Encoded Universality – Encoded Universality – an Overviewan Overview

Julia KempeUniversity of California, Berkeley

Department of Chemistry & Computer Science Division

Sponsors:

Encoded UniversalityEncoded Universality

Whaley group people:

Dave Bacon (now Caltech)Mike Hsieh (undergraduate) Julia Kempe (postdoc)Simon Myrgren (graduate student)Prof. Birgitta WhaleyJiri Vala (postdoc)Jerry Vinokurov (undergraduate)

Sponsors:

OverviewOverview

• Universal quantum computation - a bit of history• Change of paradigm• Example : Heisenberg interaction• Lie algebra formalism for encoded universal

computation• Results: Heisenberg interaction, symmetric “XY”

interaction, asymmetric “XY” with crossterms

Quantum circuitsQuantum circuits

++

= U

Barenco et al. ’95:

Single-qubit gates and CNOT generate every unitary transformation!

MantraMantra

The problemThe problem“Easy” and “hard” interactions (system-dependent)

“Easy”: intrinsic interactions “natural” to the system, easy to tune, rapid“Hard”: slower, require higher device complexity, high decoherence

System “Easy” “Hard”

Photon-qubitsSingle qubit operations

(linear optics)

Two qubit operations

(non-linearity, non-deterministic qc)

Solid state*Two qubit operations

(ex. J-gates)

Single qubit operations

(ex. local focused B-fields)

“Coherent-state” qubits1

Two qubit operations

(beam-splitter)

Single qubit operations

(non-linearity, non-det.)

*quantum dots, donor-atom nuclear spins, electron spins1 Ralph, Munroe and Milburn, 2001

Can we avoid “hard” interactions?

Almost every interaction is universal!Almost every interaction is universal!Deutsch et al.(’95), Lloyd (‘95) :

Almost Almost anyany interaction on two qubits is interaction on two qubits is universaluniversal..

In the generic sense. Does not include the most frequentes interactions.

Nature is not generic!Nature is not generic!

Hij Hji

qubit i qubit j qubit i qubit j

Change of paradigmChange of paradigm Traditionally:Traditionally: manipulate the physical system*

to produce

+ +

H1,H2,...

* Independent of system’s natural talents (fast, robust interactions) often difficult, certain gates can only be implemented with noise; high decoherence ...

Change of paradigmChange of paradigm Traditionally: manipulate the physical system*

to produce

+ +

H1,H2,...

Universal encoded computationUniversal encoded computation: interactions given by the physical system

find a way to make them universal

* Independent of system’s natural talents (fast, robust interactions) often difficult, certain gates can only be implemented with noise; high decoherence ...

HH HEncoding?Encoding?

Classical « Analogy »Classical « Analogy »

Two coins

can only flip the two coins together

« encode » « 0 »-« 1 »-

1 1

0 0flip

1 0 0 1 1 1

0 0

1 1

01

10

0 0

00 11 01 10

01

10

01

10 0

0

Encoded « coin »

Language of HamiltoniansLanguage of HamiltoniansU(t) = exp(iHt) Which interactions are universal?

Given =H1, H2,…, Hn can one generate any unitary transformation (exactly or approximatively)?

H has to generate the Lie algebra of su(N) of the unitary group SU(N)!

1) scalar multiplescalar multiple 2)

linear combinationlinear combination

3)

Lie bracketLie bracket

Ueee HitHitHit ...332211

iaHeaU )(ppibHpiaH

peee

bHaHi

}{lim 21)21(

ppiHpiHpiHpiH

p

HHi eeeee }{lim 212121 ],[

Heisenberg interactionHeisenberg interaction

• omnipresent in solid state physics (« Easy »)• is not universal: preserves the total spin of the

qubits

Lie algebra of E :Lie algebra of E :

On three qubits:

j

zyx

iijE

,,

10

01σ ,

0

0σ ,

01

10σ zyx i

i

0110 E

],[: );2(121:

)(34

1: ;:

1321323123

231312313120

HHiHEEEH

EEHEEEH

kijkji

i

HiHH

HH

],[

0],[ 0

},,{ 321 HHH su(2)

E12

E13

E23

(Pauli matrices)

The algebra The algebra L(E)L(E) of E (3 qubits) of E (3 qubits)

the algebra L3(E) splits as:

L3(E) L3(E) S1I4 S2I2

su(2)S2

Encoded qubit ?

su(2)

su(2)2

2

1000102

10 L

10001000126

11 LSimulation of all operations of one qubit (su(2)) with L3(E) on the encoded qubit !

The algebra The algebra LLnn(E)(E) of E (n qubits) of E (n qubits)the algebra Ln(E) splits as:

Ln(E) ...

...

... 12

2

)21( 0 )(

jn

n

j

n IELj

S

Commutant L’ of Ln(E) : )}( 0],[:{' ELLLAAL n

L’ is generated by (« spin » algebra su(2)) 1

n

i

nS

12

2

)21( 0' '

jn

n

jSIL

j

As a Lie algebra L’ splits into irreducible representations of

su(2).

Useful theoremUseful theoremLet S be a †-closed algebra closed under multiplication and linear

combination. Then the underlying space H is isomorphic to

such that S and its commutant S’ split as:

where M(Cd) (M(Cn)) is the algebra of all matrices on Cd (Cn).

jdj CC n Jj

Η

j

j

dn ICM

)( S

j J )( S'

j

j

j

dn CMI

J

...

...

...

1( )nM C

2( )nM C

( )knM C

Universal computation “for free”?

( ) ( )jn

jsu n M C

Useful theoremUseful theoremLet S be a †-closed algebra closed under multiplication and linear

combination. Then the underlying space H is isomorphic to

such that S and its commutant S’ split as:

where M(Cd) (M(Cn)) is the algebra of all matrices on Cd (Cn).

jdj CC n Jj

Η

j

j

dn ICM

)( S

j J )( S'

j

j

j

dn CMI

J

jj dn I

S LJj

The multiplicative algebra is not at our disposition!However the Lie algebra splits into irreducible components in the same basis:

NO!

Problem of “Encoded Universality”Problem of “Encoded Universality”

Given an ensemble of generators H with Lie algebra L(H) which splits as

can one find a component s.t. contains su(nj )?

Encode the quantum information into the corresponding sub-space.

dimension: nj

jj dn I

S LJj

jj dn ISjnS

...

...

...

1nS

2nS

knS

2nS

( ) ?jj nsu n S

Yes

D. Bacon, J. Kempe, D.P. DiVincenzo, D.A. Lidar, K.B. Whaley, “ Encoded Universality in Physical Implementations of a Quantum Computer”, Proceedings of IQC ’01, Australia

Previous Results - Heisenberg interactionPrevious Results - Heisenberg interaction

• E is universal with encoding*• introduce tensor structure, ex. blocks with 3 qubits**

*Kempe, Bacon, Lidar, Whaley, Phys. Rev. A 63:042307 (2001)

**DiVincenzo, Bacon, Kempe,Whaley, NATURE 408 (2000)

,i j i j E

efficient implementation of encoded gates:

numerical search**serial coupling - 19 operations for CNOT, 4 operations for 1-qubitparallel coupling - 7 operations for CNOT, 3 operations for 1-qubit

,exp i jiJE ti

j

exchange gate

DiVincenzo, Bacon, Kempe, Burkard, Whaley, Nature 408, 339 (2000)

Nearest neighbor exchange coupling

Tradeoffs

factor of 3 in space (encoding)

factor of ~ 10 in time

Exchange-only CNOTExchange-only CNOT

Conjoining – a new tensor structureConjoining – a new tensor structureIntroduce a cutoff that defines a single “qudit”.

In principle: ...

...

...

1nS

2nS

knS

2nS

For larger n one could find larger component with

better encoding ratio ?

knS

2log (dim ) /knr S n

Need to guarantee uniformity of quantum circuits!(“Form” of the circuit should not depend on size of problem.)Introduce cutoff -> tensor product structure.Conjoining subsystems: 2nS 2nS

knS

2

ij i j i jXY x x y y ij ijij

JH J A

encode into qutrit: 0 100

1 010

2 001

L

L

L

12 23 31operators , ,L L LA A A

12

0 1 0

. ., 1 0 0

0 0 0

Le g A

construct , (3)L LA A su i.e., 1-qutrit operations

conjoin qutrits: 0 1 9 subspaceL L D • HXY generates su(9) on this subspace

“Truncated qubit”: use and only

effectively with an ancillary qubit for gate-applications

*J. Kempe, D. Bacon, D.P. DiVincenzo, K.B. Whaley, “ Encoded universality from a single physical interaction”, in «Quantum Information and Computation»; Special Issue, Vol. 1, 2001

Anisotropic Exchange*Anisotropic Exchange*

“XY”-interaction

0 100L 1 010L

0 10L 1 01L 0

The Gates*The Gates*Gate sequences: 7 operations for single qubit operations (serial)

5 operations for Sqrt (-ZZ) (equiv. to controlled phase)

“P3”-gate:

P3() =/4

/2

-/4

-/2

/2

Truncated qubit:

xie

0 10L 1 01L

zie 0

P3(-)0

1 32x xzi iiU e e e

=

=

(Euler angles)

Two-qubit operation:

*J. Kempe and K.B.Whaley, “Exact gate-sequences for universal quantum computation using the XY-interaction alone ”, quant-ph/0112014, to appear in Phys. Rev. A

Single qubit operations:

P3(-/2)z z =

Layout – Anisotropic ExchangeLayout – Anisotropic Exchange a) triangular array

(qutrit)

b) “truncated qubit”

or

Results: Asymmetric Anisotropic Results: Asymmetric Anisotropic Exchange* Exchange*

*J. Vala and K.B. Whaley, “Encoded Universality with Generalized Anisotropic Exchange Interactions”, in preparation 2002

Poster No. 21 by Jiri Vala

Poster No. 21 by Jiri Valai j i j

ij x x x y y yA J J

i j i j i j i jij x x x y y y xy x y yx y xC J J k k

Asymmetric exchange:

Asymmetric exchange with crossterm:

Universal Encodings and Gate-SequencesUniversal Encodings and Gate-Sequences

Results: Asymmetric Anisotropic Exchange*Results: Asymmetric Anisotropic Exchange*

|000> |011>

|101>|110>

|111> |100>

|010>|001>

1 4

2 3

1 4

2 3

H4 H4

h12 H12 h12 H12

h23

H23

h23

H23

h13H13h13 H13

The total exchange Hamiltonian consists of two components:

1) symmetric, which couples the physical qubit states |01> and |10>

Hij = J ( x,i x,j + y,i y,j ) + K ( x,i y,j - y,i x,j )

2) and antisymmetric , coupling the states |00> and |11>

hij = j ( x,i x,j - y,i y,j ) + k ( x,i y,j + y,i x,j )

which both simultaneously transform pairs of code words in two code-subspaces.

code space I code space II

*Vala and Whaley, in preparation

This allows to apply similar techniques as in the symmetric XY-case!

Poster No. 21 by Jiri ValaPoster No. 21 by Jiri Vala

Is encoded universality always Is encoded universality always possible?possible?

NO!• non-interacting fermions (Valiant, Terhal&DiVincenzo, Knill ’01)• nearest-neighbor XY-interaction• linear optics quantum computation

Criterion:If a set of Hamiltonians (over n qubits) allows for (encoded) universal computation then the Lie algebra L(H) contains exponentially many linearly independent elements.

...

...

...

1nS

2nS

knS

2nS

Some component has to contain where is a polynomial function of n.( )ksu n

2log ( )knknS

ex: is not universal with any encoding.

1{ , }i i iz x xH

Continuing WorkContinuing Work

How find the gate sequences that implement the encoded one- and two-qubit gates?

Numeric search – genetic algorithms (Hsieh, Kempe)

Developed numeric tools:Preliminary results in 4-qubit encoding for Heisenberg interaction:

Results of Numerical RunsTime Exchange Population Operation Mutation

Name Parameters Coordinates Size Length Generations Rate Fitness

Gene14 Varied Fixed 60 66 500 0.05 M = 7.20e-9DoubleGene Varied Varied 60 30 150 0.05 L = 3.117

Poster No. 38 by Mike Hsieh

Poster No. 38 by Mike Hsieh

SummarySummary• Lie-algebra methods and redefinition of the tensor structure of

Hilbert space allow for universality!• Use of encoding - the implementation of one-qubit gates is

obsolete! (change of paradigm)• Heisenberg interaction is omnipresent (e.g. in solid state

physics) and easy to implement, whereas one-qubit gates are extremely hard to obtain encoded universality gives attractive qc proposals

• Evaluation of the trade-offs in space and time

Open Questions:• Find better than ad hoc ways to generate the gate sequences• General easy criteria to determine encoded power of

interaction (e.g. amount of symmetry…)• a general theory allowing for measurements and prior

entanglement (to incorporate Briegel/Rauschendorff and Nielson schemes into analysis)

ReferencesReferencesJ. Kempe and K.B.Whaley, “Exact gate-sequences for universal quantum computation using the XY-interaction alone ”, quant-ph/0112014 , to appear in Phys. Rev. A

J. Kempe, D. Bacon, D.P. DiVincenzo, K.B. Whaley, “ Encoded universality from a single physical interaction”, in «Quantum Information and Computation»; Special Issue, Vol. 1, 2001, quant-ph/0112013

D. Bacon, J. Kempe, D.P. DiVincenzo, D.A. Lidar, K.B. Whaley, “ Encoded Universality in Physical Implementations of a Quantum Computer”, Proceedings of IQC ’01, Australia, quant-ph/0102140

D.P. DiVincenzo, D. Bacon, J. Kempe, K.B. Whaley, “ Universal Quantum Computation with the Exchange Interaction”, NATURE 408, 339 (2000), quant-ph/0005116

J. Vala and K.B. Whaley, “Encoded Universality with Generalized Anisotropic Exchange Interactions”, in preparation 2002

Earlier related work on DFSs:J. Kempe, D. Bacon, D. Lidar,K.B. Whaley, Phys. Rev. A 63:042307 (2001)

D. Bacon, J. Kempe, D. Lidar, K.B. Whaley, Phys. Rev. Lett. (2000)

Conclusions/Open Questions• Encoding into sub-spaces allows to make certain

interactions universal

• Representation theory of Lie groups - powerful tool

• E and XY alone are universal - important simplification of physical implementations

Which other interactions to investigate?

General theory when interactions allow for encoded universality?

How find the gate sequences that implement the encoded one- and two-qubit gates?

• tensor product of encoded qubitsconjoined codes Bacon et al., PRL 85, 1758 (2000)

Bacon et al., quant-ph/0102140

• find entangling operations Lie algebraic analysis Kempe et al., PRA 63, 042307 (2001)

Kempe et al., JQIC (2001)

• efficient implementationnumerical search e.g. Heisenberg exchange

serial coupling - 19 operations for CNOT, 4 operations for 1-qubitparallel coupling - 7 operations for CNOT, 3 operations for 1-qubit

DiVincenzo, Bacon, Kempe, Burkard, Whaley, Nature 408, 339 (2000)

2

ij i j i jXY x x y y ij ijij

JH J A

encode into qutrit: 0 100

1 010

2 001

L

L

L

12 23 31operators , ,L L LA A A

12

0 1 0

. ., 1 0 0

0 0 0

Le g A

construct , (3)L LA A su i.e., 1-qutrit operations

conjoin qutrits: 0 1 9 subspaceL L D HXY generates su(9) on this subspace

(Kempe, Bacon, DiVincenzo, Whaley IQC’01)

ResultsResults• tensor product of encoded qubits: conjoined codes

(Bacon, Kempe, DiVincenzo, Lidar, Whaley ICQ’01)•universality: Lie algebraic analysis

(Kempe, Bacon, DiVincenzo, Whaley IQC’01)

Anisotropic Exchange Interaction:

Single-Qubit and Two-Qubit GatesSingle-Qubit and Two-Qubit Gates1) the full su(2) algebra over a single logical qubit is generated viathe commutation relations between exchange interactions over physicalqubits: e.g. [H13,H23] = i (J2 - j2) y,12 and [H12 , y,12 ] = i 2 J z,12

2) entangling two-qubit operation C(Z) results from application ofthe encoded z operation ontothe physical qubits 2-3-4 inthe triangular architecture andsingle-qubit operations

3) the commutation relations are applied via selective recoupling

4) a similar construction is valid for a general anisotropic interaction containing the cross-product terms

1 2

3 4 5

6

LOGICAL 1

2LOGICAL