Quantum Communication & Computation Using Spin Chains

Quantum Computation Part: S. C. Benjamin & S. Bose, quant-ph/0210157 (to appear in PRL)

Quantum Communication Part:S. Bose, quant-ph/0212041

Sougato BoseInstitute for Quantum Information, Caltech

& UCL, London

1D Bulk Magnets are Natural Spin Chains (Examples):

Cu (spin

½ sites)

Isotropic Heisenberg Antiferromagnet:

P. R. Hammar et. al., PRB 59, 1008 (1999).

Quantum computation using a 1D magnet

Quantum computationby applying a time varyingand inhomogeneous magneticfield to a spin chain.

Heisenberg Chain to Ising Chain Conversion:

Heisenberg

Ising

A, B = Zeeman Energies,|A-B| >> J

If

Where,

ThenImplies

Positions of Qubits & Barrier Spins

A barrier spin A qubit

Case A: When Universal Local Gates Are Possible:

The Ising interaction on each qubit is then completely cancelled atall times. Note: Both barrier spins could be in the same state (whichis easier to initialize, with periodic cancellation of Ising effects.

iε are variable energies, set to B in the passive statewhen single qubit gates are performed.

Case A: When Universal Local Gates Are Possible:

)(t1 2 3 4 5

Bt =)(εJAt +=)(ε

For a Gate between X & Y,

is changed (fast) to

Then 3 becomes resonant with 2 & 4 (2,3,4 becomea small Heisenberg chain).

Case A (Contd.)

At time

1 2 3 4 5

2 & 4 disentangle from 3.

An entangling gate between X and Y !

Use techniques of: M. J. Bremner et. al., quant-ph/0207072.J. L. Dodd et. al., PRA 65, 040301 (2002).

Case B: When Only Zeeman Energy Tuning is Possible Locally:

Method for one qubit gates:

Method for two qubit gatesCase B: When Only Zeeman Energy Tuning is Possible Locally:

Global control quantum computation schemes of Lloyd & BenjaminS. Lloyd, Science 261, 1569 (1993); S. C. Benjamin, PRL 88, 017904 (2002).

One Qubit Gates Two Qubit Gates

Case B: When No Local Ability is Present:

Control Switch of Six Settings

Control Through the Strength of a Single Field

Alice Bob

BobAlice

Quantum Communication through a Spin Chain

Avoids interfacing solid state systems with optics for the purpose of short-distance communication:

Quantum Computer

Quantum Computer

Definition of Spin-Chains:

(B) “Always On” (untunable) interactions

(A) 1D array of spins

Makes it much easier to fabricate such systems withqubit arrays (especially in solid state) than to perform arbitrary quantum computations.

s

r

H Jiji j

i j=< >∑

,

.σ σ

First consider arbitrary graphs with ferromaneticHeisenberg interactions

Initialized in the ground state0 = ≡| ... ,000 00 0

12

N

with H BB j zj

= − ∑σ ,

s

r1

2

N

At t = 0,

Ψ ( ) | .. (cos sin ) ..0 00 02

02

1 00 0= +θ θφei

1,2,..,s-1 s s+1,…,N

1 ≡

Time evolution of the spin-graph:

Ψ ( ) cos sin ( ),t e e f ti iBtj sN

j

N

= + −

=∑θ θφ

2 22

1

0 j

where, 0 = 00 0... j = 00 010 0.. ..,

j th spinf t ej s

N iHt, ( ) = −j sand

is the transition amplitude of an excitation from the s thto the j th spin due to H.

Note that only the ground & N one-excitation states of the graph are invloved (because H does not createexcitations, only propagates excitations).

s

r12

N

ρ ψ ψr r r( ) ( ) ( ) ( ) ( ( ))t P t t t P tout out= + −1 0 0

P t f tN( ) cos | ( )| sin= +2 2 2

2 2θ θ

r,s

ψθ θφ

outi iBt Nt

P te e f t( )

( )(cos sin ( ) )= +

12

02

12r r,s r

where, ,

B should be chosen so that f t f tN Nr,s r,s( ) | ( )|⇒

The graph of Heisenberg interacting spins behaves as an amplitude damping quantum channel:

M f tN0

1 00=

| ( )|r,sM f tN

1

20 10 0

= −

r,s ( ),

Fidelity averaged over the Bloch Sphere:

F t f t f tin out in= = + +∫1

412

13

16

2

πψ ρ ψ( ) ( ) ( )r,s

Nr,sN

Entanglement (Concurrence) for input of one half of a Ψ ( )+

E f tC = r,sN ( )

Exceptionally simple formulae in terms of a singletransition amplitude

We will consider two cases:A linear chain with communicating parties at opposite ends(most natural and readily implementable case):

1 NAlice Bob

A closed loop with the communicating parties at diametrically opposite ends (to compare):

Alice Bob1

NN/2

HJ

j

N

j j= −=

−

+∑2 1

1

1σ σ.

HJ J

j

N

j j N= − −=

−

+∑2 21

1

1 1σ σ σ σ. .

Eigenstates in the one excitation sector:

~ cos ( )( )m aN

m jmj

N

= − −

=

∑ π2

1 2 11

j

m N= 1,..., aN1

1= a

Nm> =1

2

(A) Linear (open chain) case:

for with and

(B) Closed chain case:

~ ( )m

Ne

iN

m j

j

N

=−

=∑1 2

1

1

π

j

m N= 1,...,for

(A Quantum Cosine Trans.)

(A Quantum Fourier Trans.)

| | | ( , ( , ))|f DCT N v tmr,sN

s r=

v t am

Nem m

i JtmN( , ) cos

( )(

cos( )

r r -1)=−

−

π π12

22

1

| | | ( , ( ))|f DFT N v tmr,sN

r-s=

v t em

i JtmN( )

cos( )

=−

22 1π

Transition amplitudes in terms of readily computable transforms:(A) Linear (open chain) case:

where

(B) Closed chain case:

where

aN1

1= a

Nm> =1

2and

Fidelity, Entanglement

Log of Scaled time

Alternative formulas in terms of Bessel functions:

f J Jt J Jt

J Jt

NN Nk

N kk

NkN k

k

N

, ( ) ( )'( ) ( ) ( ) ( )

( )

1 10

2

10

2

2 1 2 1 2

2 2

= −

+ −

≈

+=

∞

+=

∞

∑ ∑

f J Jt

J Jt

NN N k

N kk

N

/ ,( / )

( / )( )

/

( ) ( )

( )

2 12

2 2 10

2

2 1 2

2 2

= −

≈

+=

∞

∑

at the maximum near 2Jt=N

at the maximum near 2Jt=N/2

Open chain:

Closed chain:

1. Closed chain 2N ≤Open chain N

2. Can find high fidelity transfer at 2Jt=NN E Jt

N E JtC

C

= ⇒ = =

= ⇒ = =

10 2 1005 013

10 2 10017 0 06

3

4

( ) .

( ) .

⇒

Distillable

Possible Future Work

1.Sending higher D systems (Ex: 4 state systems by using up to 2 spin excitations --- with Korepin).

Q.Comm part:

2. Study Graphs which improve comm. fidelity (suggested by Preskill)

3. Direct qubit comm. (without distillation) over arbitrary distances by using spin-1 chain (--- with Thapliyal).

4. Can measurements on the chain improve transfer? (suggested byVerstraete).

Q.Compu part:

2D extensions, extensions to clusters replacing single qubits etc.--- for greater fault tolerance and robustness.