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First-time meeting • • • • • Attendance Syllabus Textbook (hardcopy or electronics) Groups Emails Introduction and Overview • From “classical physics” to “quantum mechanics” “ultraviolet catastrophe” see the video: A Brief History Of Quantum Mechanics https://www.youtube.com/watch?v=B7pACq_xWyw • Quantum mechanics is a mathematical framework or set of rules for the construction of physical theories. Computer Science • Church–Turing thesis (Alan Turing & Alonzo Church, 1936) Any algorithmic process can be simulated using a Turing machine. • Electronic computers (John von Neumann) • Transistors (William Shockley) • Moore’s law (Gordon Moore) Computer power will double for constant cost roughly once every two years. (should end in 21st century, moving to a different computing paradigm) • A classical computer can simulate a quantum computer, but it is not “efficient.” • Strong Church–Turing thesis Any algorithmic process can be simulated efficiently using a Turing machine. (People have tried to find a counter example, for instance, primality) • Randomized algorithms (an ad-hoc version of strong Church–Turing thesis) Any algorithmic process can be simulated efficiently using a probabilistic TM. • David Deutsch used quantum mechanics to derive a stronger version. Any physical system can be simulated efficiently using a quantum computer. (the proof is ongoing). Computer Science (continue) • A problem/algorithm that can be solved/executed efficiently on QC, but not on TM • Deutsch algorithm • Deutsch-Jozsa algorithm • Grover’s algorithm • Shor’s algorithm • Feynman had pointed out that there seemed to be essential difficulties in simulating quantum mechanical systems on classical computers, and suggested that building computers based on the principles of quantum mechanics would allow us to avoid those difficulties. Quantum Bits (Qubits) • Classical bits vs. Quantum bits • Qbits are mathematical objects (similarly to 0/1 in classical bits) Quantum Bits (Qubits) Measurement Note แปลง U ให้ เป็ นการหมุนในแกน X, Y, Z ใน Bloch Sphere น. 20 แค่แยกตัวประกอบได้ ไม่ได้ อธิบายว่าหมุนรอบแกน x, y, z, อย่างไร? น. 176 สูตรการหมุนรอบแกน x, y, z ใน Bloch sphere ต้ องเอาที่แยกตัวประกอบได้ มาจัดเทอมแล้ วดูเอาเอง ว่ามันคือการหมุนในแกนใดบ้ าง Multiple Qubits EPR = Einstein, Podolsky, Rosen Single Qubit Gates NOT Unitary matrix Single Qubit Gates Z gate: H gate (Hadamard): Multiple Qubit Gates Reversible / Invertible Classical XOR and NAND are irreversible or non-invertible. Qubit gates are always reversible (there are always inversed matrices). The inverse of U is the adjoint of U. Universality “Any multiple qubit logic gate may be composed from CNOT and single qubit gates.” Similarly, any multiple classical logic gate may be composed from NAND gates. Measurements in bases other than the computational basis Change basis from |0>, |1> to |+>, |-> Circuit swapping two qubits Wire = the passage of time Acyclic No FANIN No FANOUT Qubit Copying Circuit? Specific case: x = |0> or |1> Input Output (สมมติว่าถ้ าก็อปปี ก้ รณีทวั่ ๆ ไปได้ จะได้ output ดังนี ้) วงจรนี ้ก็อปปี ้ |0> หรื อ |1> ได้ Input = |0> ab = 10 Input = |1> ab = 01 output = |00> output = |11> 2 เทอมนี ้จะเท่ากันก็ต่อเมื่อ ab = 0 มี 2 กรณีคือ - หนึง่ a = 1, b = 0 หรื อ input = |0> - สอง a = 0, b = 1 หรื อ input = |1> ดังนันวงจรนี ้ ้จะก็อปปี ไ้ ด้ เฉพาะ |0> หรื อ |1> เท่านัน้ Proof pp. 532 Bell States Quantum Teleportation Sent to Bob EPR pair Alice Bob Quantum Teleportation Faster-than-light communication? ไม่ว่าจะ teleport “object” หรื อ “information” ก็ไม่มีทางเร็วไปกว่าแสงได้ Classical computations on a quantum computer Write an 8x8 matrix of Toffoli gate. Simulating NAND gate and FANOUT How to produce a random number? Quantum Parallelism Deutsch’s algorithm If f(x) = 1, y will be inverted (not). ??? Show the calculation of all the 4 cases. Deutsch-Jozsa algorithm f: {0,1}n → {0,1} f(x) is constant for all values of x. f(x) is balanced, that is, equal to 1 for exactly half of all the possible x, and 0 for the other half. f(x) is either constant or balanced. Deutsch-Jozsa algorithm x = state (1 bits) x = state (n bits) inner product x = state (n bits) ใช้ สตู รทางขวา Coeff. ของ Constant = |0..0> Balanced = other states pp. 34 The Power of Quantum Computation P = solved in polynomial time NP = verified in polynomial time PSPACE = solved in polynomial space We do not know whether P != NP PSPACE is bigger than NP NP-complete Graph isomorphism Integer Factorization BQP (bounded error quantum polynomial time) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1/3 for all instances. Stern-Gerlach Experiment ใช้ classical bit เพื่อแทน spin ในแต่ ละแกน X, Y, Z ได้ หรื อไม่ ? Video: https://www.youtube.com/watch?v=rg4Fnag4V-E Stern-Gerlach Experiment Stern-Gerlach Experiment Stern-Gerlach Experiment Further Reading http://www-bcf.usc.edu/~tbrun/Course/lecture02.pdf