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Quantum Computing & Algorithms
Loginov Oleg Department of Computational Physics Saint-Petersburg State University 2004
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Computational Algorithms
Contents Fundamentals Logic Qubit (short of quantum bit) Operators Multi-qubit systems Entangled states Quantum Circuits (Gates) Computational Algorithms Shor’s Algorithm Grover’s Algorithm
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Hilbert Space Inner product: Norm: Dual vector: Outer product:
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Qubit (short of quantum bit)
Computational basis State: Measurement non-deterministic collapse Two possible outputs (constraint)
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Operators Unitary: Tensor product For operators
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N-qubit quantum computer
Multi-qubit Systems 2-qubit QC: N-qubit quantum computer states
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Entangled states 2-qubit system Entangled state Example:
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Quantum Computer
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NOT Gate
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One-Qubit Hadamard Gate
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Multi-Qubit Hadamard Gate
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Control-NOT gate
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Conclusion Qubits have probabilistic nature
N-qubit register have 2^N basis functions Gates that are direct product of other gates do not produce entanglement. cNOT and one-qubit gates form a universal set of gates. In principle there is an infinite number unitary operators U.
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Quantum Algorithms Shor’s Algorithm (Factorization) Grover’s Algorithm
Wavelet Q-Search Extended Search Root Calculator Algorithm for Triangle Problem
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non-trivial factors of N
Factorization I non-trivial factors of N
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Factorization II Example: N = 21
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Shor #0 t n
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Shor #1 t n
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Shor #2 t n
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Shor #3 t n
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Discrete Fourier Transform
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QDFT
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QDFT 1
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Probability distribution
Before Q-DFT Probability distribution (1 register)
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Probability distribution
After QFDT Probability distribution (1 register)
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If x is not coprime to N, then use GCD(x, N), else - Shor
Example – Step 1 If x is not coprime to N, then use GCD(x, N), else - Shor
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Example – Step 2
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Example – Before QDFT 2-nd register:
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Example – After QDFT j Probability 0.41e-03 85 171 256 341 427
85 171 256 341 427 0.25e-03 0.39e-04 j 85 171 256 341 427 Probability contribution Prob(j)
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Continued Fraction Approximation
85 171 256 341 427
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Conclusion Atomic sizes Probabilistic character Speed
Classical machine – Quantum machine – timesteps timesteps Example: 300 digit code – 1E06 years 1000 digit code – 1E25 years several hours
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Search Task N states: Condition: The problem is identify the state
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Several iterations of Rotate Phase Operator and Diffusion Operator
Grover’s Algorithm Take a n-qubit register, where After n-dimension Hadamard Gate: Several iterations of Rotate Phase Operator and Diffusion Operator Measurement
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Rotate Phase Operator i 1 2 3 4 5 6 i 1 2 3 4 5 6
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Diffusion Operator i i 1 2 3 4 1 2 3 4
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Measurement Average amplitude: Addition in each step:
Exact calculations:
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Experimental Scheme in Optics
Z Rotate Gate
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Rotate Optical Gate
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CNOT optical gate control target
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Conclusion Advantage: Disadvantages: Speed instead of
Difficulty of assigning data
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Acknowledgement Prof. A.V. Tsiganov Prof. S.Y. Slavyanov
My mom and all my friends…
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