Computing many-body ground state energies and resolving electronic structure calculations are fundamental problems for fields such as quantum chemistry or condensed matter. Several quantum computing algorithms that address these problems exist, although it is often challenging to establish rigorous bounds on their performances. Here we detail three algorithms that produce approximate ground states for many-body and electronic structure problems, generalizing some previously known results for 2-local Hamiltonians. Each method comes with asymptotic bounds on the energies produced. The first one produces a separable state which improves on random product states. We test it on a spinless Hubbard model, validating numerically the theoretical result. The other two algorithms produce entangled states via shallow or deep circuits, improving on the energies of given initial states. We demonstrate their performance via numerical experiments on a 2-dimensional Hubbard model, starting from a checkerboard product state, as well as on some chemistry Hamiltonians, using the Hartree-Fock state as reference. In both cases, we show that the approximate energies produced are close to the exact ones. These algorithms provide a way to systematically improve the estimation of ground state energies and can be used stand-alone or in conjunction with existing quantum algorithms for ground states.
Computing many-body ground state energies and resolving electronic structure calculations are fundamental problems for fields such as quantum chemistry or condensed matter. Several quantum computing algorithms that address these problems exist, although it is often challenging to establish rigorous bounds on their performances. Here we detail three algorithms that produce approximate ground states for many-body and electronic structure problems, generalizing some previously known results for 2-local Hamiltonians. Each method comes with asymptotic bounds on the energies produced. The first one produces a separable state which improves on random product states. We test it on a spinless Hubbard model, validating numerically the theoretical result. The other two algorithms produce entangled states via shallow or deep circuits, improving on the energies of given initial states. We demonstrate their performance via numerical experiments on a 2-dimensional Hubbard model, starting from a checkerboard product state, as well as on some chemistry Hamiltonians, using the Hartree-Fock state as reference. In both cases, we show that the approximate energies produced are close to the exact ones. These algorithms provide a way to systematically improve the estimation of ground state energies and can be used stand-alone or in conjunction with existing quantum algorithms for ground states.
In ab-initio quantum chemistry, solving the electronic structure problem of a molecular system of interest to obtain an approximation of the ground and, in some cases, the excited state energies is a core and central task in various quantum chemistry processes. These include, but certainly not limited to, the calculation of the activation energy–that is the minimal energy required to launch a given chemical reaction–, the dipole and multipole moments which describe the polarity of the molecule, performing the optimization of the geometry of the molecule and searching for transition states. Furthermore, many quantum chemistry applications are predicated, not merely on the approximation of the ground state energy, but also on the corresponding first order derivatives. The body of literature of classical algorithms approximating the ground state energy and its first order derivative is extensively rich with a plethora of methods such as self-consistent mean-field methods, coupled-cluster methods, density matrix renormalization group and many more, each of which has intrinsic set of properties and characteristics that makes it suitable for specific tasks and cases. However, classical methods face a practically insurmountable challenge: the efficient computation of high accuracy approximation of ground state energies of large system sizes. This is fundamentally because the electronic structure problem provably belongs to the complexity class of QMA-hard problems. Which also means that it is out of reach for quantum algorithms as well in general. However, with certain assumptions that are expected to hold molecules of interest–that so far proved to be classically challenging because the required accuracy set by the standard of chemical accuracy threshold (generally held in quantum chemistry)– quantum algorithms can efficiently treat such cases. While originally theorized by Richard Feynmann as a potentially efficient way of simulating some classically intractable many-body systems, quantum computing has since proven to have a stronger and richer potential paving the way for the inception of new algorithms such as Shor’s algorithm for integer factorization, HHL for solving linear system of equation and Grover’s algorithm for constrained binary search, just to name a few. These algorithms provably exhibit an improvement rate that goes respectively exponentially and quadratically in the time complexity. Quantum chemistry is considered a prominent candidate to benefit from quantum computing given the variety of quantum algorithms put forth to solve the electronic structure problem of particular systems of interest. These quantum algorithm proposals can be broadly categorized in three main categories that span the spectrum: from noisy intermediate scale quantum (NISQ) computers to early and full scale fault tolerant quantum ((E)FTQC) computers. Firstly, the variational quantum eigensolver (VQE) family of quantum algorithms which variationally search for the ground state energy through the optimization of the energy of a parametrized state prepared in a quantum computer using an ansatz. Secondly, the quantum Krylov subspace diagonalization (QKSD) methods, which generate a subspace of the original Hilbert space and efficiently produce the lowest energy in said subspace as its output. Lastly, we have the quantum phase estimation (QPE) algorithms, which provably produce the ground state energy in Heisenberg scaling presenting hence a provable efficient solution. In this thesis, we firstly set to work on the problem of efficient Hamiltonian linear combination of unitaries (LCU) representation. This is mainly motivated by the wide-range impact an efficient compression of the Hamiltonian can potentially have on several quantum algorithms such as VQE, QPE...etc. We designed a method that efficiently compress the Hamiltonian representation in a way that is beneficially impactful across the set of the aforementioned quantum algorithms. Namely, we motivate and present in Sections 1.3 and 1.4 respectively, the regularized compressed double-factorization (RC-DF) and show how it reduces the energy measurement complexity both in terms of number of distinct observables and number of samples. Moreover, it also reduces the total depth of the quantum phase estimation circuit thanks to its low normalization factor. Secondly, given the importance of the energy derivatives, in Sections 2.1 and 2.2, we introduce and present an efficient quantum algorithm for the calculation of the analytic gradient of VQE energies of double-factorized Hamiltonians. Furthermore, in Section 2.1, we expand on the utility of RC-DF by presenting a quantum algorithm that determines the analytic gradient of energies measured with respect to RC-DF Hamiltonains. Third, in Section 3.1, we summarize the theoretical results of purification error-mitigation techniques, namely echo verification and virtual distillation. In Section 3.2, we report on the results of the experimental bench mark, of said techniques, performed on on Google’s Sycamore device to validate the theoretical scaling of error suppression and extrapolate the resources needed for classically intractable system sizes. Fourth, in Section 4 we establish the challenges of computing energy gradients in a quantum Krylov frame work. To overcome the problem, we provide, in Section 4.3, an efficient quantum algorithm for calculating energy derivative obtained through QKSD procedures. Our approach shows how the QKSD ground state can be prepared using quantum signal processing (QSP) and reduces the number of observables to O(1) per reduced density matrix (RDM). Lastly, in light of the importance of quantum RAMs due to their utility in a variety of quantum algorithms that require efficient initial state preparation, as explained in Section 5.5, we address the problematic exponential depth of the bucket-brigade QRAM (when decomposed). We present in Section 5.6 a very efficient Clifford+T decomposition of the bucket-brigade QRAM, which preserves the linear scaling of the depth of the circuit without introducing any additional ancilla.
With quantum computers on the horizon, it is essential to understand what problems they can be used to solve. One potential application is simulating quantum systems, in particular, systems with many interacting electrons. However, there is a disparity between the computational resources required to study these problems and the capabilities of modern experimental quantum devices. In this thesis, we make progress toward closing this gap by reducing the cost of quantum simulation of quantum chemistry on a number of fronts. In the first half, we develop three improved quantum simulation algorithms by approximating time evolution under the Hamiltonian using a truncated Taylor series rather than the Trotter-Suzuki decomposition. We do this first for the molecular electronic structure Hamiltonian in two representations, obtaining asymptotic improvements in terms of both the number of orbitals and the required precision. We also apply the truncated Taylor series paradigm to quantum simulation in real space, where we consider simulation of general wave functions discretized on a uniform grid. In the second half, by working with Hamiltonians tailored specifically to quantum algorithms, we demonstrate that Trotter steps of the electronic structure Hamiltonian can be simulated in exactly N depth and with only linear connectivity, where N is the number of spin-orbitals. This resolves the problem of non-locality in fermion-to-qubit mappings, and has wide-ranging applications for variational and phase estimation-based quantum simulation of quantum chemistry. We further study this algorithm on an error-corrected quantum computer using the surface code. Benchmarking it alongside other product formula approaches, we show that quantum simulation of scientifically important instances of the electronic structure problem that are classically intractable can be performed with only a few hundred thousand physical qubits and fewer than one billion costly fault-tolerant gates. Finally, we study the effect of using randomized Hamiltonians in iterative phase estimation, and how this allows us to reduce the costs of simulation as well as to exploit knowledge of the ground state in how we design our simulation algorithm.
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.
The ground state energy and the free energy of Quantum Local Hamiltonians are fundamental quantities in quantum many-body physics, however, it is QMA-Hard to estimate them in general. In this paper, we develop new techniques to find classical, additive error product-state approximations for these quantities on certain families of Quantum k-Local Hamiltonians. Namely, those which are either dense, have low threshold rank, or are defined on a sparse graph that excludes a fixed minor, building on the methods and the systems studied by Brandão and Harrow, Gharibian and Kempe, and Bansal, Bravyi and Terhal. We present two main technical contributions. First, we discuss a connection between product-state approximations of local Hamiltonians and combinatorial graph property testing. We develop a series of weak Szemerédi regularity lemmas for k-local Hamiltonians, built on those of Frieze and Kannan and others. We use them to develop constant time sampling algorithms, and to characterize the `vertex sample complexity' of the Local Hamiltonian problem, in an analog to a classical result by Alon, de la Vega, Kannan and Karpinski. Second, we build on the information-theoretic product-state approximation techniques by Brandão and Harrow, extending their results to the free energy and to an asymmetric graph setting. We leverage this structure to define families of algorithms for the free energy at low temperatures, and new algorithms for certain sparse graph families.
The low-temperature properties of interacting quantum systems are believed to require exponential resources to compute in the general case. Quantifying the extent to which such properties can be approximated using efficient algorithms remains a significant open challenge. Here, we consider the task of approximating the ground state energy of two-local quantum Hamiltonians with bounded-degree interaction graphs. Most existing algorithms optimize the energy over the set of product states. We propose and analyze a family of shallow quantum circuits that can be used to improve the approximation ratio achieved by a given product state. The algorithm takes as input an n-qubit product state with variance Var and improves its energy by an amount proportional to Var^{2}/n. In a typical case, this results in an extensive improvement in the estimated energy. We extend our results to k-local Hamiltonians and entangled initial states.
Finding the ground-state energy of electrons subject to an external electric field is a fundamental problem in computational chemistry.While the theory of QMA-completeness has been instrumental in understanding the complexity of finding ground states in many-body quantum systems, prior to this work it has been unknown whether or not the special form of the Hamiltonian for the electronic structure of molecules can be exploited to find ground states efficiently or whether the problem remains hard for this special case.We prove that the electronic-structure problem, when restricted to a fixed single-particle basis and a fixed number of electrons, is QMA-complete.In our proof, the local Hamiltonian is encoded in the choice of spatial orbitals used to discretize the electronic-structure Hamiltonian.In contrast, Schuch and Verstraete have proved hardness for the electronic-structure problem with an additional site-specific external magnetic field, but without the restriction to a fixed basis, by encoding a local Hamiltonian on qubits in the site-specific magnetic field.We also show that estimation of the energy of the lowest-energy Slater-determinant state (i.e., the Hartree-Fock state) is nondeterministic polynomial time (NP)-complete for the electronic-structure Hamiltonian in a fixed basis.