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arXiv:quant-ph/9703054 · Physical Review Letters

Simulation of Many-Body Fermi Systems on a Universal Quantum Computer

Daniel S. Abrams, Seth Lloyd

Published Sep 29, 1997• 736 citations
arXivHTMLPDF

Abstract

We provide fast algorithms for simulating many-body Fermi systems on a universal quantum computer. Both first and second quantized descriptions are considered, and the relative computational complexities are determined in each case. In order to accommodate fermions using a first quantized Hamiltonian, an efficient quantum algorithm for antisymmetrization is given. Finally, a simulation of the Hubbard model is discussed in detail.

DOI
10.1103/physrevlett.79.2586

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•Jan 1, 2009•39 citations
We provide fast algorithms for simulating many-body Fermi systems on a universal quantum computer. Both first and second quantized descriptions are considered, and the relative computational complexities are determined in each case. In order to accommodate fermions using a first quantized Hamiltonian, an efficient quantum algorithm for antisymmetrization is given. Finally, a simulation of the Hubbard model is discussed in detail.
We show in detail how the Jordan-Wigner transformation can be used to simulate any fermionic many-body Hamiltonian on a quantum computer. We develop an algorithm based on appropriate qubit gates that takes a general fermionic Hamiltonian, written as products of a given number of creation and annihilation operators, as input. To demonstrate the applicability of the algorithm, we calculate eigenvalues and eigenvectors of two model Hamiltonians, the well-known Hubbard model and a generalized pairing Hamiltonian. Extensions to other systems are discussed.
Simulating quantum many-body systems is a highly demanding task since the required resources grow exponentially with the dimension of the system. In the case of fermionic systems, this is even harder since nonlocal interactions emerge due to the antisymmetric character of the fermionic wave function. Here, we introduce a digital-analog quantum algorithm to simulate a wide class of fermionic Hamiltonians including the paradigmatic one-dimensional Fermi-Hubbard model. These digital-analog methods allow quantum algorithms to run beyond digital versions via an efficient use of coherence time. Furthermore, we exemplify our techniques with a low-connected architecture for realistic digital-analog implementations of specific fermionic models.
Simulating noninteracting fermion systems is a common task in computational many-body physics. In the absence of translational symmetries, modeling free fermions on N modes usually requires poly ( N ) computational resources. While often moderate, these costs can be prohibitive in practice when large systems are considered. We present several free fermion problems that can be solved by a quantum algorithm with substantially reduced computational costs. The memory costs are exponentially improved, poly log ( N ) . The run-time improvement, compared to the best known classical algorithms, is either exponential or significantly polynomial, depending on the geometry of the problem. The simulation of free fermion dynamics belongs to the BQP-hard complexity class (i.e., as hard as any (decision) problem that can be solved on a quantum computer). This implies (under standard assumptions) that our algorithm yields an exponential speedup for classical algorithm at least for some geometries. The key technique in our algorithm is the block encoding of objects such as correlation matrices and Green's functions into a unitary. We demonstrate how such unitaries can be efficiently realized as quantum circuits, in the context of dynamics and thermal states of tight-binding Hamiltonians. The special cases of disordered and inhomogeneous lattices, as well as large nonlattice graphs, are presented in detail. Finally, we show that our simulation algorithm generalizes to other promising targets, including free-boson systems.
While quantum computers are capable of simulating many quantum systems efficiently, the simulation algorithms must begin with the preparation of an appropriate initial state. We present a method for generating physically relevant quantum states on a lattice in real space. In particular, the present algorithm is able to prepare general pure and mixed many-particle states of any number of particles. It relies on a procedure for converting from a second-quantized state to its first-quantized counterpart. The algorithm is efficient in that it operates in time that is polynomial in all the essential descriptors of the system, the number of particles, the resolution of the lattice, and the inverse of the maximum final error. This scaling holds under the assumption that the wave function to be prepared is bounded or its indefinite integral is known and that the Fock operator of the system is efficiently simulatable.
We present a universal framework to harness fermionic ultracold atom platforms for quantum simulation, showing how variational algorithms on existing hardware can simulate many-body systems well beyond the hardware's native Hamiltonian. Our analysis provides evidence that one can quantum simulate the ground-state properties of a broad class of gapless target Hamiltonians of local observables in a quantum evolution time that grows polynomially with the inverse relative error, T∼O(poly(1/ε))T\sim O(\mathrm{poly}(1/ε))T∼O(poly(1/ε)) up to logarithmic corrections, offering an exponential speedup over na{ï}ve classical algorithms such as exact diagonalization. We provide numerical evidence and theoretical argument that this holds for energy density, density-density, and spin-spin correlations in three qualitatively distinct models -- the repulsive Hubbard model; a Hubbard model augmented with nearest-neighbor attractive interactions, which introduces the phenomenon of pairing; and the Hofstadter-Hubbard model, which introduces a gauge field and fractional quantum Hall physics. This work demonstrates quantum simulation using current fermionic platforms far beyond the models natively implemented in the hardware.
While quantum computers are capable of simulating many quantum systems efficiently, the simulation algorithms must begin with the preparation of an appropriate initial state. We present a method for generating physically relevant quantum states on a lattice in real space. In particular, the present algorithm is able to prepare general pure and mixed many-particle states of any number of particles. It relies on a procedure for converting from a second-quantized state to its first-quantized counterpart. The algorithm is efficient in that it operates in time that is polynomial in all the essential descriptors of the system, the number of particles, the resolution of the lattice, and the inverse of the maximum final error. This scaling holds under the assumption that the wave function to be prepared is bounded or its indefinite integral is known and that the Fock operator of the system is efficiently simulatable.