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Recently, there has been an increasing interest in modelling and computation of physical systems with neural networks. Hamiltonian systems are an elegant and compact formalism in classical mechanics, where the dynamics is fully determined…
The Hofstadter Q-sequence, with its simple definition, has defied all attempts at analyzing its behavior. Defined by a simple nested recurrence and an initial condition, the sequence looks approximately linear, though with a lot of noise.…
A central goal in quantum computing is the development of quantum hardware and quantum algorithms in order to analyse challenging scientific and engineering problems. Research in quantum computation involves contributions from both physics…
Specifying a computational problem requires fixing encodings for input and output: encoding graphs as adjacency matrices, characters as integers, integers as bit strings, and vice versa. For such discrete data, the actual encoding is…
We consider an operational restatement of the holographic principle, which we call the principle of asymptotic quantum tasks. Asymptotic quantum tasks are quantum information processing tasks with inputs given and outputs required on points…
A classic result due to Schaefer (1978) classifies all constraint satisfaction problems (CSPs) over the Boolean domain as being either in $\mathsf{P}$ or $\mathsf{NP}$-hard. This paper considers a promise-problem variant of CSPs called…
We study the complexity of various fundamental counting problems that arise in the context of incomplete databases, i.e., relational databases that can contain unknown values in the form of labeled nulls. Specifically, we assume that the…
A wide range of fundamental machine learning tasks that are addressed by the maximum a posteriori estimation can be reduced to a general minimum conical hull problem. The best-known solution to tackle general minimum conical hull problems…
The aim of this paper is to introduce our idea of Holonomic Quantum Computation (Computer). Our model is based on both harmonic oscillators and non-linear quantum optics, not on spins of usual quantum computation and our method is moreover…
In this article we give several new results on the complexity of algorithms that learn Boolean functions from quantum queries and quantum examples. Hunziker et al. conjectured that for any class C of Boolean functions, the number of quantum…
A holographic algorithm solves a problem in domain of size $n$, by reducing it to counting perfect matchings in planar graphs. It may simulate a $n$-value variable by a bunch of $t$ matchgate bits, which has $2^t$ values. The transformation…
The Quantum Oracle Classification (QOC) problem is to classify a function, given only quantum black box access, into one of several classes without necessarily determining the entire function. Generally, QOC captures a very wide range of…
We consider the Functional Orientation 2-Color problem, which was introduced by Valiant in his seminal paper on holographic algorithms [SIAM J. Comput., 37(5), 2008]. For this decision problem, Valiant gave a polynomial time holographic…
Holonomic quantum computation exploits the geometric evolution of eigenspaces of a degenerate Hamiltonian to implement unitary evolution of computational states. In this work we introduce a framework for performing scalable quantum…
The complexity of graph homomorphisms has been a subject of intense study [11, 12, 4, 42, 21, 17, 6, 20]. The partition function $Z_{\mathbf A}(\cdot)$ of graph homomorphism is defined by a symmetric matrix $\mathbf A$ over $\mathbb C$. We…
There has been increasing interest in methodologies that incorporate physics priors into neural network architectures to enhance their modeling capabilities. A family of these methodologies that has gained traction are Hamiltonian neural…
A polynomial Turing compression (PTC) for a parameterized problem $L$ is a polynomial time Turing machine that has access to an oracle for a problem $L'$ such that a polynomial in the input parameter bounds each query. Meanwhile, a…
State and parameter estimation, along with fault detection, are three crucial estimation problems within the control systems community. Although different approaches have been proposed for each type of problem, the modulating function…
Constraint satisfaction problems are a central pillar of modern computational complexity theory. This survey provides an introduction to the rapidly growing field of Quantum Hamiltonian Complexity, which includes the study of quantum…
A Boolean constraint satisfaction instance is a conjunction of constraint applications, where the allowed constraints are drawn from a fixed set B of Boolean functions. We consider the problem of determining whether two given constraint…