Open accessJournal Article

# Lanczos recursion on a quantum computer for the Green's function and ground state

04 Mar 2021-Physical Review A (American Physical Society)-Vol. 103, Iss: 3, pp 032404
Abstract: A state-preserving quantum counting algorithm is used to obtain coefficients of a Lanczos recursion from a single ground-state wave function on the quantum computer. This is used to compute the continued fraction representation of an interacting Green's function for use in condensed matter, particle physics, and other areas. The wave function does not need to be reprepared at each iteration. The quantum algorithm represents an exponential reduction in memory over known classical methods. An extension of the method to determining the ground state is also discussed.

Topics: Quantum algorithm (66%), Quantum computer (63%), Lanczos resampling (61%) ... read more
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Open accessPosted Content
Abstract: The method of quantum Lanczos recursion is extended to solve for multiple excitations on the quantum computer. While quantum Lanczos recursion is in principle capable of obtaining excitations, the extension to a block Lanczos routine can resolve degeneracies with better precision and only costs $\mathcal{O}(d^2)$ for $d$ excitations on top of the previously introduced quantum Lanczos recursion method. The formal complexity in applying all operators to the system at once with oblivious amplitude amplification is exponential, but this cost can be kept small to obtain the ground state by incrementally adding operators. The error of the ground state energy based on the accuracy of the Lanczos coefficients is investigated and the error of the ground state energy. It is demonstrated to scale linearly with the uncertainty of the Lanczos coefficients. Extension to non-Hermitian operators is also discussed.

Topics: Lanczos resampling (68%), Quantum computer (56%), Ground state (53%) ... read more

1 Citations

Open accessPosted Content
Abstract: Matrix product state methods are known to be efficient for computing ground states of local, gapped Hamiltonians, particularly in one dimension. We introduce the multi-targeted density matrix renormalization group method that acts on a bundled matrix product state, holding many excitations. The use of a block or banded Lanczos algorithm allows for the simultaneous, variational optimization of the bundle of excitations. The method is demonstrated on a Heisenberg model and other cases of interest. A large of number of excitations can be obtained at a small bond dimension with highly reliable local observables throughout the chain.

Topics: Matrix product state (62%), , Lanczos algorithm (58%) ... read more

1 Citations

Open accessPosted Content
Abstract: An introduction to the density matrix renormalization group is contained here, including coding examples. The focus of this code is on basic operations involved in tensor network computations, and this forms the foundation of the DMRjulia library. Algorithmic complexity, measurements from the matrix product state, convergence to the ground state, and other relevant features are also discussed. The present document covers the implementation of operations for dense tensors into the Julia language. The code can be used as an educational tool to understand how tensor network computations are done in the context of entanglement renormalization or as a template for other codes in low level languages. A comprehensive Supplemental Material is meant to be a "Numerical Recipes" style introduction to the core functions and a simple implementation of them. The code is fast enough to be used in research and can be used to make new algorithms.

Topics: Tensor (54%), , Matrix product state (53%) ... read more

Open accessJournal Article
Abstract: We present a two-step procedure called the dynamical self-energy mapping (DSEM) that allows us to find a sparse Hamiltonian representation for molecular problems. In the first part of this procedure, the approximate self-energy of a molecular system is evaluated using a low-level method and subsequently a sparse Hamiltonian is found that best recovers this low-level dynamic self-energy. In the second step, such a sparse Hamiltonian is used by a high-level method that delivers a highly accurate dynamical part of the self-energy that is employed in later calculations. The tests conducted on small molecular problems show that the sparse Hamiltonian parameterizations lead to very good total energies. DSEM has the potential to be used as a classical-quantum hybrid algorithm for quantum computing where the sparse Hamiltonian containing only O(n2) terms on a Gaussian orbital basis, where n is the number of orbitals in the system, could reduce the depth of the quantum circuit by at least an order of magnitude when compared with simulations involving a full Hamiltonian.

Topics: , Quantum computer (53%), Quantum circuit (51%)

Open accessPosted Content
Abstract: We explore the possibility to perform symmetry restoration with the variation after projection technique on a quantum computer followed by additional post-processing. The final goal is to develop configuration interaction techniques based on many-body trial states pre-optimized on a quantum computer. We show how the projection method used for symmetry restoration can prepare optimized states that could then be employed as initial states for quantum or hybrid quantum-classical algorithms. We use the quantum phase estimation and quantum Krylov approaches for the post-processing. The latter method combined with the quantum variation after projection (Q-VAP) leads to very fast convergence towards the ground-state energy. The possibility to access excited states energies is also discussed. Illustrations of the different techniques are made using the pairing hamiltonian.

Topics: Quantum computer (66%), Ground state (55%), Quantum (54%) ... read more
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42 results found

Proceedings Article
Lov K. Grover1Institutions (1)
01 Jul 1996-
Abstract: were proposed in the early 1980’s [Benioff80] and shown to be at least as powerful as classical computers an important but not surprising result, since classical computers, at the deepest level, ultimately follow the laws of quantum mechanics. The description of quantum mechanical computers was formalized in the late 80’s and early 90’s [Deutsch85][BB92] [BV93] [Yao93] and they were shown to be more powerful than classical computers on various specialized problems. In early 1994, [Shor94] demonstrated that a quantum mechanical computer could efficiently solve a well-known problem for which there was no known efficient algorithm using classical computers. This is the problem of integer factorization, i.e. testing whether or not a given integer, N, is prime, in a time which is a finite power of o (logN) . ----------------------------------------------

Topics: Quantum sort (60%), Quantum algorithm (53%), Quantum computer (52%) ... read more

5,636 Citations

Journal Article
Abstract: We review the dynamical mean-field theory of strongly correlated electron systems which is based on a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. This mapping is exact for models of correlated electrons in the limit of large lattice coordination (or infinite spatial dimensions). It extends the standard mean-field construction from classical statistical mechanics to quantum problems. We discuss the physical ideas underlying this theory and its mathematical derivation. Various analytic and numerical techniques that have been developed recently in order to analyze and solve the dynamical mean-field equations are reviewed and compared to each other. The method can be used for the determination of phase diagrams (by comparing the stability of various types of long-range order), and the calculation of thermodynamic properties, one-particle Green's functions, and response functions. We review in detail the recent progress in understanding the Hubbard model and the Mott metal-insulator transition within this approach, including some comparison to experiments on three-dimensional transition-metal oxides. We present an overview of the rapidly developing field of applications of this method to other systems. The present limitations of the approach, and possible extensions of the formalism are finally discussed. Computer programs for the numerical implementation of this method are also provided with this article.

Topics: , , Hubbard model (54%) ... read more

4,675 Citations

Open accessJournal Article
Cornelius Lanczos1Institutions (1)
Abstract: The present investigation designs a systematic method for finding the latent roots and the principal axes of a matrix, without reducing the order of the matrix. It is characterized by a wide field of applicability and great accuracy, since the accumulation of rounding errors is avoided, through the process of \"minimized iterations\". Moreover, the method leads to a well convergent successive approximation procedure by which the solution of integral equations of the Fredholm type and the solution of the eigenvalue problem of linear differential and integral operators may be accomplished.

Topics: , Inverse iteration (70%),  ... read more

3,672 Citations

Open accessJournal Article
Ulrich Schollwöck1Institutions (1)
Abstract: The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly become the method of choice for numerical studies of such systems. Its applications to the calculation of static, dynamic, and thermodynamic quantities in these systems are reviewed here. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and nonequilibrium statistical physics, and time-dependent phenomena is also discussed. This review additionally considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by the DMRG.

Topics: , Open quantum system (65%),  ... read more

2,098 Citations

Open accessJournal Article
Ferdi Aryasetiawan1, Olle Gunnarsson2Institutions (2)
Abstract: Calculations of ground-state and excited-state properties of materials have been one of the major goals of condensed matter physics. Ground-state properties of solids have been extensively investigated for several decades within the standard density functional theory. Excited-state properties, on the other hand, were relatively unexplored in ab initio calculations until a decade ago. The most suitable approach up to now for studying excited-state properties of extended systems is the Green function method. To calculate the Green function one requires the self-energy operator which is non-local and energy dependent. In this article we describe the GW approximation which has turned out to be a fruitful approximation to the self-energy. The Green function theory, numerical methods for carrying out the self-energy calculations, simplified schemes, and applications to various systems are described. Self-consistency issue and new developments beyond the GW approximation are also discussed as well as the success and shortcomings of the GW approximation.