Quantum Shift Register Circuits Mark M. Wilde arXiv: 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 cellulardeep space communication and Viterbi Algorithm is most popular technique for determining errors Applications of Shift Registers

Classical Shift Registers Store input stream sequentially Compute output streams from memory bits (D represents “delay”)

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 : Input Polynomial: 1 Output : Output Polynomial : 1 + D

Another Example Input : Input Polynomial: 1 Output : 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 StabilizerEncoded Stabilizer Laflamme et al., Physical Review Letters 77, (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 Linear-optical circuits Spin chains for state transfer

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!