We propose a quantum algorithm for inferring the molecular nuclear spin Hamiltonian from time-resolved measurements of spin-spin correlators, which can be obtained via nuclear magnetic resonance (NMR). We focus on learning the anisotropic dipolar term of the Hamiltonian, which generates dynamics that are challenging to classically simulate in some contexts. We demonstrate the ability to directly estimate the Jacobian and Hessian of the corresponding learning problem on a quantum computer, allowing us to learn the Hamiltonian parameters. We develop algorithms for performing this computation on both noisy near-term and future fault-tolerant quantum computers. We argue that the former is promising as an early beyond-classical quantum application since it only requires evolution of a local spin Hamiltonian. We investigate the example of a protein (ubiquitin) confined on a membrane as a benchmark of our method. We isolate small spin clusters, demonstrate the convergence of our learning algorithm on one such example, and then investigate the learnability of these clusters as we cross the ergodic to non-ergodic phase transition by suppressing the dipolar interaction. We see a clear correspondence between a drop in the multifractal dimension measured across many-body eigenstates of these clusters, and a transition in the structure of the Hessian of the learning cost function (from degenerate to learnable). Our hope is that such quantum computations might enable the interpretation and development of new NMR techniques for analyzing molecular structure.
We propose a quantum algorithm for inferring the molecular nuclear spin Hamiltonian from time-resolved measurements of spin-spin correlators, which can be obtained via nuclear magnetic resonance (NMR). We focus on learning the anisotropic dipolar term of the Hamiltonian, which generates dynamics that are challenging to classically simulate in some contexts. We demonstrate the ability to directly estimate the Jacobian and Hessian of the corresponding learning problem on a quantum computer, allowing us to learn the Hamiltonian parameters. We develop algorithms for performing this computation on both noisy near-term and future fault-tolerant quantum computers. We argue that the former is promising as an early beyond-classical quantum application since it only requires evolution of a local spin Hamiltonian. We investigate the example of a protein (ubiquitin) confined on a membrane as a benchmark of our method. We isolate small spin clusters, demonstrate the convergence of our learning algorithm on one such example, and then investigate the learnability of these clusters as we cross the ergodic to non-ergodic phase transition by suppressing the dipolar interaction. We see a clear correspondence between a drop in the multifractal dimension measured across many-body eigenstates of these clusters, and a transition in the structure of the Hessian of the learning cost function (from degenerate to learnable). Our hope is that such quantum computations might enable the interpretation and development of new NMR techniques for analyzing molecular structure.
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.
This research explores the integration of quantum mechanics, machine learning, and quantum computing to advance predictive modeling of nuclear magnetic resonance (NMR) parameters. The aim is to develop a hybrid quantum-enhanced machine learning model that combines the accuracy of quantum calculations with the efficiency of machine learning techniques for predicting NMR chemical shifts. The conceptual framework involves quantum mechanical calculations for accurate reference NMR parameters, supervised machine learning models trained on diverse molecular datasets, and hybrid quantum-classical algorithms to leverage quantum computing resources. A simplified numerical example demonstrates the potential of the proposed model for predicting NMR chemical shifts for small molecular systems. The results showcase the model's ability to capture underlying relationships between molecular features and NMR observables, indicating promise for larger and more complex systems. This interdisciplinary approach opens new avenues for advancing NMR spectroscopy and understanding molecular structures, dynamics, and interactions in various scientific domains. The research also discusses challenges and opportunities in integrating quantum mechanics, machine learning, and quantum computing, emphasizing the importance of diverse datasets and quantum algorithm selection. The proposed model holds significant implications for transforming NMR parameter predictions and contributing to chemistry, biochemistry, and materials science research
Quantum-information-inspired experiments in nuclear magnetic resonance spectroscopy may yield a pathway towards determining molecular structure and properties that are otherwise challenging to learn. We measure out-of-time-ordered correlators (OTOCs) [1-4] on two organic molecules suspended in a nematic liquid crystal, and investigate the utility of this data in performing structural learning tasks. We use OTOC measurements to augment molecular dynamics models, and to correct for known approximations in the underlying force fields. We demonstrate the utility of OTOCs in these models by estimating the mean ortho-meta H-H distance of toluene and the mean dihedral angle of 3',5'-dimethylbiphenyl, achieving similar accuracy and precision to independent spectroscopic measurements of both quantities. To ameliorate the apparent exponential classical cost of interpreting the above OTOC data, we simulate the molecular OTOCs on a Willow superconducting quantum processor, using AlphaEvolve-optimized [5] quantum circuits and arbitrary-angle fermionic simulation gates. We implement novel zero-noise extrapolation techniques based on the Pauli pathing model of operator dynamics [6], to repeat the learning experiments with root-mean-square error 0.05 over all circuits used. Our work highlights a computational protocol to interpret many-body echoes from nuclear magnetic systems using low resource quantum computation.
To maximize the value of fault-tolerant quantum computers, it is essential to develop concrete applications beyond well-established domains such as chemistry and condensed-matter physics. Here we construct and compile quantum algorithms to simulate the structure of atomic nuclei -- a topic that has received relatively little attention from the quantum computing community despite its similarities to the electronic structure problem in chemistry -- via effective shell-model Hamiltonians and no-core-shell-model Hamiltonians with three-body interactions derived from chiral effective field theory. Furthermore, we provide quantum resource estimates, in terms of Toffoli gate and qubit counts, for these algorithms, which, to our knowledge, are the first such estimates for fault-tolerant quantum simulation of atomic nuclei. Notably, the estimates for 32Mg and 219At shell-model Hamiltonians are comparable to recent estimates of Femoco simulations, a standard benchmark in chemistry. For no-core-shell-model Hamiltonians suitable for light nuclei (up to 40Ca or so), we find that resource requirements are significantly higher, suggesting that more bespoke strategies are required to make such simulations practicable. Throughout this work, we draw upon the similarities between nuclear and electronic structure problems, while also highlighting challenges that are specific to the former. We hope this work will spur long-term collaborations between the nuclear and quantum computing community with the ultimate goal of realizing useful nuclear simulations on quantum computers.
Advanced atomic magnetometers have made it possible to acquire nuclear magnetic resonance spectra in zero to ultralow magnetic fields. This regime carries the benefit of compact, low-cost instrumentation with reduced spin relaxation effects and the ability to probe phenomena that are inaccessible in conventional high-field experiments. A trade-off is that the resulting spectra must be interpreted using simulations that are taxing for classical computation. Working by example for small-molecule and protein spectroscopy, we demonstrate that these simulations are a promising target for fault-tolerant quantum computation. Our holistic analysis spans from input selection to the construction of explicit circuits for qubitized quantum dynamics. By maintaining parity with experimental requirements, we demonstrate how certain cases might be especially promising for early fault-tolerant architectures. Notably, valuable natural products and small proteins could be simulated with a few hundred logical qubits and less than 10 12 T gates, corresponding to days of runtime with foreseeable hardware based on spin or superconducting qubits.
This thesis is concerned with the design of numerical methods for quantum simulation and the development of improved models for quantum relaxation. Analysis is presented for the treatment of quantum systems using the density matrix formalism. This approach has been developed from the early days of quantum mechanics as a tool to describe from a statistical point of view a large number of identical quantum ensembles. Traditional methods are well established and reliable, but they perform poorly for practical simulation as the system size is scaled up. Ad hoc schemes for nuclear spin dynamics appearing in the literature can be shown to fail in certain situations. The challenge is therefore to identify efficient reduction methods for the quantum system which are also based on a rigorous foundation. The method presented in the thesis, for the time–independent Hamiltonian case, combines a quantum density matrix formalism with a procedure based on Chebyshev polynomials; application of the method to Nuclear Magnetic Resonance (NMR) spectroscopy is considered, and it is shown that the new technique outperforms existing alternatives in term of computational costs. The case of a time–dependent Hamiltonian in NMR simulation is studied as well and some splitting methods are presented. To the author’s knowledge this is the first time such methods have been applied within the NMR framework, and the numerical results show a better error–to–cost rate than traditional methods. In a separate strand of research, formulations for open quantum systems are studied and new dynamical systems approaches are considered for this problem. Motivations This thesis work is mainly focused on nuclear spin dynamics. Nuclear spin dynamics constitutes the basis for NMR, which is a very powerful spectroscopy technique that exploits the interaction between nuclear spins and magnetic fields. The same technique is used to reveal the presence of hydrogen atoms in the blood for Magnetic Resonance Imaging (MRI). Within this framework the role of simulations is extremely important, as it provides a benchmark for studies of new materials, and the development of new magnetic fields. The main computational issue is that with current software for NMR simulation it is extremely expensive to deal with systems made of more than few (7–10) spins. There is therefore a strong need to develop new algorithms capable of simulating larger systems. In recent years NMR simulations have been found to be one of the most favorable candidates for quantum computing. There are two reasons for this: nuclear quantum states maintain extremely long coherences, and it is possible to attain a very strong control on the quantum state via the application of sequences of pulses. In order to develop a proper quantum computer it is fundamental to understand how the entangled states lose coherence and relax back to equilibrium by means of external interactions. This process is described as relaxation in an open quantum system. The theory for such systems has been available for 50 years but there are still substantial limitations in the two main approaches. There are also relatively few numerical approaches for the simulation of such systems, for this reason it is important to develop numerical alternatives for the description of open quantum systems. Thesis Outline The thesis is organized as follow: the first two chapters provide background material to familiarize the reader with fundamental concepts of both quantum mechanics and nuclear spin dynamics; in this part of the thesis no new results are presented. The first chapter introduces the concept of quantum systems and the mathematical environment with which we describe those systems. We also present the main equations we need to solve to determine the dynamics of a quantum system in a statistical framework. In the second chapter we introduce the nuclear spin system, that is the physical system that has been the main reference frame in this work, for both tests and practical applications of the new algorithms. We describe how nuclear spin systems are at the basis of very important applications like NMR spectroscopy and MRI. We present in some detail the physical features of the NMR technique and the equations we need to solve to describe the dynamics of a spin system; we also focus on the relevance of numerical simulations for these systems, and consequently which must be the interest in developing new algorithms, and the major obstacles which must be overcome. In the third chapter we investigate the numerical challenges that arise in simulation of quantum systems, we describe some of the methods that have been developed in the literature, focusing on the performances and the computational costs of them, setting the new developments of this thesis in the proper research frame. We discuss one of the major issues: the evaluation of the matrix exponential. We also present the analysis we have done of a recent method called Zero Track Elimination (ZTE) that has been developed specifically for NMR simulations. This analysis shows the limitations of this method but also gives a mathematical explanation of why–and in which cases–it works. In the fourth chapter we present the main result of the thesis, the development of a new method that directly evaluates the expectation values for a quantum simulation via a different application of the well known Chebyshev expansion. We have proved that this new method can provide an excellent boost in terms of performance, with computational costs that can be reduced by a factor ten in common cases. (The results of this chapter and the new method have been presented in international conferences and recently they have been submitted for publication). We also present some attempts we have made in the application of splitting methods for the evolution of the system in a time dependent environment. To our knowledge this is the first time splitting methods have been used for NMR simulations. The results of this approach are as follows: for a particular splitting technique combined with a Lanczos iteration method it is possible to speed up the calculation by a third if compared with a Lanczos type method whilst keeping the error below a critical threshold. This last approach is still a work in progress especially in terms of developing clever ways to split the Hamiltonian. The last chapter of this thesis deals with simulation of quantum systems interacting with an external environment. After presenting the main theoretical approaches for the description of such systems we then survey several the techniques that are currently used for the numerical implementation of such theories. As a work in progress we present a considerably different new approach we have been developing aiming to overcome some of the issues that arise when treating this kind of system within usual frameworks. This is somewhat speculative work that gives rise to some new directions in the development of a numerical description for open quantum systems. We also present some numerical results. (The main core of this chapter has been presented in international conferences).