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Recent improvements in control of quantum systems make it seem feasible to finally build a quantum computer within a decade. While it has been shown that such a quantum computer can in principle solve certain small electronic structure…
In nearly every discipline, scientific computations are limited by the cost and speed of computation. For example, the best-known exact algorithms for the canonical Traveling Salesman Problem would take centuries to run on an instance of…
While concepts and tools from Theoretical Computer Science are regularly applied to, and significantly support, software development for discrete problems, Numerical Engineering largely employs recipes and methods whose correctness and…
The power of quantum computers is still somewhat speculative. While they are certainly faster than classical ones at some tasks, the class of problems they can efficiently solve has not been mapped definitively onto known classical…
Suppose we have an unknown multipartite quantum state, how can we experimentally find out whether it is genuine multipartite entangled or not? Recall that even for a bipartite quantum state whose density matrix is known, it is already…
Conformal truncation is a powerful numerical method for solving generic strongly-coupled quantum field theories based on purely field-theoretic technics without introducing lattice regularization. We discuss possible speedups for performing…
In this paper we consider the numerical approximation of the two-phase membrane (obstacle) problem by finite difference method. First, we introduce the notion of viscosity solution for the problem and construct certain discrete nonlinear…
An isomorphism between two graphs is a bijection between their vertices that preserves the edges. We consider the problem of determining whether two finite undirected weighted graphs are isomorphic, and finding an isomorphism relating them…
Many combinatorial problems can be formulated as a polynomial optimization problem that can be solved by state-of-the-art methods in real algebraic geometry. In this paper we explain many important methods from real algebraic geometry, we…
A unified model is addressed for general optimization problems in multi-scale complex systems. Based on necessary conditions and basic principles in physics, the canonical duality-triality theory is presented in a precise way to include…
Many computer vision applications need to recover structure from imperfect measurements of the real world. The task is often solved by robustly fitting a geometric model onto noisy and outlier-contaminated data. However, recent theoretical…
The first quantum technologies to solve computational problems that are beyond the capabilities of classical computers are likely to be devices that exploit characteristics inherent to a particular physical system, to tackle a bespoke…
Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the…
Quantum computers have the potential of solving certain problems exponentially faster than classical computers. Recently, Harrow, Hassidim and Lloyd proposed a quantum algorithm for solving linear systems of equations: given an $N\times{N}$…
This paper shows that, if we could examine the entire history of a hidden variable, then we could efficiently solve problems that are believed to be intractable even for quantum computers. In particular, under any hidden-variable theory…
We consider an approach to deciding isomorphism of rigid n-vertex graphs (and related isomorphism problems) by solving a nonabelian hidden shift problem on a quantum computer using the standard method. Such an approach is arguably more…
Due to their flexibility, frames of Hilbert spaces are attractive alternatives to bases in approximation schemes for problems where identifying a basis is not straightforward or even feasible. Computing a best approximation using frames,…
We introduce a new perspective on the classical Nirenberg problem of understanding the possible Gauss curvatures of metrics on $S^{2}$ conformal to the round metric. A key tool is to employ the smooth Cheeger-Gromov compactness theorem to…
Quantum algorithms can deliver asymptotic speedups over their classical counterparts. However, there are few cases where a substantial quantum speedup has been worked out in detail for reasonably-sized problems, when compared with the best…
We present a general existence proof for a wide class of non-linear elliptic equations which can be applied to problems with barrier conditions without specifying any assumptions guaranteeing the uniqueness or local uniqueness of particular…