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Physical Review A

Quantum simulation of water-molecule bond angles using an NMR quantum computer

Feng Xu, Fan Yang, Chao Wei, Xinyu Chen, Shijie Wei, Hefeng Wang, Jun Li, Tao Xin

Published Apr 15, 2024• 2 citations
Publisher

Abstract

Determining the properties of molecules, such as bond angles and bond lengths, is an important part of materials science. Quantum simulators are expected to efficiently determine them by calculating their energy spectrum. Compared with the previous single-resonant quantum eigensolver adopted in simulating water molecule [Phys. Rev. Lett. 122, 090504 (2019)], we propose a multiresonant quantum eigensolver that can search for the energy spectrum of molecules in a parallel manner. Our approach can exponentially save the cost of finding the energy spectrum by a factor of ${2}^{r}$, where $r$ is the number of ancillary qubits. As an interesting demonstration, we have designed and implemented an experiment to determine the bond angle of water molecules on a four-qubit nuclear spin quantum processor. We experimentally estimate the ground and first excitation energies and their corresponding states for effective water Hamiltonians with different H-O bond angles. The experimental results clearly show that the molecule structure with $\ensuremath{\angle}\text{H-O}\ensuremath{\sim}{100}^{\ensuremath{\circ}}$ bond angle is most stable. We believe that our approach sheds light on further applications in solving quantum chemical problems.

DOI
10.1103/physreva.109.042618

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Determining the properties of molecules, such as bond angles and bond lengths, is an important part of materials science. Quantum simulators are expected to efficiently determine them by calculating their energy spectrum. Compared with the previous single-resonant quantum eigensolver adopted in simulating water molecule [Phys. Rev. Lett. 122, 090504 (2019)], we propose a multiresonant quantum eigensolver that can search for the energy spectrum of molecules in a parallel manner. Our approach can exponentially save the cost of finding the energy spectrum by a factor of 2r{2}^{r}2r, where rrr is the number of ancillary qubits. As an interesting demonstration, we have designed and implemented an experiment to determine the bond angle of water molecules on a four-qubit nuclear spin quantum processor. We experimentally estimate the ground and first excitation energies and their corresponding states for effective water Hamiltonians with different H-O bond angles. The experimental results clearly show that the molecule structure with \ensuremath∠H-O\ensuremath∼100\ensuremath∘\ensuremath{\angle}\text{H-O}\ensuremath{\sim}{100}^{\ensuremath{\circ}}\ensuremath∠H-O\ensuremath∼100\ensuremath∘ bond angle is most stable. We believe that our approach sheds light on further applications in solving quantum chemical problems.
We compare recently proposed methods to compute the electronic state energies of the water molecule on a quantum computer. The methods include the phase estimation algorithm based on Trotter decomposition, the phase estimation algorithm based on the direct implementation of the Hamiltonian, direct measurement based on the implementation of the Hamiltonian and a specific variational quantum eigensolver, Pairwise VQE. After deriving the Hamiltonian using STO-3G basis, we first explain how each method works and then compare the simulation results in terms of gate complexity and the number of measurements for the ground state of the water molecule with different O–H bond lengths. Moreover, we present the analytical analyses of the error and the gate-complexity for each method. While the required number of qubits for each method is almost the same, the number of gates and the error vary a lot. In conclusion, among methods based on the phase estimation algorithm, the second-order direct method provides the most efficient circuit implementations in terms of the gate complexity. Moreover, Pairwise VQE serves the most practical method for near-term applications on the current available quantum computers. Finally the possibility of extending the calculation to excited states and resonances is discussed.
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It is difficult to calculate the energy levels and eigenstates of a large physical system on a classical computer because of the exponentially growing size of the Hilbert space. In this work, we experimentally demonstrate a quantum algorithm which could solve this problem via simulated resonant transitions. Using a four-qubit quantum simulator in which two qubits are used as ancillas for control and measurement, we obtain the energy spectrum of a 2-qubit low-energy effective Hamiltonian of the water molecule. The simulated transitions allow the state of the quantum simulator to transform and access large regions of the Hilbert space, including states that have no overlap with the initial state. Furthermore, we make use of this algorithm to efficiently prepare specific eigenstates on the simulator according to the measured eigenenergies.
Simulations of nuclear magnetic resonance (NMR) experiments can be an important tool for extracting information about molecular structure and optimizing experimental protocols but are often intractable on classical computers for large molecules such as proteins and for protocols such as zero-field NMR. We demonstrate the first quantum simulation of an NMR spectrum, computing the zero-field spectrum of the methyl group of acetonitrile using four qubits of a trapped-ion quantum computer. We reduce the sampling cost of the quantum simulation by an order of magnitude using compressed sensing techniques. We show how the intrinsic decoherence of NMR systems may enable the zero-field simulation of classically hard molecules on relatively near-term quantum hardware and discuss how the experimentally demonstrated quantum algorithm can be used to efficiently simulate scientifically and technologically relevant solid-state NMR experiments on more mature devices. Our work opens a practical application for quantum computation.
Standard CMOS circuits are expected to be no longer able to satisfy nowadays demand for higher performances and lower power consumption. This commits to look for new solutions. Quantum computation, whose fundamental quantity of information is the qubit, is a thoroughly new way of thinking to electronics. There are at least two main reasons for which quantum computing is expected to broaden computation horizons. First, specific algorithms can be engineered, manipulating the probability amplitudes of different states, to exploit the quantum superposition principle and achieve high parallelism. Second, it is expected to be spontaneously suitable to simulate complex quantum physical systems, as large molecules. Quantum computing requires the cooperation of several scientific fields. The leading idea has been to highlight the operations and algorithms a quantum computer must be able to perform and, then, to propose a hardware solution, providing the rigorous physical treatment mandatory for a deep understanding and for an appropriate modelling, something which is hard to find in nowadays literature. Thus, the first chapters address the quantum algorithms, focusing on Grover's search algorithm and deriving, from fundamentals, the corresponding circuit. The latter is then described in Quantum Assembly (QASM), which can be seen as the quantum equivalent of VHDL, and simulated on IBM Q Experience. The following chapters deal with a possible physical implementation of quantum processors: the encoding of qubits on nuclear spins in diamagnetic molecules. When a static magnetic field is applied to a spin-1/2 nucleus, two well-defined energy eigenstates arise, as a result of the Zeeman effect. Each of them can be associated with a qubit eigenstate. Thanks to nuclear magnetic resonance principles, the probability with which one of the two eigenstates is measured can be handled by the superposition of a radio frequency field at the resonance frequency of the nucleus. This can be interpreted as a rotation of the spin, and so of the qubit, about the RF field and enables the execution of single qubit quantum gates, since they are simple rotations of the state vector. The two-qubit CNOT gate is obtained through the interaction of spins via J-coupling. Considering that these gates constitute a universal set, every quantum algorithm can theoretically be run on an NMR processor. Different nuclei have different resonant frequencies, but the addressing of a specific spin can be achieved also in homonuclear molecules, since the resonant frequency depends also on the nuclear environment (nuclear shielding). While unwanted J-couplings can be removed thanks to refocusing, spin decoherence and relaxation phenomena force an upper limit to the allowed timescale. A main advantage of NMR is that this timescale is long, even at room temperature. Conversely, scalability is still an issue. A simple heteronuclear MATLAB model, based on the relations derived from first principles in the thesis, is proposed to prove the feasibility of NMR quantum processors. The input technological parameters, as chemical shielding and J-coupling, can be computed resorting on ORCA, or obtained from experimental data. The model, able to run a universal set of quantum gates and simple algorithms, can be useful to find an optimal operating point as a compromise between molecule physical properties and the quantities which can be controlled by nowadays NMR instrumentation.
Abstract Quantum computing leverages the quantum resources of superposition and entanglement to efficiently solve computational problems considered intractable for classical computers. Examples include calculating molecular and nuclear structure, simulating strongly interacting electron systems, and modeling aspects of material function. While substantial theoretical advances have been made in mapping these problems to quantum algorithms, there remains a large gap between the resource requirements for solving such problems and the capabilities of currently available quantum hardware. Bridging this gap will require a co-design approach, where the expression of algorithms is developed in conjunction with the hardware itself to optimize execution. Here we describe an extensible co-design framework for solving chemistry problems on a trapped-ion quantum computer and apply it to estimating the ground-state energy of the water molecule using the variational quantum eigensolver (VQE) method. The controllability of the trapped-ion quantum computer enables robust energy estimates using the prepared VQE ansatz states. The systematic and statistical errors are comparable to the chemical accuracy, which is the target threshold necessary for predicting the rates of chemical reaction dynamics, without resorting to any error mitigation techniques based on Richardson extrapolation.