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Unformatted text preview: arXiv:0804.3401v1 [quant-ph] 21 Apr 2008 Quantum Computational Complexity John Watrous Institute for Quantum Computing and School of Computer Science University of Waterloo, Waterloo, Ontario, Canada. Article outline I. Definition of the subject and its importance II. Introduction III. The quantum circuit model IV. Polynomial-time quantum computations V. Quantum proofs VI. Quantum interactive proof systems VII. Other selected notions in quantum complexity VIII. Future directions IX. References Glossary Quantum circuit. A quantum circuit is an acyclic network of quantum gates connected by wires: the gates represent quantum operations and the wires represent the qubits on which these operations are performed. The quantum circuit model is the most commonly studied model of quantum computation. Quantum complexity class. A quantum complexity class is a collection of computational problems that are solvable by a cho- sen quantum computational model that obeys certain resource constraints. For example, BQP is the quantum complexity class of all decision problems that can be solved in polynomial time by a quantum computer. Quantum proof. A quantum proof is a quantum state that plays the role of a witness or certificate to a quan- tum computer that runs a verification procedure. The quantum complexity class QMA is defined by this notion: it includes all decision problems whose yes-instances are efficiently verifiable by means of quantum proofs. Quantum interactive proof system. A quantum interactive proof system is an interaction between a verifier and one or more provers, involving the processing and exchange of quantum information, whereby the provers attempt to convince the verifier of the answer to some computational problem. 1 I Definition of the subject and its importance The inherent difficulty, or hardness , of computational problems is a fundamental concept in com- putational complexity theory. Hardness is typically formalized in terms of the resources required by different models of computation to solve a given problem, such as the number of steps of a deterministic Turing machine. A variety of models and resources are often considered, including deterministic, nondeterministic and probabilistic models; time and space constraints; and inter- actions among models of differing abilities. Many interesting relationships among these different models and resource constraints are known. One common feature of the most commonly studied computational models and resource con- straint is that they are physically motivated . This is quite natural, given that computers are physical devices, and to a significant extent it is their study that motivates and directs research on compu- tational complexity. The predominant example is the class of polynomial-time computable func- tions, which ultimately derives its relevance from physical considerations; for it is a mathematical abstraction of the class of functions that can be efficiently computed without error by physical...
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This note was uploaded on 08/08/2011 for the course COT 6600 taught by Professor Staff during the Fall '08 term at University of Central Florida.
- Fall '08
- Computer Science