Quantum Rational Transformation Using Linear Combinations of Hamiltonian Simulations

TL;DR

This paper develops real-time quantum rational transformations by expressing target matrix functions as sums of resolvents, $r(H)=\sum_k c_k (z_k - H)^{-1}$, and implementing them via linear-combination-of-Hamiltonian-simulations on quantum hardware. It introduces two complementary LCU strategies: a discrete-time approach using optimal quadrature (notably Gauss-Legendre) to realize resolvents as weighted sums of real-time evolutions, and a continuous-time approach using Gaussian ancillae to block-encode resolvents with unit-duration evolutions. The authors provide rigorous resource bounds for complex and real poles, derive Zolotarev-based rational approximants (and iterative filtering) for sharp spectral operations, and demonstrate the framework on spin models with applications to ground- and excited-state estimation via spectral filtering in ODMD. Collectively, the work offers a compact, real-time toolkit for constructing quantum rational transformations with favorable scaling and practical relevance to many-body spectra and matrix-function problems on quantum devices.

Abstract

Rational functions are exceptionally powerful tools in scientific computing, yet their abilities to advance quantum algorithms remain largely untapped. In this paper, we introduce effective implementations of rational transformations of a target operator on quantum hardware. By leveraging suitable integral representations of the operator resolvent, we show that rational transformations can be performed efficiently with Hamiltonian simulations using a linear-combination-of-unitaries (LCU). We formulate two complementary LCU approaches, discrete-time and continuous-time LCU, each providing unique strategies to decomposing the exact integral representations of a resolvent. We consider quantum rational transformation for the ubiquitous task of approximating functions of a Hermitian operator, with particular emphasis on the elementary signum function. For illustration, we discuss its application to the ground and excited state problems. Combining rational transformations with observable dynamic mode decomposition (ODMD), our recently developed noise-resilient quantum eigensolver, we design a fully real-time approach for resolving many-body spectra. Our numerical demonstration on spin systems indicates that our real-time framework is compact and achieves accurate estimation of the low-lying energies.

Figures (13)

• Figure 1: Illustration of numerical contour integration. The eigenvalues are marked as (●) and a contour, in red, encircles the negative eigenvalues. On the right, a discretization of the contour is shown with $K =8$ equidistant points on the circle $z_k = -1 + \cos(2 k \pi/8)+i\sin(2 k\pi/8)$. This leads to 6 complex poles (${\color{myblue} + }$) and 2 real poles ($\ast$), i.e., 6 resolvents to be computed for $z_k\notin \mathbb{R}$ and 2 resolvents for $z_k\in \mathbb{R}$.
• Figure 2: Discrete-time construction of a resolvent $R(z) = (z-H)^{-1}$, with a complex pole $z=a+bi$, via a quadrature rule with nodes $\{t_j\}_{j}$ and weights $\{w_j\}_{j}$, i.e., $R(z) \approx \sum_{j} w_j e^{(ia-b)t_j} e^{-iHt_j}$. On quantum hardware the Hamiltonian simulations $U(t_j) = e^{-iHt_j}$ can be performed efficiently. The circuit diagram depicts Trotterized time evolution with multiple fixed steps $\Delta t$ and a final variable step $\delta t_j$, so that $U_{\rm Trotter}(\delta t_j) U_{\rm Trotter}(\Delta t)^{M_j} \approx U(t_j)$ for $t_j = M_j \Delta t + \delta t_j$. Repeated measurements in the computational basis enable us to access scalar quantities such as $\langle \phi \vert U_{\rm Trotter}(\delta t_j) U_{\rm Trotter}(\Delta t)^{M_j} \vert \phi \rangle$, which can be collected and combined in post-processing to obtain the desired inner product $\langle \phi\vert R(z) \vert \phi \rangle$.
• Figure 3: Simulation of $e^{-iHt_j}$ by Trotterized time evolution $U_{\rm Trotter}(t) := e^{-iH_1 t}e^{-iH_2 t}$. The evolution is advanced for multiple fixed time steps $\Delta t$, common for all simulations $t_j\in \boldsymbol{t}$, and a variable shorter time step $\delta t_j<\Delta t$ dependent on $t_j$.
• Figure 4: Approximation error $\Vert R(z)-\mathcal{I_J}(z)\Vert_2$ as a function of the scaled total evolution time $\frac{T^{\rm tot}}{T^{\rm max}}$, for $z = -0.8+0.1i$ and the (scaled) MFIM Hamiltonian with $L_{\rm sys} =8$ spins. The approximation $\mathcal{I_J}(z)$ is computed using the trapezoidal (+), Legendre ($\circ$) and Laguerre ($\ast$) rule for a requested accuracy of $\epsilon$ ($10^{-3}$ left and $10^{-6}$ right) and Trotter step of $\Delta t = \frac{\epsilon}{T^{\rm max}}$.
• Figure 5: Cost metric $\frac{T^{\rm tot}}{T^{\rm max}}$ for the scaled total evolution time for quadrature rules necessary to compute an $\epsilon$-approximation to $(z-H)^{-1}$ with accuracy $\epsilon=10^{-6}$. A Trotter step of $\Delta t = \frac{\epsilon}{T^{\rm max}}$ is used to evolve the MFIM Hamiltonian with $L_{\rm sys} = 8$ spins. We consider $z=-0.8+bi$ on the left and $z=bi$ on the right, which yields $a^+=1.36$ and $a^+=1$ respectively. The Legendre rule ($\circ$) performs more efficiently than the Laguerre ($\ast$) and trapezoidal ($+$) rule. The solid line shows the predicted cost for the Legendre rule derived in \ref{['thm:1']}.

Theorems & Definitions (14)

• Theorem 1
• proof
• Theorem 2
• proof
• Lemma 1
• proof
• Corollary 1
• proof
• Corollary 2
• proof