Quantum control: A new frontier for control theory Hendra I. Nurdin Australian National University TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A

Outline of talk Motivating developments Quantum technologies Overview of quantum mechanics Quantum control strategies Linear quantum stochastic systems Concluding remarks The future

Motivating developments

Quantum technologies

Overview of quantum mechanics

Quantum control strategies

Linear quantum stochastic systems

Concluding remarks

The future

Motivating developments R. Feynman (1982): Quantum computers for simulating quantum physics D. Deutsch (1985): Theoretical model for quantum computing C. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters (1993): Quantum state teleportation P. Shor (1994): Quantum algorithm for integer factorization L. Grover (1996): Quantum algorithm for database search Quantum Information Science

R. Feynman (1982): Quantum computers for simulating quantum physics

D. Deutsch (1985): Theoretical model for quantum computing

C. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters (1993): Quantum state teleportation

P. Shor (1994): Quantum algorithm for integer factorization

L. Grover (1996): Quantum algorithm for database search

Quantum technologies Quantum computers (The Holy Grail) Quantum communication Quantum metrology Quantum control Quantum technologies

What is quantum mechanics? A physical theory to explain the behavior of objects at the atomic and very small scales Objects at the atomic and sub-atomic scale behave quite differently from our everyday experience of macroscopic objects!

A physical theory to explain the behavior of objects at the atomic and very small scales

Objects at the atomic and sub-atomic scale behave quite differently from our everyday experience of macroscopic objects!

Double slit experiment No detectors at slits Wave behavior Detectors at slits Particle behavior

Model for evolution If no one is looking, evolution is unitary Schrodinger : State :

If no one is looking, evolution is unitary

Schrodinger :

State :

Measurement model Physical quantities are Hermitian matrices. Measured values are eigenvalues of X: State change (“collapse”): Quantum measurements are invasive

Physical quantities are Hermitian matrices.

Measured values are eigenvalues of X:

State change (“collapse”):

Quantum control strategies Open loop quantum control Measurement feedback quantum control Coherent feedback quantum control

Open loop quantum control

Measurement feedback quantum control

Coherent feedback quantum control

Open loop quantum control Control with pre-designed pulse signals on an ensemble of quantum systems Particles Ensemble of quantum systems

Control with pre-designed pulse signals on an ensemble of quantum systems

Open loop quantum control Brockett, D’Alessandro, Khaneja, Rabitz and co-workers …. Gate synthesis: U G = U ( T ) for smallest time T

Brockett, D’Alessandro, Khaneja, Rabitz and co-workers ….

Gate synthesis: U G = U ( T ) for smallest time T

Measurement feedback quantum control classical controller classical control actions classical information measurement Belavkin, Wiseman, Doherty, Bouten, van Handel, James and co-workers …

Atomic magnetometry Mabuchi Lab (Stanford, formerly CalTech) magnetometry setup using cold atoms

Control using another quantum system Coherent feedback control quantum controller quantum control actions quantum information direct couplings Lloyd, Yanagisawa & Kimura, James, Nurdin & Petersen, Mabuchi

Control using another quantum system

Coherent feedback experiment Coherent control experiment, H. Mabuchi (Applied Physics, Stanford Univ.), 2008 James, Nurdin, Petersen, arXiv:quant-ph/0703150 , to appear in IEEE-TAC, 2008 Mabuchi, arXiv:0803.2007, to appear Phys. Rev. A

The rest of today’s talk Linear quantum stochastic systems Robust control Coherent LQG control Realization theory

Linear quantum stochastic systems

Robust control

Coherent LQG control

Realization theory

Linear quantum stochastic systems An (Fabry-Perot) optical cavity

An (Fabry-Perot) optical cavity

Linear quantum stochastic systems x = ( q 1 ,p 1 ,q 2 ,p 2 ,…, q n ,p n ) T A 1 = w 1 +iw 2 A 2 = w 3 +iw 4 A m =w 2m-1 +iw 2m Y 1 = y 1 + i y 2 Y 2 = y 3 + i y 4 Y m’ = y 2m’-1 + i y 2m’ S Quadratic Hamiltonian Linear coupling operator Scattering matrix S B 1 B 2 B m

Linear stochastic dynamics

Physical realizability A,B,C,D cannot be arbitrary. Assume S = I. Then the system is physically realizable if and only if

A,B,C,D cannot be arbitrary.

Assume S = I. Then the system is physically realizable if and only if

Robustness Robustness is important for quantum control systems How to design robust linear quantum controllers?

Robustness is important for quantum control systems

How to design robust linear quantum controllers?

Dynamics Plant Controller

Plant

Controller

H ∞ synthesis Plant Controller w z y u James, Nurdin, Petersen, arXiv:quant-ph/0703150 , to appear in IEEE-TAC, 2008

Coherent LQG control Attain the control objective subject to physical realizability of controller Difficult problem, even computationally! Nurdin, James, Petersen, arXiv:0711.2551, IFAC WC 2008/provisionally accepted, Automatica, 2008

Attain the control objective

subject to physical realizability of controller

Difficult problem, even computationally!

Realization theory Given A , B,C,D or S,K,R how can we build it? Non-trivial question! Quantum analogue of linear electrical network synthesis theory

Given A , B,C,D or S,K,R how can we build it?

Non-trivial question!

Quantum analogue of linear electrical network synthesis theory

Simple Illustration Optical cavity ?? What about systems with many degrees of freedom??

Linear electrical network synthesis Consider the state-space representation:

Consider the state-space representation:

Synthesis of linear quantum systems “ Divide and conquer” Wish to realize this system ( S , L , H ) ? ? ? ? ? ? Network synthesis Quantum network Input fields Output fields Input fields Output fields

“ Divide and conquer”

Cascade decomposition Nurdin, James and Doherty, arXiv:0806.4448, 2008 G 1 G 2 G 3 G n H 12 H 23 H 13 H 2n H 3n H 1n G = ( S , L , H ) A(t) y(t)

LQG design example For design example in Nurdin, James and Petersen, IFAC WC 2008

Concluding remarks Quantum control is still a relatively young multidisciplinary field Quantum control can benefit from engineering ideas and perspectives Plenty of challenging problems and work to do

Quantum control is still a relatively young multidisciplinary field

Quantum control can benefit from engineering ideas and perspectives

Plenty of challenging problems and work to do

Themes for the future Applications of linear quantum systems and networks in quantum information science (e.g., entanglement distribution, quantum repeaters …) Alternative decomposition theorems of linear quantum stochastic systems Realization of linear quantum systems in other physical domains

Applications of linear quantum systems and networks in quantum information science (e.g., entanglement distribution, quantum repeaters …)

Alternative decomposition theorems of linear quantum stochastic systems

Realization of linear quantum systems in other physical domains

That’s all folks Thank you for listening

Thank you for listening