Quantum Computation of Molecular Structure Using Data from Challenging-To-Classically-Simulate Nuclear Magnetic Resonance Experiments

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 nonergodic 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.

Popular Summary

Simulating a physical system solves a “forward problem”: given the system parameters, predict an experiment’s outcome. A related question is the “inverse problem”: given an experiment’s outcome, infer the generating system’s parameters. This is a critically important task across science, for example, to determine the product of a chemical reaction or the structure of an unknown substance. This task may be difficult when the system in question is governed by the laws of quantum mechanics and strongly correlated because of the exponential growth of the Hilbert space dimension. However, quantum computers do not suffer from this growth, as their computational state explores an identical Hilbert space.

In this work, we propose quantum-assisted Hamiltonian learning as a potential quantum computing application whenever the system to be learned has strong quantum coherence and is intractable to study classically. We find quantum algorithms for near-term and future fault-tolerant devices using the inference of molecular structure from time-resolved NMR data as an example. We find that a goldilocks region exists in the system parameter space (with deep ties to the localization transition in strongly correlated condensed matter physics): too much chaos makes the system unlearnable, too little makes it classically easy.

This work opens up a realm of potential quantum computing applications, as it applies to any physical system governed by a quantum Hamiltonian that is beyond classical to simulate. The logical followup to this work will be to find more example applications, improve the optimization and circuit methods used, and implement it on a real-world device.

Figure 1

Sketch of the “phases of learnability” of a quantum Hamiltonian. In the red region, the set of experimental data is insufficient to distinguish candidate Hamiltonians. This becomes increasingly likely for systems beyond a localization transition (red dashed line), as after this point experiments such as local correlators tend to provide little information about the system structure. In the blue region, the experimental data is sufficient to learn the system’s structure, but the data processing may be achieved classically, rendering a quantum computer unnecessary. This classical processing may either be achieved by the system being well approximated by a classically computable model, or by the experiment having sufficient control (gray dashed line) to isolate smaller subsystems. At the limit of good control, an experiment has the ability to dynamically decouple individual terms (blue dashed line), after which techniques such as those in Refs. [46, 48] can be used to estimate terms individually. A simpler limit is to simply have the ability to address individual (or small frequency regions of) spins in the system (green dashed line), which renders the system more learnable.

Figure 2

Circuits to estimate the signal ${\overline{S}}_x(t)$ (top) and the integrands ${\overline{j}}_x^n(t,s)$ (middle) and ${\overline{k}}_x^{n,m}(t,s,r)$ (bottom) that are required to calculate the first and second derivatives of our cost function $C[H]$ [Eq. (4)]. For ease of viewing, we suppress labels in the circuits themselves (see the legend for clarification). Circuits assume access to a preparation of $\rho$, and a means to simulate $U_x(t,s)$ (without control) and to implement controlled perturbations $V_n$. The desired integrand can be found to be the expectation value of the product of the indicated operators, which (in NISQ) must be read out by repeated preparation and measurement.

Figure 3

Circuit diagrams of the fault-tolerant oracles ${\mathrm{SEL}}_0$, ${\mathrm{SEL}}_1$, ${\mathrm{PREP}}_0$, and ${\mathrm{PREP}}_1$ described in this text. See the text for details. Black circles on multiqubit registers denote complex control procedures. Square boxes denote classical input to the system via quantum read-only memory (QROM) and coherent alias sampling (CAS) [15]. Subscripts are omitted from gates for ease of reading. The dashed circles on the control for $U$ in the ${\mathrm{SEL}}_a$ circuits indicate control that is only needed if the time evolution during an experiment changes between experiments.

Figure 4

Finding spin clusters within the ubiquitin protein using data from Ref. [126]. Left: a 466-spin cluster connected at a coupling of $10\mathrm{KHz}$ (the total protein has 692 H atoms). Middle top: a 238-spin subset of this cluster, connected at a coupling of $12\mathrm{KHz}$. Middle bottom: a smaller 60-spin subset of this cluster, connected at a coupling of $14\mathrm{KHz}$. Black lines and red dashed lines between spins indicate strong and medium-strength couplings within the cluster. Red dashed region in all three clusters is an eight-spin subcluster studied in this text. Right top: histogram of couplings within the 60-spin cluster and to the environment. Right bottom: an enlarged view of the histogram tail to show the distribution of the dominant couplings.

Figure 5

Learning small couplings of a six-spin cluster within ubiquitin given a fixed backbone from a noisy dataset with standard deviation ${10}^{-3}$. The system is learned from a dataset of 21 points equally spaced between 0 and 2 ms (assuming a 23.5-T background magnetic field, and a dipolar suppression of $\alpha =10$). Absolute error in individual parameters (blue faint lines) and the mean absolute error (black line) are plotted at each iteration of the conjugate gradient algorithm.

Figure 6

Plot of the multifractal dimension of small clusters of spins in the ubiquitin molecule as the dipolar term is suppressed in a background 23.5-T field. Points in the main plot are extracted from a linear fit of the mean participation entropy of the different clusters as a function of the system size, as demonstrated in the two insets for suppression factors 2 and 100. From these fits, the multifractal dimension can be estimated. The coloring of the plot indicates the expected quantum and classical learnability of the system as the dipolar term is suppressed, corresponding to the different regions in Fig. 1.

Figure 7

Eigenvalue and eigenvector participation data for Hessians of the Hamiltonian learning problem for small spin clusters in ubiquitin. [Data are taken as the set of $⟨Z_i(t)Z_j(0)⟩$ and $⟨X_i(t)X_j(0)⟩$ correlators using a set of equally spaced times between $t=0$ and $t=5$ ms in the absence of sampling noise, implying that units of the Hessian eigenvalues are arbitrary.] Top: typical (geometric mean) largest Hessian eigenvalue for different system sizes as the dipolar term is suppressed. Inset shows the typical Hessian spectrum for each suppression factor over clusters of eight spins—the light green line in the main plot is taken from the indicated cut through the inset. Bottom: typical (geometric mean) eigenvector participation [Eq. (55)] for the same dataset. Inset shows the participation across the entire Hessian spectrum for clusters of eight spins; each datapoint in the light green line in the main plot corresponds to a geometric mean over a single line in the inset.