Quantum Shift Register Circuits
description
Transcript of Quantum Shift Register Circuits
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)
• Classical Shift Register Circuits
Overview
• Examples with Classical CNOT gate
• Quantum Shift Register Circuits
• “Memory Consumption” Theorem
• Future Work
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
Classical Shift Registers
Store input stream sequentiallyCompute output streams from memory bits
(D represents “delay”)
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:
Classical Shift Register Example
Input: 1000000000000000 Input Polynomial: 1
Output: 1100000000000000 Output Polynomial: 1 + D
Another Example
Input: 1000000000000000
Input Polynomial: 1
Output: 01111111111111111
Output Polynomial: D / (1 + D)
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).
Quantum Circuit Depiction
Lattice Depiction
Brief Intro to Stabilizer Formalism
Unencoded Stabilizer Encoded Stabilizer
Laflamme et al., Physical Review Letters 77, 198-201 (1996).
Binary Vector Representation
CNOT Gate
Pauli OperatorTransformation
Binary VectorTransformation
CNOT gate with Memory
How to describe input, output, and memory?
Recursive Equations
D-Transform
Input Vector
Output Vector
Transformation
CNOT gate with more memory
Transformation
Combo Shift Register Circuits
Is it possible to simplify?
Simplified Shift Register Circuit
“Commute last gate through memory”
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)
Quantum Shift Register Circuit
“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
General Shift Register Circuit
General technique applies to
arbitrary quantum convolutional codes
Experimental Implementations?
Optical lattices of neutral atoms
Linear-optical circuits
Spin chains for state transfer
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!