Quantum ComputingQuantum Computingvia Local Controlvia Local Control

Einat FrishmanEinat Frishman

Shlomo SklarzShlomo Sklarz

David TannorDavid Tannor

““Rule” (logical operation) U(t) same for all inputRule” (logical operation) U(t) same for all input

The Schrödinger Equation:The Schrödinger Equation:

We can formally Solve:We can formally Solve:

Unitary propagator U(t) createsUnitary propagator U(t) creates mapping between mapping between (0) and (0) and (t):(t):

Hi

)0()( 0

t

Hdti

Tet

Quantum CircuitsQuantum Circuits= = Unitary TransformationsUnitary Transformations

↔T

H TT

input

ou

tput

)0()()( tUt

The Unitary Control ProblemThe Unitary Control Problem

)0(][

)0()( 0

'))'((

EU

Tet

t

dttEHi

UU((tt)) is determined by the laser field is determined by the laser field EE((··))::

UU((tt))=U=U([([EE]],t,t)) Given a desired Given a desired UU((TT)=)=OO can we find a field can we find a field

EE((··)) that produces it? that produces it? Inverse problem Inverse problem Control problem Control problem

[1] C.M. Tesch and R. de Vivie-Riedle, PRL 89, 157901 (2002)[1] C.M. Tesch and R. de Vivie-Riedle, PRL 89, 157901 (2002)[2] J.P. Palao and R. Kosloff, PRL 89, 188301 (2002)[2] J.P. Palao and R. Kosloff, PRL 89, 188301 (2002)

External laser Field E(t)

Control of a State vs. Control of Control of a State vs. Control of a Transformationa Transformation

What is usually done in quantum control:What is usually done in quantum control: - Control of a State:- Control of a State:

find find E(t)E(t) such that such that f f i i ..

Controls the evolution of Controls the evolution of oneone state state

What we have here – a harder problem !What we have here – a harder problem ! - Control of a Transformation: - Control of a Transformation:

find find E(t)E(t) such that such that ff

UUii

, ,

ff

UUii , ,

ff(n)(n)

UUiinn . .

Controls simultaneously the evolution of Controls simultaneously the evolution of allall possiblepossiblestates and phasesstates and phases

System=System=RegisterRegister++Mediating statesMediating states Two alternative realizations:Two alternative realizations:

Direct sum spaceDirect sum space Direct product spaceDirect product space

Objective:Objective: Produce Target Unitary Produce Target Unitary Transformation on register Transformation on register withoutwithout intermediate population of auxiliary intermediate population of auxiliary mediating statesmediating states

Quantum Register and Mediating Quantum Register and Mediating StatesStates

Mediating states

Register states

E(t)

E(t)

Projection onto Register Projection onto Register Separable Unitary transformation on space:Separable Unitary transformation on space:

Define Define PP a projection operator onto the a projection operator onto the

quantum register sub-manifold: quantum register sub-manifold: UURR==PPUUPP

Register states

Mediating states

MU U UR

Entire Hilbert Space

The Model:The Model: Producing Unitary Producing Unitary Transformations on the Vibrational Transformations on the Vibrational

Ground Electronic States of NaGround Electronic States of Na22R

egiste

rM

edia

ting sta

tesX1g

+

A1u+

E(t)

H=H0+Hint , Hint =( ) E*E

Definition of ConstrainedDefinition of ConstrainedUnitary Control ProblemUnitary Control Problem

System equation of motion:System equation of motion:

Control:Control: laser field laser field E(t) Objective:Objective: target unitary transformation target unitary transformation OORR

Maximize Maximize J=|Tr(OR†UR(T))|2

Constraint:Constraint: No depopulation of register No depopulation of register

Conserve Conserve C=Tr(UR†UR)

Motivation:Stimulated Raman Adiabatic Passage (STIRAP)

Bergmann et al. (1990) .Bergmann, Theuer and Shore, Rev Mod. Phys. 70, 1003 (1998).V. Malinovsky and D. J. Tannor, Phys. Rev. A 56, 4929 (1997).

S P

!

1

2

3

S

P

At each point in time:At each point in time:

Enforce constraint CEnforce constraint C dC/dt=Imag(g E(t))=0

E(t)=a g*

direction

Monotonic increaseMonotonic increase in in Objective J Objective J dJ/dt=Real(f E(t))=a Real(f g*)>0

a=Real(f g*) Sign and magnitude

Local Optimization MethodLocal Optimization Method

Re

Im

g*

f

E(t) g

Creating a Hadamard Gate in a Creating a Hadamard Gate in a Three-Level Three-Level -System-System

|1 |2

|3

E(t)

Register states

Mediating states

Femto-second pulse shaping

Registe

rM

edia

ting sta

tes

Fourier Transform on a Quantum Fourier Transform on a Quantum Register: withRegister: with (7+3) level (7+3) levelsub-manifold of Nasub-manifold of Na22; ;

w=e2i/6

[24 p.s].

111111

1

11

111

11

1

6

1

2345

2424

333

4242

5432

wwwww

wwww

www

wwww

wwwww

FT

Direct-Sum vs. Direct-Product SpaceDirect-Sum vs. Direct-Product Space(separable transformations)(separable transformations)

UR

UM

Direct Sum

U=UR UM

Direct product

U=UR UM

UR11UM …UR1n

UM

URn1UM …URnn

UM

Ion-Trap Quantum GatesIon-Trap Quantum Gates

E(t)

Atoms in linear trap

Internal states

External Center of mass modes

|e|g

|n+1 |n |n-1

|ee

|ge |eg

|gg

|n+1 |n |n-1

[1] J.I. Cirac and P. Zoller, PRL 74, 4091 (1995)[2] A. Søørensen and K. Møølmer, PRL 82, 1971 (1999)[3] T. Calarco, U. Dorner, P.S. Julienne, C.J. Williams and P. Zoller, PRA

70, 012306, (2004)

Problem: Entanglement of the Quantum register with the external modes!

Liouville-Space FormulationLiouville-Space Formulation Projection Projection PP onto register must trace out the onto register must trace out the

environment producing, in general, mixed environment producing, in general, mixed states on the register.states on the register.

Liouville space description is required!Liouville space description is required!• Space:Space: HH →→ L, L, • Density Matrix:Density Matrix: →→ ||||RR||EE

• Inner product: Inner product: Tr(Tr(††) ) →→ ||• Super Operators:Super Operators:[[H,H,]] →→ H H ||UUUU†† U U ||

• Evolution Equation:Evolution Equation:

,HUU

i

Sørensen-Mølmer SchemeSørensen-Mølmer Scheme

|n+1 |n |n-1

|ee

|ge |eg

|gg

Field internal external

Local Control (Initial) ResultsLocal Control (Initial) Resultsfor a two-qubit entangling gatefor a two-qubit entangling gate

We assumed each pulse is near-resonant with We assumed each pulse is near-resonant with one of the sidebandsone of the sidebands

We fixed the total summed intensityWe fixed the total summed intensity Results close to the Sørensen-Mølmer schemeResults close to the Sørensen-Mølmer scheme

Fields (amp,phase) and evolution of propagatorFields (amp,phase) and evolution of propagator::

SummarySummary Control of unitary propagators implies Control of unitary propagators implies simultaneouslysimultaneously controlling controlling allall possible states possible states in systemin system

We devised a We devised a Local Control methodLocal Control method to to eliminate undesired population leakageeliminate undesired population leakage

We considered two general state-space We considered two general state-space structures:structures:•Direct Sum Direct Sum E.g.:* Hadamard on a E.g.:* Hadamard on a system, system,

* SU(6)-FT on Na * SU(6)-FT on Na22 •Direct ProductDirect ProductE.g.:* Sørensen-Mølmer SchemeE.g.:* Sørensen-Mølmer Scheme

to directly produce to directly produce arbitrary2-qubit arbitrary2-qubit

gatesgates