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Quantum Computers and Decoherence: Exorcising the

Demon from the Machine

Daniel LidarChemical Physics Theory Group

Chemistry DepartmentUniversity of Toronto

BIRS WorkshopQuantum Mechanics on the Large Scale

April 13, 2003

The Problem

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The Arsenal

• Active Quantum Error Correction: Error correcting codes

• Passive Error Prevention: Decoherence-free subspaces and (noiseless) subsystems

• Dynamical Decoupling: Strong and fast “bang-bang” pulses

• Topological & Holonomic Methods: Nonabelian anyons, Toric codes, Adiabatic elimination, …

• Continuous Quantum Control: Closed-loop feedback

Underlying ParadigmAdapt decoherence-resistance method to a model of decoherenceE.g.:

•Quantum error correction: assumes local, uncorrelated errors

•Decoherence-free subspaces: assumes a symmetry in system-bath interaction

•Dynamical decoupling: assumes bath with long correlation time

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Focus on different Primary Object: Set of “Naturally” Available Interactions and

Measurements

For given proposed realization of a QC:• What are the controllable terms in the internal

Hamiltonian?• What are the possible external unitary control

options?• What are the possible measurements?

Determines options for both decoherence control and quantum computation (universality of logic gates), typically via an encoding

Interaction capable of doing both will be called “Super-Universal”

Examples of “Naturally Available” Interactions

• Electrons spin in quantum dots, nuclear spin in doped atom arrays: Heisenberg exchange interaction easily controllable, single-spin operations are hard

• Linear optics: single-photon gates easy, photon-photon interaction is hard

• Trapped ions: relative phase between lasers easily controllable, absolute phase is hard

• Superconducting flux qubits: application of local bias magnetic field hard, controllable Josephsoncoupling easy

• …

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Plan

Show how options for 1. Universal QC2. Decoherence reduction

are determined naturally by set of available and controllable interactions.

Trapped Ions

Innsbruck group

few mµQubit: two hyperfine states of trapped ion

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12 1 2 1 1 2 2( , , ) exp[ ( cos sin ) ( cos sin )]x y x yU iθ φ φ θ σ φ σ φ σ φ σ φ= + ⊗ +

Rabi freq.∝ Laser phase on ions 1,2

Natural control options• Efficient single-qubit measurements (cycling

transition)• Sorensen-Molmer gates (insensitive to heating of

ions center of mass motion)

How to avoid control of absolute phase??

Two-Qubit DFS Encoding

∴ Can generate all single DFS-qubit operations by controlling relative laser phase.Same true for controlled-phase gate between two DFS qubits

1 2

1 2

0

1

L

L

= ↓ ⊗ ↑

= ↑ ⊗ ↓

1 2 1 2exp[ ( cos( ) sin( ))]DFS i X Yθ φ φ φ φ− −

− + −a

12 1 2 1 1 2 2( , , ) exp[ ( cos sin ) ( cos sin )]x y x yU iθ φ φ θ σ φ σ φ σ φ σ φ= + ⊗ +

Same encoding protects against collective dephasing: the chief source of decoherence in trapped ions

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Collective Dephasing

Long-wavelength magnetic field B (bath) couples to spins:

( )int 1 2z zH gB σ σ == − +

1σ

02

20

gB

gB

−

1 2

1 2

1 2

1 2

↓ ↑

↑ ↓

↓ ↓

↑ ↑

1 2

1 2

=0

0

1

Encode qubit into states with :

L

L

ZM= ↓ ⊗ ↑

= ↑ ⊗ ↓0 1 is decoherence-free

L L La bψ = +

( ) ˆB t z

2σ

““A DecoherenceA Decoherence--Free Quantum Memory Using Trapped IonsFree Quantum Memory Using Trapped Ions””D. D. KielpinskiKielpinski et al., Science et al., Science 291291, 1013 (2001), 1013 (2001)

Figure 2. Decay of the DFS-encoded state (circles) and the test state (crosses) under ambient decoherence. We vary the delay time between encoding and decoding to give the ambient noise a variable time to act. Coherence data are normalized to their values for zero applied noise. The fit lines are exponential decay curves for purposes of comparison and are not theoretical predictions. The decay rate of the test state is (7.9 ± 1.5) × 103/µs, whereas the decay rate of the DFS state is (2.2 ± 0.3) × 103/µs. Because the coherence time of the DFS-encoded state is much longer than that of the test state, we see that the chief source of ambient decoherence is collective dephasing.

DFS-encoded

Bare qubits

Bare qubit:two hyperfine states of trapped 9Be+ ion

Chief decoherence sources:(i) fluctuating long-wavelength ambient magnetic fields;(ii) heating of ion CM motion during computation

DFS encoding: into pair of ions

1 2

1 2

0

1L

L

= ↓ ⊗ ↑

= ↑ ⊗ ↓

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Gate Control Option Motivates Gate Control Option Motivates Classification of all Decoherence Classification of all Decoherence Processes on Two Processes on Two QubitsQubits (Ions)(Ions)

SB DFS Leak LogicalH H H H= + +

{ , , , , , , , }LeakH XI IX YI IY XZ ZX YZ ZY B= ⊗

motional decoherence

computation

storage

{ , , , , }2 2 2DFSZI IZ XY YX XX YYH ZZ II B+ + −= ⊗

collective dephasing

{ , , }2 2 2LogicalXX YY YX XY ZI IZH BX Y Z= = =

− − −+ − −= ⊗

differential dephasing

1 2

1 2

0

1L

L

= ↓ ⊗ ↑

= ↑ ⊗ ↓immune

zσ

Can Can allall decoherence be eliminated decoherence be eliminated using just DFS encoding & using just DFS encoding & Sorensen Sorensen MolmerMolmer gates?gates?

Options:Options:Apply active quantum error correction. Apply active quantum error correction. Problem: not known how to do using only Problem: not known how to do using only Sorensen Sorensen MolmerMolmer gates.gates.Topological, Topological, HolonomicHolonomic: ??: ??Dynamical decoupling.Dynamical decoupling.

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Dynamical Decoupling (BangDynamical Decoupling (Bang--Bang)Bang)SpinSpin--echo; Carrecho; Carr--Purcell; Viola & Lloyd Phys. Rev. A Purcell; Viola & Lloyd Phys. Rev. A 5858, 2733 (1998); Byrd & , 2733 (1998); Byrd & LidarLidar, Q. Inf. Proc. , Q. Inf. Proc. 11, 19 (2002), 19 (2002)

system bath

System-bath Hamiltonian: SB S BH α αα

⊗=∑

zσ

xσ

yσ

intH

zσ

xσ

yσ

intH−

intH

Apply rapid pulsesflipping sign of Sα

xσ

yσ

zσ

intH

More general symmetrization: int

.

averaged to zero.

Satisfy very stringent time consChall

trainenge:

ts

H

Eliminating Differential Eliminating Differential DephasingDephasingUsing SM Gate in BangUsing SM Gate in Bang--Bang Bang

ModeMode1 2

1 2

0

1L

L

= ↓ ⊗ ↑

= ↑ ⊗ ↓

SBH SBH12 1 1( , , )2U πθ φ φ=− 12 1 1( , , )2U πθ φ φ=+

t

2ZI IZ− =

no differential dephasing

X−

Z−

XZX Z=−

Also holds for : Y XYX Y=−

error also eliminated2YX XYY

− −=∴

Pulse parameters not a mystery: arise from group theory, symmetrization

Time reversal (spin echo)

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SBH SBH12 1 1( , , )U θ π φ φ= 12 1 1( , , )U θ π φ φ=

t

z zσ σ⊗

Elimination of all Leakage Using Elimination of all Leakage Using SM Gate in BangSM Gate in Bang--Bang ModeBang Mode1 2

1 2

0

1L

L

= ↓ ⊗ ↑

= ↑ ⊗ ↓

LeakH =

no leakage errors

z z zLea L azk e kH Hσ σ σ σ⊗ ⊗ −=

{ , , , , , , , }LeakH XI IX YI IY XZ ZX YZ ZY B= ⊗

SM Pulses are SuperSM Pulses are Super--UniversalUniversalMethods above can be used to eliminate Methods above can be used to eliminate all dominant errors (differential all dominant errors (differential dephasingdephasing+ leakage) in a 4+ leakage) in a 4--pulse sequencepulse sequenceTo eliminate ALL twoTo eliminate ALL two--qubitqubit errors (leaving errors (leaving DFS encoding intact) need a 10DFS encoding intact) need a 10--pulse pulse sequence.sequence.Scheme entirely compatible with SMScheme entirely compatible with SM--gates gates to perform universal QC inside DFS. to perform universal QC inside DFS.

D.A.L. and L.-A. Wu, Phys. Rev. A 67, 032313 (2003). L.-A. Wu, D.A.L., Phys. Rev. Lett. 88, 207902 (2002).L.-A. Wu, M.S. Byrd, D.A.L., Phys. Rev. Lett. 89, 127901 (2002).

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Standard BB timeStandard BB time--scale assumption: scale assumption: pulses need to pulses need to be faster than fastest bath timebe faster than fastest bath time--scalescale (inverse of (inverse of bath highbath high--freq. cutoff): freq. cutoff): ~10ns~10ns for fluctuating patch for fluctuating patch potentials. Not feasible with SM pulses: potentials. Not feasible with SM pulses: 11µµss. . However, this relies on bath with However, this relies on bath with DebyeDebye--like like spectral density:spectral density:

Are the timeAre the time--scales feasible?scales feasible?

( )I ω

ωUVω

( )I ω

ωIRω

Measurements for trapped ions Measurements for trapped ions indicate indicate 1/1/ff--typetype spectrum:spectrum:

Implies much relaxed timeImplies much relaxed time--constraints (K. constraints (K. ShiokawaShiokawa & D.A.L., & D.A.L., quantquant--ph/0211081): timeph/0211081): time--scale set scale set by bath by bath lowlow--freq. cutoff. Our freq. cutoff. Our scheme then appearsscheme then appears feasible. feasible. Experimental verification welcome.Experimental verification welcome.

[ ]01log ( )tρ−

1Bk = =h

. .

0.150

20

IR

UV

Dyn Decoup

ωωω

==

=

K. Shiokawa, D.A.L., quant-ph/0211081

1/f, free

1/f, pulsed

Ohmic, free

Ohmic, pulsed

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Nanofabricated Quantum Dots

200nm

“easy” hard

Delft qubits

Natural control options• Two-spin measurements distinguishing singlet

from triplet• Heisenberg exchange gates generated from

Challenge:

Implement everything (universal QC, decoherence elimination) using only Heisenberg exchange interactions.

( )Heis

( ) controllable via applied gate voltages + global magnetic fields

i jiji j

ij

tJ

J t

H σ σ<

•=∑uuv uuv

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FourFour--QubitQubit DFS EncodingDFS Encoding

∴ Can generate all single encoded-qubit operations by controlling Heisenberg exchange interactions:

This encoding protects against collective decoherence.

( )1 0 1 1 02ij i j i js = −

12 340L

s s= ⊗

1

2

3

4

13 24a s s= ⊗

1

2

3

4

14 23b s s= ⊗

1

2

3

4

113L

a b

= +

Same is true for controlled-phase gate between two DFS qubits

1 2Z σ σ= − 1 3 1 22 1

23X σ σ σ σ

= − +

D. Bacon , J. Kempe, D.A.L. and K.B. Whaley, Phys. Rev. Lett. 85, 1758 (2000).

Collective Decoherence

T

g g g g

Collective Decoherence: set all gi equal

1

int

etc., total (pseudo-)angular momentum operators

Collective interaction:

zi

nz i

x x y y z z

S

H S B S B S B

σ=

=

=

⊗ + ⊗ + ⊗

∑

, , )

Singlets: states with zero angularDecoherence

momentum -fr

totalee

,state

(

s:

x y z

JJ S S S=r

, , coherence ( (2))only: Collective dephasing (abelian)

}: Collective de{z

x y z suSS S S

int 1y yx x z z

i i i i i iyx z

i i ini B B BH g g gσ σ σ=

= + +⊗ ⊗ ⊗∑

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Scaling Up

...

Assumption of collective decoherence less accurate the larger the number of physical qubits.

Other sources of decoherence necessarily appear.

{ }0 , 1L L

T

g g g g g g g g g g g g

⊗ ⊗ ⊗{ }0 , 1L L { }0 , 1L L

Just as in two-qubit (trapped-ion) case, all other sources can be classified as•Leave DFS invariant•Leakage•Logical errors

Can be eliminated using dynamical decoupling with Heisenberg

Dynamical Generation of Collective Dynamical Generation of Collective DecoherenceDecoherence

Details: L.-A. Wu, D.A.L., , 207902 (2002).

Requires sequence of 6 /2 pulses to create collective decoherenceconditions over blocks of 4 qubits.

Phys. Rev. Lett.

π

88

system-bath interaction (e.g., ) causeslogical errors ( ) and leakage. Leakage part can be eliminated using Heisenberg

Bilinea

, with two pu

r

lses.i j

yxi j Bσ σ

σ σπ

⊗

Details: L.-A. Wu, M.S. Byrd and D.A.L, Phys. Rev. Lett. 89, 127901 (2002).

( )1 2 1 2 1 2HeisBy rapid pulsing of

collective conditions can be created for arbitrar ly system-bath ii nn

decoherencteractear

2e

ion:

y yx x z zJH σ σ σ σ σ σ= + +

int 1

x x y y z zi i i i i i

yx zi i i

ni

x x y y z z

B B BH g g gS B S B S B

σ σ σ=

= + +⊗ ⊗ ⊗

⊗ + ⊗ + ⊗∑

→

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Heisenberg is SuperHeisenberg is Super--UniversalUniversal

Heisenberg exchange is naturally available Heisenberg exchange is naturally available interaction for spininteraction for spin--coupled Q. dots, doped atom coupled Q. dots, doped atom arrays.arrays.It alone suffices forIt alone suffices for

Universal QCUniversal QCDynamical generation of collective decoherenceDynamical generation of collective decoherenceLeakage eliminationLeakage elimination

This works in conjunction with DFS encodingThis works in conjunction with DFS encoding

Generalization and SummaryGeneralization and SummaryThe available/controllable The available/controllable interactions {interactions {HHii} are the primary } are the primary object in Q. information processingobject in Q. information processingThey define an associative algebraThey define an associative algebra

The The commutantcommutant of this algebra are of this algebra are the systemthe system--bath interactions that bath interactions that leave the system invariantleave the system invariantThis endows Hilbert space with a This endows Hilbert space with a preferred encoding: the DFSpreferred encoding: the DFSIn some cases the {In some cases the {HHii} suffice to } suffice to dynamically generate the dynamically generate the commutantcommutantfrom an arbitrary systemfrom an arbitrary system--bath bath interaction. In this case the {interaction. In this case the {HHii} are } are ““supersuper--universaluniversal””..

Heisenberg exchangeHeisenberg exchange

Group algebra of the permutation Group algebra of the permutation groupgroup

Collective decoherence processesCollective decoherence processes

The 4The 4--qubit code (for example)qubit code (for example)

Generation of collective decoherence Generation of collective decoherence from arbitrary linear systemfrom arbitrary linear system--bath bath interaction; leakage eliminationinteraction; leakage elimination

Similar conclusions seen for Heisenberg hold for anisotropic Similar conclusions seen for Heisenberg hold for anisotropic exchange models (e.g., XY, XXZ).exchange models (e.g., XY, XXZ).

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The role of the controllable The role of the controllable interactions is primary in universality interactions is primary in universality and and combattingcombatting decoherencedecoherence

Open question:Open question:Can the duality Can the duality controllablecontrollable uncontrollable uncontrollable interactions interactions

be used in quantum error correction, be used in quantum error correction, topological codes, etc.?topological codes, etc.?

Thanks

Collaborators at UC Berkeley:

Dr. Dave Bacon, Dr. Julia Kempe, Prof. Birgit Whaley

Students and postdocs at University of Toronto:

Dr. Lian-Ao Wu, Dr. Mark Byrd (Harvard), Dr. Tom Shiokawa(Maryland), Kaveh Khodjasteh

Funding:

DARPA (QuIST), NSERC, Photonics Research Ontario, Premier’s Research Excellence Award, Connaught Fund, D-Wave Systems Inc.

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Further ReadingD.A.L., I.L. Chuang & K.B. Whaley, “Decoherence-Free Subspaces for Quantum Computation”Phys. Rev. Lett. 81, 2594 (1998)

D.A.L., D. Bacon and K.B. Whaley, “Concatenating Decoherence-Free Subspaces and Quantum Error Correcting Codes”Phys. Rev. Lett. 82, 4556 (1999)

D. Bacon, J. Kempe, D.A.L. & K.B. Whaley, “Universal, Fault Tolerant Quantum Computation in Decoherence-Free Subspaces”Phys. Rev. Lett. 85, 1758 (2000)

D.A.L. and L.-A. Wu, “Reducing Constraints on Quantum Computer Design Using Encoded Selective Recoupling”Phys. Rev. Lett. 88, 017905 (2002)

L.-A. Wu and D.A.L., “Creating Decoherence-Free Subspaces Using Strong and Fast Pulses”Phys. Rev. Lett. 88, 207902 (2002)

M.S. Byrd and D.A.L., “Comprehensive Encoding and Decoupling Solution to Problems of Decoherence and Design in Solid-State Quantum Computing”Phys. Rev. Lett. 89, 047901 (2002)

L.-A. Wu, M.S. Byrd and D.A.L, “Efficient Universal Leakage Elimination for Physical and Encoded Qubits” Phys. Rev. Lett. 89, 127901 (2002)

D.A.L. and L.-A. Wu, “Encoded Recoupling and Decoupling: An Alternative to Quantum Error Correction, Applied to Trapped Ion Quantum Computation”, Phys. Rev. A 67, 032313 (2003).