Exponential improvement in quantum simulations of bosons

TL;DR

The paper addresses the challenge of simulating bosonic systems on quantum computers, where truncating the infinite Hilbert space and choosing a basis strongly influence circuit complexity. It argues that naive coordinate/Fock-basis truncations induce exponential growth in gate counts with the truncation parameter $Q$, while a universal framework that uses the quantum Fourier transform to connect coordinate and momentum bases yields polynomial scaling in $Q$ for circuit depth and cost. By applying this framework to orbifold lattice Hamiltonians, the authors show an exponential speedup in approaching the continuum limit compared to the Kogut-Susskind formulation, and they discuss the substantial resource penalties associated with gauge-invariant Hilbert spaces. The results advocate adopting the orbifold lattice and universal framework for bosonic and non-Abelian gauge theories, highlighting the potential for quantum advantage in these domains, while also noting practical caveats and the need for further cost analyses and compiler considerations.

Abstract

Hamiltonian quantum simulation of bosons on digital quantum computers requires truncating the Hilbert space to finite dimensions. The method of truncation and the choice of basis states can significantly impact the complexity of the quantum circuit required to simulate the system. For example, a truncation in the Fock basis where each boson is encoded with a register of $Q$ qubits, can result in an exponentially large number of Pauli strings required to decompose the truncated Hamiltonian. This, in turn, can lead to an exponential increase in $Q$ in the complexity of the quantum circuit. For lattice quantum field theories such as Yang-Mills theory and QCD, several Hamiltonian formulations and corresponding truncations have been put forward in recent years. There is no exponential increase in $Q$ when resorting to the orbifold lattice Hamiltonian, while we do not know how to remove the exponential complexity in $Q$ in the commonly used Kogut-Susskind Hamiltonian. Specifically, when using the orbifold lattice Hamiltonian, the continuum limit, or, in other words, the removal of the ultraviolet energy cutoff, is obtained with circuits whose resources scale like $Q$, while they scale like $\mathcal{O}(\exp(Q))$ for the Kogut-Susskind Hamiltonian: this can be seen as an exponential speed up in approaching the physical continuum limit for the orbifold lattice Hamiltonian formulation. We show that the universal framework, advocated by three of the authors (M.~H., S.~M., and E.~R.) and collaborators, provides a natural avenue to solve the exponential scaling of circuit complexity with $Q$, and it is the reason why using the orbifold lattice Hamiltonian is advantageous.

Figures (1)

• Figure 1: Fits of the number of Pauli strings $N_{\rm Pauli}$ from Table 1 of Mendicelli:2024ryt obtained by extending Table 1 of Ref. Mendicelli:2024ryt up to 14 qubits. The fit ansatz is $\frac{1}{Q}\log N_{\rm Pauli}=a+\frac{b+c\log Q}{Q}$. We can see a nonzero value of $a$ that indicates the exponential growth of $N_{\rm Pauli}$ concerning the number of qubits $Q$.