quantum-notes

quantum-notes - Quantum Computing - Lecture Notes Mark...

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Quantum Computing - Lecture Notes Mark Oskin Department of Computer Science and Engineering University of Washington Abstract The following lecture notes are based on the book Quantum Computation and Quantum In- formation by Michael A. Nielsen and Isaac L. Chuang. They are for a math-based quantum computing course that I teach here at the University of Washington to computer science grad- uate students (with advanced undergraduates admitted upon request). These notes start with a brief linear algebra review and proceed quickly to cover everything from quantum algorithms to error correction techniques. The material takes approximately 16 hours of lecture time to present. As a service to educators, the L A T E Xand Xfig source to these notes is available online from my home page: http://www.cs.washington.edu/homes/oskin . In addition, under the section “course material” from my web page, in the spring quarter/2002 590mo class you will find a sequence of homework assignments geared to computer scientists. Please feel free to adapt these notes and assignments to whatever classes your may be teaching. Corrections and expanded material are welcome; please send them by email to [email protected] . The following work is supported in part by NSF CAREER Award ACR-0133188. 1
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Contents 1 Linear Algebra (short review) 4 2 Postulates of Quantum Mechanics 5 2.1 Postulate 1: A quantum bit . . ............................ 5 2.2 Postulate 2: Evolution of quantum systems . ..................... 6 2.3 Postulate 3: Measurement . . . ............................ 7 2.4 Postulate 4: Multi-qubit systems. .......................... 8 3 Entanglement 9 4 Teleportation 11 5 Super-dense Coding 15 6 Deutsch’s Algorithm 16 6.1 Deutsch-Jozsa Algorithm . 2 0 7 Bloch Sphere 22 7.1 Phase traveling backwards through control operations. ............... 2 7 7.2 Phaseflips versus bitflips . 2 8 8 Universal Quantum Gates 29 8.1 More than two qubit controlled operations . 3 1 8.2 Other interesting gates . . . . ............................ 3 1 8 . 3 Sw a p ......................................... 3 2 2
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9 Shor’s Algorithm 33 9.1 Factoring and order-finding . . ............................ 3 3 9.2 Quantum Fourier Transform (QFT) . . . . . ..................... 3 4 9.3 Shor’s Algorithm – the easy way. .......................... 3 8 9.4 Phase estimation ................................... 3 9 9.5 Shor’s Algorithm – Phase estimation method . ................... 4 0 9.6 Continuous fraction expansion . ........................... 4 2 9.7 Modular Exponentiation . . . ............................ 4 2 10 Grover’s Algorithm 43 11 Error Correction 46 11.1 Shor’s 3 qubit bit-flip code . 4 7 11.2 Protecting phase . 5 0 11.3 7 Qubit Steane code. ................................. 5 1 11.4 Recursive error correction and the threshold theorem . ............... 5 3 3
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1 Linear Algebra (short review) The following linear algebra terms will be used throughout these notes. Z - complex conjugate if Z a b i then Z a b i ψ - vector, “ket” i.e. c 1 c 2 c n ψ - vector, “bra” i.e. c 1 c 2 c n ϕ ψ - inner product between vectors ϕ and ψ . Note for QC this is on n space not n ! Note ϕ ψ ψ ϕ Example: ϕ 2 6 i , ψ 3 4 ϕ ψ 2 6 i 3 4 6 24 i ϕ ψ - tensor product of ϕ and ψ . Also written as ϕ ψ Example: ϕ ψ 2 6 i 3 4 2 3 2 4 6 i 3 6 i 4 6 8 18 i 24 i A - complex conjugate of matrix A .
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This note was uploaded on 12/27/2011 for the course CMPSC 290h taught by Professor Chong during the Fall '09 term at UCSB.

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quantum-notes - Quantum Computing - Lecture Notes Mark...

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