Quantum Simulation of Boson-Related Hamiltonians: Techniques, Effective Hamiltonian Construction, and Error Analysis

TL;DR

This work surveys a comprehensive framework for quantum simulation of boson-related Hamiltonians, addressing boson-to-qubit mappings, initial-state preparation, ground-state techniques, and evolution via product formulas as well as qubitization. It introduces an effective Hamiltonian construction through coupled-cluster downfolding to compress bosonic problems into tractable active spaces, and provides rigorous, polylogarithmic truncation bounds for both state- and Hamiltonian-truncation across single and multiple bosonic modes, including time-dependent cases. The methods are illustrated through examples like the Holstein and spin-boson models, and extended to more complex electron-phonon and polaritonic systems, highlighting resource scaling and potential NISQ applicability with controlled errors. The findings advance provable, scalable strategies for simulating open quantum systems with bosonic environments, enabling more accurate investigations of strong system–environment interactions on quantum hardware.

Abstract

A broad spectrum of physical systems in condensed-matter and high-energy physics, vibrational spectroscopy, and circuit and cavity QED necessitates the incorporation of bosonic degrees of freedom, such as phonons, photons, and gluons, into optimized fermion algorithms for near-future quantum simulations. In particular, when a quantum system is surrounded by an external environment, its basic physics can usually be simplified to a spin or fermionic system interacting with bosonic modes. Nevertheless, troublesome factors such as the magnitude of the bosonic degrees of freedom typically complicate the direct quantum simulation of these interacting models, necessitating the consideration of a comprehensive plan. This strategy should specifically include a suitable fermion/boson-to-qubit mapping scheme to encode sufficiently large yet manageable bosonic modes, and a method for truncating and/or downfolding the Hamiltonian to the defined subspace for performing an approximate but highly accurate simulation, guided by rigorous error analysis. In this pedagogical tutorial review, we aim to provide such an exhaustive strategy, focusing on encoding and simulating certain bosonic-related model Hamiltonians, inclusive of their static properties and time evolutions. Specifically, we emphasize two aspects: (1) the discussion of recently developed quantum algorithms for these interacting models and the construction of effective Hamiltonians, and (2) a detailed analysis regarding a tightened error bound for truncating the bosonic modes for a class of fermion-boson interacting Hamiltonians.

Figures (9)

• Figure 1: System–environment interactions in an open quantum system comprising a quantum system in a complex environment can pose a challenging problem.
• Figure 2: Two-particle correlation of the quantum walkers of two indistinguishable bosons, $\Gamma_{p,q}(t) = \langle \phi_b(t) | b_p^\dagger b_q^\dagger b_q b_p|\phi_b(t)\rangle$, on 1D optical lattice with five sites. The corresponding density distribution, $\langle n_p \rangle = \langle \phi_b(t) | b_p^\dagger b_p|\phi_b(t)\rangle$, is shown in the bottom of each plot of correlation. The Hamiltonian is given in (\ref{['boson_hamiltonian']}) with $N=5$, $\mu=-0.5U$, $t=1$, and $V=0$. $|\phi_b(t)\rangle$ evolves from the initial state $|\phi_b(0)\rangle = b_2^\dagger b_4^\dagger |\rm{vac}\rangle$ through the propagator $\exp(-i \hat{H} t)$ over the time $t$ = 0.003 a.u. ($\Delta t=1.0\times 10^{-5}$ a.u.), against the positions of two-boson $p$ and $q$ for different on-site interactions $U=1$ and $U=100$.
• Figure 3: Boson (upper panels) and spin (lower panels) dynamics of a spin-boson model for different coupling regimes. The Hamiltonian is described in (\ref{['eq:Ham_SB']}) with the corresponding qubit representation in (\ref{['eq:Ham_SB_qubit']}). The decoherence is simulated through a Lindblad master equation as described in Appendix \ref{['app_a']} where the parameters $\Gamma$ and $\gamma$ account for the experimental imperfections.
• Figure 4: Circuit to prepare the state $\frac{1}{\sqrt{K}}\sum_{k=1}^K c_k |\phi_{b,k}\rangle$ with conditions (\ref{['cond']}). $R_k$ and $R'_k$ ($k=1,\cdots, K$) gates rotate the single qubit state $|0\rangle$ or $|1\rangle$ to the state $x|0\rangle + y|1\rangle$. The unitary gate $U_k$ ($k=1,\cdots, K$) acting on $|\text{vac}\rangle$ to generate $|\phi_{b,k}\rangle$. $C_{1/2}$ are two classical registers to take the projection of the ancillas $|b_{1/2}\rangle$ on the compuational basis $|0\rangle$ and $|1\rangle$. In the controlled gates, '$\circ$' denotes turning on the conditional operation when the controlled qubit is $|0\rangle$, while '$\bullet$' denotes turning on the operation when the controlled qubit is $|1\rangle$. Only when $|b_1\rangle = |1\rangle$ and $|b_2\rangle = |0\rangle$, the desired state will be prepared.
• Figure 5: The change in the ground state energy from exact diagonalization (ED) and in various orders of the PDS energy functional of a three-site Holstein model as a function of fermion-boson coupling strength.

Theorems & Definitions (8)

• Lemma 1: Short-time state truncation
• Lemma 2: Bounding bosonic number by the Lambert-W function
• proof
• Corollary 3: Long-time state truncation
• Lemma 4: Short-time Hamiltonian truncation
• Corollary 5: Long-time Hamiltonian truncation
• Theorem 6: Bosonic Hamiltonian truncation
• Theorem 7: $N$-mode bosonic Hamiltonian truncation