Calculating the ground state energy of a molecule efficiently is of great interest in quantum chemistry. The exact numerical solution of the electronic Schrodinger equation remains unfeasible for most molecules requiring approximate methods at best. In this paper we introduce the use of Quantum Community Detection performed using the D-Wave quantum annealer to reduce the molecular Hamiltonian matrix without chemical knowledge. Given a molecule represented by a matrix of Slater determinants, the connectivity between Slater determinants is viewed as a graph adjacency matrix for determining multiple communities based on modularity maximization. The resulting lowest energy cluster of Slater determinants is used to calculate the ground state energy within chemical accuracy. The details of this method are described along with demonstrating its performance across multiple molecules of interest and a bond dissociation example. This approach is general and can be used as part of electronic structure calculations to reduce the computation required.
Calculating the ground state energy of a molecule efficiently is of great interest in quantum chemistry. The exact numerical solution of the electronic Schrodinger equation remains unfeasible for most molecules requiring approximate methods at best. In this paper we introduce the use of Quantum Community Detection performed using the D-Wave quantum annealer to reduce the molecular Hamiltonian matrix without chemical knowledge. Given a molecule represented by a matrix of Slater determinants, the connectivity between Slater determinants is viewed as a graph adjacency matrix for determining multiple communities based on modularity maximization. The resulting lowest energy cluster of Slater determinants is used to calculate the ground state energy within chemical accuracy. The details of this method are described along with demonstrating its performance across multiple molecules of interest and a bond dissociation example. This approach is general and can be used as part of electronic structure calculations to reduce the computation required.
Quantum chemistry is interested in calculating ground and excited states of molecular systems by solving the electronic Schrödinger equation. The exact numerical solution of this equation, frequently represented as an eigenvalue problem, remains unfeasible for most molecules and requires approximate methods. In this paper we introduce the use of Quantum Community Detection performed using the D-Wave quantum annealer to reduce the molecular Hamiltonian matrix in Slater determinant basis without chemical knowledge. Given a molecule represented by a matrix of Slater determinants, the connectivity between Slater determinants (as off-diagonal elements) is viewed as a graph adjacency matrix for determining multiple communities based on modularity maximization. A gauge metric based on perturbation theory is used to determine the lowest energy cluster. This cluster or sub-matrix of Slater determinants is used to calculate approximate ground state and excited state energies within chemical accuracy. The details of this method are described along with demonstrating its performance across multiple molecules of interest and bond dissociation cases. These examples provide proof-of-principle results for approximate solution of the electronic structure problem using quantum computing. This approach is general and shows potential to reduce the computational complexity of post-Hartree-Fock methods as future advances in quantum hardware become available.
Slater determinants have underpinned quantum chemistry for nearly a century, yet their full potential has remained challenging to exploit. In this work, we show that a variational wavefunction composed of a few hundred optimized non-orthogonal determinants can achieve energy accuracies comparable to the state of the art. This is obtained by introducing an optimization method that leverages the quadratic dependence of the variational energy on the orbitals of each determinant, enabling an exact iterative optimization, and uses an efficient tensor-contraction algorithm to evaluate the effective Hamiltonian with a computational cost that scales as the fourth power of the number of basis functions. We benchmark the accuracy of the proposed method with exact full-configuration interaction results where available, and we achieve lower variational energies than coupled cluster (CCSD(T)) for several molecules in the double-zeta basis. Accurate description of electronic wave functions typically requires large numbers of Slater determinants. Here, the authors develop an optimization method that achieves state-of-the-art accuracy for small molecules with only few hundred determinants.
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.
Quantum annealers are an alternative approach to quantum computing, which make use of the adiabatic theorem to efficiently find the ground state of a physically realizable Hamiltonian. Such devices are currently commercially available and have been successfully applied to several combinatorial and discrete optimization problems. However, the application of quantum annealers to problems in chemistry remains a relatively sparse area of research due to the difficulty in mapping molecular systems to the Ising model Hamiltonian. In this paper, we review two different methods for finding the ground state of molecular Hamiltonians using Ising model-based quantum annealers. In addition, we compare the relative effectiveness of each method by calculating the binding energies, bond lengths, and bond angles of the H3+ and H2O molecules and mapping their potential energy curves. We also assess the resource requirements of each method by determining the number of qubits and computation time required to simulate each molecule using various parameter values. While each of these methods is capable of accurately predicting the ground state properties of small molecules, we find that they are still outperformed by modern classical algorithms and that the scaling of the resource requirements remains a challenge.
Graph partitioning (GP) applications are ubiquitous throughout mathematics, computer science, chemistry, physics, bio-science, machine learning, and complex systems. Post Moore's era supercomputing has provided us an opportunity to explore new approaches for traditional graph algorithms on quantum computing architectures. In this work, we explore graph partitioning using quantum annealing on the D-Wave 2X machine. Motivated by a recently proposed graph-based electronic structure theory applied to quantum molecular dynamics (QMD) simulations, graph partitioning is used for reducing the calculation of the density matrix into smaller subsystems rendering the calculation more computationally efficient. Unconstrained graph partitioning as community clustering based on the modularity metric can be naturally mapped into the Hamiltonian of the quantum annealer. On the other hand, when constraints are imposed for partitioning into equal parts and minimizing the number of cut edges between parts, a quadratic unconstrained binary optimization (QUBO) reformulation is required. This reformulation may employ the graph complement to fit the problem in the Chimera graph of the quantum annealer. Partitioning into 2 parts and k parts concurrently for arbitrary k are demonstrated with benchmark graphs, random graphs, and small material system density matrix based graphs. Results for graph partitioning using quantum and hybrid classical-quantum approaches are shown to be comparable to current "state of the art" methods and sometimes better.
Quantum chemistry calculations for small molecules on quantum hardware have been demonstrated to date only on universal-gate quantum computers, not quantum annealers. The latter devices are limited to finding the lowest eigenstate of the Ising Hamiltonian whereas the electronic Hamiltonian could not be mapped to the Ising form without exponential growth of the Ising Hamiltonian with the size of the system [J. Phys. Chem. B 122, 3384 (2018)]. Here we propose a novel mixed discrete-continuous optimization algorithm, which finds the lowest eigenstate of the qubit coupled cluster (QCC) method using a quantum annealer for solving a discrete part of the problem. The QCC method is a potentially exact approach for constructing the electronic wave function in the qubit space. Therefore, our methodology allows for systematically improvable quantum chemistry calculations using quantum annealears. We illustrate capabilities of our approach by calculating QCC ground electronic states for the LiH, H2O, and C6H6 molecules. C6H6 calculations involve 36 qubits and are the largest quantum chemistry calculations made on a quantum annealer (the D-Wave 2000Q system) to date. Our findings opens up a new perspective for use quantum annealers in high-throughput material discovery.