Abstract. Eigenvalue transformations, which include solving time-dependent differential equations as a special case, have a wide range of applications in scientific and engineering computation. While quantum algorithms for singular value transformations are well studied, eigenvalue transformations are distinct, especially for nonnormal matrices. We propose an efficient quantum algorithm for performing a class of eigenvalue transformations that can be expressed as a certain type of matrix Laplace transformation. This allows us to significantly extend the recently developed linear combination of Hamiltonian simulation method [D. An, J.-P. Liu, and L. Lin, Phys. Rev. Lett., 131 (2023), 150603; D. An, A. M. Childs, and L. Lin, Commun. Math. Phys. 407, 19 (2026)] to represent a wider class of eigenvalue transformations, such as powers of the matrix inverse, [Formula: see text], and the exponential of the matrix inverse, [Formula: see text]. The latter can be interpreted as the solution of a mass-matrix differential equation of the form [Formula: see text]. We demonstrate that our eigenvalue transformation approach can solve this problem without explicitly inverting [Formula: see text], thereby reducing the computational complexity.

Subspace-based signal processing techniques, such as the Estimation of Signal Parameters via Rotational Invariant Techniques (ESPRIT) algorithm, are popular methods for spectral estimation. These algorithms can achieve the so-called super-resolution scaling under low noise conditions, surpassing the well-known Nyquist limit. However, the performance of these algorithms under high-noise conditions is not as well understood. Existing state-of-the-art analysis indicates that ESPRIT and related algorithms can be resilient even for signals where each observation is corrupted by statistically independent, mean-zero noise of size$\mathcal{O}(1)$, but these analyses only show that the error$\epsilon$decays at a slow rate$\epsilon=\widetilde{\mathcal{O}}(n^{-1/2})$with respect to the cutoff frequency$n$(i.e., the maximum frequency of the measurements). In this work, we prove that under certain assumptions, the ESPRIT algorithm can attain a significantly improved error scaling$\epsilon=\widetilde{\mathcal{O}}(n^{-3/2})$, exhibiting noisy super-resolution scaling beyond the Nyquist limit$\epsilon=\mathcal{O}(n^{-1})$given by the Nyquist-Shannon sampling theorem. We further establish a theoretical lower bound and show that this scaling is optimal. Our analysis introduces novel matrix perturbation results, which could be of independent interest.

We demonstrate that with an optimally tuned scheduling function, adiabatic quantum computing (AQC) can readily solve a quantum linear system problem (QLSP) with $\mathcal{O}(\kappa~\text{poly}(\log(\kappa/\epsilon)))$ runtime, where $\kappa$ is the condition number, and $\epsilon$ is the target accuracy. This is near optimal with respect to both $\kappa$ and $\epsilon$. Our method is applicable to general non-Hermitian matrices, and the cost as well as the number of qubits can be reduced when restricted to Hermitian matrices, and further to Hermitian positive definite matrices. The success of the time-optimal AQC implies that the quantum approximate optimization algorithm (QAOA) with an optimal control protocol can also achieve the same complexity in terms of the runtime. Numerical results indicate that QAOA can yield the lowest runtime compared to the time-optimal AQC, vanilla AQC, and the recently proposed randomization method.

For 35 years, ab initio molecular dynamics (AIMD) has been the method of choice for modeling complex atomistic phenomena from first principles. However, most AIMD applications are limited by computational cost to systems with thousands of atoms at most. We report that a machine learning based simulation protocol (Deep Potential Molecular Dynamics), while retaining ab initio accuracy, can simulate more than 1 nanosecond-long trajectory of over 100 million atoms per day, using a highly optimized code (GPU DeePMD-kit) on the Summit supercomputer. Our code can efficiently scale up to the entire Summit supercomputer, attaining 91 PFLOPS in double precision (45.5% of the peak) and 162/275 PFLOPS in mixed-single/half precision. The great accomplishment of this work is that it opens the door to simulating unprecedented size and time scales with ab initio accuracy. It also poses new challenges to the next-generation supercomputer for a better integration of machine learning and physical modeling.

Real-time time-dependent density functional theory (rt-TDDFT) with hybrid exchange-correlation functional has wide-ranging applications in chemistry and material science simulations. However, it can be thousands of times more expensive than a conventional ground state DFT simulation, and hence is limited to small systems. In this paper, we accelerate hybrid functional rt-TDDFT calculations using the parallel transport gauge formalism, and the GPU implementation on Summit. Our implementation can efficiently scale to 786 GPUs for a large system with 1536 silicon atoms, and the wall clock time is only 1.5 hours per femtosecond. This unprecedented speed enables the simulation of large systems with more than 1000 atoms using rt-TDDFT and hybrid functional.

Given a sparse matrix A, the selected inversion algorithm is an efficient method for computing certain selected elements of A-1. These selected elements correspond to all or some nonzero elements of the LU factors of A. In many ways, the types of matrix updates performed in the selected inversion algorithm are similar to those performed in the LU factorization, although the sequence of operations is different.

We describe an efficient parallel implementation of the selected inversion algorithm for distributed memory computer systems, which we call PSelInv. The PSelInv method computes selected elements of a general sparse matrix A that can be decomposed as A = LU , where L is lower triangular and U is upper triangular. The implementation described in this article focuses on the case of sparse symmetric matrices. It contains an interface that is compatible with the distributed memory parallel sparse direct factorization SuperLU_DIST. However, the underlying data structure and design of PSelInv allows it to be easily combined with other factorization routines, such as PARDISO. We discuss general parallelization strategies such as data and task distribution schemes. In particular, we describe how to exploit the concurrency exposed by the elimination tree associated with the LU factorization of A . We demonstrate the efficiency and accuracy of PSelInv by presenting several numerical experiments. In particular, we show that PSelInv can run efficiently on more than 4,000 cores for a modestly sized matrix. We also demonstrate how PSelInv can be used to accelerate large-scale electronic structure calculations.

We develop a method for improving the parallel scalability of computations that involve asynchronous task execution. We apply this method to the recently developed parallel selected inversion algorithm [Jacquelin, Lin and Yang 2014], named PSelInv, on massively parallel distributed memory machines. In the PSelInv method, we compute selected elements of the inverse of a sparse matrix A that can be decomposed as A = LU, where L is lower triangular and U is upper triangular. Computing these selected elements of A-1 requires restricted collective communications among a subset of processors within each column or row communication group created by a block cyclic distribution of L and U. We describe how this type of restricted collective communication can be implemented using asynchronous point-to-point MPI communications combined with a binary tree based data propagation scheme. Because multiple restricted collective communications may take place at the same time, we need to use a heuristic to prevent processors participating in multiple collective communications from receiving too many messages. This heuristic allows us to reduce communication load imbalance and improve the overall scalability of the selected inversion algorithm. For instance, when 6, 400 processors are used, we observe that the use of this heuristic leads to over 5x speedup for a number of test matrices. It also mitigates the performance variability introduced by an inhomogeneous network topology.

We describe an efficient implementation of an algorithm for computing selected elements of a general sparse symmetric matrix A that can be decomposed as A = LDLT , where L is lower triangular and D is diagonal. Our implementation, which is called SelInv , is built on top of an efficient supernodal left-looking LDLT factorization of A . We discuss how computational efficiency can be gained by making use of a relative index array to handle indirect addressing. We report the performance of SelInv on a collection of sparse matrices of various sizes and nonzero structures. We also demonstrate how SelInv can be used in electronic structure calculations.