Variational Quantum Eigensolver Research Articles

Sketching phase diagrams using low-depth variational quantum algorithms

Abstract Mapping out phase diagrams of quantum systems using classical simulations can be challenging or intractable due to the computational resources required to simulate even small quantum systems far away from the thermodynamic limit. We investigate using quantum computers and the Variational Quantum Eigensolver (VQE) for this task. In contrast to the task of preparing the exact ground state using VQE, sketching phase diagrams might require less quantum resources and accuracy, because low fidelity approximations to the ground state may be enough to correctly identify different phases. We used classical numerical simulations of low-depth VQE circuits to compute order parameters for four well-studied spin and fermion models which represent a mix of 1D and 2D, and exactly-solvable and classically hard systems. We find that it is possible to predict the location of phase transitions up to reasonable accuracy using states produced by VQE even when their overlap with the true ground state is small. Further, we introduce a model-agnostic predictor of phase transitions based on the speed with which the VQE energy improves with respect to the circuit depth, and find that in some cases this is also able to predict phase transitions.

Extension of Exactly-Solvable Hamiltonians Using Symmetries of Lie Algebras.

Exactly solvable Hamiltonians that can be diagonalized by using relatively simple unitary transformations are of great use in quantum computing. They can be employed for the decomposition of interacting Hamiltonians either in Trotter-Suzuki approximations of the evolution operator for the quantum phase estimation algorithm or in the quantum measurement problem for the variational quantum eigensolver. One of the typical forms of exactly solvable Hamiltonians is a linear combination of operators forming a modestly sized Lie algebra. Very frequently, such linear combinations represent noninteracting Hamiltonians and thus are of limited interest for describing interacting cases. Here, we propose an extension in which the coefficients in these combinations are substituted by polynomials of the Lie algebra symmetries. This substitution results in a more general class of solvable Hamiltonians, and for qubit algebras, it is related to the recently proposed noncontextual Pauli Hamiltonians. In fermionic problems, this substitution leads to Hamiltonians with eigenstates that are single Slater determinants but with different sets of single-particle states for different eigenstates. The new class of solvable Hamiltonians can be measured efficiently using quantum circuits with gates that depend on the result of a midcircuit measurement of the symmetries.

Information flow in parameterized quantum circuits

In this work, we introduce a new way to quantify information flow in quantum systems, especially for parameterized quantum circuits (PQCs). We use a graph representation of the circuits and propose a new distance metric using the mutual information between gate nodes. We then present an optimization procedure for variational algorithms using paths based on the distance measure. We explore the features of the algorithm by means of the variational quantum eigensolver, in which we compute the ground state energies of the Heisenberg model. In addition, we employ the method to solve a binary classification problem using variational quantum classification. From numerical simulations, we show that our method can be successfully used for optimizing the PQCs primarily used in near-term algorithms. We further note that information-flow based paths can be used to improve convergence of existing stochastic gradient based methods.

Range-separated density functional theory using multiresolution analysis and quantum computing.

Quantum computers are expected to outperform classical computers for specific problems in quantum chemistry. Such calculations remain expensive, but costs can be lowered through the partition of the molecular system. In the present study, partition was achieved with range-separated density functional theory (RS-DFT). The use of RS-DFT reduces both the basis set size and the active space size dependence of the ground state energy in comparison with the use of wave function theory (WFT) alone. The utilization of pair natural orbitals (PNOs) in place of canonical molecular orbitals (MOs) results in more compact qubit Hamiltonians. To test this strategy, a basis-set independent framework, known as multiresolution analysis (MRA), was employed to generate PNOs. Tests were conducted with the variational quantum eigensolver for a number of molecules. The results show that the proposed approach reduces the number of qubits needed to reach a target energy accuracy.

Quantum Equation of Motion with Orbital Optimization for Computing Molecular Properties in Near-Term Quantum Computing.

Determining the properties of molecules and materials is one of the premier applications of quantum computing. A major question in the field is how to use imperfect near-term quantum computers to solve problems of practical value. Inspired by the recently developed variants of the quantum counterpart of the equation-of-motion (qEOM) approach and the orbital-optimized variational quantum eigensolver (oo-VQE), we present a quantum algorithm (oo-VQE-qEOM) for the calculation of molecular properties by computing expectation values on a quantum computer. We perform noise-free quantum simulations of BeH2 in the series of STO-3G/6-31G/6-31G* basis sets and of H4 and H2O in 6-31G using an active space of four electrons and four spatial orbitals (8 qubits) to evaluate excitation energies, electronic absorption, and, for twisted H4, circular dichroism spectra. We demonstrate that the proposed algorithm can reproduce the results of conventional classical CASSCF calculations for these molecular systems.

Demonstration of the Effectiveness of the Cascaded Variational Quantum Eigensolver Using the Jastrow Ansatz for Molecular Calculations

Demonstration of the Effectiveness of the Cascaded Variational Quantum Eigensolver Using the Jastrow Ansatz for Molecular Calculations

Fast Emulation of Fermionic Circuits with Matrix Product States.

We describe a matrix product state (MPS) extension for the Fermionic Quantum Emulator (FQE) software library. We discuss the theory behind symmetry-adapted MPSs for approximating many-body wave functions of spin-1/2 Fermions, and we present an open-source, MPS-enabled implementation of the FQE interface (MPS-FQE). The software uses the open-source pyblock3 and block2 libraries for most elementary tensor operations, and it can largely be used as a drop-in replacement for FQE that allows for more efficient but approximate emulation of larger Fermionic circuits. Finally, we show several applications relevant to both near-term and fault-tolerant quantum algorithms where approximate emulation of larger systems is expected to be useful: characterization of state preparation strategies for quantum phase estimation, the testing of different variational quantum eigensolver ansätze, the numerical evaluation of Trotter errors, and the simulation of general quantum dynamics problems. In all these examples, approximate emulation with MPS-FQE allows us to treat systems that are significantly larger than those accessible with a full statevector emulator.

Quantum Davidson algorithm for excited states

Excited state properties play a pivotal role in various chemical and physical phenomena, such as charge separation and light emission. However, the primary focus of most existing quantum algorithms has been the ground state, as seen in quantum phase estimation and the variational quantum eigensolver (VQE). Although VQE-type methods have been extended to explore excited states, these methods grapple with optimization challenges. In contrast, the quantum Krylov subspace (QKS) method has been introduced to address both ground and excited states, positioning itself as a cost-effective alternative to quantum phase estimation. However, conventional QKS methodologies depend on a pre-generated subspace through real or imaginary-time evolutions. This subspace is inherently expansive and can be plagued with issues like slow convergence or numerical instabilities, often leading to relatively deep circuits. Our research presents an economic QKS algorithm, which we term the quantum Davidson (QDavidson) algorithm. This innovation hinges on the iterative expansion of the Krylov subspace and the incorporation of a pre-conditioner within the Davidson framework. By using the residues of eigenstates to expand the Krylov subspace, we manage to formulate a compact subspace that aligns closely with the exact solutions. This iterative subspace expansion paves the way for a more rapid convergence in comparison to other QKS techniques, such as the quantum Lanczos. Using quantum simulators, we employ the novel QDavidson algorithm to delve into the excited state properties of various systems, spanning from the Heisenberg spin model to real molecules. Compared to the existing QKS methods, the QDavidson algorithm not only converges swiftly but also demands a significantly shallower circuit. This efficiency establishes the QDavidson method as a pragmatic tool for elucidating both ground and excited state properties on quantum computing platforms.

Vibrational ADAPT-VQE: Critical points lead to problematic convergence.

Quantum chemistry is one of the most promising applications for which quantum computing is expected to have a significant impact. Despite considerable research in the field of electronic structure, calculating the vibrational properties of molecules on quantum computers remains a relatively unexplored field. In this work, we develop a vibrational Adaptive Derivative-Assembled Pseudo-Trotter Variational Quantum Eigensolver (vADAPT-VQE) formalism based on an infinite product representation (IPR) of anti-Hermitian excitation operators of the Full Vibrational Configuration Interaction (FVCI) wavefunction, which allows for preparing eigenstates of vibrational Hamiltonians on quantum computers. In order to establish the vADAPT-VQE algorithm using the IPR, we study the exactness of disentangled Unitary Vibrational Coupled Cluster (dUVCC) theory and show that dUVCC can formally represent the FVCI wavefunction in an infinite expansion. To investigate the performance of the vADAPT-VQE algorithm, we numerically study whether the vADAPT-VQE algorithm generates a sequence of operators that may represent the FVCI wavefunction. Our numerical results indicate frequent appearance of critical points in the wavefunction preparation using vADAPT-VQE. These results imply that one may encounter diminishing usefulness when preparing vibrational wavefunctions on quantum computers using vADAPT-VQE and that additional studies are required to find methods that can circumvent this behavior.

Efficient and Robust Parameter Optimization of the Unitary Coupled-Cluster Ansatz.

The variational quantum eigensolver (VQE) framework has been instrumental in advancing near-term quantum algorithms. However, parameter optimization remains a significant bottleneck for VQE, requiring a large number of measurements for successful algorithm execution. In this paper, we propose sequential optimization with approximate parabola (SOAP) as an efficient and robust optimizer specifically designed for parameter optimization of the unitary coupled-cluster ansatz on quantum computers. SOAP leverages sequential optimization and approximates the energy landscape as quadratic functions, minimizing the number of energy evaluations required to optimize each parameter. To capture parameter correlations, SOAP incorporates the average direction from previous iterations into the optimization direction set. Numerical benchmark studies on molecular systems demonstrate that SOAP achieves significantly faster convergence and greater robustness to noise compared with traditional optimization methods. Furthermore, numerical simulations of up to 20 qubits reveal that SOAP scales well with the number of parameters in the ansatz. The exceptional performance of SOAP is further validated through experiments on a superconducting quantum computer using a 2-qubit model system.

Applications of quantum circuit learning model using particle-number-conserving state on quantum chemical calculations

Although the variational quantum eigensolver is a typical quantum algorithm utilized in near-term quantum devices, many measurements are required in an iterative closed feedback loop between the classical and quantum computers to obtain sufficient accuracy. In this study, we attempt to construct a quantum circuit learning model to infer potential energy surfaces and atomic forces without using the iterative loop to optimize parameters for every bond length. To realize a high, accurate inference performance, measurement is introduced in the middle of the circuit. When the proposed quantum circuit learning model is applied to the H2 molecule, the energies and atomic forces can be estimated with high accuracy in a single feed-forward calculation with varying bond lengths. A nonlinear relation between outcomes from data-encoding layer and inputs can also be used to eliminate the data-encoding layer, allowing for our quantum circuit model with lower learning costs by reducing the number of parameters to be optimized. Our model can be also extended to six-qubit systems, such as the H3 molecule, and to the water molecule with two internal degrees of freedom. Finally, we use the IBM Quantum backend to perform inference with a real quantum computer and show the effect of noise on the actual quantum device.

Absence of Barren Plateaus in Finite Local-Depth Circuits with Long-Range Entanglement.

Ground state preparation is classically intractable for general Hamiltonians. On quantum devices, shallow parametrized circuits can be effectively trained to obtain short-range entangled states under the paradigm of variational quantum eigensolver, while deep circuits are generally untrainable due to the barren plateau phenomenon. In this Letter, we give a general lower bound on the variance of circuit gradients for arbitrary quantum circuits composed of local 2-designs. Based on our unified framework, we prove the absence of barren plateaus in training finite local-depth circuits (FLDC) for the ground states of local Hamiltonians. FLDCs are allowed to be deep in the conventional circuit depth to generate long-range entangled ground states, such as topologically ordered states, but their local depths are finite, i.e., there is only a finite number of gates acting on individual qubits. This characteristic sets FLDC apart from shallow circuits: FLDC in general cannot be classically simulated to estimate local observables efficiently by existing tensor network methods in two and higher dimensions. We validate our analytical results with extensive numerical simulations and demonstrate the effectiveness of variational training using the generalized toric code model.

Self-Consistent Field Approach for the Variational Quantum Eigensolver: Orbital Optimization Goes Adaptive.

We present a self-consistent field (SCF) approach within the adaptive derivative-assembled problem-tailored ansatz variational quantum eigensolver (ADAPT-VQE) framework for efficient quantum simulations of chemical systems on near-term quantum computers. To this end, our ADAPT-VQE-SCF approach combines the idea of generating an ansatz with a small number of parameters, resulting in shallow-depth quantum circuits with a direct minimization of an energy expression that is correct to second order with respect to changes in the molecular orbital basis. Our numerical analysis, including calculations for the transition-metal complex ferrocene [Fe (C5H5)2], indicates that convergence in the self-consistent orbital optimization loop can be reached without a considerable increase in the number of two-qubit gates in the quantum circuit by comparison to a VQE optimization in the initial molecular orbital basis. Moreover, the orbital optimization can be carried out simultaneously within each iteration of the ADAPT-VQE cycle. ADAPT-VQE-SCF thus allows us to implement a routine analogous to the complete active space SCF, a cornerstone of state-of-the-art computational chemistry, in a hardware-efficient manner on near-term quantum computers. Hence, ADAPT-VQE-SCF paves the way toward a paradigm shift for quantitative quantum-chemistry simulations on quantum computers by requiring fewer qubits and opening up for the use of large and flexible atomic orbital basis sets in contrast to earlier methods that are predominantly based on the idea of full active spaces with minimal basis sets.

The variational quantum eigensolver self-consistent field method within a polarizable embedded framework.

We formulate and implement the Variational Quantum Eigensolver Self Consistent Field (VQE-SCF) algorithm in combination with polarizable embedding (PE), thereby extending PE to the regime of quantum computing. We test the resulting algorithm, PE-VQE-SCF, on quantum simulators and demonstrate that the computational stress on the quantum device is only slightly increased in terms of gate counts compared to regular VQE-SCF. On the other hand, no increase in shot noise was observed. We illustrate how PE-VQE-SCF may lead to the modeling of real chemical systems using a simulation of the reaction barrier of the Diels-Alder reaction between furan and ethene as an example.

A versatile single-photon-based quantum computing platform

AbstractQuantum computing aims at exploiting quantum phenomena to efficiently perform computations that are unfeasible even for the most powerful classical supercomputers. Among the promising technological approaches, photonic quantum computing offers the advantages of low decoherence, information processing with modest cryogenic requirements, and native integration with classical and quantum networks. So far, quantum computing demonstrations with light have implemented specific tasks with specialized hardware, notably Gaussian boson sampling, which permits the quantum computational advantage to be realized. Here we report a cloud-accessible versatile quantum computing prototype based on single photons. The device comprises a high-efficiency quantum-dot single-photon source feeding a universal linear optical network on a reconfigurable chip for which hardware errors are compensated by a machine-learned transpilation process. Our full software stack allows remote control of the device to perform computations via logic gates or direct photonic operations. For gate-based computation, we benchmark one-, two- and three-qubit gates with state-of-the art fidelities of 99.6 ± 0.1%, 93.8 ± 0.6% and 86 ± 1.2%, respectively. We also implement a variational quantum eigensolver, which we use to calculate the energy levels of the hydrogen molecule with chemical accuracy. For photon native computation, we implement a classifier algorithm using a three-photon-based quantum neural network and report a six-photon boson sampling demonstration on a universal reconfigurable integrated circuit. Finally, we report on a heralded three-photon entanglement generation, a key milestone toward measurement-based quantum computing.

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.

Many-Body-Expansion Based on Variational Quantum Eigensolver and Deflation for Dynamical Correlation.

In this work, we utilize the framework of many-body expansion (MBE) to decompose electronic structures into fragments by incrementing virtual orbitals, aiming to accurately solve the ground and excited state energies of each fragment using the variational quantum eigensolver and deflation algorithms. While our approach is primarily based on unitary coupled cluster singles and doubles (UCCSD) and its generalization, we also introduce modifications and approximations to conserve quantum resources in MBE by partially generalizing the UCCSD operator and neglecting the relaxation of the reference states. As a proof of concept, we investigate the potential energy surfaces for the bond-breaking processes of the ground state of two molecules (H2O and N2) and calculate the ground and excited state energies of three molecules (LiH, CH+, and H2O). The results demonstrate that our approach can, in principle, provide reliable descriptions in all the tests including strongly correlated systems when appropriate approximations are chosen. Additionally, we perform model simulations to investigate the impact of shot noise on the total MBE energy and show that precise energy estimation is crucial for lower-order MBE fragments.

Optimizing Shot Assignment in Variational Quantum Eigensolver Measurement.

Variational quantum eigensolvers (VQEs) show promise for tackling complex quantum chemistry challenges and realizing quantum advantages. However, in VQE, the measurement step encounters difficulties due to errors in objective function evaluation, e.g., the energy of a quantum state. While increasing the number of measurement shots can mitigate measurement errors, this approach leads to higher costs. Strategies for shot assignment have been investigated, allowing for the allocation of varying shot numbers to different Hamiltonian terms and reducing measurement variance through term-specific insights. In this paper, we introduce a dynamic approach, the Variance-Preserved Shot Reduction (VPSR) method. This technique strives to minimize the total number of measurement shots while preserving the variance of measurements throughout the VQE process. Our numerical experiments on H2 and LiH molecular ground states demonstrate the effectiveness of VPSR in achieving VQE convergence with a notably lower shot count.

Cascaded variational quantum eigensolver algorithm

Cascaded variational quantum eigensolver algorithm