On the NP-completeness of the Hartree-Fock method for translationally invariant systems

Whitfield, James D.; Zimborás, Zoltán

doi:10.1063/1.4903453

Cited by 7 publications

(10 citation statements)

References 25 publications

(30 reference statements)

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“…In previous work, arbitrary spin-glasses have been mapped to instances of the Hartree-Fock problem 32 . In addition to other works 33,34 , this shows that the Hartree-Fock problem is difficult in the worst case setting. The complexity class of non-deterministic polynomial time (NP) problems are the set of problems 35 that can be solved efficiently with a hint 36 .…”

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confidence: 64%

Limitations of Hartree-Fock with quantum resources

Gulania,

Whitfield

2020

Preprint

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The Hartree-Fock problem provides the conceptual and mathematical underpinning of a large portion of quantum chemistry. As efforts in quantum technology aim to enhance computational chemistry algorithms, the fundamental Hartree-Fock problem is a natural target. While quantum computers and quantum simulation offer many prospects for the future of modern chemistry, the Hartree-Fock problem is not a likely candidate. We highlight this fact from a number of perspectives including computational complexity, practical examples, and the full characterization of the energy landscapes for simple systems.

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confidence: 64%

Limitations of Hartree-Fock with quantum resources

Gulania,

Whitfield

2020

Preprint

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“…Solving the Hartree-Fock (HF) method, which intends to find a single Slater determinant that best approximates the true wave function, has been shown to be NP-complete 65 . The NP-completeness has also been proven to persist even if one restricts to translationally invariant systems 213 .…”

Section: Complexity Theory Of the Electronic Structure Problemmentioning

confidence: 99%

“…Approximating Kohn-Sham potential in TDDFT 217 Hartree-Fock on general quantum systems 65 Universal functional in DFT 65 Hartree-Fock on translationally invariant systems 213 Bose-Hubbard model 210…”

Section: Complexity Theory Of the Electronic Structure Problemmentioning

confidence: 99%

Quantum Chemistry in the Age of Quantum Computing

Cao,

Romero,

Olson

et al. 2018

Preprint

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Practical challenges in simulating quantum systems on classical computers have been widely recognized in the quantum physics and quantum chemistry communities over the past century. Although many approximation methods have been introduced, the complexity of quantum mechanics remains hard to appease. The advent of quantum computation brings new pathways to navigate this challenging complexity landscape. By manipulating quantum states of matter and taking advantage of their unique features such as superposition and entanglement, quantum computers promise to efficiently deliver accurate results for many important problems in quantum chemistry such as the electronic structure of molecules.In the past two decades significant advances have been made in developing algorithms and physical hardware for quantum computing, heralding a revolution in simulation of quantum systems. This article is an overview of the algorithms and results that are relevant for quantum chemistry. The intended audience is both quantum chemists who seek to learn more about quantum computing, and quantum computing researchers who would like to explore applications in quantum chemistry.

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“…However, all methods hitherto devised face the insurmountable challenge of escalating computational resource requirements as the calculation is extended either to higher accuracy or to larger systems. Computational complexity in electronic structure calculations [5,6,7] suggests that these restrictions are an inherent difficulty associated with simulating quantum systems.Electronic structure algorithms developed for quantum computers provide a new promising route to advance the field of electronic structure calculations for large systems [8,9]. Recently, there has been an attempt at using an adiabatic quantum computing model -as is implemented on the D-Wave machine -to perform electronic structure calculations [10].…”

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confidence: 99%

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confidence: 99%

Electronic Structure Calculations and the Ising Hamiltonian

2017

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The exact solution of the Schrödinger equation for atoms, molecules and extended systems continues to be a "Holy Grail" problem for the field of atomic and molecular physics since inception. Recently, breakthroughs have been made in the development of hardwareefficient quantum optimizers and coherent Ising machines capable of simulating hundreds of interacting spins through an Ising-type Hamiltonian. One of the most vital questions associated with these new devices is: "Can these machines be used to perform electronic structure calculations?" In this study, we discuss the general standard procedure used by these devices and show that there is an exact mapping between the electronic structure Hamiltonian and the Ising Hamiltonian. The simulation results of the transformed Ising Hamiltonian for H2, He2, HeH + , and LiH molecules match the exact numerical calculations. This demonstrates that one can map the molecular Hamiltonian to an Ising-type Hamiltonian which could easily be implemented on currently available quantum hardware.The determination of solutions to the Schrödinger equation is fundamentally difficult as the dimensionality of the corresponding Hilbert space increases exponentially with the number of particles in the system, requiring a commensurate increase in computational resources. Modern quantum chemistry -faced with difficulties associated with solving the Schrödinger equation to chemical accuracy (∼1 kcal/mole) -has largely become an endeavor to find approximate methods. A few products of this effort from the past few decades include methods such as: ab initio, Density Functional, Density Matrix, Algebraic, Quantum Monte Carlo and Dimensional Scaling [1,2,3,4]. However, all methods hitherto devised face the insurmountable challenge of escalating computational resource requirements as the calculation is extended either to higher accuracy or to larger systems. Computational complexity in electronic structure calculations [5,6,7] suggests that these restrictions are an inherent difficulty associated with simulating quantum systems.Electronic structure algorithms developed for quantum computers provide a new promising route to advance the field of electronic structure calculations for large systems [8,9]. Recently, there has been an attempt at using an adiabatic quantum computing model -as is implemented on the D-Wave machine -to perform electronic structure calculations [10]. The fundamental concept behind the adiabatic quantum computing (AQC) method is to define a problem Hamiltonian, H P , engineered to have its ground state encode the solution of a corresponding computational problem. The system is initialized in the ground state of a beginning Hamiltonian, H B , which is easily solved classically. The system is then allowed to evolve adiabatically as: The largest scale implementation of AQC to date is by D-Wave Systems [11,12]. In the case of the DWave device, the physical process undertaken which acts as an adiabatic evolution is more broadly called quantum annealing (QA). The quantum processor...

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On the NP-completeness of the Hartree-Fock method for translationally invariant systems

Cited by 7 publications

References 25 publications

Limitations of Hartree-Fock with quantum resources

Limitations of Hartree-Fock with quantum resources

Quantum Chemistry in the Age of Quantum Computing

Electronic Structure Calculations and the Ising Hamiltonian

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