Download ppt - University of Toronto Physics

Trapped Ions and the Cluster State
Paradigm of Quantum Computing
Universität Ulm, 21 November 2005
Daniel F. V. JAMES
Department of Physics, University of Toronto,
60, St. George St., Toronto,
Ontario M5S 1A7, CANADA
Email: [email protected]
Standard Paradigm for Quantum Computing
You can do ANYTHING if you can do the following things
with initialized qubits:
• Unitary operations on any individual qubit:
A+ B1  A' + B '1
U
• Two qubit gates such as the “Controlled Z gate”
a + b1  c1 + d11 
a + b1  c1 - d11
• Projective measurement of each qubit:
i.e.
A0+ B1  0 (probability P0=|A|2)
OR A0+ B1  1 (probability P1=|B|2)
Z
DiVincenzo’s Five Commandments*
1. Scalable physical system with well-characterized
qubits.
2. Ability to initialize the state of the qubits in some
fiducial state.
3. Long (relative) decoherence times, much longer
than gate-operation time.
4. Universal set of quantum gates (e.g. arbitrary one
qubit operations + CNOT with any two qubits).
5. Qubit-specific measurement capability.
*D. P. DiVinzenco, Fortschr. Phys. 48 (2000) 771-783
DiVincenzo’s Criteria
1.
S
ch cab
ar le
ac , w
te e
riz ll2.
ed
In
itia
qu
liz
bit
3.
at
s
ion
Lo
tim ng
es de
co
4.
he
Un
re
nc
ive
e
rs
a
lg
5.
at
M
es
ea
su
re
m
en
t
Roadmap Traffic-Light Diagram
(Apr 2004) - updated
• watch these spaces
NMR
Experimental
reality
Trapped Ion
Theoretical
Possibility
Neutral Atom
No known
viable
approach
Optical
Solid State
Superconducting
Cavity QED
Do we need a 6th Commandment?
• Shor’s Algorithm is the the “killer app”.
• State of the Factoring Art with Conventional Computers:
RSA-155 (512 bits) factored on a distributed network with a
number field sieve in 3.7 months (9.0 106 sec) [1].
• Quantum factoring (without error correction) of a N-bit
number requires ~ 544 N3 two qubit quantum gates [2].
9.0 106
544 512
3
 123 sec
• Sixth Commandment: for quantum computers to be useful,
quantum gates need to take less than 1 microsecond.

[1] Factorization of RSA-155, www.rsasecurity.com/rsalabs/challenges/factoring/rsa155.html
[2] R. J. Hughes, D. F. V. James, E. H. Knill, R Laflamme and A. G. Petschek, Phys. Rev. Lett.
77, 3240 (1996), eq.(7).
What’s the Speed Limit for Trapped Ions?
• Time for 2-ion logic gates is limited by need to resolve
different oscillation modes in frequency [1]:
Tgate ftrap 2
  1

 2.6

• Trapping frequency is limited by the need to spatially
resolve individual ions with the laser [2]:
1/ 3



2
q
ions
  F

2 
4 M
 0 ions 2ftrap  
• Bottom line: you’re limited to ~10 MHz
 Appl. Phys. B 66, 181 (1998).
[1] D. F. V. James,
[2] R. J. Hughes, D. F. V. James, E. H. Knill, R Laflamme and A. G. Petschek,
Phys. Rev. Lett. 77, 3240 (1996).
It gets worse...
• Gates in scalable (multi-trap) architectures have fivesteps:
1. Extract two ions from “storage” trap.
2. Move ions to “logic” trap.
3. Sympathetic cooling.
4. Perform logic gate.
5. Return ions to “storage” trap.
Moving Trapped Ions Quickly
1 2 1
2
2 ˆ
ˆ
ˆ
H
p  m x  s t 
2m
2
displacement of trap center


• Solution:
t   e i t Dˆ  t  0
 t   i
t
m
s t e it dt 

2 0
displacement operator
Fidelity of the Ground State after motion:

2 

t
1

F t   exp  2  sÝt e it dt  

 4
0


width of the ground state 
2m
s t 

L


T<<1/
10/19
Are cluster states the answer?*
Definitions:
• Number of qubits in a circuit = breadth, m
• Number of gates in a circuit = depth, n
Claim:
For any quantum circuit there exists a pure state (m,n)
such that:
• (m,n) involves O(m.n) qubits
• (m,n) can be prepared with poly(m.n) resources
• Local measurement in an appropriate basis + feed
forward simulates the quantum circuit.
*R. Raussendorff and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001); (also unpublished notes
by M. A. Nielsen).
Circuit Identities
• 1. Transport Circuit:
 
Rz()

H
XmH Rz() 
Z
• 2. Discard Circuit

 

Z


m
H

 


m

Zm
Circuit Identities
• 3. Indirect Entangling Gate:


H

H

Z
Z
H
c
H
d
Z




Zd
Z
Zc
3x4 Cluster State
1.UA
2.UB
3.UC
• Each circle represents a qubit.
• Prepare each qubit in state |0.
• Perform Hadamard Gate (AKA pulse) on each qubit.
• Perform Controlled-Z between neighbors.
Notation: Unitary UA followed by measurement;
then UB followed by measurement,
then UC followed by measurement.
Single Unitary
 
Rz()
Rx()
U

1. H Rz()

H

H

H
2. H Rz(’)
Rz()
m1
H
Rz(’)
Z
Z
H
m2
Remember the Circuit Identities:
• 1. Transport Circuit:
 
Rz()

H
Z
H
m
XmH Rz() 
Single Unitary
 
Rz()
Rx()
U

1. H Rz()
 

2. H Rz(’)
Rz()
H

’=(-1)m1
m1
H
Rz(’)
Z
H
Z
H
m2
Xm2H Rz(’)Xm1H Rz() 
output becomes Rx()Rz()
Simple 2 Qubit Circuit
 
 
U
Z
U
This needs a 4 x7 Cluster State:
1.I
1.I
1.I
1.I
1.I
1.I
1.I
1.I
1.I
1.I
1.I
1.I
Step 1: measure indicated qubits and correct for discard
Remember the Circuit Identities Again
• 2. Discard Circuit

 

Z




 

Zm
m
 shifts on some of
(so we’ll need to correct for the phase
the qubits)

You get this Cluster State:
2.HRz()
3.HRz(’)
4.H
4.H
2.H
3.H
Step 2 &3: perform single qubit unitary as before
Step 4: Measurement on linking qubits to perform two qubit
gate operation
Remember this one?


H

H

Z
Z
H
c
H
d
Z


 

Zd
Z
Zc
2.HRz()
3.HRz(’)
5.H
6.H
7.H
8.H
6.H
7.HRz()
8.HRz(’)
4.H
4.H
2.H
3.H
5.H
Step 5&6: propagate the quantum information
Step 7&8: perform second unitary
Implications
• Quantum Computing is reduced to initially creating
a big-ass entangled state, then local unitatries and
measurement.
• This is a natural for optical quantum computing.
• What about trapped ions?
- Number of Controlled Z gates reduced to 4 total!
- Trap configuration can be optimized for cluster state creation
- Will need a lot more ions
- Basis requirements (read out and fast feed-forward) already
demonstrated in teleportation experiment.
- Can measurement be fast enough?