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An Introduction to Quantum Computation Sandy Irani Department of Computer Science University of California, Irvine
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“Simulating Physics with Computers” Richard Feynman – Keynote Talk, 1 st Conference on Physics and Computation, MIT, 1981
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“Simulating Physics with Computers” Richard Feynman – Keynote Talk, 1 st Conference on Physics and Computation, MIT, 1981 Is it possible to build computers that use the laws of quantum mechanics to compute?
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Quantum Superposition √2√2 1 + √2√2 1
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√2√2 1 - √2√2 1 -
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Implementations of a “Qubit” Energy level of an atom Spin orientation of an electron Polarization of a photon. NMR, Ion traps,…
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Information: 1 Bit Example (Schrodinger’s Cat) Classical Information: – A bit is in state 0 or state 1 Classical Information with Uncertainty – Bit is 0 with probability p 0 – Bit is 1 with probability p 1 – State (p 0, p 1 ) Quantum Information – State is a superposition over states 0 and 1 – State is ( 0, 1 ) where 0, 1 are complex. X=0 OR X=1
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Information: 1 Bit Example (Schrodinger’s Cat) Classical Information: – A bit is in state 0 or state 1 Classical Information with Uncertainty – Bit is 0 with probability p 0 – Bit is 1 with probability p 1 – State (p 0, p 1 ) Quantum Information – State is a superposition over states 0 and 1 – State is ( 0, 1 ) where 0, 1 are complex. X=1 with prob p 1 X=0 OR X=0 with prob p 0 X=1
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Information: 1 Bit Example (Schrodinger’s Cat) Classical Information: – A bit is in state 0 or state 1 Classical Information with Uncertainty – State (p 0, p 1 ) Quantum Information – State is partly 0 and partly 1 – State is ( 0, 1 ) where 0, 1 are complex. X=1 with prob p 1 X=0 OR X=0 with prob p 0 11 0 X=1
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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 2 n possibilities – (p 0, p 1, …., p 2 n -1 ) Quantum Information – State is a superposition over 2 n possibilities – ( 0, 1,…, 2 n -1 ), where a is complex X = 011
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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 2 n possibilities – (p 0, p 1, …., p 2 n -1 ) Quantum Information – State is a superposition over 2 n possibilities – ( 0, 1,…, 2 n -1 ), where a is complex p 000 p 001 p 010 p 011 p 100 p 101 p 110 p 111
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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 2 n possibilities – (p 0, p 1, …., p 2 n -1 ) Quantum Information – State is a superposition over 2 n possibilities – ( 0, 1,…, 2 n -1 ), where a is complex 000 001 010 011 100 101 110 111
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A quantum kilobyte of data (8192 qubits) Encodes 2 8192 complex numbers 2 8192 ~ 10 2466 (Number of atoms in the universe ~ 10 82)
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State of n qubits ( 0,…, 2 n -1 ) stores 2 n complex numbers: Rich in information How to use it? How to access it?
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Quantum Measurement State of n qubits ( 0,…, 2 n -1 ) If all n qubits are examined: – Outcome is 010 with probability | 010 | 2. 000 001 010 011 100 101 110 111
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Quantum Measurement State of n qubits ( 0,…, 2 n -1 ) If all n qubits are examined: – Outcome is 010 with probability | 010 | 2. – The measurement causes the state of the system to change: » The state “collapses” to 010
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Ingredients in Computation Store information about a problem to be solved Manipulate the information to solve the problem Read out an answer
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Computer Circuits AND OR AND OR AND NOT 0/1 Input Output
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Quantum Circuits U1U1 U2U2 U3U3 U4U4 U5U5 U6U6 U7U7 U8U8 M Time Input Read the output by measurement
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Interference [Image from www.thehum.info, due to Dr. Glen MacPherson]www.thehum.info
![Interference [Image from due to Dr. Glen MacPherson]](http://images.slideplayer.com./32/9850778/slides/slide_21.jpg "Interference [Image from due to Dr. Glen MacPherson]")
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[Figure from gwoptics.org]
![[Figure from gwoptics.org]](http://images.slideplayer.com./32/9850778/slides/slide_22.jpg "[Figure from gwoptics.org]")
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Quantum Gates: An Example Operation H: H Input:Output: Input:Output:
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Qubit measurement Measure: With probability Outcome = With probability Outcome =
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Qubit measurement Measure: With probability Outcome = With probability Outcome =
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Quantum Interference: An Example |0 + |1 |0 - |1
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Quantum Interference: An Example |0 + |1 |0 - |1 Apply Gate H
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Quantum Interference: An Example |0 + |1 |0 - |1 (|0 +|1 ) Apply Gate H
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Quantum Interference: An Example |0 + |1 |0 - |1 (|0 +|1 )(|0 -|1 ) + Apply Gate H
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Quantum Interference: An Example |0 + |1 |0 - |1 (|0 +|1 )(|0 -|1 ) + Result: |0 Apply Gate H
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Quantum Interference: An Example |0 + |1 |0 - |1 (|0 +|1 )(|0 -|1 ) + Result: |0 Apply Gate H Apply Gate H
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Quantum Interference: An Example |0 + |1 |0 - |1 (|0 +|1 )(|0 -|1 ) + Result: |0 Apply Gate H Apply Gate H (|0 +|1 )
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Quantum Interference: An Example |0 + |1 |0 - |1 (|0 +|1 )(|0 -|1 ) + Result: |0 Apply Gate H Apply Gate H (|0 +|1 )(|0 -|1 ) -
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Quantum Interference: An Example |0 + |1 |0 - |1 (|0 +|1 )(|0 -|1 ) + Result: |0 Apply Gate H Apply Gate H (|0 +|1 )(|0 -|1 ) - Result: |1
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Quantum Circuits U1U1 U2U2 U3U3 U4U4 U5U5 U6U6 U7U7 U8U8 M 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.
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Factoring Given a positive integer, find its prime factorization. 24 = 2 x 2 x 2 x 3 Input Output
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Factoring RSA-210 = 2452466449002782119765176635730880184 6702678767833275974341445171506160083 0038587216952208399332071549103626827 1916798640797767232430056005920356312 4656121846581790410013185929961993381 7012149335034875870551067
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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.
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Computer Science Quantum Mechanics Classical Simulation Of Quantum Systems Algorithm Design For Quantum Circuits Physical Implementation Of Quantum Computers Quantum Cryptography Quantum Information Theory Quantum Complexity Theory
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Quantum Computers for Simulation in Physics
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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.
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