The computational complexity of density functional theory

Density functional theory is a successful branch of numerical simulations of quantum systems. While the foundations are rigorously defined, the universal functional must be approximated resulting in a `semi'-ab initio approach. The search for improved functionals has resulted in hundreds of functionals and remains an active research area. This chapter is concerned with understanding fundamental limitations of any algorithmic approach to approximating the universal functional. The results based on Hamiltonian complexity presented here are largely based on \cite{Schuch09}. In this chapter, we explain the computational complexity of DFT and any other approach to solving electronic structure Hamiltonians. The proof relies on perturbative gadgets widely used in Hamiltonian complexity and we provide an introduction to these techniques using the Schrieffer-Wolff method. Since the difficulty of this problem has been well appreciated before this formalization, practitioners have turned to a host approximate Hamiltonians. By extending the results of \cite{Schuch09}, we show in DFT, although the introduction of an approximate potential leads to a non-interacting Hamiltonian, it remains, in the worst case, an NP-complete problem.Comment: Contributed chapter to "Many-Electron Approaches in Physics, Chemistry and Mathematics: A Multidisciplinary View

Quantum number preserving ansätze and error mitigation studies for the variational quantum eigensolver

Computational chemistry has advanced rapidly in the last decade on the back of the progress of increased performance in CPU and GPU based computation. The prediction of reaction properties of varying chemical compounds in silico promises to speed up development in, e.g., new catalytic processes to reduce energy demand of varying known industrial used reactions. Theoretical chemistry has found ways to approximate the complexity of the underlying intractable quantum many-body problem to various degrees to achieve chemically accurate ab initio calculations for various, experimentally verified systems. Still, in theory limited by fundamental complexity theorems accurate and reliable predictions for large and/or highly correlated systems elude computational chemists today. As solving the Schrödinger equation is one of the main use cases of quantum computation, as originally envisioned by Feynman himself, computational chemistry has emerged as one of the applications of quantum computers in industry, originally motivated by potential exponential improvements in quantum phase estimation over classical counterparts. As of today, most rigorous speed ups found in quantum algorithms are only applicable for so called error-corrected quantum computers, which are not limited by local qubit decoherence in the length of the algorithms possible. Over the last decade, the size of available quantum computing hardware has steadily increased and first proof of concepts of error-correction codes have been achieved in the last year, reducing error rates below the individual error rates of qubits comprising the code. Still, fully error-corrected quantum computers in sizes that overcome the constant factor in speed up separating classical and quantum algorithms in increasing system size are a decade or more away. Meanwhile, considerable efforts have been made to find potential quantum speed ups of non-error corrected quantum systems for various applications in the noisy intermediate-scale quantum (NISQ) era. In chemistry, the variational quantum eigensolver (VQE), a family of classical-quantum hybrid algorithms, has become a topic of interest as a way of potentially solving computational chemistry problems on current quantum hardware. The main contributions of this work are: extending the VQE framework with two new potential ansätze, (1) a maximally dense first-order trotterized ansatz for the paired approximation of the electronic structure Hamiltonian, (2) a gate fabric with many favourable properties like conserving relevant quantum numbers, locality of individual operations and potential initialisation strategies mitigating plateaus of vanishing gradient during optimisation. (3) Contributions to one of largest and most complex VQE to date, including the aforementioned ansatz in paired approximation, benchmarking different error-mitigation techniques to achieve accurate results, extrapolating performance to give perspective on what is needed for NISQ devices having potential in competing with classical algorithms and (4) Simulations to find optimal ways of measuring Hamiltonians in this error-mitigated framework. (5) Furthermore a simulation of different purification error mitigation techniques and their combination under different noise models and a way of efficiently calibrating for coherent noise for one of them is part of this manuscript. We discuss the state of VQE after almost a decade after its introduction and give an outlook on computational chemistry on quantum computers in the near future

Computational Complexity in Electronic Structure

In quantum chemistry, the price paid by all known efficient model chemistries is either the truncation of the Hilbert space or uncontrolled approximations. Theoretical computer science suggests that these restrictions are not mere shortcomings of the algorithm designers and programmers but could stem from the inherent difficulty of simulating quantum systems. Extensions of computer science and information processing exploiting quantum mechanics has led to new ways of understanding the ultimate limitations of computational power. Interestingly, this perspective helps us understand widely used model chemistries in a new light. In this article, the fundamentals of computational complexity will be reviewed and motivated from the vantage point of chemistry. Then recent results from the computational complexity literature regarding common model chemistries including Hartree-Fock and density functional theory are discussed.Comment: 14 pages, 2 figures, 1 table. Comments welcom

Introduction to Quantum Algorithms for Physics and Chemistry

In this introductory review, we focus on applications of quantum computation to problems of interest in physics and chemistry. We describe quantum simulation algorithms that have been developed for electronic-structure problems, thermal-state preparation, simulation of time dynamics, adiabatic quantum simulation, and density functional theory.Comment: 44 pages, 5 figures; comments or suggestions for improvement are welcom

Simulating chemistry using quantum computers

The difficulty of simulating quantum systems, well-known to quantum chemists, prompted the idea of quantum computation. One can avoid the steep scaling associated with the exact simulation of increasingly large quantum systems on conventional computers, by mapping the quantum system to another, more controllable one. In this review, we discuss to what extent the ideas in quantum computation, now a well-established field, have been applied to chemical problems. We describe algorithms that achieve significant advantages for the electronic-structure problem, the simulation of chemical dynamics, protein folding, and other tasks. Although theory is still ahead of experiment, we outline recent advances that have led to the first chemical calculations on small quantum information processors.Comment: 27 pages. Submitted to Ann. Rev. Phys. Che

Sea Spray Aerosol: Where Marine Biology Meets Atmospheric Chemistry.

Atmospheric aerosols have long been known to alter climate by scattering incoming solar radiation and acting as seeds for cloud formation. These processes have vast implications for controlling the chemistry of our environment and the Earth's climate. Sea spray aerosol (SSA) is emitted over nearly three-quarters of our planet, yet precisely how SSA impacts Earth's radiation budget remains highly uncertain. Over the past several decades, studies have shown that SSA particles are far more complex than just sea salt. Ocean biological and physical processes produce individual SSA particles containing a diverse array of biological species including proteins, enzymes, bacteria, and viruses and a diverse array of organic compounds including fatty acids and sugars. Thus, a new frontier of research is emerging at the nexus of chemistry, biology, and atmospheric science. In this Outlook article, we discuss how current and future aerosol chemistry research demands a tight coupling between experimental (observational and laboratory studies) and computational (simulation-based) methods. This integration of approaches will enable the systematic interrogation of the complexity within individual SSA particles at a level that will enable prediction of the physicochemical properties of real-world SSA, ultimately illuminating the detailed mechanisms of how the constituents within individual SSA impact climate

Exploring evolutionary stability in a concurrent artificial chemistry

Multi-level selection has proven to be an affective mean to provide resistance against parasites for catalytic networks (Cronhjort and Blomberg, 1997). One way to implement these multi-level systems is to group molecules into several distinct compartments (cells) which are capable of cellular division (where an offspring cell replaces another cell). In such systems parasitized cells decay and are ultimately displaced by neighboring healthy cells. However in relatively small cellular populations, it is also possible that infected cells may rapidly spread parasites throughout the entire cellular population. In which case, group selection may fail to provide resistance to parasites. In this paper, we propose a concurrent artificial chemistry (AC) which has been implemented on a cluster of computers where each cell is running on a single CPU. This multi-level selectional artificial chemistry called the Molecular Classifier Systems was based on the Holland broadcast language. An attribute inherent to such a concurrent system is that the computational complexity of the molecular species contained in a reactor may now affect the fitness of the cell. This molecular computational cost may be regarded as the chemical activation energy necessary for a reaction to occur. Such a property is often not considered in typical Artificial Life models. Our experimental results obtained with this system suggest that this activation energy property may improve the resistance to parasites for catalytic networks. This work highlights some of the benefits that could be obtained using a concurrent architecture on top of computational efficiency. We first briefly present the Molecular Classifier Systems, this is then followed by a description of the multi-level concurrent model. Finally we discuss the benefits of using this multi-level concurrent model to enhance evolutionary stability for catalytic networks in our AC

Direct Application of the Phase Estimation Algorithm to Find the Eigenvalues of the Hamiltonians

The eigenvalue of a Hamiltonian, $\mathcal{H}$, can be estimated through the phase estimation algorithm given the matrix exponential of the Hamiltonian, $exp(-i\mathcal{H})$. The difficulty of this exponentiation impedes the applications of the phase estimation algorithm particularly when $\mathcal{H}$ is composed of non-commuting terms. In this paper, we present a method to use the Hamiltonian matrix directly in the phase estimation algorithm by using an ancilla based framework: In this framework, we also show how to find the power of the Hamiltonian matrix-which is necessary in the phase estimation algorithm-through the successive applications. This may eliminate the necessity of matrix exponential for the phase estimation algorithm and therefore provide an efficient way to estimate the eigenvalues of particular Hamiltonians. The classical and quantum algorithmic complexities of the framework are analyzed for the Hamiltonians which can be written as a sum of simple unitary matrices and shown that a Hamiltonian of order $2^n$ written as a sum of $L$ number of simple terms can be used in the phase estimation algorithm with $(n+1+logL)$ number of qubits and $O(2^anL)$ number of quantum operations, where $a$ is the number of iterations in the phase estimation. In addition, we use the Hamiltonian of the hydrogen molecule as an example system and present the simulation results for finding its ground state energy.Comment: 10 pages, 3 figure