Holonomic quantum Holonomic quantum computation in computation in

decoherence-free subspacesdecoherence-free subspaces

Lian-Ao WuLian-Ao Wu

Center for Quantum Information and Quantum Control

In collaboration with Polao Zanardi and Daniel Lidar

BackgroundBackground:: Decoherence-free subspace (DFS): Decoherence-free subspace (DFS):

symmetry-aided protection of quantum symmetry-aided protection of quantum information, against such as information, against such as collective noisecollective noise

Holonomic (adiabatic) quantum computation Holonomic (adiabatic) quantum computation (HQC): all-geometrical Quantum Information (HQC): all-geometrical Quantum Information Processing strategy, robust against Processing strategy, robust against operational errorsoperational errors

Question is:Question is: Can we combine the advantages of the two?Can we combine the advantages of the two? Bringing together the best of two worlds....!!Bringing together the best of two worlds....!!

First recall Univerisal Quantum Computation (requires)

Have 2-level product space (qubits), prepare initial state

2 2| in ( ) , where : 0 & 1nin CC

Have universal set of gates(gates can take one state to arbitrary state): e.g. 1-qubit X, Z gates for each qubit plus CPHASE gates.Gates usually are evolution operators given by

Measure the final state

| |final inU

/dU dt iHU

Introduce DFS Decoherence-Free Subspace (DFS) IDecoherence? a quantum information processor (system) cannot be isolated from its environment (Bath),

exp( )1 2| 0 | | 0 | |1 |iHtB B B

due to the interaction of the system with its bath, i.e.

S B SBH H H H where( )x y z

i i i i i ii

SBH X B Y B Z B ( ) S BH H System (Bath) Hamiltonian

Introduce DFS Decoherence-Free Subspace (DFS) II

Invariant subspaces if there is symmetry in He.g. collective dephasing 0 and y x z z

i i iB B B B z

SB ii

H B Z For example, subspace spanned by |01> and |10> will be invariant. (|0001>,|0010>,|0100> & |1000>)

1 2

1 2

01 ( ) 01 =0

10 ( ) 10 =0

exp

01 10

exp(( ) exp( ))

zSB

S

zSB

B

a b

i

H B Z Z

H t

H B Z Z

iHt B iH t B

Therefore, for arbitrary

B-S interaction not harmful for the system

Control time-dependent periodical Hamiltonian through M parameters

where 1 2( ( ), ( ),..., ( ))MH t t t

(0) ( )i i T

Introduce HQCHolonomic Quantum Computation (HQC) I

T- period, evolution operator:

0

( ) exp( ( ) )t

U t T i H d

( ( )) ( ( ))n i iE t n t and

HQC is based on the adiabatic theorem, which shows

0exp( ( ( )) )exp( (( ) ( ( )) ))

t

n iii t ti E dtt nt

if H has non-degenerate eigenvalues

If start with an eigenstate of H(t), the system will stick on it but 2 phases

, 1 min | |r s r sT E E under condition:

One is interested in the case at time T, if start with

The Berry phase is all geometrical, independent of speed of parameters

1 2 3 1 2 3( ( ), ( ), ( )) ( ) ( ) ( )H t t t t X t Y t Z

0exp( ( (

(0) ( (0))

( exp) (0) )) )( )T

n i

i

i

n

T i E t dt

Then

Introduce HQCHolonomic Quantum Computation (HQC) II

Example,

The Berry phase is the solid angle swept out by the vector

1 2 3( ) ( ( ), ( ), ( ))t t t t

Allow operational error, as long as solid angle same, the Berry phase same.

( ) exp( ) (0)T i

Geometrical Phase Gate: if ( ( )) 0n iE t

Dark Eigen State:

Using dark state to generate all-geometrical phase gate:

sin ( 1 2 2 1 ) cos ( 3 2 2 3 )i iZH e e

( ( )) 0 ( ( )) 0i iD E t H t D A State with or

Introduce HQCHolonomic Quantum Computation (HQC) III

We need 4 states |0>, |1>,|2> & |3> for 1 particle, a controllable Hamiltonian in terms of parameters (t) and (t):

The dark state:

cos 1 sin 3iD e The Hamiltonian does nothing on |0> and add a Berry phase on state |1> after evolution from 0 to T if (0)=0 If we define our qubit by |0> & |1>

0 0

1 1ie

Phase gate 1 0

0 ie

Phase gate matrix

Using dark state to generate all-geometrical X gate: eiX/2

sin ( 2 2 ) cos ( 3 2 2 3 )

1 ( 0 1 )2

i iXH e e

where

Introduce HQCHolonomic Quantum Computation (HQC) V

In the 4-state space by |0>, |1>,|2> & |3> for one particle , we need a controllable Hamiltonian

The dark state:

cos sin 3iD e

The Hamiltonian will do nothing on |+> and add a Berry phase for state |-> after evolution from 0 to T.

0 cos / 2 0 sin / 2 1

1 cos / 2 1 sin / 2 0i

i

e i

X gate

Using dark state to generate all-geometrical 2-qubit gate

4 sin ( 11 21 11 21 ) cos ( 31 21 21 31 )i iH e e

Introduce HQCHolonomic Quantum Computation (HQC) V

We have 16 states for 2 particles, |00>, |01>, |02>,…. Chose a controllable Hamiltonian,

The dark state:

cos 11 sin 31iD e

The Hamiltonian will do nothing on |00>,|01>& |10> and add a Berry phase for state |11> after evolution from 0 to T.

00 00

01 01

10 10

11 11ie

2 qubit gate

Use dark states to generate all-geometrical universal set of gates, 1 qubit Z, X gate & 2-qubit CPHASE gate (by controlling Hz, Hx and H4)

A brief Sum-Up of HQCHolonomic Quantum Computation (HQC) VI

For a dark state, the wave function at T

Using this relation to perform phase gates.

( ) exp( ) (0)T i

In above cases, is half of solid angle swept outby the vector ()

We have to use ancillas |2> and |3> for each qubitwhen make gates. We pay more price. We need to Have 4-dimensional working space to support 1 qubit.

Come to our workCome to our work

A Decoherence-Free Subspace as working spaceA Decoherence-Free Subspace as working space

If interaction between system and bath isIf interaction between system and bath is

1 2 3 4ˆ ˆ( )SBH BZ B Z Z Z Z

4 qubit DFS: 4 qubit DFS: C=spanC=span{ |1000>, |0100>, { |1000>, |0100>, |0010>, |0001>}|0010>, |0001>}

C is a DFS against collective dephasing, willC is a DFS against collective dephasing, willbe our working space to support encoded logical qubitbe our working space to support encoded logical qubit

1 1 1( ); ( ); ( )2 2 2

x y zlm l m l m lm l m l m lm m lR X X YY R X Y Y X R Z Z

Time-dependent controllable Hamiltonian Time-dependent controllable Hamiltonian SetSet

[ , ] 0; , ,lmR Z x y z

Every eigenspace of Z is invariant under Every eigenspace of Z is invariant under action ofaction of lmR

Assume that the system dynamics is Assume that the system dynamics is generated by the Hamiltoniangenerated by the Hamiltonian ( )x x y y

ml ml ml mll m

H J R J R

where parameters are dynamically where parameters are dynamically controllablecontrollable..

010 | 001mlinl mlD J J e

0lmnH D

Dark-states in the DFSDark-states in the DFS

In the basis In the basis {|100>,|010>,|001>}{|100>,|010>,|001>} for for qubits l, m and n, the above Hamiltonian qubits l, m and n, the above Hamiltonian has a dark statehas a dark state

SatisfyingSatisfying

( cos sin )x y xlmn ml ml ml ml ml nl mlH J R R J R

Turn on the parameters in such a way to Turn on the parameters in such a way to getget

One-qubit geometrical gatesOne-qubit geometrical gates

cos 1 sin | 0100iL

D e

24 24 34[sin ( cos sin ) cos ]x y xZH J R R R

Turn on the parameters in such a way to Turn on the parameters in such a way to getgetActing only on 2, 3 and 4 qubit statesActing only on 2, 3 and 4 qubit states||1000>1000>43214321, |0100>, |0100>43214321 and and |0010>|0010>43214321 nothing nothing on on |0001>|0001>43214321 .. Define the logic qubit supported by state Define the logic qubit supported by state ||0>0>LL=|0001>=|0001> and and |1>|1>LL=|0010>=|0010>. . Dark state in qubits 2,3 and 4Dark state in qubits 2,3 and 4

/ 21 1

0 0

ZiL L

L L

e

Adiabatically changing parameters a Adiabatically changing parameters a Berry Phase for |1>Berry Phase for |1>LL

cos sin | 0100iL

D e

24 14 24 1434[sin ( cos sin ) cos ]

2 2

x x y yx

XR R R RH J R

xxgategate

Turn on the parameters in such a way to Turn on the parameters in such a way to getget

wherewhere, ,

24 1434( , ) 0

2

x y x yx

L

R RR

/ 2XiL L

L L

e

( 1 0 ) / 2L L L

Adiabatically changing parameters a Adiabatically changing parameters a Berry Phase forBerry Phase for |->|->LL

Easy to proveEasy to proveDark state isDark state is

0 cos 0 sin 14 4

1 cos 1 sin 04 4

X XL L L

X XL L L

i

i

Two-qubit geometrical gatesTwo-qubit geometrical gates

2 2cos 1 sin | 0100iLL

D e

4 24 24 68 68 34 78[sin ( cos sin )( cos sin ) cos ]x y x y x xH J R R R R R R

Suppose that one can engineer the four-Suppose that one can engineer the four-body interactionbody interaction

Dark state isDark state is

/ 211 11

10 10

01 01

00 00

PiL L

L L

L L

L L

e

Adiabatically changing parameters a Adiabatically changing parameters a Berry Phase for |11>Berry Phase for |11>LL

Compare Compare General HQCGeneral HQC with with HQC in HQC in DFSDFS General

HQC HQC in DFS4D working space: |0>, |1>, |2>, |3>

4D working space: |0001>, |0010>,|0100>,|1000>

Qubit by |0> and |1>Logic qubit by |0>L=|

0001> and |1>L =|0010>Controllable

Hamiltonian Hx, Hz and H4

Experimental implementation: depend on the system

Hx, Hz and H4 have same matrix representations but acting different spaces

Interesting to note X, Z need only 2-body interaction

Robust against operational error

Robust: operational error and collective dephasing

ImplementationsSpin-based quantum dot proposals

( ) ( )[ ( ) (1 ( )) ]y xij ij ij ij ij i jH t J t t R t R Z Z

One qubit Hamiltonians achievable: confining potential, pulse shaping (Stepanenko etal al 2003,2004)

11tan

),( ZX HH

Ion Traps),( ZX HH Sorensen-Moelmer scheme (two lasers control)(Kielpinski et al 2002)

4H Realizable as well! SM over two pairs of trapped ionsgeometrical gates already realized (Leibfried et al 2003)

SummarySummary

We have discussed how to merge We have discussed how to merge together together universal universal HQC & DFS by using HQC & DFS by using dark-states in decoherence-free dark-states in decoherence-free subspace against collective dephasing. subspace against collective dephasing. The scheme can be extended to the The scheme can be extended to the cases against general collective noise.cases against general collective noise.

LA Wu, PZ, DA Lidar, PRL 95, 130501 (2005)

Thank you for the attention!

geometrical phase factor is precisely the holonomy in a Hermitian line bundle since the adiabatic theorem naturally defines a connection in such a bundle.