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Acquaintance strategy for block-diagonal Hamiltonian with Nb = 4 and nκ = 10. ‘${K}_{n}^{4}$’ indicates a four-complete swap network on n qubits, i.e., one that acquaints the $\left(\genfrac{}{}{0.0pt}{}{n}{4}\right)$ subsets of four qubits with each other. The other gates are double bipartite swap networks, explained in figures D2 and D3.
Source publication
All-electron electronic structure methods based on the linear combination of atomic orbitals method with Gaussian basis set discretization offer a well established, compact representation that forms much of the foundation of modern correlated quantum chemistry calculations—on both classical and quantum computers. Despite their ability to describe e...
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Citations
... It is worth pointing out that such "diagonal" basis sets also deliver significant advantage for quantum computing applications. 47,48 All the same, the original shortcoming of non-adaptivity was partially overcome in the work of White and Stoudenmire 45 by introducing curvilinear coordinates while maintaining the diagonal approximation. The deformation used in this work is of separable or multi-sliced (cf. ...
In this article, we combine the periodic sinc basis set with a curvilinear coordinate system for electronic structure calculations. This extension allows for variable resolution across the computational domain, with higher resolution close to the nuclei and lower resolution in the inter-atomic regions. We address two key challenges that arise while using basis sets obtained by such a coordinate transformation. First, we use pseudospectral methods to evaluate the integrals needed to construct the Hamiltonian in this basis. Second, we demonstrate how to construct an appropriate coordinate transformation by solving the Monge–Ampére equation using a new approach that we call the cyclic Knothe–Rosenblatt flow. The solution of both of these challenges enables mean-field calculations at a cost that is log-linear in the number of basis functions. We demonstrate that our method approaches the complete basis set limit faster than basis sets with uniform resolution. We also emphasize how these basis sets satisfy the diagonal approximation, which is shown to be a consequence of the pseudospectral method. The diagonal approximation is highly desirable for the solution of the electronic structure problem in many frameworks, including mean field theories, tensor network methods, quantum computing, and quantum Monte Carlo.
... For example, in Ref. [30] the orbitals from a Gaussian basis set were contracted through intrinsic contraction. Moreover, in contrast to this work, where multiwavelets in a discontinuous Galerkin fashion are used to represent orbitals that form the second quantized Hamiltonian, Ref. [31] investigated a direct construction in the numerical basis. ...
In electronic structure theory, variational computing offers a valuable paradigm for the approximation of electronic ground states. However, for historical reasons, this principle is mostly restricted to model-chemistries in pre-defined fixed basis sets. Especially in quantum computation, these model-chemistries are far from an accurate description of the initial electronic Hamiltonian. Throwing down the gauntlet, we here demonstrate a fully variational approach to the electronic structure problem by variationally optimizing the orbitals representing the second quantized Hamiltonian alongside a quantum circuit that generates the many-electron wavefunction. To this end, the orbitals are represented within an adaptive multi-wavelet format, guaranteeing numerical precision. We then showcase explicit numerical protocols and highlight the quantum circuit's effects on determining the optimal orbital basis.
... We highlight that the Gausslets were designed in part for deployment within tensor network approaches to electronic structure, where they nonetheless enjoy advantages over PWs in maintaining locality of entanglement and allowing for efficient representation of the many-body Hamiltonian. It is worth pointing out that such 'diagonal' basis sets also deliver significant advantage for quantum computing applications 47,48 . ...
In this article, we combine the periodic sinc basis set with a curvilinear coordinate system for electronic structure calculations. This extension allows for variable resolution across the computational domain, with higher resolution close to the nuclei and lower resolution in the inter-atomic regions. We address two key challenges that arise while using basis sets obtained by such a coordinate transformation. First, we use pseudospectral methods to evaluate the integrals needed to construct the Hamiltonian in this basis. Second, we demonstrate how to construct an appropriate coordinate transformation by solving the Monge-Amp\`ere equation using a new approach that we call the cyclic Knothe-Rosenblatt flow. The solution of both of these challenges enables mean-field calculations at a cost that is log-linear in the number of basis functions. We demonstrate that our method approaches the complete basis set limit faster than basis sets with uniform resolution. We also emphasize how these basis sets satisfy the diagonal approximation, which is shown to be a consequence of the pseudospectral method. The diagonal approximation is highly desirable for the solution of the electronic structure problem in many frameworks, including mean field theories, tensor network methods, quantum computing, and quantum Monte Carlo.
... We expect our approach to have a broad impact on quantum algorithms research. In particular, it could provide a practical way to boost the performance of quantum-simulation algorithms, such as quantum chemistry simulations [51][52][53], where standard methods are used for discretization. Our approach also provides a practical way to accurately estimate the condition number for large-size systems, which is taken as input to many quantum algorithms [3,5]. ...
Partial differential equations (PDEs) are ubiquitous in science and engineering. Prior quantum algorithms for solving the system of linear algebraic equations obtained from discretizing a PDE have a computational complexity that scales at least linearly with the condition number of the matrices involved in the computation. For many practical applications, scales polynomially with the size N of the matrices, rendering a polynomial-in-N complexity for these algorithms. Here we present a quantum algorithm with a complexity that is polylogarithmic in N but is independent of for a large class of PDEs. Our algorithm generates a quantum state that enables extracting features of the solution. Central to our methodology is using a wavelet basis as an auxiliary system of coordinates in which the condition number of associated matrices is independent of N by a simple diagonal preconditioner. We present numerical simulations showing the effect of the wavelet preconditioner for several differential equations. Our work could provide a practical way to boost the performance of quantum-simulation algorithms where standard methods are used for discretization.
... A series of recent works has focused on reducing the gate complexity of Hamiltonian simulation for chemistry applications [24,[31][32][33][34][35][36][37][38][39][40][41]. These papers propose methods for reducing gate complexity by approximations that utilize the Hamiltonian alone. ...
It is crucial to reduce the resources required to run quantum algorithms and simulate physical systems on quantum computers due to coherence time limitations. With regards to Hamiltonian simulation, a significant effort has focused on building efficient algorithms using various factorizations and truncations, typically derived from the Hamiltonian alone. We introduce a new paradigm for improving Hamiltonian simulation and reducing the cost of ground state problems based on ideas recently developed for classical chemistry simulations. The key idea is that one can find efficient ways to reduce resources needed by quantum algorithms by making use of two key pieces of information: the Hamiltonian operator and an approximate ground state wavefunction. We refer to our algorithm as the (SQuISH). By performing our scheme iteratively, one can drive SQuISH to create an accurate wavefunction using a truncated, resource-efficient Hamiltonian. Utilizing this truncated Hamiltonian provides an approach to reduce the gate complexity of ground state calculations on quantum hardware. As proof of principle, we implement SQuISH using configuration interaction for small molecules and coupled cluster for larger systems. Through our combination of approaches, we demonstrate how SQuISH performs on a range of systems, the largest of which would require more than 200 qubits to run on quantum hardware. Though our demonstrations are on a series of electronic structure problems, our approach is relatively generic and hence likely to benefit additional applications where the size of the problem Hamiltonian creates a computational bottleneck.
... In a recent work [5], we have introduced a Discontinuous Galerkin (DG) inspired basis set to interpolate between the compact but dense description of the Hamiltonian through GTOs/Gaussian-based molecular orbitals, and the sparse description by means of a primitive basis set. In the DG formalism, the ERI tensor takes a block-diagonal form (vide infra); hence, quantum simulations performed in the DG basis enjoy the same reduced asymptotic scaling as in the primitive basis set. ...
... In [5], the computational domain is partitioned into a set of rectangular subdomains, which is ideally suited for describing quasi-1D systems such as hydrogen chains. The rectangular partitioning strategy is also consistent with the use of the DG basis functions for density functional theory (DFT)-based calculations [6,7], and will be referred to as the DG-Rectangular method (or DG-R for short). ...
... The DG methodology systematically compress an active-space basis set {ϕ } into a set of orthonormal basis functions partitioned into spatial elements, i.e., the DG elements. For a detailed introduction to the DG methodology, we refer the reader to Sect. 3 in Ref. [5]. Consider each DG element to be a rectangular region in the supercell, i.e., DG-R. ...
To circumvent a potentially dense two-body interaction tensor and obtain lower asymptotic costs for quantum simulations of chemistry, the discontinuous Galerkin (DG) basis set with a rectangular partitioning strategy was recently introduced [McClean et al, New J. Phys. 22, 093015, 2020]. We propose and numerically scrutinize a more general DG basis set construction based on a Voronoi decomposition with respect to the nuclear coordinates. This allows the construction of DG basis sets for arbitrary molecular and crystalline configurations. We here employ the planewave dual basis set as primitive basis set in the supercell model; as a set of grid-based nascent delta functions, the planewave dual functions provide sufficient flexibility for the Voronoi partitioning. The presented implementation of this DG-Voronoi approach is in Python and solely based on PySCF. We numerically investigate the performance, at the mean-field and correlated level of theory for quasi-1D, quasi-2D and fully 3D systems, and exemplify the application to crystalline systems.
... Also the number of basis functions will increase rapidly with the threshold and one will most likely have to resort to direct diagonalization. Second, our algorithm has similarities with the discontinuous Galerkin method 81,90 and it is possible to obtain a set of Galerkin basis that are localized to a given Voronoi partition. This will have a significant advantage that the fitting functions will also be perfectly localized to an atomic domain and thus the cubic scaling of the QRCP step and the evaluation of the W (R g , R g ) will be reduced to linear and quadratic scaling respectively. ...
In this article we present an algorithm to efficiently evaluate the exchange matrix in periodic systems when Gaussian basis set with pseudopotentials are used. The usual algorithm for evaluating exchange matrix scales cubically with the system size because one has to perform fast Fourier transforms (FFT). Here we introduce an algorithm that retains the cubic scaling but reduces the prefactor significantly by eliminating the need to do FFTs during each exchange build. This is accomplished by representing the products of Gaussian basis function using a linear combination of an auxiliary basis the number of which scales linearly with the size of the system. We store the potential due to these auxiliary functions in memory which allows us to obtain the exchange matrix without the need to do FFT, albeit at the cost of additional memory requirement. Although the basic idea of using auxiliary functions is not new, our algorithm is cheaper due to a combination of three ingredients: (a) we use robust Pseudospectral method that allows us to use a relatively small number of auxiliary basis to obtain high accuracy (b) we use occ-RI exchange which eliminates the need to construct the full exchange matrix and (c) we use the (interpolative separable density fitting) ISDF algorithm to construct these auxiliary basis that are used in the robust pseudospectral method. The resulting algorithm is accurate and we note that the error in the final energy decreases exponentially rapidly with the number of auxiliary functions. Finally, we find that the wall time of 62 s and 118 s are needed for a single build of the Coulomb and exchange matrices respectively on 16 nodes for a LiH unit cell that contains 1024 electrons and 4864 basis functions.
... Over the last decade, KSSOLV has be-come a useful research and teaching tool for studying electronic structures of molecules and solids and developing new methods for solving the Kohn-Sham problem, as evidenced by an increasing number of publications that use KSSOLV to perform numerical experiments required to demonstrate improved convergence or better accuracy of new theoretical methods and numerical algorithms. Examples of such developments include tensor hypercontraction algorithm [4], improved generalized Davidson algorithm [5], elliptic preconditioner in self-consistent field iteration [6], low rank approximation in G 0 W 0 calculations [7], perturbation theory [8], quantum embedding theory [9], quantum computation [10], linear-response time-dependent density functional theory [11,12], phonon calculations [13], proximal gradient method [14], trace-penalty minimization method [15]. ...
KSSOLV (Kohn-Sham Solver) is a MATLAB toolbox for performing Kohn-Sham density functional theory (DFT) calculations with a plane-wave basis set. KSSOLV 2.0 preserves the design features of the original KSSOLV software to allow users and developers to easily set up a problem and perform ground-state calculations as well as to prototype and test new algorithms. Furthermore, it includes new functionalities such as new iterative diagonalization algorithms, k-point sampling for electron band structures, geometry optimization and advanced algorithms for performing DFT calculations with local, semi-local, and hybrid exchange-correlation functionals. It can be used to study the electronic structures of both molecules and solids. We describe these new capabilities in this work through a few use cases. We also demonstrate the numerical accuracy and computational efficiency of KSSOLV on a variety of examples.
... Over the last decade, KSSOLV has be-come a useful research and teaching tool for studying electronic structures of molecules and solids and developing new methods for solving the Kohn-Sham problem, as evidenced by an increasing number of publications that use KSSOLV to perform numerical experiments required to demonstrate improved convergence or better accuracy of new theoretical methods and numerical algorithms. Examples of such developments include tensor hypercontraction algorithm [4], improved generalized Davidson algorithm [5], elliptic preconditioner in self-consistent field iteration [6], low rank approximation in G 0 W 0 calculations [7], perturbation theory [8], quantum embedding theory [9], quantum computation [10], linear-response time-dependent density functional theory [11,12], phonon calculations [13], proximal gradient method [14], trace-penalty minimization method [15]. ...
KSSOLV (Kohn-Sham Solver) is a MATLAB toolbox for performing Kohn-Sham density functional theory (DFT) calculations with a plane-wave basis set. KSSOLV 2.0 preserves the design features of the original KSSOLV software to allow users and developers to easily set up a problem and perform ground-state calculations as well as to prototype and test new algorithms. Furthermore, it includes new functionalities such as new iterative diagonalization algorithms, k-point sampling for electron band structures, geometry optimization and advanced algorithms for performing DFT calculations with local, semi-local, and hybrid exchange-correlation functionals. It can be used to study the electronic structures of both molecules and solids. We describe these new capabilities in this work through a few use cases. We also demonstrate the numerical accuracy and computational efficiency of KSSOLV on a variety of examples.
Program summary
Program title: Kohn-Sham Solver 2.0 (KSSOLV 2.0)
CPC Library link to program files: https://doi.org/10.17632/pp8vgvfcv4.1
Developer's repository link: https://bitbucket.org/berkeleylab/kssolv2.0/src/release/
Licensing provisions: BSD 3-clause
Programming language:: MATLAB
Nature of problem: KSSOLV2.0 is used to perform Kohn-Sham density functional theory based electronic structure calculations to study chemical and material properties of molecules and solids. The key problem to be solved is a constrained energy minimization problem, which can also be formulated as a nonlinear eigenvalue problem.
Solution method: The KSSOLV 2.0 implements both the self-consistent field (SCF) iteration with a variety of acceleration strategies and a direct constrained minimization algorithms. It is written completely in MATLAB and uses MATLAB's object oriented programming features to make it easy to use and modify.
... [31][32][33] To circumvent the limitation of the exact diagonalization approach, various classes of solvers have been proposed, including a class of selected CI methods, [34][35][36][37] full CI quantum Monte Carlo, 38,39 density matrix renormalization group (DMRG) 40,41 , and many-body expanded full CI. 42,43 An impressive example to demonstrate the success of these methods is the employment of the DMRG method to solve a large CAS problem of (113,76) for the [MoFe 7 S 9 C] FeMo cofactor (FeMoco) system, an iron-sulfur cluster in nitrogenase MoFe protein. 44 In materials science, Green's function (GF) methods [45][46][47] are widely used to describe quasiparticle properties, such as the band structure and optical absorption spectra, for correlated materials. ...
... Considering their success in electronic structure method, the potential use of many real-space methods in quantum computing is worth exploiting. 113,114 The appearance of new strong-correlation electronic structure methods, such as DMRG and tensor network methods, has renewed the interest in developing alternative basis function approaches. The strong locality feature of basis functions is essential to exploit the low-entanglement nature of electronic interactions, which is also critical for developing low-depth quantum algorithms of quantum chemistry. ...
... Overall, the diagonal representation and strict locality make gausslet naturally suitable for describing highly entangled systems with lower gate complexity in quantum simulations of quantum chemistry. 113 ...
The advantages of quantum computers are believed to significantly change the research paradigm of chemical and materials sciences, where computational characterization and theoretical design play an increasingly important role. It is especially desirable to solve the electronic structure problem, a central problem in chemistry and materials science, efficiently and accurately with well-designed quantum algorithms. Various quantum electronic-structure algorithms have been proposed in the literature. In this article, we briefly review recent progress in this direction with a special emphasis on the basis sets and boundary conditions. Compared to classical electronic structure calculations, there are new considerations in choosing a basis set in quantum algorithms. For example, the effect of the basis set on the circuit complexity is very important in quantum algorithm design. Electronic structure calculations should be performed with an appropriate boundary condition. Simply using a wave function ansatz designed for molecular systems in a material system with a periodic boundary condition may lead to significant errors. Artificial boundary conditions can be used to partition a large system into smaller fragments to save quantum resources. The basis sets and boundary conditions are expected to play a crucial role in electronic structure calculations on future quantum computers, especially for realistic systems.
