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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