Related papers: Optimal measurements for the dihedral hidden subgr…
This paper studies the quantum computational complexity of the discrete logarithm (DL) and related group-theoretic problems in the context of generic algorithms -- that is, algorithms that do not exploit any properties of the group…
It is known that unambiguous discrimination among non-orthogonal but linearly independent quantum states is possible with a certain probability of success. Here, we consider a variant of that problem. Instead of discriminating among all of…
We study the convergence rate of the optimal quantization for a probability measure sequence $(\mu_{n})_{n\in\mathbb{N}^{*}}$ on $\mathbb{R}^{d}$ converging in the Wasserstein distance in two aspects: the first one is the convergence rate…
We consider the problem of quantum state certification, where we are given the description of a mixed state $\sigma \in \mathbb{C}^{d \times d}$, $n$ copies of a mixed state $\rho \in \mathbb{C}^{d \times d}$, and $\varepsilon > 0$, and we…
We describe a group theoretic analysis of Shor's algorithm and other related hidden subgroup problems in mathematics and relate these to symmetries of molecular and condensed phase assemblies. By recasting Shor's algorithm through the lens…
Fidelity is the standard measure for quantifying the similarity between two quantum states. It is equal to the square of the minimum Bhattacharyya coefficient between the probability distributions induced by quantum measurements on the two…
We present the view of quantum algorithms as a search-theoretic problem. We show that the Fourier transform, used to solve the Abelian hidden subgroup problem, is an example of an efficient elimination observable which eliminates a constant…
We give an algorithm for the hidden subgroup problem for the dihedral group $D_N$, or equivalently the cyclic hidden shift problem, that supersedes our first algorithm and is suggested by Regev's algorithm. It runs in $\exp(O(\sqrt{\log…
Quantum process tomography, the task of estimating an unknown quantum channel, is a central problem in quantum information theory. A long-standing open question is to determine the optimal number of uses of an unknown channel required to…
A broad class of hybrid quantum-classical algorithms known as "variational algorithms" have been proposed in the context of quantum simulation, machine learning, and combinatorial optimization as a means of potentially achieving a quantum…
The hidden subgroup problem (HSP) provides a unified framework to study problems of group-theoretical nature in quantum computing such as order finding and the discrete logarithm problem. While it is known that Fourier sampling provides an…
Random walk algorithms are crucial for sampling and approximation problems in statistical physics and theoretical computer science. The mixing property is necessary for Markov chains to approach stationary distributions and is facilitated…
We consider the problem of discrimination between two pure quantum states. It is well known that the optimal measurement under both the error-probability and log-loss criteria is a projection, while under an ``erasure-distortion'' criterion…
It has been known for some time that graph isomorphism reduces to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum computation are obtained by solving instances of the HSP. A common feature of the…
Identification of nonorthogonal quantum states without error is crucial for various applications in quantum information technology, as well as the foundations of quantum physics. Theoretical studies have proposed measurements that maximize…
We formulate and analyze a stochastic threshold group testing problem motivated by biological applications. Here a set of $n$ items contains a subset of $d \ll n$ defective items. Subsets (pools) of the $n$ items are tested -- the test…
Graph clustering is the problem of identifying sparsely connected dense subgraphs (clusters) in a given graph. Proposed clustering algorithms usually optimize various fitness functions that measure the quality of a cluster within the graph.…
Optimal quantization for mixed distributions has emerged as a compelling area of study. In this work, we have focused on a mixed distribution formed from two uniform distributions with partially overlapping supports. For this class of…
We consider the quantitative group testing problem where the objective is to identify defective items in a given population based on results of tests performed on subsets of the population. Under the quantitative group testing model, the…
This thesis aims to establish notions of symmetry for quantum states and channels as well as describe algorithms to test for these properties on quantum computers. Ideally, the work will serve as a self-contained overview of the subject. We…