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Explore the fundamentals of quantum computation based on Richard Feynman's groundbreaking keynote talk in 1981. Delve into the concept of quantum superposition and implementations of qubits, comparing classical and quantum information representation. Understand quantum measurement, computation ingredients, circuits, gates, interference, and more for harnessing the power of quantum mechanics in computing.
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An Introduction to Quantum Computation Sandy Irani Department of Computer Science University of California, Irvine
• “Simulating Physics with Computers” Richard Feynman – Keynote Talk, 1st Conference on Physics and Computation, MIT, 1981
• “Simulating Physics with Computers” Richard Feynman – Keynote Talk, 1st Conference on Physics and Computation, MIT, 1981 Is it possible to build computers that use the laws of quantum mechanics to compute?
Quantum Superposition 1 1 + √2 √2
Quantum Superposition 1 1 - - √2 √2
Implementations of a “Qubit” • Energy level of an atom • Spin orientation of an electron • Polarization of a photon. • NMR, Ion traps,…
Information: 1 Bit Example (Schrodinger’s Cat) • Classical Information: – A bit is in state 0 or state 1 OR • Classical Information with Uncertainty – Bit is 0 with probability p0 – Bit is 1 with probability p1 – State (p0, p1) X=1 X=0 • Quantum Information – State is a superposition over states 0 and 1 – State is (a0, a1) where a0, a1 are complex.
Information: 1 Bit Example (Schrodinger’s Cat) • Classical Information: – A bit is in state 0 or state 1 OR • Classical Information with Uncertainty – Bit is 0 with probability p0 – Bit is 1 with probability p1 – State (p0, p1) X=1 X=0 X=1 X=0 with prob p1 with prob p0 • Quantum Information – State is a superposition over states 0 and 1 – State is (a0, a1) where a0, a1 are complex.
Information: 1 Bit Example (Schrodinger’s Cat) • Classical Information: – A bit is in state 0 or state 1 OR X=1 X=0 • Classical Information with Uncertainty – State (p0, p1) X=1 X=0 with prob p1 with prob p0 • Quantum Information – State is partly 0 and partly 1 a a1 + a + a0 – State is (a0, a1) where a0, a1 are complex.
Information: n Bit Example • Classical Information: – State of n bits specified by a string x in {0,1}n • Classical Information with Uncertainty – State described by probability distribution over 2npossibilities – (p0, p1, …., p2n-1) X = 011 • Quantum Information – State is a superposition over 2npossibilities – (a0, a1,…, a2n-1), where a is complex
Information: n Bit Example p000 • Classical Information: – State of n bits specified by a string x in {0,1}n p001 • Classical Information with Uncertainty – State described by probability distribution over 2npossibilities – (p0, p1, …., p2n-1) p010 p011 p100 p101 • Quantum Information – State is a superposition over 2npossibilities – (a0, a1,…, a2n-1), where a is complex p110 p111
Information: n Bit Example a a000 + a + a001 + a + a010 + a + a011 + a + a100 + a + a101 + a + a110 + a + a111 • Classical Information: – State of n bits specified by a string x in {0,1}n • Classical Information with Uncertainty – State described by probability distribution over 2npossibilities – (p0, p1, …., p2n-1) • Quantum Information – State is a superposition over 2npossibilities – (a0, a1,…, a2n-1), where a is complex
• A quantum kilobyte of data (8192 qubits) Encodes 28192complex numbers 28192~ 102466 (Number of atoms in the universe ~ 1082)
• State of n qubits (a0,…, a2n-1) stores 2n complex numbers: Rich in information How to use it? How to access it?
Quantum Measurement • State of n qubits (a0,…, a2n-1) a a000 + a + a001 + a + a010 + a + a011 + a + a100 + a + a101 + a + a110 + a + a111 • If all n qubits are examined: – Outcome is 010 with probability |a a010|2.
Quantum Measurement • State of n qubits (a0,…, a2n-1) • If all n qubits are examined: – Outcome is 010 with probability |a a010|2. – The measurement causes the state of the system to change: »The state “collapses” to 010
Ingredients in Computation • Store information about a problem to be solved • Manipulate the information to solve the problem • Read out an answer
Computer Circuits 0/1 0/1 AND NOT AND 0/1 0/1 0/1 OR OR 0/1 0/1 0/1 0/1 OR AND AND 0/1 0/1 Output Input
Quantum Circuits 0 U1 U6 U5 U2 M U4 U7 U3 U8 0 Time Input Read the output by measurement
Interference [Image from www.thehum.info, due to Dr. Glen MacPherson]
Quantum Gates: An Example H • Operation H: 1 1 0 + 0 1 Input: Output: 2 2 1 1 1 0 1 Input: Output: 2 2
Qubit measurement Measure: 1 1 1 1 + 0 1 0 1 2 2 2 2 2 1 1 2 1 1 With probability With probability 2 2 2 2 0 1 Outcome = Outcome =
Qubit measurement Measure: 1 1 1 1 + 0 1 0 1 2 2 2 2 2 1 1 2 1 1 With probability With probability 2 2 2 2 0 Outcome = 1 Outcome =
Quantum Interference: An Example |0 - |1 |0 + |1
Quantum Interference: An Example |0 - |1 |0 + |1 Apply Gate H
Quantum Interference: An Example |0 - |1 |0 + |1 Apply Gate H (|0 +|1 )
Quantum Interference: An Example |0 - |1 |0 + |1 Apply Gate H (|0 +|1 ) (|0 -|1 ) +
Quantum Interference: An Example |0 - |1 |0 + |1 Apply Gate H (|0 +|1 ) (|0 -|1 ) + Result: |0
Quantum Interference: An Example |0 - |1 |0 + |1 Apply Gate H Apply Gate H (|0 +|1 ) (|0 -|1 ) + Result: |0
Quantum Interference: An Example |0 - |1 |0 + |1 Apply Gate H Apply Gate H (|0 +|1 ) (|0 -|1 ) + Result: |0 (|0 +|1 )
Quantum Interference: An Example |0 - |1 |0 + |1 Apply Gate H Apply Gate H (|0 +|1 ) (|0 -|1 ) + - Result: |0 (|0 +|1 ) (|0 -|1 )
Quantum Interference: An Example |0 - |1 |0 + |1 Apply Gate H Apply Gate H (|0 +|1 ) (|0 -|1 ) + - Result: |0 (|0 +|1 ) (|0 -|1 ) Result: |1
Quantum Circuits 0 U1 U6 U5 U2 M U4 U7 U3 U8 0 Quantum Algorithms Manipulate data so that negative interference causes wrong answers to have small amplitude and right answers to have high amplitude, so that when we measure output, we are likely to get the right answer.
Factoring • Given a positive integer, find its prime factorization. • 24 = 2 x 2 x 2 x 3 Output Input
Factoring • RSA-210 = 2452466449002782119765176635730880184 6702678767833275974341445171506160083 0038587216952208399332071549103626827 1916798640797767232430056005920356312 4656121846581790410013185929961993381 7012149335034875870551067
Can Quantum Computers Be Built? • Key challenge: prevent decoherence (interaction with the environment). • Can factor N=15 on a quantum computer • Larger problems will require quantum error correcting codes.
Computer Science <-> Quantum Mechanics Classical Simulation Of Quantum Systems Physical Implementation Of Quantum Computers Algorithm Design For Quantum Circuits Quantum Information Theory Quantum Cryptography Quantum Complexity Theory
My Research • Design efficient algorithms on a quantum (or classical) computer that will provably compute properties of a quantum system. – For what kinds of systems is this possible? – Or: give mathematical evidence that there is no efficient way to solve this problem.
