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!