Solving for molecular excited states remains one of the key challenges of modern quantum chemistry. Traditional methods are constrained by existing computational capabilities, limiting the complexity of the molecules that can be studied or the accuracy of the results that can be obtained. Several quantum computing methods have been suggested to address this limitation. However, these typically have hardware requirements which may not be achieved in the near term. We propose a variational quantum machine learning based method to determine molecular excited states aiming at being as resilient as possible to the defects of early Noisy Intermediate Scale Quantum (NISQ) computers and demonstrate an implementation for H2 on IBMQ. Our method uses a combination of two parametrized quantum circuits, working in tandem, combined with a Variational Quantum Eigensolver (VQE) to iteratively find the eigenstates of a molecular Hamiltonian.
The negative hydrogen ion is the first three body quantum problem whose ground state energy is calculated using the `Chandrasekhar Wavefunction' that accounts for the electron-electron correlation. Solving multi-body systems is a daunting task in quantum mechanics as it includes choosing a trial wavefunction and the calculation of integrals for the system that becomes almost impossible for systems with three or more particles. This difficulty can be addressed by quantum computers. They have emerged as a tool to address different electronic structure problems with remarkable efficiency. They ha…
Quantum computers have the potential to transform the ways in which we tackle some important problems. The efforts by companies like Google, IBM and Microsoft to construct quantum computers have been making headlines for years. Equally important is the challenge of translating problems into a state that can be fed to these machines. Because quantum computers are in essence controllable quantum systems, the problems that most naturally map to them are those of quantum mechanics. Quantum chemistry has seen particular success in the form of the variational quantum eigensolver (VQE) algorithm, whi…
One of the most promising applications of noisy intermediate-scale quantum computers is the simulation of molecular Hamiltonians using the variational quantum eigensolver. We show that encoding symmetries of the simulated Hamiltonian in the VQE ansatz reduces both classical and quantum resources compared to other, widely available ansatze. Through simulations of the H$_2$ molecule, we verify that these improvements persist in the presence of noise. This simulation is performed with IBM software using noise models from real devices. We also demonstrate how these techniques can be used to find m…
Quantum computers are anticipated to transcend classical supercomputers for computationally intensive tasks by exploiting the principles of quantum mechanics. However, the capabilities of the current generation of quantum devices are limited due to noise or errors, and therefore implementation of error mitigation and/or correction techniques is pivotal to reliably process quantum algorithms. In this work, we have performed a comparative analysis of the error mitigation capability of the [[4,2,2]] quantum error-detecting code (QEC method), duplicate circuit technique, and the Bayesian read-out …
Quantum algorithms can be used to perform unsupervised machine learning tasks like data clustering by mapping the distance between data points to a graph optimization problem (i.e. MAXCUT) and finding optimal solution through energy minimization using hybrid quantum classical methods. Taking advantage of the IBM runtime environment, we benchmark the performance of the "Warm-Start" (ws) variant of Quantum Approximate Optimization Algorithm (QAOA) versus the standard implementation of QAOA and the variational quantum eigensolver (VQE) for unstructured clustering problems using real world dataset…
The ground and excited state calculations at key geometries, such as the Frank-Condon (FC) and the conical intersection (CI) geometries, are essential for understanding photophysical properties. To compute these geometries on noisy intermediate-scale quantum devices, we proposed a strategy that combined a chemistry-inspired spin-restricted ansatz and a new excited state calculation method called the variational quantum eigensolver under automatically-adjusted constraints (VQE/AC). Unlike the conventional excited state calculation method, called the variational quantum deflation, the VQE/AC doe…
Variational quantum eigensolver (VQE) is promising to show quantum advantage on near-term noisy-intermediate-scale quantum (NISQ) computers. One central problem of VQE is the effect of noise, especially the physical noise on realistic quantum computers. We study systematically the effect of noise for the VQE algorithm, by performing numerical simulations with various local noise models, including the amplitude damping, dephasing, and depolarizing noise. We show that the ground state energy will deviate from the exact value as the noise probability increase and normally noise will accumulate as…
Hybrid quantum-classical embedding methods for correlated materials simulations provide a path towards potential quantum advantage. However, the required quantum resources arising from the multi-band nature of $d$ and $f$ electron materials remain largely unexplored. Here we compare the performance of different variational quantum eigensolvers in ground state preparation for interacting multi-orbital embedding impurity models, which is the computationally most demanding step in quantum embedding theories. Focusing on adaptive algorithms and models with 8 spin-orbitals, we show that state prepa…
The characterization of observables, expressed via Hermitian operators, is a crucial task in quantum mechanics. For this reason, an eigensolver is a fundamental algorithm for any quantum technology. In this work, we implement a semi-autonomous algorithm to obtain an approximation of the eigenvectors of an arbitrary Hermitian operator using the IBM quantum computer. To this end, we only use single-shot measurements and pseudo-random changes handled by a feedback loop, reducing the number of measures in the system. Due to the classical feedback loop, this algorithm can be cast into the reinforce…
Quantum computers have an exponential speed-up advantage over classical computers. One of the most prominent utilities of quantum computers is their ability to study complex quantum systems in various fields using quantum computational algorithms. Quantum computational algorithms can be used to study cosmological systems and how they behave with variations in the different parameters of the system. Here, we use the variational quantum eigensolver (VQE) algorithm to simulate the Hawking radiation phenomenon. VQE algorithm is a combination of quantum and classical computation methods used to obt…
Quantum computers are expected to be highly beneficial for chemistry simulations, promising significant improvements in accuracy and speed. The most prominent algorithm for chemistry simulations on NISQ devices is the Variational Quantum Eigensolver (VQE). It is a hybrid quantum-classical algorithm which calculates the ground state energy of a Hamiltonian based on parametrized quantum circuits, while a classical optimizer is used to find optimal parameter values. However, quantum hardware is affected by noise, and it needs to be understood to which extent it can degrade the performance of the …
Variational quantum eigensolver (VQE) is a promising algorithm suitable for near-term quantum machines. VQE aims to approximate the lowest eigenvalue of an exponentially sized matrix in polynomial time. It minimizes quantum resource requirements both by co-processing with a classical processor and by structuring computation into many subproblems. Each quantum subproblem involves a separate state preparation terminated by the measurement of one Pauli string. However, the number of such Pauli strings scales as $N^4$ for typical problems of interest--a daunting growth rate that poses a serious li…
Quantum computers promise to efficiently solve important problems that are intractable on a conventional computer. Quantum computational algorithms have the potential to be an exciting new way of studying quantum cosmology. In quantum cosmology, we learn about the dynamics of the universe without constructing a complete theory of quantum gravity. Since the universal wavefunction exists in an infinite-dimensional superspace over all possible 3D metrics and modes of matter configurations, we take minisuperspaces for our work by constraining the degrees of freedom to particular 3D metrics and uni…
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 favourable 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 quasi-dynamical unitary evolution. We conduct…
The variational quantum eigensolver (VQE) is one of the most promising algorithms to find eigenvalues and eigenvectors of a given Hamiltonian on noisy intermediate-scale quantum (NISQ) devices. A particular application is to obtain ground or excited states of molecules. The practical realization is currently limited by the complexity of quantum circuits. Here we present a novel approach to reduce quantum circuit complexity in VQE for electronic structure calculations. Our algorithm, called ClusterVQE, splits the initial qubit space into subspaces (qubit clusters) which are further distributed …
As quantum computing approaches its first commercial implementations, quantum simulation emerges as a potentially ground-breaking technology for several domains, including Biology and Chemistry. However, taking advantage of quantum algorithms in Quantum Chemistry raises a number of theoretical and practical challenges at different levels, from the conception to its actual execution. We go through such challenges in a case study of a quantum simulation for the hydrogen (H2) and lithium hydride (LiH) molecules, at an actual commercially available quantum computer, the IBM Q. The former molecule …
This paper investigates the application of quantum computing technology to airline gate-scheduling quadratic assignment problems (QAP). We explore the quantum computing hardware architecture and software environment required for porting classical versions of these type of problems to quantum computers. We discuss the variational quantum eigensolver and the inclusion of space-efficient graph coloring to the Quadratic Unconstrained Binary Optimization (QUBO). These enhanced quantum computing algorithms are tested with an 8 gate and 24 flight test case using both the IBM quantum computing simulat…
Combinatorial optimization models a vast range of industrial processes aiming at improving their efficiency. In general, solving this type of problem exactly is computationally intractable. Therefore, practitioners rely on heuristic solution approaches. Variational quantum algorithms are optimization heuristics that can be demonstrated with available quantum hardware. In this case study, we apply four variational quantum heuristics running on IBM's superconducting quantum processors to the job shop scheduling problem. Our problem optimizes a steel manufacturing process. A comparison on 5 qubit…
Variational quantum algorithms have shown promise in numerous fields due to their versatility in solving problems of scientific and commercial interest. However, leading algorithms for Hamiltonian simulation, such as the Variational Quantum Eigensolver (VQE), use fixed preconstructed ansatzes, limiting their general applicability and accuracy. Thus, variational forms---the quantum circuits that implement ansatzes ---are either crafted heuristically or by encoding domain-specific knowledge. In this paper, we present an Evolutionary Variational Quantum Eigensolver (EVQE), a novel variational alg…
The social worker scheduling problem is a class of combinatorial optimization problems that combines scheduling with routing issues. These types of problems with classical computing can only be solved, in the best of cases, in an approximate way and significantly when the input data does not grow considerably. Today, the focus on the quantum computer should no longer be only on its enormous computing power, but also on the use of its imperfection for this era, (Noisy Intermediate-Scale Quantum (NISQ)) to create a powerful optimization and learning device that uses variational techniques. We ha…
The theory of stochastic processes impacts both physical and social sciences. At the molecular scale, stochastic dynamics is ubiquitous because of thermal fluctuations. The Fokker-Plank-Smoluchowski equation models the time evolution of the probability density of selected degrees of freedom in the diffusive regime and it is therefore a workhorse of physical chemistry. In this paper we report the development and implementation of a Variational Quantum Eigensolver procedure to solve the Fokker-Planck-Smoluchowski eigenvalue problem. We show that such an algorithm, typically adopted to address qu…
We propose an efficient method for simultaneously optimizing both the structure and parameter values of quantum circuits with only a small computational overhead. Shallow circuits that use structure optimization perform significantly better than circuits that use parameter updates alone, making this method particularly suitable for noisy intermediate-scale quantum computers. We demonstrate the method for optimizing a variational quantum eigensolver for finding the ground states of Lithium Hydride and the Heisenberg model in simulation, and for finding the ground state of Hydrogen gas on the IB…
Estimation of deuteron binding energy with renormalization group-based effective interactions using the variational quantum eigensolver
Indian Institute of Technology Madras（印度马德拉斯理工学院）；Center for Quantum Information, Communication and Computing (CQuICC), Indian Institute of Technology Madras（印度马德拉斯理工学院量子信息、通信与计算中心）
In this paper we develop a quantum algorithm to realize finite temperature simulation on a quantum computer. As quantum computers use real-time evolution we did not use the imaginary time methods popular on classical algorithms. Instead, we implemented a real-time thermo field dynamics formalism, which has the added benefit of being able to compute quantities that are both time- and temperature-dependent. To implement thermo field dynamics we apply a unitary transformation to discrete quantum mechanical operators to make new Hamiltonians with encoded temperature dependence. The method works no…
The intractability of deterministic solutions in solving $\mathcal{NP}$-Hard Combinatorial Optimisation Problems (COP) is well reported in the literature. One mechanism for overcoming this difficulty has been the use of efficient COP non-deterministic approaches. However, with the advent of quantum technology, these modern devices' potential to overcome this tractability limitation requires exploration. This paper juxtaposes classical and quantum optimisation algorithms' performance to solve two common COP, the Travelling Salesman Problem (TSP) and the Quadratic Assignment Problem (QAP). Two a…
We study the effective matrix model for for gauge fields and fermions on a quantum computer. We use the Variational Quantum Eigensolver (VQE) using IBM QISKit for the effective matrix model for SU(2) and SU(3) including fermions in the fundamental representation. For SU(2) we study the effects of finite temperature and nonzero chemical potential. In all cases we find excellent agreement with the classical computation.
We present error mitigation (EM) techniques for noisy intermediate-scale quantum computers (QC) based on density matrix purification and perturbative corrections to the target energy. We incorporate this scheme into the variational quantum eigensolver (VQE) and demonstrate chemically-accurate ground state energy calculations of various alkali metal hydrides using IBM quantum computers. Both the density matrix purification improvements and the perturbative corrections require only meager classical computational resources, and are conducted exclusively as post-processing of the measured density …
The emerging field of quantum simulation of many-body systems is widely recognized as a very important application of quantum computing. A crucial step towards its realization in the context of many-electron systems requires a rigorous quantum mechanical treatment of the different interactions. In this pilot study, we investigate the physical effects beyond the mean-field approximation, known as electron correlation, in the ground state energies of atomic systems using the classical-quantum hybrid variational quantum eigensolver (VQE) algorithm. To this end, we consider three isoelectronic spe…
We study the Z3 gauge theory with fermions on the quantum computer using the Variational Quantum Eigensolver (VQE) algorithm with IBM QISKit software. Using up to 9 qubits we are able to obtain accurate results for the ground state energy. Introducing nonzero chemical potential we are able to determine the Equation of State (EOS) for finite density on the quantum computer. We discuss possible realizations of quantum advantage for this system over classical computers with regards to finite density simulations and the fermion sign problem.
Algorithmic discrepancy theory seeks efficient algorithms to find those two-colorings of a set that minimize a given measure of coloring imbalance in the set, its {\it discrepancy}. The {\it Euclidean discrepancy} problem and the problem of balancing covariates in randomized trials have efficient randomized algorithms based on the Gram-Schmidt walk (GSW). We frame these problems as quantum Ising models, for which variational quantum algorithms (VQA) are particularly useful. Simulating an example of covariate-balancing on an IBM quantum simulator, we find that the variational quantum eigensolve…
Quantum simulations of molecular systems on quantum computers often employ minimal basis sets of Gaussian orbitals. In comparison with more realistic basis sets, quantum simulations employing minimal basis sets require fewer qubits and quantum gates, but yield results of lower accuracy. A natural strategy to achieve more accurate results is to increase the basis set size, which in turn requires increasing the number of qubits and quantum gates. Here we explore the use of intrinsic atomic orbitals (IAOs) in quantum simulations of molecules, to improve the accuracy of energies and properties at …
Quantum computing has demonstrated the potential to revolutionize our understanding of nuclear, atomic, and molecular structure by obtaining forefront solutions in non-relativistic quantum many-body theory. In this work, we show that quantum computing can be used to solve for the structure of hadrons, governed by strongly-interacting relativistic quantum field theory. Following our previous work on light unflavored mesons as a relativistic bound-state problem within the nonperturbative Hamiltonian formalism, we present the numerical calculations on simulated quantum devices using the basis lig…
