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We derive an intuitive and novel method to represent nodes in a graph with special unitary operators, or quantum operators, which does not require parameter training and is competitive with classical methods on scoring similarity between…
We describe a quantum algorithm for preparing states that encode solutions of non-homogeneous linear partial differential equations. The algorithm is a continuous-variable version of matrix inversion: it efficiently inverts differential…
Near-term quantum devices generally suffer from shallow circuit depth and hence limited expressivity due to noise and decoherence. To address this, we propose tensor-network-assisted parametrized quantum circuits, which concatenate a…
We introduce two kinds of quantum algorithms to explore microcanonical and canonical properties of many-body systems. The first one is a hybrid quantum algorithm that, given an efficiently preparable state, computes expectation values in a…
We design quantum compression algorithms for parametric families of tensor network states. We first establish an upper bound on the amount of memory needed to store an arbitrary state from a given state family. The bound is determined by…
We establish efficient approximate counting algorithms for several natural problems in local lemma regimes. In particular, we consider the probability of intersection of events and the dimension of intersection of subspaces. Our approach is…
We introduce a variational algorithm based on the quantum alternating operator ansatz (QAOA) for the approximate solution of computationally hard counting problems. Our algorithm, dubbed VQCount, is based on the equivalence between random…
Operator learning based on neural operators has emerged as a promising paradigm for the data-driven approximation of operators, mapping between infinite-dimensional Banach spaces. Despite significant empirical progress, our theoretical…
In optimization, one of the well-known classical algorithms is power iterations. Simply stated, the algorithm recovers the dominant eigenvector of some diagonalizable matrix. Since numerous optimization problems can be formulated as an…
Matrix scaling and matrix balancing are two basic linear-algebraic problems with a wide variety of applications, such as approximating the permanent, and pre-conditioning linear systems to make them more numerically stable. We study the…
This paper studies the polynomial optimization problem whose feasible set is a union of several basic closed semialgebraic sets. We propose a unified hierarchy of Moment-SOS relaxations to solve it globally. Under some assumptions, we prove…
Without large quantum computers to empirically evaluate performance, theoretical frameworks such as the quantum statistical query (QSQ) are a primary tool to study quantum algorithms for learning classical functions and search for quantum…
Given a finite metric space $(X\cup Y, \mathbf{d})$ the $k$-median problem is to find a set of $k$ centers $C\subseteq Y$ that minimizes $\sum_{p\in X} \min_{c\in C} \mathbf{d}(p,c)$. In general metrics, the best polynomial time algorithm…
This paper introduces a Moment-Quaternion-Sum-of-Squares (QSOS) hierarchy for solving a class of quaternion polynomial optimization problems. This hierarchy is formulated directly in the quaternion domain and consists of a sequence of…
Quantum algorithms for solving noisy linear problems are reexamined, under the same assumptions taken from the existing literature. The findings of this work include on the one hand extended applicability of the quantum Fourier transform to…
Kernelization algorithms in the context of Parameterized Complexity are often based on a combination of reduction rules and combinatorial insights. We will expose in this paper a similar strategy for obtaining polynomial-time approximation…
The computational cost of exact methods for quantum simulation using classical computers grows exponentially with system size. As a consequence, these techniques can only be applied to small systems. By contrast, we demonstrate that quantum…
In a recent preprint by Deutsch et al. [1995] the authors suggest the possibility of polynomial approximability of arbitrary unitary operations on $n$ qubits by 2-qubit unitary operations. We address that comment by proving strong lower…
The treatment of two-dimensional random walks in the quarter plane leads to Markov processes which involve semi-infinite matrices having Toeplitz or block Toeplitz structure plus a low-rank correction. Finding the steady state probability…
Loading functions into quantum computers represents an essential step in several quantum algorithms, such as quantum partial differential equation solvers. Therefore, the inefficiency of this process leads to a major bottleneck for the…