Chemistry on Quantum Computers with Virtual Quantum Subspace Expansion

Nonadiabatic Molecular Dynamics with Fermionic Subspace-Expansion Algorithms on Quantum Computers

We introduce a novel computational framework for excited-state molecular quantum dynamics simulations driven by quantum-computing-based electronic-structure calculations. This framework leverages the fewest-switches surface-hopping method for simulating the nuclear dynamics and calculates the required excited-state transition properties with different flavors of the quantum subspace expansion and quantum equation-of-motion algorithms. We apply our method to simulate the collision reaction between a hydrogen atom and a hydrogen molecule. For this system, we critically compare the accuracy and efficiency of different quantum subspace expansion and equation-of-motion algorithms and show that only methods that can capture both weak and strong electron correlation effects can properly describe the nonadiabatic effects that tune the reactive event.

Quantum Algorithmic Approach to Multiconfigurational Valence Bond Theory: Insights from Interpretable Circuit Design

Efficient ways to prepare Fermionic ground states on quantum computers are in high demand, and different techniques have been developed over the past few years. Despite having a vast set of methods, it is still unclear which method performs well for which system. In this work, we combine interpretable circuit designs with an effective basis approach in order to optimize a multiconfigurational valence bond wave function. Based on selected model systems, we show how this leads to an explainable performance. We demonstrate that the developed methodology outperforms related methods in terms of the size of the effective basis, as well as individual quantum resources for the involved circuits.

Measurement Optimization Techniques for Excited Electronic States in Near-Term Quantum Computing Algorithms

The variational quantum eigensolver (VQE) remains one of the most popular near-term quantum algorithms for solving the electronic structure problem. Yet, for its practicality, the main challenge to overcome is improving the quantum measurement efficiency. Numerous quantum measurement techniques have been developed recently, but it is unclear how these state-of-the-art measurement techniques will perform in extensions of VQE for obtaining excited electronic states. Assessing the measurement techniques' performance in the excited state VQE is crucial because the measurement requirements in these extensions are typically much greater than in the ground state VQE, as one must measure the expectation value of multiple observables in addition to that of the electronic Hamiltonian. Here, we adapt various measurement techniques to two widely used excited state VQE algorithms: multistate contraction and quantum subspace expansion. Then, the measurement requirements of each measurement technique are numerically compared. We find that the best methods for multistate contraction are ones utilizing Hamiltonian data and wave function information to minimize the number of measurements. In contrast, randomized measurement techniques are more appropriate for quantum subspace expansion, with many more observables of vastly different energy scales to measure. Nevertheless, when the best possible measurement technique for each excited state VQE algorithm is considered, significantly fewer measurements are required in multistate contraction than in quantum subspace expansion.

Quantum self-consistent equation-of-motion method for computing molecular excitation energies, ionization potentials, and electron affinities on a quantum computer

Near-term quantum computers are expected to facilitate material and chemical research through accurate molecular simulations. Several developments have already shown that accurate ground-state energies for small molecules can be evaluated on present-day quantum devices. Although electronically excited states play a vital role in chemical processes and applications, the search for a reliable and practical approach for routine excited-state calculations on near-term quantum devices is ongoing. Inspired by excited-state methods developed for the unitary coupled-cluster theory in quantum chemistry, we present an equation-of-motion-based method to compute excitation energies following the variational quantum eigensolver algorithm for ground-state calculations on a quantum computer. We perform numerical simulations on H2, H4, H2O, and LiH molecules to test our quantum self-consistent equation-of-motion (q-sc-EOM) method and compare it to other current state-of-the-art methods. q-sc-EOM makes use of self-consistent operators to satisfy the vacuum annihilation condition, a critical property for accurate calculations. It provides real and size-intensive energy differences corresponding to vertical excitation energies, ionization potentials and electron affinities. We also find that q-sc-EOM is more suitable for implementation on NISQ devices as it is expected to be more resilient to noise compared with the currently available methods.

Improving the accuracy of the variational quantum eigensolver for molecular systems by the explicitly-correlated perturbative [2]R12correction

We provide an integration of the universal, perturbative explicitly correlated [2]_R12-correction in the context of the Variational Quantum Eigensolver (VQE). This approach is able to increase the accuracy of the underlying reference method significantly while requiring no additional quantum resources. The proposed approach only requires knowledge of the one- and two-particle reduced density matrices (RDMs) of the reference wavefunction; these can be measured after having reached convergence in the VQE. This computation comes at a cost that scales as the sixth power of the number of electrons. We explore the performance of the VQE + [2]_R12 approach using both conventional Gaussian basis sets and our recently proposed directly determined pair-natural orbitals obtained by multiresolution analysis (MRA-PNOs). Both Gaussian orbital and PNOs are investigated as a potential set of complementary basis functions in the computation of [2]_R12. In particular the combination of MRA-PNOs with [2]_R12 has turned out to be very promising - persistently throughout our data, this allowed very accurate simulations at a quantum cost of a minimal basis set. Additionally, we found that the deployment of PNOs as complementary basis can greatly reduce the number of complementary basis functions that enter the computation of the correction at a complexity.

Wave Function Adapted Hamiltonians for Quantum Computing

The use of the variational quantum eigensolver (VQE) for quantum chemistry is one of the most promising applications for noisy intermediate-scale quantum (NISQ) devices. A major limitation is represented by the need to build compact and shallow circuit ansatzes having the variational flexibility to catch the complexity of the electronic structure problem. To alleviate this drawback, we introduce a modified VQE scheme in which the form of the molecular Hamiltonian is adapted to the circuit ansatz through an optimization procedure. Exploiting the invariance of the Hamiltonian by molecular orbital rotations, we can optimize it using gradients that can be calculated without significant computational overload. The proposed method, named Wavefunction Adapted Hamiltonian Through Orbital Rotation (WAHTOR), has been applied to small molecules in numerical state vector simulations. The results demonstrate that, at variance with standard VQE, the method is less dependent on circuit topology and less prone to be trapped into high-energy local minima. It is able to recover a significant amount of electron correlation even with only empirical ansatzes with shallow circuit depth. Noisy calculations demonstrate the robustness and feasibility of the proposed methodology and indicate the hardware requirements to effectively apply the procedure using forthcoming NISQ devices.

Fermi.jl: A Modern Design for Quantum Chemistry

Approximating molecular wave functions involves heavy numerical effort; therefore, codes for such tasks are written completely or partially in efficient languages such as C, C++, and Fortran. While these tools are dominant throughout quantum chemistry packages, the efficient development of new methods is often hindered by the complexity associated with code development. In order to ameliorate this scenario, some software packages take a dual approach where a simpler, higher-level language, such as Python, substitutes the traditional ones wherever performance is not critical. Julia is a novel, dynamically typed, programming language that aims to solve this two-language problem. It gained attention because of its modern and intuitive design, while still being highly optimized to compete with "low-level" languages. Recently, some chemistry-related projects have emerged exploring the capabilities of Julia. Herein, we introduce the quantum chemistry package Fermi.jl, which contains the first implementations of post-Hartree-Fock methods written in Julia. Its design makes use of many Julia core features, including multiple dispatch, metaprogramming, and interactive usage. Fermi.jl is a modular package, where new methods and implementations can be easily added to the existing code. Furthermore, it is designed to maximize code reusability by relying on general functions with specialized methods for particular cases. The feasibility of the project is explored through evaluating the performance of popular ab initio methods. It is our hope that this project motivates the usage of Julia within the community and brings new contributions into Fermi.jl.

Quantum-classical hybrid algorithm for the simulation of all-electron correlation

While chemical systems containing hundreds to thousands of electrons remain beyond the reach of quantum devices, hybrid quantum-classical algorithms present a promising pathway toward a quantum advantage. Hybrid algorithms treat the exponentially scaling part of the calculation-the static correlation-on the quantum computer and the non-exponentially scaling part-the dynamic correlation-on the classical computer. While a variety of algorithms have been proposed, the dependence of many methods on the total wave function limits the development of easy-to-use classical post-processing implementations. Here, we present a novel combination of quantum and classical algorithms, which computes the all-electron energy of a strongly correlated molecular system on the classical computer from the 2-electron reduced density matrix (2-RDM) evaluated on the quantum device. Significantly, we circumvent the wave function in the all-electron calculations by using density matrix methods that only require input of the statically correlated 2-RDM. Although the algorithm is completely general, we test it with two classical density matrix methods, the anti-Hermitian contracted Schrödinger equation (ACSE) and multiconfiguration pair-density functional theories, using the recently developed quantum ACSE method for simulating the statically correlated 2-RDM. We obtain experimental accuracy for the relative energies of all three benzyne isomers and thereby demonstrate the ability of the developed algorithm to achieve chemically relevant and accurate results on noisy intermediate-scale quantum devices.

Fermionic Partial Tomography via Classical Shadows

We propose a tomographic protocol for estimating any k-body reduced density matrix (k-RDM) of an n-mode fermionic state, a ubiquitous step in near-term quantum algorithms for simulating many-body physics, chemistry, and materials. Our approach extends the framework of classical shadows, a randomized approach to learning a collection of quantum-state properties, to the fermionic setting. Our sampling protocol uses randomized measurement settings generated by a discrete group of fermionic Gaussian unitaries, implementable with linear-depth circuits. We prove that estimating all k-RDM elements to additive precision ϵ requires on the order of (n/k)k^{3/2}log(n)/ϵ^{2} repeated state preparations, which is optimal up to the logarithmic factor. Furthermore, numerical calculations show that our protocol offers a substantial improvement in constant overheads for k≥2, as compared to prior deterministic strategies. We also adapt our method to particle-number symmetry, wherein the additional circuit depth may be halved at the cost of roughly 2-5 times more repetitions.

Quantum HF/DFT-embedding algorithms for electronic structure calculations: Scaling up to complex molecular systems

In the near future, material and drug design may be aided by quantum computer assisted simulations. These have the potential to target chemical systems intractable by the most powerful classical computers. However, the resources offered by contemporary quantum computers are still limited, restricting the simulations to very simple molecules. In order to rapidly scale up to more interesting molecular systems, we propose the embedding of the quantum electronic structure calculation into a classically computed environment obtained at the Hartree-Fock (HF) or density functional theory (DFT) level of theory. This result is achieved by constructing an effective Hamiltonian that incorporates a mean field potential describing the action of the inactive electrons on a selected Active Space (AS). The ground state of the AS Hamiltonian is then determined by means of the variational quantum eigensolver algorithm. We show that with the proposed HF and DFT embedding schemes, we can obtain significant energy corrections to the reference HF and DFT calculations for a number of simple molecules in their strongly correlated limit (the dissociation regime) as well as for systems of the size of the oxirane molecule.

Quantum Information and Algorithms for Correlated Quantum Matter

Discoveries in quantum materials, which are characterized by the strongly quantum-mechanical nature of electrons and atoms, have revealed exotic properties that arise from correlations. It is the promise of quantum materials for quantum information science superimposed with the potential of new computational quantum algorithms to discover new quantum materials that inspires this Review. We anticipate that quantum materials to be discovered and developed in the next years will transform the areas of quantum information processing including communication, storage, and computing. Simultaneously, efforts toward developing new quantum algorithmic approaches for quantum simulation and advanced calculation methods for many-body quantum systems enable major advances toward functional quantum materials and their deployment. The advent of quantum computing brings new possibilities for eliminating the exponential complexity that has stymied simulation of correlated quantum systems on high-performance classical computers. Here, we review new algorithms and computational approaches to predict and understand the behavior of correlated quantum matter. The strongly interdisciplinary nature of the topics covered necessitates a common language to integrate ideas from these fields. We aim to provide this common language while weaving together fields across electronic structure theory, quantum electrodynamics, algorithm design, and open quantum systems. Our Review is timely in presenting the state-of-the-art in the field toward algorithms with nonexponential complexity for correlated quantum matter with applications in grand-challenge problems. Looking to the future, at the intersection of quantum information science and algorithms for correlated quantum matter, we envision seminal advances in predicting many-body quantum states and describing excitonic quantum matter and large-scale entangled states, a better understanding of high-temperature superconductivity, and quantifying open quantum system dynamics.