Over the last century, a large number of physical and mathematical developments paired with rapidly advancing technology have allowed the field of quantum chemistry to advance dramatically. However, the lack of computationally efficient methods for the exact simulation of quantum systems on classical computers presents a limitation of current computational approaches. We report, in detail, how a set of pre-computed molecular integrals can be used to explicitly create a quantum circuit, i.e. a sequence of elementary quantum operations, that, when run on a quantum computer, obtains the energy of a molecular system with fixed nuclear geometry using the quantum phase estimation algorithm. We extend several known results related to this idea and discuss the adiabatic state preparation procedure for preparing the input states used in the algorithm. With current and near future quantum devices in mind, we provide a complete example using the hydrogen molecule of how a chemical Hamiltonian can be simulated using a quantum computer.
Over the last century, a large number of physical and mathematical developments paired with rapidly advancing technology have allowed the field of quantum chemistry to advance dramatically. However, the lack of computationally efficient methods for the exact simulation of quantum systems on classical computers presents a limitation of current computational approaches. We report, in detail, how a set of pre-computed molecular integrals can be used to explicitly create a quantum circuit, i.e. a sequence of elementary quantum operations, that, when run on a quantum computer, obtains the energy of a molecular system with fixed nuclear geometry using the quantum phase estimation algorithm. We extend several known results related to this idea and discuss the adiabatic state preparation procedure for preparing the input states used in the algorithm. With current and near future quantum devices in mind, we provide a complete example using the hydrogen molecule of how a chemical Hamiltonian can be simulated using a quantum computer.
methods for the exact simulation of quantum systems on classical computers presents a limitation of current computational approaches. We report, in detail, how a set of pre-computed molecular integrals can be used to explicitly create a quantum circuit, i.e. a sequence of elementary quantum operations, that, when run on a quantum computer, to obtain the energy of a molecular system with xed nuclear geometry using the quantum phase estimation algorithm. We extend several known results related to this idea and discuss the adia- batic state preparation procedure for preparing the input states used in the algorithm. With current and near future quantum devices in mind, we provide a complete example using the hydrogen molecule, of how a chemical Hamiltonian can be simulated using a quantum computer.
Over the last century, ingenious physical and mathematical insights paired with rapidly advancing technology have allowed the field of quantum chemistry to advance dramatically. However, efficient methods for the exact simulation of quantum systems on classical computers do not exist. The present paper reports an extension of one of the authors’ previous work [Aspuru-Guzik et al., Science 309 p. 1704, (2005)] where it was shown that the chemical Hamiltonian can be efficiently simulated using a quantum computer. In particular, we report in detail how a set of molecular integrals can be used to create a quantum circuit that allows the energy of a molecular system with fixed nuclear geometry to be extracted using the phase estimation algorithm proposed by Abrams and Lloyd [Phys. Rev. Lett. 83 p. 5165, (1999)]. We extend several known results related to this idea and present numerical examples of the state preparation procedure required in the algorithm. With future quantum devices in mind, we provide a complete example using the Hydrogen molecule, of how a chemical Hamiltonian can be simulated using a quantum computer. Some of the results we present here represent an extension of our recent collaboration appearing in B. P. Lanyon et al. [Nature Chem., advance online publication, doi: 10.1038/nchem.483 (2010)].
Determining properties of molecules and materials is one of the premier applications of quantum computing. A major question in the thesis is: how might we use long- and near-term quantum computers to solve problems of chemical interest? First, we give an overview of the standard model in electronic structure theory. Electronic structure methods have been improved significantly in the last decades; however, it is still a non-trivial task to fit experimental results for complex molecular systems. Second, we introduce quantum computation and quantum algorithms as an alternative device to solve the electronic structure problem. In the remaining part of the thesis, we describe the research projects. The thesis covers two (rather different) research directions: software and hardware projects. For the software part, we contribute with two new algorithms: Phase-Estimation Interval Target Energy Readout (PHILTER) and Recursive Variational Series Estimation (RVSE), where both are designed to be run on a quantum digital computer and extract materials and molecular properties. The PHILTER algorithm is a modification of the Quantum Phase Estimation and is designed to compute excited state energies within a target energy-interval for cases where good approximations for the states in the target energy-interval are either unknown or hard to prepare. For the RVSE algorithm, we pose the question: how might we use imperfect near-term quantum computers to solve problems of practical value? In this work, we introduce a method to compute expectation values of general functions of a Hamiltonian with respect to an input state using low depth quantum circuits suitable for noisy intermediate-scale quantum computers. For the hardware part, we propose a new chemical quantum computer by using molecular electronics as a basis for quantum computing. In this work, we pose the question: can scattering of electrons in molecules produce quantum gates? We construct one-qubit gates using one-electron scattering in molecules, and two-qubit controlled-phase gates using electron-electron scattering along metallic leads. We show initial applications of the framework by illustrating one-qubit gates using the molecular electronic structure of molecular hydrogen as a baseline model.
Quantum chemistry models atoms and molecules using quantum mechanics, with the electronic structure problem at its core. This problem, which determines the energy and properties of electrons in a molecule, is exponentially hard to solve exactly on classical computers. To approximate solutions, effective models introduce layers of simplifications. However, these approximations can fail for strongly correlated systems, necessitating alternative approaches. Quantum computers, capable of processing quantum information natively, offer a promising path to simulate molecular systems beyond classical reach, with potential applications in catalysis, materials science, and photochemistry. Despite rapid advancements in quantum hardware, practical quantum computations remain challenging due to limited resources. This thesis optimizes quantum resource usage in computational chemistry, introducing novel embedding techniques and exploring applications beyond ground-state energies. Chapter 1 establishes the foundation by reviewing quantum chemistry methods, from Hartree-Fock and Density Functional Theory (DFT) to active space approaches. It also provides an introduction to quantum computing for chemistry, focusing on near-term algorithms. The following chapters present novel methods aimed at reducing quantum resource requirements and tackling problems classical methods struggle with. Chapter 2 introduces a technique to reduce the
λ-norm of the electronic structure Hamiltonian, a key measure of quantum resource demand. By transforming molecular orbitals, this method enhances sampling efficiency in Noisy Intermediate-Scale Quantum (NISQ) algorithms and reduces runtime in Fault-Tolerant Quantum Computing (FTQC). Localized orbitals outperform canonical ones, and an orbital-optimization scheme is proposed to minimize the
λ-norm. Benchmarks on hydrogen chains, FeMoco, and Ruthenium complexes demonstrate its effectiveness. Chapter 3 presents the FragPT2 framework for simulating large systems with multiple interacting fragments. This method embeds each fragment in the mean field of others and applies multi-reference perturbation theory to capture inter-fragment correlations such as dispersion and charge transfer efficiently. The approach reduces computational overhead and enables scalable simulations, as shown in applications to \ce{N2} dimers, aromatic dimers, and butadiene. Chapter 4 focuses on photochemical systems, where multiple electronic states must be treated equally. The state-averaged orbital-optimized variational quantum eigensolver (SAOOVQE) is extended with analytical energy gradients and non-adiabatic couplings, enabling geometry optimizations and simulations of non-adiabatic dynamics on quantum hardware. This is demonstrated by locating the conical intersection in formaldimine, highlighting SAOOVQE’s potential for studying photoisomerization in biomolecules. Building on this, Chapter 5 develops a quantum algorithm to detect conical intersections, where the Born-Oppenheimer approximation breaks down. These intersections are characterized by a discrete Berry phase (0 or
π), representing their topological nature. The algorithm extracts this phase as a single bit of information, ensuring resilience to noise. Analytical proof and numerical validation using a toy molecular model illustrate its effectiveness for studying light-perception biochemistry and related photochemical processes. Finally, Chapter 6 explores machine learning density functionals trained on noisy quantum data. While DFT is computationally efficient, its accuracy depends on approximated exchange-correlation functionals. This work proposes a neural network-based approach to learn the exact functional from quantum data, showing that training on noisy outputs from NISQ devices can mitigate sampling errors. Applications to the Fermi-Hubbard model and other systems demonstrate its potential to generalize and provide accurate results under noise. This thesis advances quantum computational chemistry by optimizing quantum resource usage, developing new embedding strategies, and expanding the scope of quantum simulations beyond ground-state properties. These contributions pave the way for practical quantum algorithms that could revolutionize molecular simulations in the near future.
Abstract In this chapter, we present quantum algorithms to approximate Hamiltonian eigenstates. We start discussing the quantum phase estimation algorithms, an important subroutine that allows us to approximate eigenpairs of a Hamiltonian operator given an initial guess and a quantum circuit implementing the time evolution operator. We then present the adiabatic state preparation, a technique to transform the ground state of a Hamiltonian into the ground state of another Hamiltonian. We describe two important heuristic methods, the quantum approximate optimization algorithm, and the variational quantum eigensolver. The second part of the chapter discusses concrete applications in the field of quantum chemistry. After introducing the Born-Oppenheimer Hamiltonian in second quantization and the Hartree-Fock method, we illustrate how a chemical problem can be mapped onto a quantum computer. We conclude the chapter with the simulation of the hydrogen molecule in a minimal basis with the iterative phase estimation algorithm.