Quantum Computers and Decoherence: Exorcising the

Demon from the Machine

Daniel Lidar

Chemical Physics Theory Group

Chemistry Department

University of Toronto

BIRS WorkshopQuantum Mechanics on the Large Scale

April 13, 2003

The Problem

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 Paradigm

Adapt decoherence-resistance method to a model of decoherence

E.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

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 Josephson coupling easy

• …

Plan

Show how options for

1. Universal QC

2. Decoherence reduction

are determined naturally by set of available and controllable interactions.

Trapped Ions

Innsbruck group

few mQubit: two hyperfine states of trapped ion

12 1 2 1 1 2 2( , , ) exp[ ( cos sin ) ( cos sin )]x y x yU i

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( ))]

DFSi X Y

12 1 2 1 1 2 2( , , ) exp[ ( cos sin ) ( cos sin )]x y x yU i

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

Collective Dephasing

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

int 1 2z zH gB

1

0

2

2

0

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 b

( ) ˆB t z

2

““A Decoherence-Free Quantum Memory Using Trapped Ions”A Decoherence-Free Quantum Memory Using Trapped Ions”D. Kielpinski et al., Science D. Kielpinski 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

1

L

L

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

SB DFS Leak LogicalH H H H

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

motional decoherence

computation

storage

{ , , , , }2 2 2DFS

ZI IZ XY YX XX YYH ZZ II B

collective dephasing

{ , , }2 2 2Logical

XX YY YX XY ZI IZH BX Y Z

differential dephasing

1 2

1 2

0

1

L

L

immune

z

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

Topological, Holonomic: ??Topological, Holonomic: ??

Dynamical decoupling.Dynamical decoupling.

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

system bath

System-bath Hamiltonian: SB S BH

z

x

y

intH

z

x

y

intH

intH

Apply rapid pulsesflipping sign of S

x

y

z

intH

More general symmetrization: int

.

averaged to zero.

Satisfy very stringent time consChall

trainenge:

ts

H

Eliminating Differential Dephasing Eliminating Differential Dephasing Using SM Gate in Bang-Bang Using SM Gate in Bang-Bang

ModeMode1 2

1 2

0

1

L

L

SBH

SBH12 1 1

( , , )2

U 12 1 1

( , , )2

U

t

2ZI IZ

no differential dephasing

X

Z

XZX Z

Also holds for : Y XYX Y

error also eliminated2

YX XYY

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

Time reversal (spin echo)

SBH

SBH12 1 1

( , , )U 12 1 1

( , , )U

t

z z

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

1 2

0

1

L

L

LeakH

no leakage errors

z z zLea L azk e kH H

{ , , , , , , , }Leak

H XI IX YI IY XZ ZX YZ ZY B

SM Pulses are Super-UniversalSM Pulses are Super-UniversalMethods above can be used to eliminate Methods above can be used to eliminate all dominant errors (differential dephasing all dominant errors (differential dephasing + leakage) in a 4-pulse sequence+ leakage) in a 4-pulse sequence

To eliminate ALL two-qubit errors (leaving To eliminate ALL two-qubit errors (leaving DFS encoding intact) need a 10-pulse DFS encoding intact) need a 10-pulse sequence.sequence.

Scheme entirely compatible with SM-gates Scheme entirely compatible with SM-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).

Standard BB time-scale assumption: Standard BB time-scale assumption: pulses need to pulses need to be faster than fastest bath time-scalebe faster than fastest bath time-scale (inverse of (inverse of bath high-freq. cutoff): bath high-freq. cutoff): ~10ns~10ns for fluctuating patch for fluctuating patch potentials. Not feasible with SM pulses: potentials. Not feasible with SM pulses: 1μs1μs. . However, this relies on bath with Debye-like However, this relies on bath with Debye-like spectral density:spectral density:

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

( )I

UV

( )I

IR

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

Implies much relaxed time-Implies much relaxed time-constraints (K. Shiokawa & D.A.L., constraints (K. Shiokawa & D.A.L., quant-ph/0211081): time-scale set quant-ph/0211081): time-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 ( )t

1Bk

. .

0.1

50

20

IR

UV

Dyn Decoup

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

1/f, free

1/f, pulsed

Ohmic, free

Ohmic, pulsed

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

����������������������������

Four-Qubit DFS EncodingFour-Qubit DFS Encoding

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

This encoding protects against collective decoherence.

10 1 1 0

2ij i j i js

12 340

Ls 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 1 3 1 22 1

23X

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

n

z i

x x y y z z

S

H S B S B S B

, , )

Singlets: states with zero angularDecoherence

momentum -fr

totalee

, state

(

s:

x y z

J

J S S S

, , coherence ( (2))

only: Collective dephasing (abelian)}: Collective de{

z

x y z su

SS S S

int 1y yx x z z

i i i i i iyx z

i i in

iB B BH g g g

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 decoherence

conditions over blocks of 4 qubits.

Phys. Rev. Lett.

88

system-bath interaction (e.g., ) causes

logical errors ( ) and leakage.

Leakage part can be eliminated using Heisenberg

Bilinea

, with two pu

r

lses.i j

yxi j B

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

int 1

x x y y z zi i i i i i

yx zi i i

n

i

x x y y z z

B B BH g g g

S B S B S B

Heisenberg is Super-UniversalHeisenberg is Super-Universal

Heisenberg exchange is naturally available Heisenberg exchange is naturally available interaction for spin-coupled Q. dots, doped atom interaction for spin-coupled Q. dots, doped atom arrays.arrays.

It alone suffices forIt alone suffices for Universal QCUniversal QC Dynamical generation of collective decoherenceDynamical generation of collective decoherence Leakage eliminationLeakage elimination

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

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

The commutant of this algebra are The commutant of this algebra are the system-bath interactions that the system-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 commutant dynamically generate the commutant from an arbitrary system-bath from an arbitrary system-bath interaction. In this case the {interaction. In this case the {HHii} are } are “super-universal”.“super-universal”.

Heisenberg exchangeHeisenberg exchange

Group algebra of the permutation Group algebra of the permutation groupgroup

Collective decoherence processesCollective decoherence processes

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

Generation of collective decoherence Generation of collective decoherence from arbitrary linear system-bath from arbitrary linear system-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).

The role of the controllable The role of the controllable interactions is primary in universality interactions is primary in universality and combatting decoherenceand combatting decoherence

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.

Further Reading D.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).