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arXiv:2210.07985 · Physical Review A

Simulating scalar field theories on quantum computers with limited resources

Andy C. Y. Li, Alexandru Macridin, S. Mrenna, Panagiotis Spentzouris

Published Mar 3, 2023• 16 citations
arXivHTMLPDF

Abstract

We present a quantum algorithm for implementing ${\ensuremath{\phi}}^{4}$ lattice scalar field theory on qubit computers. The field is represented in the discretized field amplitude basis. The number of qubits and elementary gates required by the implementation of the evolution operator is proportional to the lattice size. The algorithm allows efficient ${\ensuremath{\phi}}^{4}$ state preparation for a large range of input parameters in both the normal and broken-symmetry phases. The states are prepared using a combination of variational and adiabatic evolution methods. First, the ground state of a local Hamiltonian, which includes the ${\ensuremath{\phi}}^{4}$ self-interaction, is prepared using short variational circuits. Next, this state is evolved by switching on the coupling between the lattice sites adiabatically. The parameters defining the local Hamiltonian are adjustable and constitute the input of our algorithm. We present a method to optimize these parameters in order to reduce the adiabatic time required for state preparation. For preparing broken-symmetry states, the adiabatic evolution problems caused by crossing the phase transition critical line and by the degeneracy of the broken-symmetry ground state can be addressed using an auxiliary external field which gradually turns off during the adiabatic process. We show that the time dependence of the external field during the adiabatic evolution is important for addressing the broken-symmetry ground state degeneracy. The adiabatic time dependence on the inverse error tolerance can be reduced from quadratic to linear by using a field strength that decreases exponentially in time relative to one that decreases linearly.

DOI
10.1103/physreva.107.032603

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Quantum state preparation is crucial for quantum algorithms, and improved state preparation methods can reduce resource requirements by orders of magnitude. In this work, we introduce a state preparation method for lattice Hamiltonians on digital quantum computers inspired by nucleation. In our method, the exact ground state of a small lattice Hamiltonian is prepared in a quantum circuit, then a series of growth stages are performed via Trotterized adiabatic evolution to increase the size of the lattice to the desired model. Using the adiabatic theorem and Trotter error bounds, we provide sufficient conditions on the total evolution time and number of Trotter steps required for nucleation to prepare an initial state with desired fidelity. In addition, we introduce optimized nucleation which considers variational optimization on top of the nucleation structure. For a two-dimensional Ising model, we show that optimized nucleation can outperform a current leading method for state preparation on digital quantum computers.
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We present two quantum algorithms based on evolution randomization, a simple variant of adiabatic quantum computing, to prepare a quantum state ∣x⟩\vert x \rangle∣x⟩ that is proportional to the solution of the system of linear equations Ax⃗=b⃗A \vec{x}=\vec{b}Ax=b. The time complexities of our algorithms are O(κ2log⁡(κ)/ϵ)O(\kappa^2 \log(\kappa)/\epsilon)O(κ2log(κ)/ϵ) and O(κlog⁡(κ)/ϵ)O(\kappa \log(\kappa)/\epsilon)O(κlog(κ)/ϵ), where κ\kappaκ is the condition number of AAA and ϵ\epsilonϵ is the precision. Both algorithms are constructed using families of Hamiltonians that are linear combinations of products of AAA, the projector onto the initial state ∣b⟩\vert b \rangle∣b⟩, and single-qubit Pauli operators. The algorithms are conceptually simple and easy to implement. They are not obtained from equivalences between the gate model and adiabatic quantum computing. They do not use phase estimation or variable-time amplitude amplification, and do not require large ancillary systems. We discuss a gate-based implementation via Hamiltonian simulation and prove that our second algorithm is almost optimal in terms of κ\kappaκ. Like previous methods, our techniques yield an exponential quantum speedup under some assumptions. Our results emphasize the role of Hamiltonian-based models of quantum computing for the discovery of important algorithms.
Quantum simulation of quantum field theory is a flagship application of quantum computers that promises to deliver capabilities beyond classical computing. The realization of quantum advantage will require methods that can accurately predict error scaling as a function of the resolution and parameters of the model and that can be implemented efficiently on quantum hardware. In this paper, we address the representation of lattice bosonic fields in a discretized field amplitude basis, develop methods to predict error scaling, and present efficient qubit implementation strategies. A low-energy subspace of the bosonic Hilbert space, defined by a boson occupation number cutoff, can be represented with exponentially good accuracy by a low-energy subspace of a finite-size Hilbert space. The finite representation construction and the associated errors are directly related to the accuracy of the Nyquist-Shannon sampling and the finite Fourier transforms of the boson number states in the field and the conjugate-field bases. We analyze the relation between the boson mass, the discretization parameters used for wave function sampling, and the finite representation size. Numerical simulations of small size ${\mathrm{\ensuremath{\Phi}}}^{4}$ problems demonstrate that the boson mass optimizing the sampling of the ground state wave function is a good approximation to the optimal boson mass yielding the minimum low-energy subspace size. However, we find that accurate sampling of general wave functions does not necessarily result in accurate representation. We develop methods for validating and adjusting the discretization parameters to achieve more accurate simulations.