Shorter width truncated Taylor series for Hamiltonian dynamics simulations

TL;DR

The paper tackles the resource bottleneck of simulating Hamiltonian dynamics with truncated Taylor series by introducing a shorter-width quantum circuit that preserves the asymptotic gate cost but greatly reduces ancilla qubit counts. It achieves this via a two-ancilla architecture, mid-circuit measurements, and parallelization to implement $U_{\tau,K}=\sum_{k=0}^{K} \widetilde{\beta}_k \widetilde{H}^k$ with $\widetilde{H}=(-i/\|\alpha\|_1)\sum_{\ell} \alpha_{\ell} H_{\ell}$, reducing qubits from $K+K\lceil\log L\rceil+n$ to $\lceil\log K\rceil+\lceil\log L\rceil+n$. By reexpressing multi-controlled $H^k$ blocks as a set of singly-controlled blocks using a logarithmic-depth decomposition, the method preserves the $\mathcal{O}(KT)$ time scaling while lowering qubit overhead. The authors provide a concrete circuit $\ ilde{W}$, analyze the postselection success probability $p_{\tilde{W}}$, and demonstrate equivalent performance to the original Berry scheme in a 1D Ising model, with resource analyses and runtime considerations supported by Guppy-based simulations. They also discuss improvements via BLISS to reduce the $\ell_1$-norm of Hamiltonian coefficients and potential amplification strategies, highlighting practical trade-offs when mid-circuit measurements are time comparable to gate operations. Overall, the work offers a viable path to scalable Hamiltonian dynamics simulation on near-term fault-tolerant architectures by dramatically reducing ancilla requirements without sacrificing asymptotic efficiency.

Abstract

As established in the seminal work by Berry et al.[1], expanding the time evolution operator using truncated Taylor series (up to some order $K$) makes a good candidate for simulating Hamiltonian dynamics. Here, we adapt the method but present an alternative quantum circuit that maintains an equivalent asymptotic elementary gate cost but has an exponentially reduced number of ancilla qubits. This is realized by utilizing mid-circuit measurements (with early abort-and-restart of circuit execution), and transforming a series of multi-controlled$(H^k)$ to a series of singly-controlled$(H^{k'})$, where $H$ is a linear combination of unitaries and $k, k'$ are integers. The proposed circuit utilizes a total of $\lceil \log(K) \rceil + \lceil \log(L) \rceil +n$ qubits, where $L$ is the number of terms in the Hamiltonian and $n$ is the system qubit size. Our shorter width circuit with mid-measurements protocol is implemented and evaluated using the programming language Guppy[2,3].

Figures (7)

• Figure 1: $\bm k$ LCU blocks of $\bm{\widetilde{H}}$ for $\bm{\widetilde{H}^k}$. Postselection is performed after every LCU $U_{\widetilde{H}}$. An unsuccessful postselection aborts and restarts the circuit.
• Figure 2: Shorter width truncated Taylor series expansion circuit. $\widetilde{W}$ for $e^{-iH\tau}$ (up to order $K$). The $W_{\widetilde{H}^k}$ operator is defined in Fig. \ref{['fig:Hk_k_blocks']}. $\kappa = \lceil \log (K+1) \rceil$ qubits are used to P REPARE the Taylor series coefficients and $K$ singly-controlled LCUs of $\widetilde{H}$ to implement the S ELECT($\widetilde{V}$).
• Figure 3: Success probability of simulating$\bm{U_{\tau,K}}$. A 1D-four-lattice-site Ising model Hamiltonian, $H=J\sum_{\langle i,j\rangle}^n Z_i Z_j + h\sum_i^nX_i$, is used with interaction strength $J=1.0$ a.u., external magnetic field $h=0.5$ a.u., and $\tau=0.05$ a.u. All non-numerical data points are obtained using (statevector) simulations, in the absence of any error model, of the circuits defined by $\widetilde{W}$ (with mid-circuit measurements) and $W$ sketched in Fig. \ref{['fig:U_lcu_berry1']} (with deferred measurements). For $\widetilde{W}$, a successful shot means having measured $K$$|\bar{0}\rangle_\ell$ before $|\bar{0}\rangle_k$ at the end. The numerical data set is obtained by calculating the right-hand side of Eq. (\ref{['eq:success_probability_U']}) via matrix multiplication.
• Figure 4: Average two-qubit gate counts in $\bm W$ ($\circ$) and $\bm{\widetilde{W}}$ ($\star$) circuits. The LCU circuits are compiled and run using the multiplexor synthesis technique demonstrated in Ref. sze2025hamiltoniandynamicssimulationusing. The $\star$ data points also include the number of mid-circuit measurements. The 1D-Ising model Hamiltonian, $H=J\sum_{\langle i,j\rangle}^n Z_i Z_j + h\sum_i^nX_i$, with varying lattice sites $n_{sites}$ are used. $W$ circuit utilizes linear amount of resources with increasing $K$, while it is piecewise linear in $\widetilde{W}$. $K$ values that have the equal $\lceil \log(K+1) \rceil$ use the same number of $K$ controlled-$U_{\widetilde{H}}$. Thus, no cost reduction, more expensive in fact, for $K$ satisfying $\log(K+1) < \lceil \log(K+1) \rceil$ if $\widetilde{W}$ circuit is employed.
• Figure 5: LCU circuit $\bm W$ for $\bm{U_{\tau,K}}$, where the Taylor expansion coefficients are encoded in unary as described in Ref. Berry_etal_2015. $B_k$ and $B_k^\dagger$ operators prepare the $K+1$ rescaled Taylor expansion coefficients using $K$ qubits. $\hbox{P{\scriptsize REP}}_H$ and $\hbox{P{\scriptsize REP}}^\dagger_H$ prepare the Hamiltonian coefficients $\alpha_\ell$. The $\hbox{S\scriptsize ELECT}$ operator consists of $K$ singly-controlled-$H$'s. A successful postselection on $K + K\lceil \log L \rceil$ qubits at the end of the circuit leads to $\frac{1}{s}U_{\tau,K}|\psi\rangle$, where $s=\lVert\widetilde{\beta}\rVert_{1}$ . The boxed subcircuit (without the single controls on $H$) performs the product of $K$ block encodings of $H$, where the end result upon successful postselection on $K$ prepare registers leads to $\widetilde{H}^K|\psi\rangle = \frac{H^K}{\lVert\alpha\rVert_{1}^{2K}}|\psi\rangle$.