Quantum-classical Algorithm Research Articles

Enquiring Electronic Structure Using Quantum Computers: Hands on Qiskit

Solving the electronic structure for multi-electronic systems is a hard problem. Even for small atoms and molecules, approximations have to be made in order to solve numerically the Schrödinger equation. Although different methods have been developed to take into account electron correlations, their computational cost reduces their feasibility. Quantum simulation provides an alternative to traditional computational methods for enquiring the electronic structure of molecules. Specifically, the Variational Quantum Eigensolver (VQE) algorithm provides a hybrid quantum-classical algorithm for the implementation on current near term quantum devices. In this work, we explore the implementation of VQE on Qiskit for calculating the ground-state energy of diatomic Hydrogen molecule.

Symmetry enhanced variational quantum spin eigensolver

The variational quantum-classical algorithms are the most promising approach for achieving quantum advantage on near-term quantum simulators. Among these methods, the variational quantum eigensolver has attracted a lot of attention in recent years. While it is very effective for simulating the ground state of many-body systems, its generalization to excited states becomes very resource demanding. Here, we show that this issue can significantly be improved by exploiting the symmetries of the Hamiltonian. The improvement is even more effective for higher energy eigenstates. We introduce two methods for incorporating the symmetries. In the first approach, called hardware symmetry preserving, all the symmetries are included in the design of the circuit. In the second approach, the cost function is updated to include the symmetries. The hardware symmetry preserving approach indeed outperforms the second approach. However, integrating all symmetries in the design of the circuit could be extremely challenging. Therefore, we introduce hybrid symmetry preserving method in which symmetries are divided between the circuit and the classical cost function. This allows to harness the advantage of symmetries while preventing sophisticated circuit design.

Improved Algorithms of Quantum Imaginary Time Evolution for Ground and Excited States of Molecular Systems.

Quantum imaginary time evolution (QITE) is a recently proposed quantum-classical hybrid algorithm that is guaranteed to reach the lowest state of systems. In this study, we present several improvements on QITE, mainly focusing on molecular applications. We analyze the derivation of the underlying QITE equation order-by-order and suggest a modification that is theoretically well founded. Our results clearly indicate the soundness of the here-derived equation, enabling a better approximation of the imaginary time propagation by a unitary. We also discuss how to accurately estimate the norm of an imaginary-time-evolved state and applied it to excited-state calculations using the quantum Lanczos algorithm. Finally, we propose the folded-spectrum QITE scheme as a straightforward extension of QITE for general excited-state simulations. The effectiveness of all these developments is illustrated by simulations with or without noise effect, offering further insights into quantum algorithms for imaginary time evolution.

Integration of Simulated Quantum Annealing in Parallel Tempering and Population Annealing for Heterogeneous-Profile QUBO Exploration

Simulated Quantum Annealing (SQA) is a heuristic algorithm which can solve Quadratic Unconstrained Binary Optimization (QUBO) problems by emulating the exploration of the solution space done by a quantum annealer. It mimics the quantum superposition and tunnelling effects through a set of correlated replicas of the spins system representing the problem to be solved and performing Monte Carlo steps. However, the effectiveness of SQA over a classical algorithm strictly depends on the cost/energy profile of the target problem. In fact, quantum annealing only performs well in exploring functions with high and narrow peaks, while classical annealing is better in overcoming flat and wide energy-profile barriers. Unfortunately, real-world problems have a heterogeneous solution space and the probability of success of each solver depends on the size of the energy profile region compatible with its exploration mechanism. Therefore, significant advantages could be obtained by exploiting hybrid solvers, which combine SQA and classical algorithms. This work proposes four new quantum-classical algorithms: Simulated Quantum Parallel Tempering (SQPT), Simulated Quantum Population Annealing (SQPA), Simulated Quantum Parallel Tempering - Population Annealing v1 (SQPTPA1) and Simulated Quantum Parallel Tempering - Population Annealing v2 (SQPTPA2). They are obtained by combining SQA, Parallel Tempering (PT), and Population Annealing (PA). Their results are compared with those provided by SQA, considering benchmark QUBO problems, characterized by different profiles. Even though this work is preliminary, the obtained results are encouraging and prove hybrid solvers’ potential in solving a generic optimization problem.

Quantum Dynamic Mode Decomposition Algorithm for High-Dimensional Time Series Analysis

The dynamic mode decomposition (DMD) algorithm is a widely used factorization and dimensionality reduction technique in time series analysis. When analyzing high-dimensional time series, the DMD algorithm requires extremely large amounts of computational power. To accelerate the DMD algorithm, we propose a quantum-classical hybrid algorithm that we call the quantum dynamic mode decomposition (QDMD) algorithm. Given a time series X ∈ R n × ( m + 1) with n ≫ m , the QDMD algorithm first executes quantum singular value decomposition on a matrix related to X and obtains a quantum state containing the main singular values and singular vectors of the decomposed matrix, then performs a low-sampling-frequency process on the obtained quantum state and computes the low-dimensional projection of the DMD operator through the sampling results. Finally, the algorithm computes the DMD eigenvalues and prepares the amplitude-encoding states of the DMD modes using the obtained classical information and X . Considering the main variables, the complexity of the QDMD algorithm is O ~ M m polylog n / ϵ , where M = O ~ m 3 / ϵ 2 denotes the number of samples. Compared with the classical DMD algorithm, which has complexity O ~ n m 2 log 1 / ϵ , the QDMD algorithm provides an exponential acceleration of n , at the cost of greater dependence on M and ϵ . We test the effects of M on the QDMD algorithm in the specific application scenarios of data denoising, scene background extraction, and fluid dynamics analysis. We determined that the QDMD algorithm requires only a small number of samples M in specific applications, further demonstrating the quantum advantage of the QDMD algorithm in high-dimensional data analysis.

Variational Quantum Eigensolver with Mutual Variance-Hamiltonian Optimization

The zero-energy variance principle can be exploited in variational quantum eigensolvers for solving general eigenstates but its capacity for obtaining a specified eigenstate, such as ground state, is limited as all eigenstates are of zero energy variance. We propose a variance-based variational quantum eigensolver for solving the ground state by searching in an enlarged space of wavefunction and Hamiltonian. With a mutual variance-Hamiltonian optimization procedure, the Hamiltonian is iteratively updated to guild the state towards to the ground state of the target Hamiltonian by minimizing the energy variance in each iteration. We demonstrate the performance and properties of the algorithm with numeral simulations. Our work suggests an avenue for utilizing guided Hamiltonian in hybrid quantum-classical algorithms.

Avoiding barren plateaus via transferability of smooth solutions in a Hamiltonian variational ansatz

Parametrized quantum circuits inspired by adiabatic quantum computation often suffer from vanishing gradients, hindering the trainability of hybrid quantum-classical algorithms. Here, the authors put forward a strategy to overcome this by reusing parameters and iterating from small to large systems.

Variational Quantum Simulation of Valence-Bond Solids

We introduce a hybrid quantum-classical variational algorithm to simulate ground-state phase diagrams of frustrated quantum spin models in the thermodynamic limit. The method is based on a cluster-Gutzwiller ansatz where the wave function of the cluster is provided by a parameterized quantum circuit whose key ingredient is a two-qubit real XY gate allowing to efficiently generate valence-bonds on nearest-neighbor qubits. Additional tunable single-qubit Z- and two-qubit ZZ-rotation gates allow the description of magnetically ordered and paramagnetic phases while restricting the variational optimization to the U(1) subspace. We benchmark the method against the J1−J2 Heisenberg model on the square lattice and uncover its phase diagram, which hosts long-range ordered Neel and columnar anti-ferromagnetic phases, as well as an intermediate valence-bond solid phase characterized by a periodic pattern of 2×2 strongly-correlated plaquettes. Our results show that the convergence of the algorithm is guided by the onset of long-range order, opening a promising route to synthetically realize frustrated quantum magnets and their quantum phase transition to paramagnetic valence-bond solids with currently developed superconducting circuit devices.

A Preprocessing Perspective for Quantum Machine Learning Classification Advantage in Finance Using NISQ Algorithms.

Quantum Machine Learning (QML) has not yet demonstrated extensively and clearly its advantages compared to the classical machine learning approach. So far, there are only specific cases where some quantum-inspired techniques have achieved small incremental advantages, and a few experimental cases in hybrid quantum computing are promising, considering a mid-term future (not taking into account the achievements purely associated with optimization using quantum-classical algorithms). The current quantum computers are noisy and have few qubits to test, making it difficult to demonstrate the current and potential quantum advantage of QML methods. This study shows that we can achieve better classical encoding and performance of quantum classifiers by using Linear Discriminant Analysis (LDA) during the data preprocessing step. As a result, the Variational Quantum Algorithm (VQA) shows a gain of performance in balanced accuracy with the LDA technique and outperforms baseline classical classifiers.

A hybrid quantum-classical Hamiltonian learning algorithm

A hybrid quantum-classical Hamiltonian learning algorithm

A hybrid quantum-classical generative adversarial networks algorithm based on inherited layerwise learning with circle-connectivity circuit

A hybrid quantum-classical generative adversarial networks algorithm based on inherited layerwise learning with circle-connectivity circuit

Application of a variational hybrid quantum-classical algorithm to heat conduction equation and analysis of time complexity

The prosperous development of both hardware and algorithms for quantum computing (QC) potentially prompts a paradigm shift in scientific computing in various fields. As an increasingly active topic in QC, the variational quantum algorithm leads a promising tool for solving partial differential equations on noisy intermediate scale quantum devices. Although a clear perspective on the advantages of QC over classical computing techniques for specific mathematical and physical problems exists, applications of QC in computational fluid dynamics to solve practical flow problems, though promising, are still at the early stage of development. To explore QC in practical simulation of flow problems, this work applies a variational hybrid quantum-classical algorithm, namely the variational quantum linear solver (VQLS), to resolve the heat conduction equation through finite difference discretization of the Laplacian operator. Details of the VQLS implementation are discussed by various test instances of linear systems. The effect of the number of shots on the accuracy is studied, which reveals a logarithmic relationship. Furthermore, the heuristic scaling of the VQLS with the precision ε, the number of qubits n and the condition number k validates its time complexity reported in the literature. In addition, the successful state vector simulations of the heat conduction equation in one and two dimensions demonstrate the validity of the present VQLS-based algorithm by proof-of-concept results. Finally, the heuristic scaling for the heat conduction problem indicates that the time complexity of the present approach is logarithmically dependent on the precision ε and linearly dependent on the number of qubits n.

Tensor Network Quantum Virtual Machine for Simulating Quantum Circuits at Exascale

The numerical simulation of quantum circuits is an indispensable tool for development, verification, and validation of hybrid quantum-classical algorithms intended for near-term quantum co-processors. The emergence of exascale high-performance computing (HPC) platforms presents new opportunities for pushing the boundaries of quantum circuit simulation. We present a modernized version of the Tensor Network Quantum Virtual Machine (TNQVM) that serves as the quantum circuit simulation backend in the eXtreme-scale ACCelerator (XACC) framework. The new version is based on the scalable tensor network processing library ExaTN (Exascale Tensor Networks). It provides multiple configurable quantum circuit simulators that perform either an exact quantum circuit simulation via the full tensor network contraction or an approximate simulation via a suitably chosen tensor factorization scheme. Upon necessity, stochastic noise modeling from real quantum processors is incorporated into the simulations by modeling quantum channels with Kraus tensors. By combining the portable XACC quantum programming frontend and the scalable ExaTN numerical processing backend, we introduce an end-to-end virtual quantum development environment that can scale from laptops to future exascale platforms. We report initial benchmarks of our framework, which include a demonstration of the distributed execution, incorporation of quantum decoherence models, and simulation of the random quantum circuits used for the certification of quantum supremacy on Google’s Sycamore superconducting architecture.

Quantum algorithms for Schrieffer-Wolff transformation

The Schrieffer-Wolff transformation aims to solve degenerate perturbation problems and give an effective Hamiltonian that describes the low-energy dynamics of the exact Hamiltonian in the low-energy subspace of unperturbed Hamiltonian. This unitary transformation decoupling the low-energy and high-energy subspaces for the transformed Hamiltonian can be realized by quantum circuits. We give a fully quantum algorithm for realizing the SW transformation. We also propose a hybrid quantum-classical algorithm for finding the effective Hamiltonian on NISQ hardware, where a general cost function is used to indicate the decoupling degree. Numerical simulations without or with noise and experiments on quantum computer ibmq_manila are implemented for a Heisenberg chain model. Our results verify the algorithm and show its validity on near-term quantum computers.

Quantum Gaussian filter for exploring ground-state properties

Filter methods realize a projection from a superposed quantum state onto a target state, which can be efficient if two states have sufficient overlap. Here we propose a quantum Gaussian filter (QGF) with the filter operator being a Gaussian function of the system Hamiltonian. A hybrid quantum-classical algorithm feasible on near-term quantum computers is developed, which implements the quantum Gaussian filter as a linear combination of Hamiltonian evolution at various times. Remarkably, the linear combination coefficients are determined classically and can be optimized in the postprocessing procedure. Compared to the existing filter algorithms whose coefficients are given in advance, our method is more flexible in practice under given quantum resources with the help of postprocessing on classical computers. We demonstrate the quantum Gaussian filter algorithm for the quantum Ising model with numeral simulations under noises. We also propose an alternative full quantum approach that implements a QGF with an ancillary continuous-variable mode.

Hybrid Quantum-Classical Computing for Future Network Optimization

Future communication networks will require increased flexibility, scalability, and data computation capabilities to adequately respond to the growing number of service demands. Advanced mixed-integer network resource optimization models and algorithms are required to meet these requirements. The purpose of this article is to introduce a hybrid quantum-classical computing framework for addressing future network resource optimization issues. We begin by discussing the fundamentals of quantum computing and its parallelism. Following that, we discuss in detail the hybrid quantum-classical computing paradigm. Then, we discuss the potential applications of the proposed paradigm for network resource optimization, including network function virtualization (NFV), multi-access edge computing/fog/cloud computing, and cloud radio access networks (C-RANs). Finally, we discuss the difficulties associated with the design and implementation of hybrid quantum-classical algorithms.

Parallel quantum chemistry on noisy intermediate-scale quantum computers

A novel parallel hybrid quantum-classical algorithm for the solution of the quantum-chemical ground-state energy problem on gate-based quantum computers is presented. This approach is based on the reduced density-matrix functional theory (RDMFT) formulation of the electronic structure problem. For that purpose, the density-matrix functional of the full system is decomposed into an indirectly coupled sum of density-matrix functionals for all its subsystems using the adaptive cluster approximation to RDMFT. The approximations involved in the decomposition and the adaptive cluster approximation itself can be systematically converged to the exact result. The solutions for the density-matrix functionals of the effective subsystems involves a constrained minimization over many-particle states that are approximated by parametrized trial states on the quantum computer similarly to the variational quantum eigensolver. The independence of the density-matrix functionals of the effective subsystems introduces a new level of parallelization and allows for the computational treatment of much larger molecules on a quantum computer with a given qubit count. In addition, for the proposed algorithm techniques are presented to reduce the qubit count, the number of quantum programs, as well as its depth. The new approach is demonstrated for Hubbard-like systems on IBM quantum computers based on superconducting transmon qubits.

Measurements as a roadblock to near-term practical quantum advantage in chemistry: Resource analysis

Recent advances in quantum computing devices have brought attention to hybrid quantum-classical algorithms like the Variational Quantum Eigensolver (VQE) as a potential route to practical quantum advantage in chemistry. However, it is not yet clear whether such algorithms, even in the absence of device error, could actually achieve quantum advantage for systems of practical interest. We have performed an exhaustive analysis to estimate the number of qubits and number of measurements required to compute the combustion energies of small organic molecules and related systems to within chemical accuracy of experimental values using VQE. We consider several key modern improvements to VQE, including low-rank factorizations of the Hamiltonian. Our results indicate that although these techniques are useful, they will not be sufficient to achieve practical quantum computational advantage for our molecular set, or for similar molecules. This suggests that novel approaches to operator estimation leveraging quantum coherence, such as Enhanced Likelihood Functions [arxiv:2006.09350, arxiv:2006.09349], may be required.

Adaptive construction of shallower quantum circuits with quantum spin projection for fermionic systems

Quantum computing is a promising approach to harnessing strong correlation in molecular systems; however, current devices only allow for hybrid quantum-classical algorithms with a shallow circuit depth, such as the variational quantum eigensolver (VQE). In this paper, we report the importance of the Hamiltonian symmetry in constructing VQE circuits adaptively (ADAPT-VQE). This treatment often violates symmetry, thereby deteriorating the convergence of fidelity to the exact solution and ultimately resulting in deeper circuits. We demonstrate that spin-symmetry projection can provide a simple yet effective solution to this problem, by keeping the quantum state in the correct symmetry space, to reduce the overall gate operations. To further investigate the role of spin-symmetry in computing molecular properties with ADAPT-VQE, we have derived the analytical derivative of symmetry-projected VQE energy. Our illustrative calculations reveal the significance of preserving symmetry in providing accurate dipole moments and geometries with variational approximations.

Quantum Orbital Minimization Method for Excited States Calculation on a Quantum Computer.

We propose a quantum-classical hybrid variational algorithm, the quantum orbital minimization method (qOMM), for obtaining the ground state and low-lying excited states of a Hermitian operator. Given parametrized ansatz circuits representing eigenstates, qOMM implements quantum circuits to represent the objective function in the orbital minimization method and adopts a classical optimizer to minimize the objective function with respect to the parameters in ansatz circuits. The objective function has an orthogonality constraint implicitly embedded, which allows qOMM to apply a different ansatz circuit to each input reference state. We carry out numerical simulations that seek to find excited states of H2, LiH, and a toy model consisting of four hydrogen atoms arranged in a square lattice in the STO-3G basis with UCCSD ansatz circuits. Comparing the numerical results with existing excited states methods, qOMM is less prone to getting stuck in local minima and can achieve convergence with more shallow ansatz circuits.