Approximating ground and excited state energies on a quantum computer

Hadfield, Stuart; Papageorgiou, Anargyros

doi:10.1007/s11128-015-0927-y

Cited by 1 publication

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“…As the best classical algorithms known for this problem have costs that grow exponentially in d, our quantum algorithm gives an exponential speedup. The results of this chapter can also be found in [112].…”

Section: Introductionmentioning

confidence: 84%

“…In this chapter, we present an entirely new approach for approximating a constant number of low-order eigenvalues. Our approach applies to a general class of eigenvalue problems [112]. We illustrate our results by considering the time-independent Schrödinger equation with a number of degrees of freedom d, under weaker assumptions than those of [182,183].…”

Section: Introductionmentioning

confidence: 99%

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Quantum Algorithms for Scientific Computing and Approximate Optimization

Hadfield

2018

Preprint

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In this thesis we study the application of quantum computers to computational problems in science and engineering, and to combinatorial optimization problems. We outline the results below.Algorithms for scientific computing require modules, i.e., building blocks, implementing elementary numerical functions that have well-controlled numerical error, are uniformly scalable and reversible, and that can be implemented efficiently. We derive quantum algorithms and circuits for computing square roots, logarithms, and arbitrary fractional powers, and derive worst-case error and cost bounds. We describe a modular approach to quantum algorithm design as a first step towards numerical standards and mathematical libraries for quantum scientific computing.A fundamental but computationally hard problem in physics is to solve the time-independent Schrödinger equation. This is accomplished by computing the eigenvalues of the corresponding Hamiltonian operator. The eigenvalues describe the different energy levels of a system. The cost of classical deterministic algorithms computing these eigenvalues grows exponentially with the number of system degrees of freedom. The number of degrees of freedom is typically proportional to the number of particles in a physical system. We show an efficient quantum algorithm for approximating a constant number of low-order eigenvalues of a Hamiltonian using a perturbation approach.We apply this algorithm to a special case of the Schrödinger equation and show that our algorithm succeeds with high probability, and has cost that scales polynomially with the number of degrees of freedom and the reciprocal of the desired accuracy. This improves and extends earlier results on quantum algorithms for estimating the ground state energy.We consider the simulation of quantum mechanical systems on a quantum computer. We show EpigraphThe importance of accelerating approximating and computing mathematics by factors like 10,000 or more, lies not only in that one might thereby do in 10,000 times less time problems which one is now doing...but rather in that one will be able to handle problems which are considered completely unassailable at present. The projected device, or rather the species of which it is to be the first representative, is so radically new that many of its uses will become clear only after it has been put into operation, and after we have adjusted our mathematical habits and ways of thinking to its existence and possibilities. Furthermore, these uses which are not, or not easily, predictable now, are likely to be the most important ones...because they are farthest removed from what is now feasible.

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Section: Introductionmentioning

confidence: 84%