We introduce a method for representing the low-energy subspace of a bosonic field theory on the qubit space of digital quantum computers. This discretization leads to an exponentially precise description of the subspace of the continuous theory thanks to the Nyquist-Shannon sampling theorem. The method makes the implementation of quantum algorithms for purely bosonic systems as well as fermion-boson interacting systems feasible. We present algorithmic circuits for computing the time evolution of these systems. The complexity of the algorithms scales polynomially with the system size. The algorithm is a natural extension of the existing quantum algorithms for simulating fermion systems in quantum chemistry and condensed-matter physics to systems involving bosons and fermion-boson interactions and has a broad variety of potential applications in particle physics, condensed matter, etc. Due to the relatively small amount of additional resources required by the inclusion of bosons in our algorithm, the simulation of electron-phonon and similar systems can be placed in the same near-future reach as the simulation of interacting electron systems. We benchmark our algorithm by implementing it for a two-site Holstein polaron problem on an Atos Quantum Learning Machine quantum simulator. The polaron quantum simulations are in excellent agreement with the results obtained by exact diagonalization.
Qubit, operator, and gate resources required for the digitization of lattice $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ scalar field theories onto quantum computers are considered, building upon the foundational work by Jordan et al. [Quantum Inf. Comput. 14, 1014 (2014); Science 336, 1130 (2012)], with a focus towards noisy intermediate-scale quantum devices. The Nyquist-Shannon sampling theorem, introduced in this context by Macridin et al. [Phys. Rev. A 98, 042312 (2018)] building on the work of Somma [Quantum Inf. Comput. 16, 1125 (2016)], provides a guide with which to evaluate the efficacy of two field-space bases, the eigenstates of the field operator, as used by Jordan et al., and eigenstates of a harmonic oscillator, to describe (
0+1)- and (
d+1)-dimensional scalar field theory. We show how techniques associated with improved actions, which are heavily utilized in lattice QCD calculations to systematically reduce lattice-spacing artifacts, can be used to reduce the impact of the field digitization in $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$, but are found to be inferior to a complete digitization improvement of the Hamiltonian using a quantum Fourier transform. When the Nyquist-Shannon sampling theorem is satisfied, digitization errors scale as $|log|log|{\ensuremath{\epsilon}}_{\mathrm{dig}}|||\ensuremath{\sim}{n}_{Q}$ (number of qubits describing the field at a given spatial site) for the low-lying states, leaving the familiar power-law lattice-spacing and finite-volume effects that scale as $|log|{\ensuremath{\epsilon}}_{\mathrm{latt}}||\ensuremath{\sim}{N}_{Q}$ (total number of qubits in the simulation). For localized (delocalized) field-space wave functions, it is found that ${n}_{Q}\ensuremath{\sim}4(7)$ qubits per spatial lattice site are sufficient to reduce theoretical digitization errors below error contributions associated with approximation of the time-evolution operator and noisy implementation on near-term quantum devices. Only classical computing resources have been used to obtain the results presented in this work.
To simulate bosons on a qubit- or qudit-based quantum computer, one has to regularize the theory by truncating infinite-dimensional local Hilbert spaces to finite dimensions. In the search for practical quantum applications, it is important to know how big the truncation errors can be. In general, it is not easy to estimate errors unless we have a good quantum computer. In this paper, we show that traditional sampling methods on classical devices, specifically Markov Chain Monte Carlo, can address this issue for a rather generic class of bosonic systems with a reasonable amount of computational resources available today. As a demonstration, we apply this idea to the scalar field theory on a two-dimensional lattice, with a size that goes beyond what is achievable using exact diagonalization methods. This method can be used to estimate the resources needed for realistic quantum simulations of bosonic theories, and also, to check the validity of the results of the corresponding quantum simulations.
Abstract To simulate bosons on a qubit- or qudit-based quantum computer, one has to regularize the theory by truncating infinite-dimensional local Hilbert spaces to finite dimensions. In the search for practical quantum applications, it is important to know how big the truncation errors can be. In general, it is not easy to estimate errors unless we have a good quantum computer. In this paper, we show that traditional sampling methods on classical devices, specifically Markov Chain Monte Carlo, can address this issue for a rather generic class of bosonic systems with a reasonable amount of computational resources available today. As a demonstration, we apply this idea to the scalar field theory on a two-dimensional lattice, with a size that goes beyond what is achievable using exact diagonalization methods. This method can be used to estimate the resources needed for realistic quantum simulations of bosonic theories, and also, to check the validity of the results of the corresponding quantum simulations.
Qubit, operator and gate resources required for the digitization of lattice
λϕ4 scalar field theories onto quantum computers in the NISQ era are considered, building upon the foundational work by Jordan, Lee and Preskill. The Nyquist-Shannon sampling theorem, introduced in this context by Macridin, Spetzouris, Amundson and Harnik building on the work of Somma, provides a guide with which to evaluate the efficacy of two field-space bases, the eigenstates of the field operator, as used by Jordan, Lee and Preskill, and eigenstates of a harmonic oscillator, to describe
0+1- and
1+1-dimensional scalar field theory. We show how techniques associated with improved actions, which are heavily utilized in Lattice QCD calculations to systematically reduce lattice-spacing artifacts, can be used to reduce the impact of the field digitization in
λϕ4, but are found to be inferior to a complete digitization-improvement of the Hamiltonian using a Quantum Fourier Transform. When the Nyquist-Shannon sampling theorem is satisfied, digitization errors scale as
∣log∣log∣ϵdig∣∣∣∼nQ (number of qubits describing the field at a given spatial site) for the low-lying states, leaving the familiar power-law lattice-spacing and finite-volume effects that scale as
∣log∣ϵlatt∣∣∼NQ (total number of qubits in the simulation). We find that fewer than
nQ=10 qubits per spatial lattice site are sufficient to reduce digitization errors below noise levels expected in NISQ-era quantum devices for both localized and delocalized field-space wavefunctions. For localized wavefunctions,
nQ=4 qubits are likely to be sufficient for calculations requiring modest precision.
Quantum many-body systems involving bosonic modes or gauge fields have infinite-dimensional local Hilbert spaces which must be truncated to perform simulations of real-time dynamics on classical or quantum computers. To analyze the truncation error, we develop methods for bounding the rate of growth of local quantum numbers such as the occupation number of a mode at a lattice site, or the electric field at a lattice link. Our approach applies to various models of bosons interacting with spins or fermions, and also to both abelian and non-abelian gauge theories. We show that if states in these models are truncated by imposing an upper limit Λ on each local quantum number, and if the initial state has low local quantum numbers, then an error at most ϵ can be achieved by choosing Λ to scale polylogarithmically with ϵ−1 , an exponential improvement over previous bounds based on energy conservation. For the Hubbard-Holstein model, we numerically compute a bound on Λ that achieves accuracy ϵ , obtaining significantly improved estimates in various parameter regimes. We also establish a criterion for truncating the Hamiltonian with a provable guarantee on the accuracy of time evolution. Building on that result, we formulate quantum algorithms for dynamical simulation of lattice gauge theories and of models with bosonic modes; the gate complexity depends almost linearly on spacetime volume in the former case, and almost quadratically on time in the latter case. We establish a lower bound showing that there are systems involving bosons for which this quadratic scaling with time cannot be improved. By applying our result on the truncation error in time evolution, we also prove that spectrally isolated energy eigenstates can be approximated with accuracy ϵ by truncating local quantum numbers at Λ=polylog(ϵ−1) .