Session N49: Quantum Algorithms for Electronic Structure Problems

Towards simulating molecular excited states on quantum computers

Simulation of molecular electronically excited states presents a unique opportunity for quantum computers. Excited states are often strongly correlated making accurate calculations from sophisticated quantum chemistry methods challenging even for relatively small molecules. Since it is expected that quantum computers will be most useful for solving strongly correlated problems, molecular excited state processes may be one of the first problems to take advantage of the quantum computing era. In this talk, I will outline several methods developed for computing excited states on a quantum computer. I will then present our team’s development of the quantum self-consistent equation-of-motion (q-sc-EOM) method and some of the new advances we are making in its implementation for small molecules. q-sc-EOM energies satisfy important relationships corresponding to the “killer condition” and “size-intensivity/extensivity” and, at the same time, q-sc-EOM is more robust to noise compared with current state-of-the-art methods.   

Quantum Equation of Motion in Active Spaces for Computing Molecular Excitation Energies 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 might we use imperfect near-term quantum computers to solve problems of practical value? Inspired by the recently developed variants of the quantum counterpart of equation-of-motion (qEOM) approach and the orbital optimized variational quantum eigensolver (oo-VQE), we present a quantum algorithm for the calculation of molecular excitation energies and excited states using the active space approximation.   

Excited-CAFQA: A classical simulation bootstrap for the variational estimation of molecular excited states.

Variational Quantum Algorithms (VQAs) are iterative algorithms suited to implementation on current-era quantum devices. VQAs employ classical optimization to minimize cost functions evaluated on quantum circuits. However, the extent to which VQAs manage noise is often insufficient for quantum chemistry applications. One method of improving VQAs is through accurate ansatz initialization. The CAFQA (Clifford Ansatz For Quantum Accuracy) protocol runs a discrete search through a classically simulatable subset of the entire state space to find a desirable initialization. Prior work has evaluated CAFQA applied to the Variational Quantum Eigensolver (VQE), a VQA that computes grounds states of a Hamiltonian. Motivated by CAFQA's success, we propose Excited-CAFQA initialization for Variational Quantum Deflation (VQD), a quantum algorithm that extends VQE by allowing the computation of excited states. VQD recursively computes excited states, by constraining the kth state to be orthogonal to the previous k−1 computed energy states via a penalty term appended to the standard VQE cost function. Just as with VQE, the VQD cost function can be efficiently computed classically for the states considered in the discrete CAFQA search, allowing for the discrete CAFQA optimizer to find good initial parameters for each energy level computation. Preliminary evaluation shows that Excited-CAFQA achieves 90 to 99+% accuracy across a variety of bond lengths and excited states for H2 and HeH+ molecular systems.   

Fault-tolerant quantum algorithm for symmetry-adapted perturbation theory

We investigate the calculation of binding energies using the first-order SAPT formalism on a fault-tolerant device, a crucial task in drug discovery. Observable-specific tensor factorization techniques can significantly reduce the algorithm's runtime. We assess the necessary qubit resources, required circuit depth, expected accuracy, and specific steps needed to implement such an algorithm down to the gate level. We identify two significant bottlenecks: the eigenstate reflection subroutine and the 1-norm of the SAPT observable. Future work should consider developing a framework for second-order SAPT energy contributions, studying resource requirements of basis sets with diffuse functions, and exploring alternative techniques. As quantum algorithms become more efficient, this work will be a crucial starting point for drug design and materials research.   

Quantum chemistry on a budget: resource estimation and measurement optimization for near-term VQE

Finding molecular groundstates using the variational quantum eigensolver (VQE) is one of the main proposed use-cases of NISQ devices. Yet significant hurdles remain to develop practical, useful implementations for existing quantum computing hardware. In particular, the number of electronic excitations varies quadratically with the number of active electrons, leading to deep quantum circuits even for simple molecules. Canonical VQE have been extended to schemes that adaptively select the excitations most contributing to the final groundstate. However, even given an effective scheme for finding these excitations, optimizing their parameters remains computationally expensive and noisy due to the large measurement overhead required to compute the expectation value of realistic target Hamiltonians.

A variety of schemes have been proposed to try to lower the number of measurements needed to perform this optimization. In this work, we use finite-shots simulation to estimate the benefit of each of these and their feasibility for a variety of “benchmark” molecular systems.   

Efficient quantum Monte Carlo algorithm on quantum computers with robust Matchgate shadows

Solving the electronic structure problem of molecules and solids to high accuracy is one of the big challenges in quantum chemistry and condensed matter physics. The rapid emergence and development of quantum computers offer a promising route to systematically tackle this problem. Recent work by Huggins, et al [1] introduced a hybrid quantum-classical quantum Monte Carlo (QC-QMC) algorithm using Clifford classical shadows to compute the ground state of a Fermionic Hamiltonian. This approach displayed inherent noise resilience and the potential for improved accuracy compared to its purely classical counterpart. Nevertheless, the use of Clifford shadows introduces an exponentially scaling post-processing cost. In this work, we investigate an improved QC-QMC scheme utilizing the recently developed Matchgate shadows technique [2], which removes the exponential bottleneck. We also formulate a robust variant of Matchgate shadows, with connections to randomized benchmarking. Finally, we observe that even without the robust protocol, the use of Matchgate shadows is still noise resilient-but with a much more subtle origin than in the case of Clifford classical shadows. Our work strengthens the outlook for the QC-QMC approach, and the viability of achieving practical quantum advantage in the NISQ regime.   

Mapping the Hubbard model to self-consistent spin models: a quantum-embedded Jordan-Wigner transformation

This work introduces a quantum-embeded Jordan-Wigner transformation which preserves the ground-state properties of fermionic Hamiltonians via the resolution of the representability problem within a variational many-body ansatz. As an example, the multi-orbital Hubbard model is mapped to a self-consistent local quantum spin model, utilizing the exact evaluation of the generalized Gutzwiller-Baeryswil wave-function in d=infty. This transformation reveals the order parameter of the Mott transition as the in-plane magnetization of the local quantum spin model, and the associated Landau theory is derived. Furthermore, the resulting local quantum spin model is amenable to solution via a quantum computer, thus presenting a pragmatic hybrid quantum-classical algorithm for scrutinizing the multi-orbital Hubbard model in thermodynamic limit.   

Sampling based quantum training of arbitrary spin-graphs

Machine Learning algorithms trainable on a quantum device are fast emerging as

a robust alternative to the classical ones in a wide variety of tasks as they exploit

the power of quantum devices. Of particular interest are spin-graphs which has been

recently proven to offer an expressive and efficient neural-network based representation

of quantum states of a given system.1–3 In this work, we show that arbitrary such

graphs can be trained on a quantum device using a sampling technique that is capable of

demonstrating quantum advantage. Besides, the algorithm is linear scaling with system

size in qubit resources and gate-requirements. To demonstrate its efficacy, we exemplify

the protocol on a wide-variety systems ranging from strongly correlated molecules to

important class of spin models which forms the basis of quantum magnetism in 1D and

2D. Apart from such wide trainability, the results are in reasonable agreement with

those obtained from conventional means which serves to explicate that our algorithm

can offer a promising alternative to traditional standards even using noisy near-term

devices.   

Intrinsic Error Mitigation in the Cascaded Variational Quantum Eigensolver

We present results obtained using the Cascaded Variational Quantum Eigensolver (CVQE) algorithm.  The results indicate the parameter optimization process within CVQE automatically and substantially mitigates the errors introduced during quantum circuit executions.  This is possible because all of the errors are introduced before the optimization process begins in the CVQE algorithm.  In contrast, these errors are introduced at every iteration of the optimization process in other Variational Quantum Eigensolver (VQE) algorithms.  Furthermore, because the entire ansatz in the VQE algorithm is implemented in a quantum circuit, the solution will necessarily be sensitive to quantum errors.  In CVQE, only a portion of the ansatz is implemented in the quantum circuit and the variational parameters are not implemented in this circuit.  Thus, the variational parameters are able to freely evolve to compensate for initial errors during the circuit executions.  Consequently, the CVQE algorithm could be capable of simulating systems within a certain error tolerance that would otherwise be impossible using present-day quantum computers.   

Surrogate Optimization for Quantum Circuits

VAriational quantum Eigensolvers are touted as a near-term algoirthm capable of impacting many applicaitons. However, the potential has yet ot be realized with few claims of quantum advantage and high resource estimates mainly due to the need for optimization in the presence of noise. Finding algorithms and methods capable to improve the convergence is essential to acclerate the capabilities of near-term hardware for VQE or more broad applications of hybrid methods in which optimization is required. To this goal we look to use modern approaches recently developed in circuit simulations and stochastic classical optimization that can be combined in a surrogate optimization approach to quantum circuits. Using an approximate state vector simulator, we efficiently calculate an approximate Hessian, fed as an input for a detailed quantum circuit simulator. We demonstrate the capabilities of such an approach with and without sampling noise. We also show that this method outperforms Powell in the precense of quantum circuit shot noise by a factor of 2-4   

Schrödinger as a Quantum Programmer: Estimating Entanglement via Steering

Quantifying entanglement is an important task by which the resourcefulness of a quantum state can be measured. Here we develop a quantum algorithm that tests for and quantifies the separability of a general bipartite state, by making use of the quantum steering effect, the latter originally discovered by Schrödinger. Our separability test consists of a distributed quantum computation involving two parties: a computationally limited client, who prepares a purification of the state of interest, and a computationally unbounded server, who tries to steer the reduced systems to a probabilistic ensemble of pure product states. To design a practical algorithm, we replace the role of the server by a combination of parameterized unitary circuits and classical optimization techniques to perform the necessary computation. The result is a variational quantum steering algorithm (VQSA), which is a modified separability test that is better suited for the capabilities of quantum computers available today. We then simulate our VQSA on noisy quantum simulators and find favorable convergence properties on the examples tested. We also develop semidefinite programs, executable on classical computers, that benchmark the results obtained from our VQSA. Our findings here thus provide a meaningful connection between steering, entanglement, quantum algorithms, and quantum computational complexity theory. They also demonstrate the value of a parameterized mid-circuit measurement in a VQSA and represent a first-of-its-kind application for a distributed VQA.   

The Adjoint Is All You Need: Characterizing Barren Plateaus in Quantum Ansätze

Using tools from the representation theory of compact Lie groups, we formulate a theory of Barren Plateaus for parameterized quantum circuits whose observables lie in their dynamical Lie algebra (DLA), a setting that we term Lie Algebra Supported Ansatz (LASA). A large variety of commonly used ansätze such as the Hamiltonian Variational Ansatz, Quantum Alternating Operator Ansatz, and many equivariant quantum neural networks are LASAs. In particular, our theory provides for the first time the ability to compute the variance of the gradient of the cost function for a non-trivial, subspace uncontrollable family of quantum circuits, the quantum compound ansätze. We rigorously prove that the variance of the gradient of the cost function, under Haar initialization, scales inversely with the dimension of the DLA, which agrees with existing numerical observations. Lastly, we include potential extensions for handling cases when the observable lies outside of the DLA and the implications of our results.