH, and KH diatomic hydride molecules using the VQE. TiH was chosen as a relatively simple chemical system that incorporates d orbitals and strong electron correlation. Because current VQE implementations on existing quantum hardware are limited by qubit error rates, the number of qubits available, and the allowable gate depth, recent studies using it have focused on chemical systems involving s and p block elements. Through VQE + UCCSD calculations of TiH, we evaluate the near‐term feasibility of modeling a molecule with d‐orbitals on real quantum hardware. We demonstrate that the inclusion of d‐orbitals and the use of the UCCSD ansatz, which are both necessary to capture the correct TiH physics, dramatically increase the cost of this problem. We estimate the approximate error rates necessary to model TiH on current quantum computing hardware using VQE + UCCSD and show them to likely be prohibitive until significant improvements in hardware and error correction algorithms are available.

Solving the electronic structure problem is a notorious challenge in quantum chemistry and material science. Variational quantum eigensolver (VQE) is a promising hybrid classical-quantum algorithm for finding the lowest-energy configuration of a molecular system. However, it typically requires many qubits and quantum gates with substantial quantum circuit depth to accurately represent the electronic wave function of complex structures. Here, we propose an alternative approach to solve the electronic structure problem using VQE with a single qudit. Our approach exploits a high-dimensional orbital angular momentum state of a heralded single photon and notably reduces the required quantum resources compared to conventional multi-qubit-based VQE. We experimentally demonstrate that our single-qudit-based VQE can efficiently estimate the ground state energy of hydrogen (H2) and lithium hydride (LiH) molecular systems corresponding to two- and four-qubit systems, respectively. We believe that our scheme opens a pathway to perform a large-scale quantum simulation for solving more complex problems in quantum chemistry and material science.

The possibility of adopting quantum computing in the field of financial services can bring a new breakthrough in portfolio management since the optimization of this method requires large numbers for the solution that exceed the capabilities of classical computing algorithms. This study focuses on using a hybrid quantum-classical computational method known as the Variational Quantum Eigensolver (VQE) in optimizing portfolio selection. The VQE aims at finding the best allocation of the portfolio; the best weights that correspond to minimum risk and maximum return as predicted by the modern portfolio theory. The study assesses the suitability and speed of VQE in solving matters involving portfolio management tasks such as handling vast data and nonlinear constraints which are hard to solve through other optimization methods. Thus, under the theoretical framework of quantum computing, the study seeks to address some of the following challenges: computational time and scalability in portfolio optimization. The VQE algorithm work on different portfolio configurations; their performance in comparison with classical techniques; special attention is paid to the rate of convergence and the achievement of the optimal solution. Based on the results of this work we can conclude that there is a considerable literature on quantum finance and that this field can benefit from utilization of quantum algorithms as a tool to improve decision-making in the financial market. That is why the study proves that quantum optimization techniques, to wit VQE, can provide better solutions when it comes to portfolio management within the sphere of finance.

The variational quantum eigensolver (VQE) stands as a prominent quantum-classical hybrid algorithm for near-term quantum computers to obtain the ground states of molecular Hamiltonians in quantum chemistry. However, due to the noncommutativity of the Pauli operators in the Hamiltonian, the number of measurements required on quantum computers increases significantly as the system size grows, which may hinder practical applications of VQE. In this work, we present a protocol termed (JBM-VQE) to reduce the number of measurements and speed up the VQE algorithm. Our method employs joint Bell measurements, enabling the simultaneous measurement of the absolute values of all expectation values of Pauli operators present in the Hamiltonian. In the course of the optimization, JBM-VQE estimates the absolute values of the expectation values of the Pauli operators for each iteration by the joint Bell measurement, while the signs of them are measured less frequently by the conventional method to measure the expectation values. Our approach is based on the empirical observation that the signs do not often change during optimization. We illustrate the speed-up of JBM-VQE compared to conventional VQE by numerical simulations for finding the ground states of molecular Hamiltonians of small molecules, and the speed-up of JBM-VQE at the early stage of the optimization becomes increasingly pronounced in larger systems. Our approach based on the joint Bell measurement is not limited to VQE and can be utilized in various quantum algorithms whose cost functions are expectation values of many Pauli operators. Published by the American Physical Society 2024

The variational quantum eigensolver (VQE) algorithm, designed to calculate the energy of molecular ground states on near‐term quantum computers, requires specification of symmetries that describe the system, for example, spin state and number of electrons. This opens the possibility of using VQE to obtain excited states as the lowest‐energy solutions of a given set of symmetries. In this paper, the performances of various unitary coupled cluster (UCC) ansätze applied to VQE calculations on excited states are investigated using quantum circuits designed to represent single reference and multireference wavefunctions to calculate energy curves with respect to variations in the molecular geometry. These ansätze include standard Unitary Coupled Cluster Singles and Doubles (UCCSD), as well as modified versions of Unitary Coupled Cluster Generalized Singles and Doubles (UCCGSD) and k‐UpCCGSD, which are engineered to tackle excited states without undesired spin symmetry crossover to lower states during VQE optimization. These studies are carried out on a range of systems, including H2, H3, H4, NH, OH+, CH2, and , covering examples of spin singlet, doublet, and triplet molecular ground states with single and multireference excited states. In most cases, our calculations are in excellent agreement with results from full‐configuration interaction calculations on classical machines, thus showing that the VQE algorithm is capable of calculating the lowest excited state at a certain symmetry, including multireference closed‐ and open‐shell states, by setting appropriate restrictions on the excitations considered in the cluster operator and appropriate constraints in the qubit register encoding the starting mean field state.