The procedure for simulating the nuclear magnetic resonance spectrum linked to the spin system of a molecule for a certain nucleus entails diagonalizing the associated Hamiltonian matrix. As the dimensions of said matrix grow exponentially with respect to the spin system's atom count, the calculation of the eigenvalues and eigenvectors marks the performance of the overall process. The aim of this paper is to provide a formalization of the complete procedure of the simulation of a spin system's NMR spectrum while also explaining how to diagonalize the Hamiltonian matrix with a quantum computer, thus enhancing the overall process's performance. Two well-known quantum algorithms for calculating the eigenvalues of a matrix are analyzed and put to the test in this context: quantum phase estimation and the variational quantum eigensolver. Additionally, we present simulated results for the later approach while also addressing the hypothetical noise found in a physical quantum computer.
The procedure for simulating the nuclear magnetic resonance spectrum linked to the spin system of a molecule for a certain nucleus entails diagonalizing the associated Hamiltonian matrix. As the dimensions of said matrix grow exponentially with respect to the spin system's atom count, the calculation of the eigenvalues and eigenvectors marks the performance of the overall process. The aim of this paper is to provide a formalization of the complete procedure of the simulation of a spin system's NMR spectrum while also explaining how to diagonalize the Hamiltonian matrix with a quantum computer, thus enhancing the overall process's performance. Two well-known quantum algorithms for calculating the eigenvalues of a matrix are analyzed and put to the test in this context: quantum phase estimation and the variational quantum eigensolver. Additionally, we present simulated results for the later approach while also addressing the hypothetical noise found in a physical quantum computer.
Quantum computing opens up new possibilities for the simulation of many-body nuclear systems. As the number of particles in a many-body system increases, the size of the space if the associated Hamiltonian increases exponentially. This presents a challenge when performing calculations on large systems when using classical computing methods. By using a quantum computer, one may be able to overcome this difficulty thanks to the exponential way the Hilbert space of a quantum computer grows with the number of quantum bits (qubits). Our aim is to develop quantum computing algorithms which can reproduce and predict nuclear structure such as level schemes and level densities. As a sample Hamiltonian, we use the Lipkin-Meshkov-Glick model. We use an efficient encoding of the Hamiltonian onto many-qubit systems, and have developed an algorithm allowing the full excitation spectrum of a nucleus to be determined with a variational algorithm capable of implementation on today's quantum computers with a limited number of qubits. Our algorithm uses the variance of the Hamiltonian, ⟨H2⟩−⟨H⟩2, as a cost function for the widely-used variational quantum eigensolver (VQE). In this work we present a variance based method of finding the excited state spectrum of a small nuclear system using a quantum computer, using a reduced-qubit encoding method.
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.
The simulation of the spectra measured in nuclear magnetic resonance (NMR) spectroscopy experiments is a computationally non-trivial problem which, due to its natural interpretation as a quantum spin problem, maps in a straightforward way to a quantum computer. As such, it represents a problem for which such a device may provide some practical advantage over traditional computing methods. In order to understand the extent to which such problems may indeed provide examples of useful quantum advantage, it is important to understand the limitations of existing classical simulation methods. In this work, we benchmark our own classical solver designed to study such problems. This solver uses a clustering approximation to achieve a resource scaling which is linear in the total number of nuclear spins in a given molecule, for a fixed cluster size. The success of such an approximation would present a stark repudiation to the common claim that such problems require an exponential scaling of resources, the very claim which makes simulating an NMR spectra a candidate for quantum advantage. Our benchmarking results indicate that our approximation performs well throughout, and even somewhat beyond, the more typical experimental regimes. We discuss what implications this may have for future efforts to demonstrate quantum advantage in the context of NMR.
Quantum ground-state problems are computationally hard problems for general many-body Hamiltonians; there is no classical or quantum algorithm known to be able to solve them efficiently. Nevertheless, if a trial wavefunction approximating the ground state is available, as often happens for many problems in physics and chemistry, a quantum computer could employ this trial wavefunction to project the ground state by means of the phase estimation algorithm (PEA). We performed an experimental realization of this idea by implementing a variational-wavefunction approach to solve the ground-state problem of the Heisenberg spin model with an NMR quantum simulator. Our iterative phase estimation procedure yields a high accuracy for the eigenenergies (to the 10⁻⁵ decimal digit). The ground-state fidelity was distilled to be more than 80%, and the singlet-to-triplet switching near the critical field is reliably captured. This result shows that quantum simulators can better leverage classical trial wave functions than classical computers.
The Variational Quantum Algorithms (VQAs) are hybrid quantum-classical algorithms and they can be used in the Nosiy Intermadiate Scale Quantum (NISQ) devises. The Variational Quantum Eigensolver (VQE) was suggested as a first VQA. VQE is based on the variational method of quantum mechanics and it is used to find the ground state energy of a quantum system. In this study, VQE is used for the analysis of NMR spectra for the AB and AB2 spin systems. The frequencies and the spin coupling values are obtained from the sample spectra for these spin systems. Then the Hamiltonians are written in terms of pauli spin operators and transformed into a suitable forms for quantum computer. By employing VQE the ground state energies are obtained for the related spin systems. They are found to be in good agreement with the results obtained from the known variation method.
Simulating nuclear magnetic resonance (NMR) spectra is a promising application of quantum simulation. Using Quantinuum's System Model H2, a trapped-ion quantum computer, we demonstrate an end-to-end, large-scale digital NMR simulation of a classically challenging benchmark molecule, 1,2-di-tert-butyl-diphosphane. We implement a hardware-efficient reduction of the nuclear-spin Hamiltonian, enabling Trotterized real-time evolution of an effective 21-spin model with tailored error suppression to reduce the effects of device noise. The reconstructed liquid-state proton NMR spectrum agrees with classical reference calculations and reproduces key spectroscopic features that previous quantum hardware demonstrations did not capture. Given the widespread use of NMR in chemical analysis and industrial research, these results advance digital quantum simulation of NMR spectra towards practical quantum utility.