Quantum Shift Register Circuits
description
Transcript of Quantum Shift Register Circuits
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Quantum ShiftRegister Circuits
Mark M. Wilde
arXiv:0903.3894
National Institute of Standards and Technology,Wednesday, June 10, 2009
To appear in Physical Review A
(from a company in Northern Virginia)
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• Classical Shift Register Circuits
Overview
• Examples with Classical CNOT gate
• Quantum Shift Register Circuits
• “Memory Consumption” Theorem
• Future Work
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Shift Registers and Convolutional Coding techniques have application in
cellular deep space communicationand
Viterbi Algorithm is most popular technique for determining errors
Applications of Shift Registers
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Classical Shift Registers
Store input stream sequentiallyCompute output streams from memory bits
(D represents “delay”)
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Mathematical RepresentationInput stream is a binary sequence
Output stream is a binary sequence
Convolve input stream with system functionto get output stream:
Can also represent input stream as a polynomial
And same for output stream
Multiply input with system functionto get output polynomial:
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Classical Shift Register Example
Input: 1000000000000000 Input Polynomial: 1
Output: 1100000000000000 Output Polynomial: 1 + D
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Another Example
Input: 1000000000000000
Input Polynomial: 1
Output: 01111111111111111
Output Polynomial: D / (1 + D)
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What is a quantum shift register?A quantum shift register circuit
acts on a set of input qubits and memory qubits,
outputs a set of output qubits and updated memory qubits,
and feeds the memory back into the device for the next cycle
(similar to the operation of a classical shift register).
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Quantum Circuit Depiction
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Lattice Depiction
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Brief Intro to Stabilizer Formalism
Unencoded Stabilizer Encoded Stabilizer
Laflamme et al., Physical Review Letters 77, 198-201 (1996).
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Binary Vector Representation
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CNOT Gate
Pauli OperatorTransformation
Binary VectorTransformation
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CNOT gate with Memory
How to describe input, output, and memory?
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Recursive Equations
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D-Transform
Input Vector
Output Vector
Transformation
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CNOT gate with more memory
Transformation
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Combo Shift Register Circuits
Is it possible to simplify?
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Simplified Shift Register Circuit
“Commute last gate through memory”
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Example of a Code
Check matrix of a CSS quantum convolutional code
Use Grassl-Roetteler algorithm to decompose as
CNOT(3,2)(1+1/D)
CNOT(1,2)(D)
CNOT(1,3)(1+D)
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Quantum Shift Register Circuit
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“CSS Shift Register Memory” Theorem
Given a description of a quantum convolutional code,
how large of a quantum memory do we need to implement?
Proof uses induction and exhaustively considers all the waysthat CNOT gates can combine
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General Shift Register Circuit
General technique applies to
arbitrary quantum convolutional codes
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Experimental Implementations?
Optical lattices of neutral atoms
Linear-optical circuits
Spin chains for state transfer
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Current Directions
Extend Memory Consumption Theoremto arbitrary quantum convolutional codes
Study the Entanglement Structure of statesthat are input to a quantum shift register circuit
(Area Laws should apply here)
THANK YOU!