By Michel Le Bellac
Quantum info and computation is a swiftly increasing and cross-disciplinary topic. This publication offers a self-contained creation to the sector for physicists, mathematicians and laptop scientists who need to know extra approximately this intriguing topic. After a step by step creation to the quantum bit (qubit) and its major houses, the writer provides the required historical past in quantum mechanics. The middle of the topic, quantum computation, is illustrated via a close therapy of 3 quantum algorithms: Deutsch, Grover and Shor. the ultimate chapters are dedicated to the actual implementation of quantum desktops, together with the latest facets, comparable to superconducting qubits and quantum dots, and to a quick account of quantum details. Written at a degree appropriate for undergraduates in actual sciences, no earlier wisdom of quantum mechanics is believed, and in simple terms easy notions of physics are required. The ebook contains many brief routines, with recommendations on hand to teachers via [email protected]
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Additional resources for A Short Introduction to Quantum Information and Quantum Computation
An NNF is decomposable (called a DNNF) iff each of its conjunctions is decomposable; see Figure 1(b). We use vars(N ) to denote the set of variables mentioned by an NNF node N . Figure 1(a) depicts an example vtree. Given an internal node v in a vtree for variables Z, we use v l and v r to refer to its left and right children, use vars(v) to denote the set of variables at or below v in the tree. A DNNF respects a vtree iff every and-node has exactly two children N l and N r , and we have vars(N l ) ⊆ vars(v l ) and vars(N r ) ⊆ vars(v r ) for some vtree node v.
We then have: f (Z) = g(X) ∧ h(Y) ∧ e(X, Y). Moreover, the (X, Y) interaction function is given by: fXY = e ∨ ¬(h ∧ ∃X(g ∧ e)) ∨ ¬(g ∧ ∃Y(h ∧ e)). This proposition spells out precisely the difference between cutset clauses and interaction functions. 4 Hence, while current decomposition methods consider only the cutset clauses e, one also needs to account for the components ¬(h ∧ ∃X(g ∧ e)) and ¬(g ∧ ∃Y(h ∧ e)), which may lead to a decomposition whose size is exponentially smaller. Consider the following CNF for an example: f (Z) = X2 ∧ X3 ∧ .
We adopt the following decompositions: For f o (Z) : f o (X) ∧ f e (Y), f e (X) ∧ f o (Y) For f e (Z) : f e (X) ∧ f e (Y), f o (X) ∧ f o (Y) Since these decompositions utilize only the odd and even parity functions, they can be applied recursively. As a result, we have that, at each vtree node v, (i) only two distinct functions (f o , f e over vars(v)) are needed and (ii) the size of any decomposition is exactly two. Therefore, by Theorem 2, for every vtree node v, Algorithm 1 only constructs a constant number of DNNF nodes.