Category: Uncategorized
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Probabilistic Alterations
Probabilistic methods are extremely useful in combinatorial problems to prove an existence of certain configurations. The general approach is to take a finite probability space of configurations then to apply linearity of expectations to find the expected value of some random variable. This likely non-integral expected value is used to come up with configurations with…
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Cutting Edges
A cutting edge refers to an edge of a graph that uniquely connects two components of a graph \(G\). More precisely, a cutting edge \(e \in G\) must satisfy \(C(G) < C(G-e)\) where the function \(C(G)\) denotes the number of components of \(G\). Proposition. An edge is a cutting edge if and only if it…
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Why the Hamiltonian?
When I took a course in classical mechanics, I was confronted with the two mystery equations of Hamiltonian Formalism. In this post, I want to derive the two equations abiding by the subtle dependencies between quantities and the partial derivative relations. Denote the coordinates as \(q_i\) and momentum as \(p_i\). Occasionally, we will drop the…
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Grover’s Algorithm Pt 1
We have investigated methods to learn about a given mystery quantum oracle. The Deutsch-Jozsa algorithm determines whether or not the function \(f:\{0, 1\}^n \rightarrow \{0, 1\}\) is well balanced and the Bernstein-Vazirani algorithm yields the unique string that satisfies \(f(x) = 1\). The oracle presented in the Bernstein-Vazirani algorithm provides ample information about the function…
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Superdense Coding and Quantum Teleportation
When studying quantum systems, we naturally seek the connection between classical systems. A natural discourse is to seek the link between classical bits and qubits. In particular, we seek the answers for two questions. First, is it possible to transfer a classical bit using a quantum channel? Second, is it possible to copy an arbitrary…