introduction to qis ii
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Introduction to QIS IIJames Amundson2018/2019 Fermilab Academic LecturesNovember 15, 2018
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Resources
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• There are many lectures, and even textbooks, freely available on the web• Scott Aaronson (SA)– Computer Scientist at UT Austin, formerly MIT– https://www.scottaaronson.com/blog/?p=3943
• John Preskill (JP)– Physicist at Caltech, well known in HEP– http://www.theory.caltech.edu/people/preskill/ph229/
• Michael A. Nielsen & Isaac L. Chuang (MNIC)– Textbook: Quantum Computation and Quantum Information– http://csis.pace.edu/ctappert/cs837-18spring/QC-textbook.pdf
• Quantum Simulators– Tremendously useful and interesting– Many available– Adam Lyon will walk you through one of them in the next series of lectures
Quantum Computing Resources
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Calculations with Qubits
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• Representation of a single qubit• Operations on single qubits• Operations on two qubits• Universal gate set• Classical computation on quantum computers• Quantum Fourier transform• No-cloning theorem
Overview
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• The difference between a classical bit and a qubit is that a qubit can be in a superposition of states
• Normalization condition
• Without loss of generality
• Drop the unobservable phase
Bloch Sphere (MNIC)
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• Represent our state by
• Define an operator
• Now we have
• X is the quantum NOT gate
• Like so many things in life, it all boils down to linear algebra
A Gate Operation (MNIC)
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• Z gate– which leaves |0> unchanged, and flips the sign of |1> to give −|1>
• Hadamard gate
More Single-Qubit Gate Operations (MNIC)
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• The Pauli matrices
• Hadamard, phase and T (pi/8)
Useful Set of Single-Qubit Gates (MNIC)
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• CNOT– A quantum gate with two input qubits, known as the control qubit and target qubit,
respectively
– If the control is |1>, the target is flipped. Otherwise, nothing happens.
Multi-Qubit Gates (MNIC)
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• Classical computing– {AND, OR, NOT} can calculate an arbitrary classical function
• Quantum computing– Any multiple qubit logic gate may be composed from CNOT and single qubit gates
• {CNOT, H, S}?– No! Not universal– Any circuit made of these gates can be simulated in polynomial time on a classical computer
• {CNOT, T, S}?– Yes! Universal– Many other possibilities
Universal Gate Sets (SA)
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• Classical Boolean gates such as NAND are irreversible• Single-qubit gates are reversible• The three-qubit Toffoli gate is reversible– Two control bits– One target bit can be used to simulate NAND
• Quantum computers can do anything classical computers can do
Classical Computing on Quantum Computers
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n.b.: basis vector
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• Start with the discrete Fourier transform
• quantum Fourier transform
• … which is equivalent to
– where the amplitudes yk are the discrete Fourier transform of the xks• The quantum Fourier transform can be rewritten as
Quantum Fourier Transform (MNIC)
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• Define a unitary transformation Rk:
• The following circuit performs the transform
• additional swap operations are required to restore the order of the qubits
A Quantum Fourier Transform Circuit (MNIC)
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• Conveniently qubits are upside-down compared to the general figure
Concrete example for three qubits
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• (Discrete) Fourier transform of N = 2n numbers– Classical number of steps
– Quantum number of steps
• Exponential savings
What Did We Gain? (MNIC)
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• Want to copy from to • Initial system
• Unitary cloning operator U
• Now consider a second pure state. and apply U to both
• Take inner product of two equations
• States must be either orthogonal or zero. No universal cloning operator can exist.
No-cloning Theorem (MNIC)
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