Quantum Shift
Register Circuits
Mark M. Wilde
(from a company in Northern Virginia)
arXiv:0903.3894
To appear in Physical Review A
National Institute of Standards and Technology,
Wednesday, June 10, 2009
Overview
• Classical Shift Register Circuits
• Examples with Classical CNOT gate
• Quantum Shift Register Circuits
• “Memory Consumption” Theorem
• Future Work
Applications of Shift Registers
Shift Registers and Convolutional Coding techniques
have application in
cellular and
deep space communication
Viterbi Algorithm is most popular technique for determining errors
Classical Shift Registers
(D represents “delay”)
Compute
output
from memory bits
Store
input streams
stream sequentially
Mathematical Representation
Input stream is a binary sequence
Output stream is a binary sequence
Convolve input stream with system function
to get output stream:
Can also represent input stream as a polynomial
And same for output stream
Multiply input with system function
to get output polynomial:
Classical Shift Register Example
Input: 1000000000000000
Output: 1100000000000000
Input Polynomial: 1
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 Operator
Transformation
Binary Vector
Transformation
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 ways
that CNOT gates can combine
General Shift Register Circuit
General technique applies to
arbitrary quantum convolutional codes
Experimental Implementations?
Optical lattices of
neutral atoms
Spin chains for
state transfer
Linear-optical circuits
Current Directions
Extend Memory Consumption Theorem
to arbitrary quantum convolutional codes
Study the Entanglement Structure of states
that are input to a quantum shift register circuit
(Area Laws should apply here)
THANK YOU!