Variational Quantum Simulation for Quantum Chemistry

Hückel molecular orbital theory on a quantum computer: A scalable system-agnostic variational implementation with compact encoding

Hückel molecular orbital (HMO) theory provides a semi-empirical treatment of the electronic structure in conjugated π-electronic systems. A scalable system-agnostic execution of HMO theory on a quantum computer is reported here based on a variational quantum deflation (VQD) algorithm for excited state quantum simulation. A compact encoding scheme is proposed here that provides an exponential advantage over the direct mapping and allows for quantum simulation of the HMO model for systems with up to 2n conjugated centers with n qubits. The transformation of the Hückel Hamiltonian to qubit space is achieved by two different strategies: an iterative refinement transformation and the Frobenius-inner-product-based transformation. These methods are tested on a series of linear, cyclic, and hetero-nuclear conjugated π-electronic systems. The molecular orbital energy levels and wavefunctions from the quantum simulation are in excellent agreement with the exact classical results. However, the higher excited states of large systems are found to suffer from error accumulation in the VQD simulation. This is mitigated by formulating a variant of VQD that exploits the symmetry of the Hamiltonian. This strategy has been successfully demonstrated for the quantum simulation of C60 fullerene containing 680 Pauli strings encoded on six qubits. The methods developed in this work are easily adaptable to similar problems of different complexity in other fields of research.

Applications of noisy quantum computing and quantum error mitigation to "adamantaneland": a benchmarking study for quantum chemistry

The field of quantum computing has the potential to transform quantum chemistry. The variational quantum eigensolver (VQE) algorithm has allowed quantum computing to be applied to chemical problems in the noisy intermediate-scale quantum (NISQ) era. Applications of VQE have generally focused on predicting absolute energies instead of chemical properties that are relative energy differences and that are most interesting to chemists studying a chemical problem. We address this shortcoming by constructing a molecular benchmark data set in this work containing isomers of C10H16 and carbocationic rearrangements of C10H15+, calculated at a high-level of theory. Using the data set, we compared noiseless VQE simulations to conventionally performed density functional and wavefunction theory-based methods to understand the quality of results. We also investigated the effectiveness of a quantum state tomography-based error mitigation technique in applications of VQE under noise (simulated and real). Our findings reveal that the use of quantum error mitigation is crucial in the NISQ era and advantageous to yield almost noiseless quality results.

Experimental quantum natural gradient optimization in photonics

Variational quantum algorithms (VQAs) combining the advantages of parameterized quantum circuits and classical optimizers, promise practical quantum applications in the noisy intermediate-scale quantum era. The performance of VQAs heavily depends on the optimization method. Compared with gradient-free and ordinary gradient descent methods, the quantum natural gradient (QNG), which mirrors the geometric structure of the parameter space, can achieve faster convergence and avoid local minima more easily, thereby reducing the cost of circuit executions. We utilized a fully programmable photonic chip to experimentally estimate the QNG in photonics for the first time, to the best of our knowledge. We obtained the dissociation curve of the He-H⁺ cation and achieved chemical accuracy, verifying the outperformance of QNG optimization on a photonic device. Our work opens up a vista of utilizing QNG in photonics to implement practical near-term quantum applications.

Power of the Sine Hamiltonian Operator for Estimating the Eigenstate Energies on Quantum Computers

Quantum computers have been shown to have tremendous potential in solving difficult problems in quantum chemistry. In this paper, we propose a new classical-quantum hybrid method, named as power of sine Hamiltonian operator (PSHO), to evaluate the eigenvalues of a given Hamiltonian (Ĥ). In PSHO, for any reference state |φ0⟩, the normalized energy of the sinn(H^τ)|φ0⟩ state can be determined. With the increase of the power, the initial reference state can converge to the eigenstate with the largest |sin(Eiτ)| value in the coefficients of the expansion of |φ0⟩, and the normalized energy of the sinn(H^τ)|φ0⟩ state converges to E_i. The ground- and excited-state energies of a Hamiltonian can be determined by taking different τ values. The performance of the PSHO method is demonstrated by numerical calculations of the H₄ and LiH molecules. Compared with the current popular variational quantum eigensolver method, PSHO does not need to design the ansatz circuits and avoids the complex nonlinear optimization problems. PSHO has great application potential in near-term quantum devices.

Adiabatic state preparation of correlated wave functions with nonlinear scheduling functions and broken-symmetry wave functions

Adiabatic state preparation (ASP) can generate the correlated wave function by simulating the time evolution of wave function under the time-dependent Hamiltonian that interpolates the Fock operator and the full electronic Hamiltonian. However, ASP is inherently unsuitable for studying strongly correlated systems, and furthermore practical computational conditions for ASP are unknown. In quest for the suitable computational conditions for practical applications of ASP, we performed numerical simulations of ASP in the potential energy curves of N₂, BeH₂, and in the C_2v quasi-reaction pathway of the Be atom insertion to the H₂ molecule, examining the effect of nonlinear scheduling functions and the ASP with broken-symmetry wave functions with the S² operator as the penalty term, contributing to practical applications of quantum computing to quantum chemistry. Eventually, computational guidelines to generate the correlated wave functions having the square overlap with the complete-active space self-consistent field wave function close to unity are discussed.

Orthogonal State Reduction Variational Eigensolver for the Excited-State Calculations on Quantum Computers

Variational quantum eigensolver (VQE) is a promising method for ground-state calculations on current noisy intermediate-scale quantum computers. However, the research progress of excited-state calculations on quantum computers is relatively slow. In order to extend the framework of VQE for excited-state calculations, we propose a new algorithm, orthogonal state reduction variational eigensolver (OSRVE), to determine the energies of excited states. Theoretical derivations prove that the optimized state in the OSRVE method can ensure the energy minimum and orthogonality constraint simultaneously, and OSRVE is also applicable for the degenerate state. The performance of OSRVE is demonstrated by numerical calculations of the H₄ and H₂O molecules. Compared with other excited-state calculation algorithms, OSRVE has obvious advantages in calculating lower-order excited states. This work extends the VQE algorithm to excited-state calculations, and OSRVE can be implemented on near-term quantum computers.

Quantum computing of Hückel molecular orbitals of π-electron systems

In order to demonstrate an applicability of quantum computing to fundamental electronic structure problems of molecules, we describe the Hückel Hamiltonian matrix in terms of quantum gates and obtain the orbital energies of fundamental π-electron molecules (C₂H₄, C₃H₄, C₄H₄, C₄H₆, and C₆H₆) using a superconducting-qubit-type quantum computer (ibm_kawasaki) with a post-selection error mitigation method. We show that the orbital energies are obtained with sufficiently high accuracy and small uncertainties and that characteristic features of the electronic structure of the π-electron molecules can be extracted by quantum computing in a straightforward manner.

Variational quantum eigensolver simulations with the multireference unitary coupled cluster ansatz: a case study of the C2v quasi-reaction pathway of beryllium insertion to H2 molecule

Variational quantum eigensolver (VQE)-based quantum chemical calculations have been extensively studied as a computational model using noisy intermediate-scale quantum devices. VQE uses a parametrized quantum circuit defined through an “ansatz”...

Bayesian phase difference estimation: a general quantum algorithm for the direct calculation of energy gaps

Quantum computers can perform full configuration interaction (full-CI) calculations by utilising the quantum phase estimation (QPE) algorithms including Bayesian phase estimation (BPE) and iterative quantum phase estimation (IQPE). In these quantum algorithms, the time evolution of wave functions for atoms and molecules is simulated conditionally with an ancillary qubit as the control, which make implementation to real quantum devices difficult. Also, most of the problems in chemistry discuss energy differences between two electronic states rather than total energies themselves, and thus direct calculations of energy gaps are promising for future applications of quantum computers to real chemistry problems. In the race of finding efficient quantum algorithms to solve quantum chemistry problems, we test a Bayesian phase difference estimation (BPDE) algorithm, which is a general algorithm to calculate the difference of two eigenphases of unitary operators in the several cases of the direct calculations of energy gaps between two electronic states on quantum computers, including vertical ionisation energies, singlet-triplet energy gaps, and vertical excitation energies. In the BPDE algorithm, state preparation is carried out conditionally on the ancillary qubit, and the time evolution of the wave functions in superposition of two electronic states are executed unconditionally. Based on our test, we conclude that BPDE is capable of computing the energy gap with an accuracy similar to BPE without controlled-time evolution simulations and with the smaller number of iterations in Bayesian optimisations.

A quantum algorithm for spin chemistry: a Bayesian exchange coupling parameter calculator with broken-symmetry wave functions

The Heisenberg exchange coupling parameter J (H = -2J S i · S j ) characterises the isotropic magnetic interaction between unpaired electrons, and it is one of the most important spin Hamiltonian parameters of multi-spin open shell systems. The J value is related to the energy difference between high-spin and low-spin states, and thus computing the energies of individual spin states are necessary to obtain the J values from quantum chemical calculations. Here, we propose a quantum algorithm, B̲ayesian ex̲change coupling parameter calculator with b̲roken-symmetry wave functions (BxB), which is capable of computing the J value directly, without calculating the energies of individual spin states. The BxB algorithm is composed of the quantum simulations of the time evolution of a broken-symmetry wave function under the Hamiltonian with an additional term j S 2, the wave function overlap estimation with the SWAP test, and Bayesian optimisation of the parameter j. Numerical quantum circuit simulations for H2 under a covalent bond dissociation, C, O, Si, NH, OH+, CH2, NF, O2, and triple bond dissociated N2 molecule revealed that the BxB can compute the J value within 1 kcal mol-1 of errors with less computational costs than conventional quantum phase estimation-based approaches.

Electronic structure with direct diagonalization on a D-wave quantum annealer

Quantum chemistry is regarded to be one of the first disciplines that will be revolutionized by quantum computing. Although universal quantum computers of practical scale may be years away, various approaches are currently being pursued to solve quantum chemistry problems on near-term gate-based quantum computers and quantum annealers by developing the appropriate algorithm and software base. This work implements the general Quantum Annealer Eigensolver (QAE) algorithm to solve the molecular electronic Hamiltonian eigenvalue-eigenvector problem on a D-Wave 2000Q quantum annealer. The approach is based on the matrix formulation, efficiently uses qubit resources based on a power-of-two encoding scheme and is hardware-dominant relying on only one classically optimized parameter. We demonstrate the use of D-Wave hardware for obtaining ground and excited electronic states across a variety of small molecular systems. The approach can be adapted for use by a vast majority of electronic structure methods currently implemented in conventional quantum-chemical packages. The results of this work will encourage further development of software such as qbsolv which has promising applications in emerging quantum information processing hardware and has expectation to address large and complex optimization problems intractable for classical computers.

Speedup in classical simulation of Gaussian boson sampling

Gaussian boson sampling is an alternative model for demonstrating quantum computational supremacy, where squeezed states are injected into every input mode, instead of applying single photons as in the case of standard boson sampling. Here by analyzing numerically the computational costs, we establish a lower bound for achieving quantum computational supremacy for a class of Gaussian boson-sampling problems. Specifically, we propose a more efficient method for calculating the transition probabilities, leading to a significant reduction of the simulation costs. Particularly, our numerical results indicate that one can simulate up to 18 photons for Gaussian boson sampling at the output subspace on a normal laptop, 20 photons on a commercial workstation with 256 cores, and about 30 photons for supercomputers. These numbers are significantly smaller than those in standard boson sampling, suggesting that Gaussian boson sampling could be experimentally-friendly for demonstrating quantum computational supremacy.

A probabilistic spin annihilation method for quantum chemical calculations on quantum computers

A probabilistic spin annihilation method based on the quantum phase estimation algorithm is presented for quantum chemical calculations on quantum computers. This approach can eliminate more than one spin component from the spin contaminated wave functions by single operation. Comparison with the spin annihilation operation on classical computers is given.

Improving the Efficiency of the Multireference Driven Similarity Renormalization Group via Sequential Transformation, Density Fitting, and the Noninteracting Virtual Orbital Approximation

This study examines several techniques to improve the efficiency of the linearized multireference driven similarity renormalization group truncated to one- and two-body operators [MR-LDSRG(2)]. We propose a sequential MR-LDSRG(2) [sq-MR-LDSRG(2)] scheme, in which one-body substitutions are folded exactly into the Hamiltonian. This new approach is combined with density fitting (DF) to reduce the storage cost of two-electron integrals. To further avoid storage of large four-index intermediates, we propose a noninteracting virtual orbital (NIVO) approximation of the Baker-Campbell-Hausdorff series that neglects commutators terms with three and four virtual indices. The NIVO approximation reduces the computational prefactor of the MR-LDSRG(2), bringing it closer to that of coupled cluster with singles and doubles (CCSD). We test the effect of the DF and NIVO approximations on the MR-LDSRG(2) and sq-MR-LDSRG(2) methods by computing properties of eight diatomic molecules. The diatomic constants obtained by DF-sq-MR-LDSRG(2)+NIVO are found to be as accurate as those from the original MR-LDSRG(2) and coupled cluster theory with singles, doubles, and perturbative triples. Finally, we demonstrate that the DF-sq-MR-LDSRG(2)+NIVO scheme can be applied to chemical systems with more than 550 basis functions by computing the automerization energy of cyclobutadiene with a quintuple-ζ basis set. The predicted automerization energy is found to be similar to the value computed with Mukherjee's state-specific multireference coupled cluster theory with singles and doubles.

Universal bound on sampling bosons in linear optics and its computational implications

In linear optics, photons are scattered in a network through passive optical elements including beam splitters and phase shifters, leading to many intriguing applications in physics, such as Mach-Zehnder interferometry, the Hong-Ou-Mandel effect, and tests of fundamental quantum mechanics. Here we present the fundamental limit in the transition amplitudes of bosons, applicable to all physical linear optical networks. Apart from boson sampling, this transition bound results in many other interesting applications, including behaviors of Bose-Einstein condensates (BEC) in optical networks, counterparts of Hong-Ou-Mandel effects for multiple photons, and approximating permanents of matrices. In addition, this general bound implies the existence of a polynomial-time randomized algorithm for estimating the transition amplitudes of bosons, which represents a solution to an open problem raised by Aaronson and Hance (Quantum Inf Comput 2012; 14: 541-59). Consequently, this bound implies that computational decision problems encoded in linear optics, prepared and detected in the Fock basis, can be solved efficiently by classical computers within additive errors. Furthermore, our result also leads to a classical sampling algorithm that can be applied to calculate the many-body wave functions and the S-matrix of bosonic particles.

Quantum chemistry on quantum computers: quantum simulations of the time evolution of wave functions under the S2 operator and determination of the spin quantum number S

Quantum computers have an enormous impact on quantum chemical calculations. Approaches to calculate the energies of atoms and molecules on quantum computers by utilizing quantum phase estimation (QPE) and the variational quantum eigensolver (VQE) have been well documented, and dozens of methodological improvements to decrease computational costs and to mitigate errors have been reported until recently. However, the possible methodological implementation of observables on quantum computers such as calculating the spin quantum numbers of arbitrary wave functions, which is a crucial issue in quantum chemistry, has been discussed less. Here, we propose a quantum circuit to simulate the time evolution of wave functions under an S2 operator, exp(-iS2t)|Ψ, and integrate it into the QPE circuit enabling us to determine the spin quantum number of the arbitrary wave functions. We demonstrate that the spin quantum numbers of up to three spins can be determined by only one qubit measurement in QPE.