Efficient Simulation of Pre-Born-Oppenheimer Dynamics on a Quantum Computer

TL;DR

This work develops a first-principles quantum algorithm for direct pre-Born–Oppenheimer dynamics on a real-space grid, treating electrons and nuclei on equal footing. It introduces a swap-network block-encoding framework to achieve linear scaling in particle number for the Coulomb potential and a kinetic-energy encoding that reduces the 1-norm, enabling time evolution with a Toffoli cost of order $ ilde{ ext{O}}( ext{η}^3 t)$. The full Hamiltonian is realized as a controlled sum of potential and kinetic block-encodings, leveraging qubitization and quantum signal processing to bound errors. Resource estimates for industrially relevant reactions show substantial improvements over prior benchmarks, including $8.72 imes10^{9}$ Toffolis per femtosecond for NH$_3$+BF$_3$ on about 1.3k qubits, highlighting the practical potential of fault-tolerant quantum chemistry beyond the Born–Oppenheimer regime.

Abstract

In this work, we present a quantum algorithm for direct first-principles simulation of electron-nuclear dynamics on a first-quantized real-space grid. Our algorithm achieves best-in-class efficiency for block-encoding the pre-Born-Oppenheimer molecular Hamiltonian by harnessing the linear scaling of swap networks for implementing the quadratic number of particle interactions, while using a novel alternating sign implementation of the Coulomb interaction that exploits highly optimized arithmetic routines. We benchmark our approach for a series of scientifically and industrially relevant chemical reactions. We demonstrate over an order-of-magnitude reduction in costs compared to previous state-of-the-art for the $\rm NH_3+BF_3$ reaction, achieving a Toffoli cost of $8.7\times10^{9}$ per femtosecond using $1362$ logical qubits (system + ancillas). Our results significantly lower the resources required for fault-tolerant simulations of photochemical reactions, while providing a suite of algorithmic primitives that are expected to serve as foundational building blocks for a broader class of quantum algorithms.

Figures (22)

• Figure 1: Quantum simulation of chemical reactions on a real-space grid. The Coulomb interaction $1/r$ is implemented as a linear combination of unitaries using the alternating sign technique: an auxiliary register $\ket{m}$ is prepared in an equal superposition over $M$ values and flagged with $\pm1$ phases depending on the value of $r$, such that the resulting sum approximates $1/r$ up to $\mathcal{O}(1/M)$ accuracy. The flagging routine is efficiently implemented using quantum arithmetic.
• Figure 2: Controlled implementation of Coulomb term $U^{(V)}_{12}$ in Eq. \ref{['eq:U_12']}. Here we have defined all the operations inside the blue box as $\mathcal{U}_{\rm arithmetic}$, with the qubit register labelled as $\ket{0}_{\mathcal{F}}$ carrying the information of the flagging procedure. See \ref{['app:circuits']} for more details on the circuit directives. Note that the ancilla qubits from the subtraction by $M^2$ are kept alive and uncomputed when applying $\mathcal{U}_{\rm arithmetic}^\dagger$ with measurement and phase. All qubits registers not shown in the output have been returned to all-zeros state.
• Figure 3: Controlled block-encoding of kinetic energy of single particle $U_1^{(T)}$ in \ref{['eq:T1']}, with squaring routine implementing $\mathtt{SQ}\ket{q}\rightarrow (2q^2-1)\ket{q}$. See \ref{['app:circuits']} for a detailed explanation of all circuit directives.
• Figure 4: $\textup{SEL}$ operator for block-encoding full Hamiltonian in \ref{['eq:ham_sel']}. The block-encoding $U_{12}^{(V)}$ is shown in \ref{['fig:U_V']}, while $U_1^{(T)}$ corresponds to \ref{['fig:T1']}. The $\ket{\cdot}_{\mathcal{N}}$ register flags nuclear-nuclear interactions for the optimized potential energy implementation.
• Figure 5: a) Toffoli cost decomposition per walk operator for different reactions. b) Toffoli cost decomposition of $U^{(V)}_{12}$ operation implementing $1/r$ interaction for $\rm NH_3+BF_3$ reaction. Note that cost profiles are extremely similar for different reactions.

Theorems & Definitions (48)

• Theorem 1: Block-encoding with linear scaling
• Theorem 2: Time evolution
• Lemma 3: One-dimensional swap network block-encoding
• proof
• Lemma 4: Two-dimensional swap network block-encoding
• proof
• Lemma 5: Absolute value in Two's Complement Representation
• proof
• Proposition 6
• proof