Related papers: From Hilbert's Tenth Problem to Quantum Speedup: E…
Solving a quadratic nonlinear system of equations (QNSE) is a fundamental, but important, task in nonlinear science. We propose an efficient quantum algorithm for solving $n$-dimensional QNSE. Our algorithm embeds QNSE into a…
We present two quantum interior point methods for semidefinite optimization problems, building on recent advances in quantum linear system algorithms. The first scheme, more similar to a classical solution algorithm, computes an inexact…
We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Hoyer, and Tapp, and imply an O(N^{3/4} log N) quantum upper…
This paper generalizes recent advances on quadratic manifold (QM) dimensionality reduction by developing kernel methods-based nonlinear-augmentation dimensionality reduction. QMs, and more generally feature map-based nonlinear corrections,…
To date, research in quantum computation promises potential for outperforming classical heuristics in combinatorial optimization. However, when aiming at provable optimality, one has to rely on classical exact methods like integer…
Grover's algorithm is a primary algorithm offered as evidence that quantum computers can provide an advantage over classical computers. It involves an "oracle" specified for a given application whose structure is not part of the formal…
This paper considers the quantum query complexity of {\it $\eps$-biased oracles} that return the correct value with probability only $1/2 + \eps$. In particular, we show a quantum algorithm to compute $N$-bit OR functions with…
We give a quantum speedup for solving the canonical semidefinite programming relaxation for binary quadratic optimization. This class of relaxations for combinatorial optimization has so far eluded quantum speedups. Our methods combine…
We investigate infinitary wellfounded systems for linear logic with fixed points, with transfinite branching rules indexed by some closure ordinal $\alpha$ for fixed points. Our main result is that provability in the system for some…
Continuous search problems (CSPs), which involve finding solutions within a continuous domain, frequently arise in fields such as optimization, physics, and engineering. Unlike discrete search problems, CSPs require navigating an…
We describe a quantum algorithm based on an interior point method for solving a linear program with $n$ inequality constraints on $d$ variables. The algorithm explicitly returns a feasible solution that is $\varepsilon$-close to optimal,…
Building on the blueprint from Goemans and Williamson (1995) for the Max-Cut problem, we construct a polynomial-time approximation algorithm for orthogonally constrained quadratic optimization problems. First, we derive a semidefinite…
Coherent quantum rollout for sequential decision problems requires a unitary simulator: randomness must live in explicit quantum registers, and basis-state selectors must be mapped to actions reversibly. With branch-dependent valid actions,…
In this paper, we give quantum algorithms for two fundamental computation problems: solving polynomial systems over finite fields and optimization where the arguments of the objective function and constraints take values from a finite field…
The mean of a random variable can be understood as a linear functional on the space of probability distributions. Quantum computing is known to provide a quadratic speedup over classical Monte Carlo methods for mean estimation. In this…
Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and…
Nonlinear boolean equation systems play an important role in a wide range of applications. Grover's algorithm is one of the best-known quantum search algorithms in solving the nonlinear boolean equation system on quantum computers. In this…
We propose a novel optimization scheme designed to find optimally correctable subspace codes for a known quantum noise channel. To each candidate subspace code we first associate a universal recovery map, as if the code was perfectly…
We develop a pivot-shifted Carleman linearization framework for quantum algorithms solving quadratic nonlinear ordinary differential equations. By shifting the dynamics by a pivot state prior to Carleman lifting, and combining this with a…
The hypercomputers compute functions or numbers, or more generally solve problems or carry out tasks, that cannot be computed or solved by a Turing machine. Several numerical simulations of a possible hypercomputational algorithm based on…