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Simulating high-accuracy nuclear motion Hamiltonians in discrete variable representation using Walsh-Hadamard QROM with fault-tolerant quantum computers

Michał Szczepanik, Ákos Nagy, Emil Żak

TL;DR

This work presents a fault-tolerant quantum algorithm for simulating rovibrational spectra by embedding a general curvilinear KEO with a non-SOP PES in a discrete-variable representation. The key innovation is a Walsh–Hadamard QROM-based block-encoding workflow that loads PES data efficiently, enabling high-accuracy energy estimates via quantum phase estimation. A hybrid mixed FBR–DVR representation is shown to offer exponential reductions in quantum resources (qubits and T-gates) relative to traditional approaches, with substantial improvements demonstrated for water and larger polyatomic models. The proposed framework also includes strategies to reduce QROM load costs, including d-sparse block encoding with fused oracles and diagonal-operator techniques, and provides extensive resource estimates across representative molecular systems. Overall, the method suggests a scalable quantum route to high-accuracy rovibrational dynamics and spectroscopy, with memory and time advantages that become more pronounced as system dimensionality grows, albeit with significant hardware requirements to realize the asymptotic gains.

Abstract

We present a quantum algorithm for simulating rovibrational Hamiltonians on fault-tolerant quantum computers. The method integrates exact curvilinear kinetic energy operators and general-form potential energy surfaces expressed in a hybrid finite-basis/discrete-variable representation. The Hamiltonian is encoded as a unitary quantum circuit using a quantum read-only memory construction based on the Walsh--Hadamard transform, enabling high-accuracy quantum phase estimation of rovibrational energy levels. Our technique provides asymptotic reductions in both logical-qubit count and T-gate complexity that are exponential in the number of atoms and at least polynomial in the total Hilbert-space size, relative to existing block-encoding techniques based on linear combinations of unitaries. Compared with classical variational methods, it offers exponential memory savings and polynomial reductions in time complexity. The quantum volume required for computing the rovibrational spectrum of water can be reduced by up to $10^{5}$ times compared with other quantum methods, increasing to at least $10^{6}$ for a 30-dimensional (12-atom) model system. For this case with a six-body coupled potential, estimating spectroscopic-accuracy energy levels would require about three months on a $1~\mathrm{MHz}$ fault-tolerant quantum processor with fewer than 300 logical qubits, versus over 30,000 years on the fastest current classical supercomputer. These estimates are approximate and subject to technological uncertainties, and realizing the asymptotic advantage will require substantial quantum resources and continued algorithmic progress.

Simulating high-accuracy nuclear motion Hamiltonians in discrete variable representation using Walsh-Hadamard QROM with fault-tolerant quantum computers

TL;DR

This work presents a fault-tolerant quantum algorithm for simulating rovibrational spectra by embedding a general curvilinear KEO with a non-SOP PES in a discrete-variable representation. The key innovation is a Walsh–Hadamard QROM-based block-encoding workflow that loads PES data efficiently, enabling high-accuracy energy estimates via quantum phase estimation. A hybrid mixed FBR–DVR representation is shown to offer exponential reductions in quantum resources (qubits and T-gates) relative to traditional approaches, with substantial improvements demonstrated for water and larger polyatomic models. The proposed framework also includes strategies to reduce QROM load costs, including d-sparse block encoding with fused oracles and diagonal-operator techniques, and provides extensive resource estimates across representative molecular systems. Overall, the method suggests a scalable quantum route to high-accuracy rovibrational dynamics and spectroscopy, with memory and time advantages that become more pronounced as system dimensionality grows, albeit with significant hardware requirements to realize the asymptotic gains.

Abstract

We present a quantum algorithm for simulating rovibrational Hamiltonians on fault-tolerant quantum computers. The method integrates exact curvilinear kinetic energy operators and general-form potential energy surfaces expressed in a hybrid finite-basis/discrete-variable representation. The Hamiltonian is encoded as a unitary quantum circuit using a quantum read-only memory construction based on the Walsh--Hadamard transform, enabling high-accuracy quantum phase estimation of rovibrational energy levels. Our technique provides asymptotic reductions in both logical-qubit count and T-gate complexity that are exponential in the number of atoms and at least polynomial in the total Hilbert-space size, relative to existing block-encoding techniques based on linear combinations of unitaries. Compared with classical variational methods, it offers exponential memory savings and polynomial reductions in time complexity. The quantum volume required for computing the rovibrational spectrum of water can be reduced by up to times compared with other quantum methods, increasing to at least for a 30-dimensional (12-atom) model system. For this case with a six-body coupled potential, estimating spectroscopic-accuracy energy levels would require about three months on a fault-tolerant quantum processor with fewer than 300 logical qubits, versus over 30,000 years on the fastest current classical supercomputer. These estimates are approximate and subject to technological uncertainties, and realizing the asymptotic advantage will require substantial quantum resources and continued algorithmic progress.
Paper Structure (75 sections, 2 theorems, 208 equations, 21 figures, 28 tables)

This paper contains 75 sections, 2 theorems, 208 equations, 21 figures, 28 tables.

Table of Contents

  1. Introduction
  2. Classical computing rovibrational calculations and their limitations
  3. Kinetic energy operator.
  4. Potential Energy Surface.
  5. Hamiltonian representation.
  6. Eigenvalue problem.
  7. Largest classical computations.
  8. Curse of dimensionality in rovibrational calculations
  9. Key challenges in high-accuracy nuclear motion simulations.
  10. Motivation for First-Principles Quantum Simulation.
  11. Opportunities for Quantum Advantage
  12. Potential Energy Surface
  13. Density of States
  14. Coupling Strength
  15. Quantum computing techniques for simulating nuclear motion dynamics: state-of-the-art and this work's contribution
  16. ...and 60 more sections

Key Result

Theorem A.1

Assume that $\Theta$ is Lipschitz continuous with Lipschitz constant, $K_\Theta$. Let $\underline{n} \mathrel{\vcenter{\hbox{.}\hbox{.}}}=$ n_1, n_2, …, n_D $\in \mathbbm{Z}_+^D$ and Then, for all $\underline{n}^\prime \mathrel{\vcenter{\hbox{.}\hbox{.}}}=$ n_1^, n_2^, …, n_D^$\in \mathbbm{Z}_+^D$, with $n_a \leqslant n_a^\prime$, we have that

Figures (21)

  • Figure 1: Circuit block encoding the KEO in DVR-FBR representation as given in \ref{['eq:KEODVR']}. The controlled version of $\mathcal{B}\mathopen{}\mathclose{\left[\boldsymbol{P}^{FBR}_i\right]$, $\mathcal{B}\mathopen{}\mathclose{\left[\boldsymbol{G}}^{DVR}_i\right]$ and $\mathcal{B}\mathopen{}\mathclose{\left[\boldsymbol{P}}^{FBR}_j\right]^\dag$ means that if register $a'$ is in state $\ket{i-1}}_{a'_1}\ket{j-1}_{a'_2}$ the block encoding of $\boldsymbol{P}^{FBR}_i$, $\boldsymbol{g}_{ij}^{DVR}$ and $\mathopen{}\mathclose{\left(\boldsymbol{P}_j^{FBR}\right)^\dag$ is executed. Note that here $\boldsymbol{T}}$ denotes the DVR unitary and not the $T=Z^{\frac{1}{4}}$ gate.
  • Figure 2: a) Comparison of quantum volume (T-gate times qubit count) for block encoding rovibrational Hamiltonians at $J=0$ for water (3D), methane (9D) and model molecules ($21D$,$30D$,$51D$). Shown are: our present work using the FBR-DVR form of the Hamiltonian with WH-QROM (red filled circles), present work using LCU + FBR (purple squares), present work using SELECT-SWAP QROM for DVR-FBR Hamiltonian (blue squares). b) Respective T-gate count for QPE. $L=6$ PES is used for all cases except methane where $L=D=9$ and water where $L=D=3$.
  • Figure 3: Sketch of complexity regimes in rovibrational calculations. The horizontal axis denotes the number of dimensions (the number of internal coordinates), and the vertical axis corresponds to a ladder of levels of theories with increasing accuracy. The memory-intractable regime is defined as computation requiring classical memory beyond the standard capacity of classical HPC architectures ($\approx$ 10 TB RAM), while complexity-intractable regions correspond to the number of FLOPS that exceed a realistic time-computer time execution with today's CPUs and GPUs. The boundaries shown in the figure are only approximate and for concept illustration purposes only. Selected classical computing methods are located within the plot as well a general region where other known quantum techniques apply (cf. Table \ref{['tab:comparison-qc']} and \ref{['tab:other-compare']}). HOKEO denotes harmonic-oscillator KEO, Iter denotes iterative eigensolver (few eigenvalues), Dir denotes direct eigensolver (many eigenvalues), Watson denotes rectilinear Watson Hamiltonian Watson1968, PB means pruned/contracted basis and non-BO is full-non-Born-Oppenheimer theory.
  • Figure 4: a) Schematic representation for the segment construction of the DVR oracle matrix. Basis set (column) index is represented as $q=wF+v$, where $w=0,1,2,...,\frac{N}{F}-1$ and $v=0,1,2,...,F-1$ for the purpose of splitting the column space in the DVR matrix into $\frac{N}{F}$ segments. Columns in each segment are constructed simultaneously in the ascending and descending horizontal direction following the three-step recursion given in eq. \ref{['eq:recursion']}; b) Color map encoding values of the DVR transformation matrix for the Gauss-Hermite quadrature with $N=128$ elements. Figure adapted from ref. plis25.
  • Figure 5: Quantum circuit representing initialization unitary for the DVR oracle defined in eq. \ref{['eq:uinit']}. $\hat{D}_{N,m}(A)$ denotes QROM encoding data represented by function $A$ for $N$ arguments and output stored in $m$ qubits. Here $T_{p\tilde{q}}$ denotes the DVR matrix element for the $\tilde{q}$'th column, where $\tilde{q}=wF+\frac{F}{2}$. The column index $q$ is represented as $q=wF+v$. Adapted from ref. plis25.
  • ...and 16 more figures

Theorems & Definitions (4)

  • Theorem A.1
  • proof
  • Corollary A.2
  • Remark A.3