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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 computing via local control

Quantum Computingvia Local Control

Einat Frishman

Shlomo Sklarz

David Tannor

slide2

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
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
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

quantum register and mediating states

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

projection onto register

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
the model producing unitary transformations on the vibrational ground electronic states of na 2

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 constrained unitary control problem
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)
slide9

2

WS

WP

3

1

Motivation:

Stimulated Raman Adiabatic Passage (STIRAP)

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).

local optimization method

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

creating a hadamard gate in a three level l system

|3

E(t)

|2

|1

Mediating states

Register states

Creating a Hadamard Gate in a Three-Level L-System
fourier transform on a quantum register with 7 3 level sub manifold of na 2

Mediating states

Register

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

w=e2pi/6

[24 p.s.]

direct sum vs direct product space separable transformations

UR11UM …UR1nUM

  

URn1UM…URnnUM

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

UR

UM

  • Direct Sum

U=URUM

  • Direct product

U=URUM

ion trap quantum gates

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
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:
s rensen m lmer scheme

|ee

w2

w1

|eg

|ge

w1

w2

|gg

Sørensen-Mølmer Scheme

Field internalexternal

|n+1|n|n-1

local control initial results for a two qubit entangling gate
Local Control (Initial) Resultsfor a two-qubit entangling gate

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
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