Quantum fast-forwarding fermion-boson interactions via the polaron transform

TL;DR

This work identifies the efficient unitary transformation enabling fast-forwarded evolution of the fermion-boson interaction term, yielding an interaction-picture based simulation algorithm with complexity polylogarithmic in $\Lambda$.

Abstract

Simulating interactions between fermions and bosons is central to understanding correlated phenomena, yet these systems are inherently difficult to treat classically. Previous quantum algorithms for fermion-boson models exhibit computation costs that scale polynomially with the bosonic truncation parameter, $Λ$. In this work we identify the efficient unitary transformation enabling fast-forwarded evolution of the fermion-boson interaction term, yielding an interaction-picture based simulation algorithm with complexity polylogarithmic in $Λ$. We apply this transformation to explicitly construct an efficient quantum algorithm for the Hubbard-Holstein model and discuss its generalisation to other fermion-boson interacting models. This approach yields an important asymptotic improvement in the dependence on the bosonic cutoff and establishes that, for certain models, fermion-boson interactions can be simulated with resources comparable to those required for purely fermionic systems.

Figures (9)

• Figure 1: High-level circuit diagram for the interaction picture algorithm for quantum simulation of $\mathrm{e}^{-\mathrm{i}Ht}$ where $H$ is the Hubbard-Holstein model, the piece denoted is $\mathrm{e}^{-\mathrm{i}H_0t/r} \mathcal{T} \left[\mathrm{e}^{-\mathrm{i}\int_{0}^{t/r} V(s) ds}\right]$, which must be repeated $r$ times to evolve by time $t$. $\bigotimes_{i=1}^N\ket{\psi_{f_i}}\otimes\ket{\psi_{b_i}}$ denotes the $N$ site system register where each site has a single qubit to represent the fermionic degree of freedom and a $\lceil \log_2(\Lambda) \rceil$ qubit register to hold the bosonic degree of freedom truncated at $\Lambda$. $\tilde{V}= \mathcal{D} V \mathcal{D}^\dag$ and $\tilde{H}_0= \mathcal{D} H_0 \mathcal{D}^\dag$ denote the polaron transformed Hamiltonian terms given in Eq. \ref{['eq:1QtildeV']} and Eq. \ref{['eqn:form_of_h0']}, respectively. The time-ordered exponential is approximated by the Dyson expansion truncated at commutators of depth $K$ requiring a $K$-qubit ancilla register to label the integrals in the expansion, each ancilla qubit requires a $\log_2 L$ ancilla register to facilitate the approximation of the integral by a $L$-term sum. Using the polaron transform $\mathcal{D}$ we can efficiently implement evolution under the bosonic, diagonal fermionic and the bosonic-fermionic interaction directly, denoted $\tilde{H}_0$ whereas the evolution under the rest of the fermionic term is block encoded requiring an ancillary register labeled $\ket{c}$. See kan2025optimized for more detailed implementation of the PREP and control structure to implement this truncated Dyson series expansion.
• Figure 2: Circuit to simulate the evolution of a diagonal Hamiltonian $H = \sum_x d(x) \ketbra{x}{x}$ where the diagonal elements can be efficiently computed. First the diagonal elements are computed in superposition onto an ancillary register with $U_{d_k}$, and the $k$-qubit phase gate $R_k$ is applied to the ancilla register effectively implementing $\ket{x}\ket{d_k(x)} \rightarrow \mathrm{e}^{-\mathrm{i}t d_k(x)}\ket{x}\ket{d_k(x)}$. Finally the ancilla is uncomputed, disentangling it with the system register, $\mathrm{e}^{-\mathrm{i}td_k(x)}\ket{x}\ket{d_k(x)} \rightarrow \mathrm{e}^{-\mathrm{i}td_k(x)}\ket{x}\ket{0}$. All together this can be implemented with gate complexity $\textrm{poly}(k) + k \mathcal{O}(\log_2(1/\epsilon') + \log_2 k)$, where $k = \left\lceil \log_2 \frac{1.053|t|}{\epsilon'} \right\rceil$ (assuming $\epsilon' \leq 0.1$), $U_{d_k}$ takes $\textrm{poly}(k)$ and each rotation synthesis takes $\mathcal{O}(\log_2(1/\epsilon') + \log_2 k)$ gates.
• Figure 3: The circuit implementation of the polaron transform (Eqn. \ref{['eqn:app dis1Q']}) in first quantisation. There are $i\in\{1,N\}$ sites each with a fermionic and bosonic degree of freedom. The fermionic register at site $i$, $\ket{\psi_\text{fermion}}_i = \ket{\rho_i}\otimes \ket{n_{i\downarrow}}$ consists of two qubits where $\ket{\rho_i}$ holds the parity of the fermionic state and $\ket{n_{i\downarrow}}$ the down spin occupation. The bosonic register at site $i$, $\ket{x_i}_i$, consists of $b_M:=\log_2(M)$ qubits and holds the position coordinate. $\mathrm{QFT_{x\rightarrow p}}$ denotes the centered quantum fourier transform that converts the bosonic position to momentum $\mathrm{QFT}_{x\rightarrow p}\ket{x_1}_1 = \ket{p_1}_1$- it is implemented by first shifting the computational basis with a $R_Z(\pi)$ gate on the most significant bit, followed by a normal QFT and unshifting the basis. We take $M$ to be a power of $2$ so that the QFT requires $4b_M\log_2(b_M/\epsilon_{QFT})$ T gates park2024tcountoptimizationapproximatequantum. Controlled on the site's fermionic register, $(-1)^{1+ n_{i\downarrow}}p_i$ is recursively added to a $(b_N + b_M)$-ancilla register. Using the implementation from Gidney_2018, a $(b_N+b_M)$-bit adder requires $4(b_N+b_M)+\mathcal{O}(1)$ T gates, and using Barenco_1995 controlling such an adder on a single qubit increases this count to $16(b_N+b_M)+\mathcal{O}(1)$ T gates. $(b_M + b_N)$ rotations are then performed across the ancilla to apply the correct phase before $U_{1QD}$ is uncomputed. This cumulative structure for computing the phase saves a multiplicative factor of $N$.
• Figure 4: The controlled walk operator $c-W(K)$ in terms of prepare and select oracles.
• Figure 5: Implementation for $\textrm{PREP}$ and $\textrm{SELECT}$ for the generator of the displacement operator in second quantisation. The $\textrm{PREP}$ acts on $K+2$ temporary ancilla qubits.