Topological Quantum Computation Parsa Bonderson Microsoft Station - - PowerPoint PPT Presentation
Measurement-Only Topological Quantum Computation Parsa Bonderson Microsoft Station Q DIAS Theoretical Physics Seminar August 21, 2008 work done in collaboration with: Mike Freedman and Chetan Nayak arXiv:0802.0279 (PRL 08) and
work done in collaboration with: Mike Freedman and Chetan Nayak arXiv:0802.0279 (PRL „08) and arXiv:0808.1933
Microsoft Station Q DIAS Theoretical Physics Seminar August 21, 2008
two dimensional systems:
condensates, quantum loop gases/string nets?
providing naturally (“topologically protected”) fault- tolerant hardware.
quantum computation?
(unitary braided tensor categories)
Describe quasiparticle braiding statistics in gapped two dimensional systems.
Finite set of anyonic charges:
c ab
N
Fusion multiplicities are integers specifying the dimension of the fusion and splitting spaces
c c ab
ab c c ab V
V ,
C
c b a , ,
Fusion rules: Unique “vacuum” charge, denoted has trivial fusion and braiding with all particles.
I
Hilbert space construct from state vectors associated with fusion/splitting channels of anyons. Expressed diagrammatically: Inner product:
' cc
c ab c
ab
Can be non-Abelian if there are multiple fusion channels c
: rules Fusion ) 1 , , (a.k.a. , , : types Particle ? ruthenates , insulators al topologic honeycomb, Kitaev
d Slingerlan and (PB FQH? 2LL
and
Read
FQH
SU
anyons Ising
2 1 5 12 2 5 2
n n I = =
I I I I I
5 12 3
I I I I
c
1
c a a a a
(Kitaev, Preskill, Freedman, Larsen, Wang)
a a
1
1
c
1
c a a a a
(Bonesteel, et. al.)
(Kitaev, Preskill, Freedman, Larsen, Wang)
time
a a
(must be supplemented)
c
1
c a a a a
(Bonesteel, et. al.)
(Kitaev, Preskill, Freedman, Larsen, Wang)
Topological Charge Measurement
time
a a
e.g. loop operator measurements in lattice models, energy splitting measurement
c
c c
a b
c
e.g. 2PC FQH, and Anyonic Mach-Zehnder (idealized, not FQH)
Asymptotically characterized as projection of the target‟s anyonic charge AND decoherence of anyonic charge entanglement between the interior and exterior
(for projective measurement)
Entanglement Resource: maximally entangled anyon pair
Want to teleport: Form:
=
23 1
; , I a a
and perform Forced Measurement on anyons 12
a a
a
a a a
=
23 1
; , I a a
(projective)
a a a
( :
12 I
a a a
1
e
2
f
(
12
1
e
(
23
2
f
(
12 I
a a a
1
e
a a a
2
f
a a a
I
(projective)
a a a
( :
12 I
a a a
1
e
2
f
(
12
1
e
(
23
2
f
(
12 I
(projective)
a a a
=
3 12
; , I a a
“Success” occurs with probability for each repeat try.
2
1
a
d
( :
12 I
23 1
; , I a a
( ( :
12 12 I I
a a
a a
a a a a
I
a a a a
) 23 ( I
) 13 ( I
) 34 ( I
) 23 ( I
a a a a a a a a
) 23 ( I
I
) 13 ( I
) 34 ( I
) 23 ( I
a a a a a a a a
) 23 ( I
) 13 ( I
) 34 ( I
) 23 ( I
I
a a a a a a a a
) 23 ( I
) 13 ( I
) 34 ( I
) 23 ( I
I
a a a a a a a a
=
) 23 ( ) 13 ( ) 34 ( ) 23 ( ) 14 ( I I I I
R
c
1
c a a a a
Topological Charge Measurement
time
a a
c
1
c a a a a
Topological Charge Measurement
a a
Topological Charge Measurement
e.g. 2PC FQH, and Anyonic Mach-Zehnder (idealized, not FQH)
Asymptotically characterized as projection of the target‟s anyonic charge AND decoherence of anyonic charge entanglement between the interior and exterior
int
For a inside the interferometer and b outside:
c
int
int
int
Using Interferometric Measurements is similar but more complicated, requiring the density matrix description. The resulting “forced measurement” procedure must include an additional measurement (of 8 or fewer anyons, i.e. still bounded size) in each teleportation attempt to ensure the overall charge of the topological qubits being acted upon remains trivial. Note: For the Ising model TQC qubits, interferometric measurements are projective.
(in FQH)
(needs one gate supplement)
Dn=5/2 ~ 600 mK
(braiding = Clifford group)
(from any gates)
(2 anyon measurements)
distinguishing I and
(precise phase calibration)
(needs one gate supplement)
(see Bonesteel, et. al.)
(from entangling gates)
(2,4,8 anyon measurements)
distinguishing I and
(amplitude of interference)