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Quantum mechanics is widely regarded as a complete theory, yet we argue it is a tractable projection of a deeper, computationally-inaccessible classical variational structure. By analyzing the coupled partial differential equations of the…
The thesis deals with Quantum Algorithms for solving Hard Constrained Optimization Problems. It shows how quantum computers can solve difficult everyday problems such as finding the best schedule for social workers or the path of a robot…
A large literature specifies conditions under which the information complexity for a sequence of numerical problems defined for dimensions $1, 2, \ldots$ grows at a moderate rate, i.e., the sequence of problems is tractable. Here, we focus…
Solving optimisation problems is a promising near-term application of quantum computers. Quantum variational algorithms leverage quantum superposition and entanglement to optimise over exponentially large solution spaces using an…
Trading fidelity for scale enables approximate classical simulators such as matrix product states (MPS) to run quantum circuits beyond exact methods. A control parameter, the so-called bond dimension $\chi$ for MPS, governs the allocated…
We study QPT (quasi-polynomial tractability) in the worst case setting for linear tensor product problems defined over Hilbert spaces. We assume that the domain space is a reproducing kernel Hilbert space so that function values are well…
Open quantum systems are ubiquitous in the physical sciences, with widespread applications in the areas of chemistry, condensed matter physics, material science, optics, and many more. Not surprisingly, there is significant interest in…
The advent of quantum computers, operating on entirely different physical principles and abstractions from those of classical digital computers, sets forth a new computing paradigm that can potentially result in game-changing efficiencies…
Estimating the ground state energy of a multiparticle system with relative error $\e$ using deterministic classical algorithms has cost that grows exponentially with the number of particles. The problem depends on a number of state…
We propose and analyze a set of variational quantum algorithms for solving quadratic unconstrained binary optimization problems where a problem consisting of $n_c$ classical variables can be implemented on $\mathcal O(\log n_c)$ number of…
The intensive pursuit for quantum advantage in terms of computational complexity has further led to a modernized crucial question: {\it When and how will quantum computers outperform classical computers?} The next milestone is undoubtedly…
Functions with singularities are notoriously difficult to approximate with conventional approximation schemes. In computational applications, they are often resolved with low-order piecewise polynomials, multilevel schemes, or other types…
While rigorous quantum dynamical simulations of many-body systems are extremely difficult (or impossible) due to the exponential scaling with dimensionality, corresponding classical simulations completely ignore quantum effects.…
General relativity and quantum mechanics are incompatible at the Planck scale. This contention can be examined if a quantum computer is set to operate at a rate that exceeds the classical limit of one operation per Planck volume-time, or…
Quantum algorithms and complexity have recently been studied not only for discrete, but also for some numerical problems. Most attention has been paid so far to the integration problem, for which a speed-up is shown by quantum computers…
We argue that the complex numbers are an irreducible object of quantum probability. This can be seen in the measurements of geometric phases that have no classical probabilistic analogue. Having complex phases as primitive ingredient…
Quasi-Monte Carlo (QMC) methods for high dimensional integrals over unit cubes and products of spheres are well-studied in literature. We study QMC tractability of integrals of functions defined over the product of $m$ copies of the simplex…
We give an overview of certain aspects of tractability analysis of multivariate problems. This paper is not intended to give a complete account of the subject, but provides an insight into how the theory works for particular types of…
Quantum computers are expected to revolutionize our ability to process information. The advancement from classical to quantum computing is a product of our advancement from classical to quantum physics -- the more our understanding of the…
Consider a multi-class labelling problem, where the labels can take values in $[k]$, and a predictor predicts a distribution over the labels. In this work, we study the following foundational question: Are there notions of multi-class…