Quantum Communication & Computation Using Spin...
Transcript of Quantum Communication & Computation Using Spin...
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).
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 ,
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
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.