quantum computing via local control einat frishman shlomo sklarz david tannor
TRANSCRIPT
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