Quantum variational embedding for ground-state energy problems: sum of squares and cluster selection

We introduce a sum-of-squares SDP hierarchy approximating the ground-state energy from below for quantum many-body problems, with a natural quantum embedding interpretation. We establish the connections between our approach and other variational methods for lower bounds, including the variational embedding, the RDM method in quantum chemistry, and the Anderson bounds. Additionally, inspired by the quantum information theory, we propose efficient strategies for optimizing cluster selection to tighten SDP relaxations while staying within a computational budget. Numerical experiments are presented to demonstrate the effectiveness of our strategy. As a byproduct of our investigation, we find that quantum entanglement has the potential to capture the underlying graph of the many-body Hamiltonian

Quantum approximate Bayesian computation for NMR model inference

Recent technological advances may lead to the development of small scale quantum computers capable of solving problems that cannot be tackled with classical computers. A limited number of algorithms has been proposed and their relevance to real world problems is a subject of active investigation. Analysis of many-body quantum system is particularly challenging for classical computers due to the exponential scaling of Hilbert space dimension with the number of particles. Hence, solving problems relevant to chemistry and condensed matter physics are expected to be the first successful applications of quantum computers. In this paper, we propose another class of problems from the quantum realm that can be solved efficiently on quantum computers: model inference for nuclear magnetic resonance (NMR) spectroscopy, which is important for biological and medical research. Our results are based on the cumulation of three interconnected studies. Firstly, we use methods from classical machine learning to analyze a dataset of NMR spectra of small molecules. We perform a stochastic neighborhood embedding and identify clusters of spectra, and demonstrate that these clusters are correlated with the covalent structure of the molecules. Secondly, we propose a simple and efficient method, aided by a quantum simulator, to extract the NMR spectrum of any hypothetical molecule described by a parametric Heisenberg model. Thirdly, we propose an efficient variational Bayesian inference procedure for extracting Hamiltonian parameters of experimentally relevant NMR spectra

The Quantum Many-Body Problem: Methods and Analysis

This dissertation concerns the quantum many-body problem, which is the problem of predicting the properties of systems of several quantum particles from the first principles of quantum mechanics. Included under this umbrella are various problems of fundamental importance in quantum chemistry, condensed matter physics, and materials science. Of particular interest is the electronic structure problem, the problem of determining the state of the electrons in a system with fixed atomic nuclei. Since direct numerical solution of the many-body Schrödinger equation is intractable even for systems of moderate size, a diverse array of approximate methods has been developed. The broad goals of this dissertation are to improve the mathematical understanding of certain widely-used approximations, as well as to propose new methods. Roughly speaking, we consider three (overlapping) categories of methods: Green's function methods, embedding methods, and variational methods.One can understand Green's function methods in terms of many-body perturbation theory, which computes series expansions of physical quantities about a non-interacting reference system. These expansions can be expressed graphically in terms of Feynman diagrams, which can in turn be reorganized, in some cases, into an expansion in terms of so-called bold diagrams. Green's function methods can be specified by choosing a subset of bold diagrams to approximate the sum. At the same time, such methods can be understood in terms of an object known as the Luttinger-Ward (LW) functional, which admits a representation in terms of the bold diagrams. Many aspects of these constructions are purely formal, and indeed the existence of the fermionic LW functional as a single-valued functional has recently been called into question. To contribute to the understanding of these issues, we provide rigorous proofs of the combinatorial construction and analytic interpretation of the bold diagrams in the simplified setting of a classical field theory. In this setting we also provide a rigorous non-perturbative construction of the LW functional via convex duality and prove several key properties, including continuity up to the boundary of its domain and asymptotics in the limit of large interaction.Quantum embedding methods, meanwhile, view a large system as being composed of smaller fragments that are treated with high accuracy and embedded in the larger system in a mutually consistent way. Inspired by a connection between the boundary analysis of the LW functional and embedding, we perform similar analysis for the 1-RDM theory for fermionic systems, which is also developed via convex duality, illustrating a relation to fermionic embedding methods such as the density matrix embedding theory (DMET).Another embedding method of note is the dynamical mean-field theory (DMFT), which is at the same time a Green's function method that can be understood in terms of the LW functional. DMFT relies on the solution of impurity problems, which specify the embedding of an interacting system into a non-interacting bath. Underlying DMFT is a result about the sparsity pattern of the self-energy matrix for impurity problems, which to our knowledge has not been proved in the literature. We provide a rigorous proof of this result in various classical and quantum settings. We go on to investigate the fermionic DMFT in depth, identifying the key mathematical structures that appear in the algorithmic loop for solving it and using these to prove the well-posedness of this loop, in a certain sense.Finally, we introduce a suite of new approaches to the quantum many-body problem that provide variational lower bounds to the ground-state energy. These methods, which combine the themes of convexity and embedding, are based on novel convex relaxations of the variational principles for the ground-state energies of many-body systems. To begin, we recover a second-quantized version of the formalism of strictly correlated electrons (SCE), which yields an exact expression for the exchange-correlation functional in Kohn-Sham density functional theory in the limit of infinite Coulomb repulsion in terms of the solution of a multi-marginal optimal transport problem. We introduce a semidefinite relaxation method for approximately solving this problem and obtaining a lower bound for the ground-state energy. The ideas underlying this relaxation are generalized considerably, outside the context of SCE, to yield much tighter lower bounds, which we validate numerically for both quantum spin systems and fermionic systems. We also describe how these relaxation methods can be interpreted as embedding methods via convex duality

Increasing the representation accuracy of quantum simulations of chemistry without extra quantum resources

Proposals for near-term experiments in quantum chemistry on quantum computers leverage the ability to target a subset of degrees of freedom containing the essential quantum behavior, sometimes called the active space. This approximation allows one to treat more difficult problems using fewer qubits and lower gate depths than would otherwise be possible. However, while this approximation captures many important qualitative features, it may leave the results wanting in terms of absolute accuracy (basis error) of the representation. In traditional approaches, increasing this accuracy requires increasing the number of qubits and an appropriate increase in circuit depth as well. Here we introduce a technique requiring no additional qubits or circuit depth that is able to remove much of this approximation in favor of additional measurements. The technique is constructed and analyzed theoretically, and some numerical proof of concept calculations are shown. As an example, we show how to achieve the accuracy of a 20 qubit representation using only 4 qubits and a modest number of additional measurements for a simple hydrogen molecule. We close with an outlook on the impact this technique may have on both near-term and fault-tolerant quantum simulations

Resource-Efficient Chemistry on Quantum Computers with the Variational Quantum Eigensolver and the Double Unitary Coupled-Cluster Approach.

Applications of quantum simulation algorithms to obtain electronic energies of molecules on noisy intermediate-scale quantum (NISQ) devices require careful consideration of resources describing the complex electron correlation effects. In modeling second-quantized problems, the biggest challenge confronted is that the number of qubits scales linearly with the size of the molecular basis. This poses a significant limitation on the size of the basis sets and the number of correlated electrons included in quantum simulations of chemical processes. To address this issue and enable more realistic simulations on NISQ computers, we employ the double unitary coupled-cluster (DUCC) method to effectively downfold correlation effects into the reduced-size orbital space, commonly referred to as the active space. Using downfolding techniques, we demonstrate that properly constructed effective Hamiltonians can capture the effect of the whole orbital space in small-size active spaces. Combining the downfolding preprocessing technique with the variational quantum eigensolver, we solve for the ground-state energy of H2, Li2, and BeH2 in the cc-pVTZ basis using the DUCC-reduced active spaces. We compare these results to full configuration-interaction and high-level coupled-cluster reference calculations