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Phase in Quantum Computing

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Main concepts of computing illustrated with simple examples

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Quantum Theory Made Easy 0 1 Classical p0p0 p1p1 probabilities Quantum a0a0 a1a1 0 1 amplitudes p 0 +p 1 =1|a 0 | 2 +|a 1 | 2 =1 bit qubit p i is a real numbera i is a complex number Prob(i)=p i Prob(i)=|a i | 2

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Quantum Theory Made Easy Classical Evolution 00 01 10 11 00 01 10 11 Quantum Evolution stochastic matrix 00 01 10 11 00 01 10 11 transition probabilitiestransition amplitudes unitary matrix

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Interference 0 1 0 1 measure 0 50% 1 measure 0 50% 1 100% 0% 100% 0% qubit input

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Interfering Pathways 100% H 50% 50% C 50% H 10% 90% 20% 80% 15% H 85% C 1.0 H 0.707 0.707 C 0.707 H 0.707 -0.707 0.707 0.0 H 1.0 C Always addition!Subtraction! Classical Quantum

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Superposition Qubits a0a0 a1a1 0 1 amplitudes a i is a complex number 1 √2√2 ( | + | ) Schrödinger’s Cat

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Classical versus quantum computers

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Some differences between classical and quantum computers superposition Hidden properties of oracles

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Randomised Classical Computation versus Quantum Computation Deterministic Turing machine Probabilistic Turing machine

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Probabilities of reaching states

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Formulas for reaching states

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Relative phase, destructive and constructive inferences Destructive interference Constructive interference

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Most quantum algorithms can be viewed as big interferometry experiments Equivalent circuits

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The “eigenvalue kick-back” concept

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There are also some other ways to introduce a relative phase

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The “eigenvalue kick-back” concept Now we know that the eigenvalue is the same as relative phase

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The “eigenvalue kick- back” concept illustrated for DEUTSCH

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The “shift operation” as a generalization to Deutsch’s Tricks

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Change of controlled gate in Deutsch with Controlled-Ushift gate

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Now we deal with new types of eigenvalues and eigenvectors

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The general concept of the answer encoded in phase

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Shift operator allows to solve Deutsch’s problem with certainty

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Controlling amplitude versus controlling phase

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Exercise for students

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Dave Bacon Lawrence Ioannou Sources used

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