Quantum circuit and mapping algorithms for wavepacket dynamics: case study of anharmonic hydrogen bonds in protonated and hydroxide water clusters

TL;DR

This paper tackles quantum nuclear dynamics in multidimensional systems by presenting two complementary quantum algorithmic strategies: a Hamiltonian-mapping protocol that translates the Born-Oppenheimer molecular Hamiltonian ${\cal H}_{Mol}$ into an ion-trap Ising Hamiltonian ${\cal H}_{IT}$ (exact for $N=3$ qubits, approximate for larger $N$) and a Quantum Shannon Decomposition (QSD) framework that recasts the unitary propagator into a universal-gate circuit (in principle exact for any $N$). The authors contrast analog spin-lattice simulation with digital circuit decomposition, and validate both approaches on proton-transfer problems in H$_5$O$_2^+$ and H$_3$O$_2^-$, achieving good agreement with classical wavepacket dynamics and vibrational frequencies within a few tenths of a wavenumber. Key methodological advances include exploiting a block-structured Ising form via a q-sphere/Hamming-space basis partition, transforming the nuclear Hamiltonian with Givens rotations to align with Ising blocks, and implementing QSD to enable scalable, near-optimal quantum circuits. The work demonstrates the potential of quantum computing to model quantum nuclear effects in anharmonic hydrogen bonds and provides a foundation for extending these techniques to higher dimensions, with practical implications for understanding proton-transfer processes in water clusters and related systems.

Abstract

The accurate computational study of wavepacket nuclear dynamics is considered to be a classically intractable problem, particularly with increasing dimensions. Here we present two algorithms that, in conjunction with other methods developed by us, will form the basis for performing quantum nuclear dynamics in arbitrary dimensions. For one algorithm, we present a direct map between the Born-Oppenheimer Hamiltonian describing the wavepacket time-evolution and the control parameters of a spin-lattice Hamiltonian that describes the dynamics of qubit states in an ion-trap quantum computer. This map is exact for three qubits, and when implemented, the dynamics of the spin states emulate those of the nuclear wavepacket. However, this map becomes approximate as the number of qubits grow. In a second algorithm we present a general quantum circuit decomposition formalism for such problems using a method called the Quantum Shannon Decomposition. This algorithm is more robust and is exact for any number of qubits, at the cost of increased circuit complexity. The resultant circuit is implemented on IBM's quantum simulator (QASM) for 3-7 qubits. In both cases the wavepacket dynamics is found to be in good agreement with the classical result and the corresponding vibrational frequencies obtained from the wavepacket density time-evolution, are in agreement to within a few tenths of a wavenumbers.

Figures (9)

• Figure 1: A generalized Bloch sphere for an arbitrary number of qubits, along with the classification of basis states shown using red and blue colors.
• Figure 2: Recursive block structure of the Ising Hamiltonian in Eq. (\ref{['HIT-gen']}) for three (a) and four (b) qubits. The upper triangular portion of the Hamiltonian matrix is shown (excluding the diagonal). The computational basis is partitioned into odd, $\left\{ {\bf S^+}^{2n-1} \ket{00\cdots}\right\}$, and even, $\left\{ {\bf S^+}^{2n} \ket{00\cdots}\right\}$, spans of the total spin raising operators. The interaction between states, $\ket{i}$ and $\ket{j}$ is the $ij^{th}$ matrix element of the ion trap Hamiltonian. For example in (b), $\bra{0101} {\cal H}_{IT}$$\ket{1111} = J_{13}^{x} - J_{13}^{y}$. The off-diagonal block that couples the odd and even spans of the total spin-raising operators are marked in green. Zero coupling is represented with a "dot".
• Figure 3: The classification of the Givens-transformed grid bases, and the permuted computational bases in the q-sphere representation. Note that the Givens rotations result in symmetric and anti-symmetric combinations of pairs of symmetrically located grid basis states. The map between the transformed Hamiltonians results from a map between the corresponding blocks of the basis of representation.
• Figure 4: An outline of the mapping algorithm: The algorithm converts the Born-Oppenheimer potential surface and kinetic energy terms in a quantum-nuclear problem to a set of controllable ion-trap parameters, $\left\{ \left\{ B_i^z \right\}; \left\{ J_{ij}^x, J_{ij}^y, J_{ij}^z \right\} \right\}$, and facilitates the dynamical evolution of quantum states in an ion trap.
• Figure 5: Decomposition of a 3-qubit unitary, U, into one and two-qubit gates. The decomposition involves alternate layers of CSD (red) and VDW (blue). M-Rz(Ry) are $N$ and $N-1$ qubit multiplexed Rz(Ry) gates which can be further decomposed into a set of CNOT and Rz(Ry) gates. The ultimate layer (gray) involves the decomposition into single qubit gates.