Quantum Computing via Local Control

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Quantum Computing via Local Control. Einat Frishman Shlomo Sklarz David Tannor. Quantum Circuits = Unitary Transformations. output. input. Y. T. T. H. T. “ Rule ” (logical operation) U(t) same for all input The Schr ö dinger Equation: We can formally Solve:

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Quantum Computingvia Local Control

Einat Frishman

Shlomo Sklarz

David Tannor

Quantum Circuits =

Unitary Transformations

output

input

Y

T

T

H

T

• “Rule” (logical operation) U(t) same for all input
• The Schrödinger Equation:
• We can formally Solve:
• Unitary propagator U(t) creates

mapping between y(0) and y(t):

The Unitary Control Problem

External laser Field E(t)

• U(t) is determined by the laser field E(·):U(t)=U([E],t)
• Given a desired U(T)=O can we find a field E(·) that produces it?
• Inverse problem  Control problem

[1] C.M. Tesch and R. de Vivie-Riedle, PRL 89, 157901 (2002)

[2] J.P. Palao and R. Kosloff, PRL 89, 188301 (2002)

Control of a State vs. Control of a Transformation
• What is usually done in quantum control:

- Control of a State:find E(t) such that

Yf  Yi .

Controls the evolution of one state

• What we have here – a harder problem !

- Control of a Transformation: find E(t) such that

Yf(1)= U Yi(1) ,

Yf(2)= U Yi(2) ,

 

Yf(n)= U Yi(n) .

Controls simultaneously the evolution of allpossible

states and phases

E(t)

E(t)

Quantum Register and Mediating States
• System=Register+Mediating states
• Two alternative realizations:

Direct sum spaceDirect product space

• Objective: Produce Target Unitary Transformation on register without intermediate population of auxiliary mediating states

Mediating states

Register states

U

UR

Entire

Hilbert Space

Register states

Mediating states

Projection onto Register
• Separable Unitary transformation on space:
• Define Pa projection operator onto the quantum register sub-manifold: UR=PUP

A1Su+

E(t)

Mediating states

X1Sg+

Register

The Model: Producing Unitary Transformations on the Vibrational Ground Electronic States of Na2

H=H0+Hint , Hint= ( )

mE

mE*

Definition of ConstrainedUnitary Control Problem
• System equation of motion:
• Control: laser field E(t)
• Objective: target unitary transformation ORMaximize J=|Tr(OR†UR(T))|2
• Constraint: No depopulation of register Conserve C=Tr(UR†UR)

2

WS

WP

3

1

Motivation:

WS

WP

!

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).

Im

g*

f

Re

g

E(t)

Local Optimization Method

At each point in time:

• Enforce constraint CdC/dt=Imag(g E(t))=0

 E(t)=a g*

 direction

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

 Sign and magnitude

|3

E(t)

|2

|1

Mediating states

Register states

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

Mediating states

Register

Fourier Transform on a Quantum Register: with (7+3) levelsub-manifold of Na2;

w=e2pi/6

[24 p.s.]

UR11UM …UR1nUM

  

URn1UM…URnnUM

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

UR

UM

• Direct Sum

U=URUM

• Direct product

U=URUM

Atoms in linear trap

Internal states

|ee

|e

|g

E(t)

w2

w1

|n+1|n|n-1

|eg

|ge

|n+1|n|n-1

External Center of mass modes

w1

w2

|gg

Ion-Trap Quantum Gates

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

[1] J.I. Cirac and P. Zoller, PRL 74, 4091 (1995)

[2] A. Sørensen and K. Mølmer, PRL 82, 1971 (1999)

[3] T. Calarco, U. Dorner, P.S. Julienne, C.J. Williams and P. Zoller, PRA 70, 012306, (2004)

Liouville-Space Formulation
• Projection P onto register must trace out the environment producing, in general, mixed states on the register.
• Liouville space description is required!
• Space:H → L,
• Density Matrix:r → |r =|rR|rE,
• Inner product: Tr(r†s) → r|s
• Super Operators:[H,r] → H |r, UrU† U |r
• Evolution Equation:

|ee

w2

w1

|eg

|ge

w1

w2

|gg

Sørensen-Mølmer Scheme

Field internalexternal

|n+1|n|n-1

Fields (amp,phase) and evolution of propagator:

• We assumed each pulse is near-resonant with one of the sidebands
• We fixed the total summed intensity
• Results close to the Sørensen-Mølmer scheme
Summary
• Control of unitary propagators implies simultaneously controlling all possible states in system
• We devised a Local Control method to eliminate undesired population leakage
• We considered two general state-space structures:
• Direct Sum E.g.:* Hadamard on a L system, * SU(6)-FT on Na2
• Direct ProductE.g.:* Sørensen-Mølmer Scheme to directly produce arbitrary2-qubit gates