1. Adiabatic Gate Teleportation
& Its Applications
Dave Bacon
Department of Computer Science & Engineering
University of Washington, Seattle, WA USA
[joint work with Steve Flammia (Perimeter),
Alice Neels (Washington), Andrew Landahl (Sandia Labs)]
13. Alternative History
Imagine: no transistors, no integrated circuits...
Could we build a computer out of unreliable components?
J. von Neumann, “Probabilistic Logics and the Synthesis of
Reliable Organism from Unreliable Components” 1956
14. Von Neumann’s Solution
1x 1x
2x 1y M
3x 1z
4x 2x M
5x 2y M
1y
2z M
2y 3x
3
My 3y M π M
4y 3z
5y 4x M
4y M
1z 4z
2z M
3z 5x
4z 5y M
5z 5z
gate error correction
Theorem (von Neumann): A circuit of g gates
can be made to fail with probability ε using
O(g log (g/ε) ) gates which fail with probability
p, assuming p<pth (vN estimated pth ~ 1%)
15. Engineering? We Don’t Need
Your Stickin’ Engineering
There are distinct physical reasons why robust
classical computation is possible
16. Kitaev’s Idea
“Magnetism arise from spins of individual atoms. Each spin is
quite sensitive to thermal fluctuations. But the spins interact
with each other and tend to be oriented in the same
direction. If some spin flips to the opposite direction, the
interaction forces it to flip back to the direction of other
spins. This process is quite similar to the standard error
correction procedure for the repetition code. We may say
that errors are being corrected at the physical level. Can we
propose something similar in the quantum case? Yes, but
it is not so simple.”
Alexei Kitaev
Kitaev, “Fault-tolerant quantum computation by anyons” (1997, published 2003)
17. Topological Quantum
Computing
non-abelian Anyons used to build a
universal quantum computer
Kitaev, “Fault-tolerant quantum computation by anyons” (1997, published 2003)
Freedman, “Quantum computation and the localization of modular functors” (2001)
18. Topological Quantum
Computing
non-abelian Anyons used to build a
universal quantum computer
√1 √1
H= √1
2 2
1
− √2
quantum computing
2
using representations
of braid group
Kitaev, “Fault-tolerant quantum computation by anyons” (1997, published 2003)
Freedman, “Quantum computation and the localization of modular functors” (2001)
21. Controversy
Is topological quantum computing fault-tolerant?
gate
2x
1x 1x
1y M
error correction
3x 1z
4x 2x M
5x 2y M
1y
2z M
2y 3x
3
My 3y M π M
4y 3z
5y 4x M
4y M
1z 4z
2z M
3z 5x
4z 5y M
5z 5z
Theorem (von Neumann): A circuit of g gates
can be made to fail with probability ε using
O(g log (g/ε) ) gates which fail with probability
p, assuming p<pth (vN estimated pth ~ 1%)
22. Controversy
Is topological quantum computing fault-tolerant?
gate
2x
1x 1x
1y M
error correction
3x 1z
If we use this definition of fault-tolerant, M
4x 2x
5x 2y M
1y
2z M
then I believe the answer is NO!
2y 3x
3
My 3y M π M
4y 3z
5y 4x M
4y M
1z 4z
2z M
3z 5x
4z 5y M
5z 5z
Theorem (von Neumann): A circuit of g gates
can be made to fail with probability ε using
O(g log (g/ε) ) gates which fail with probability
p, assuming p<pth (vN estimated pth ~ 1%)
23. Controversy
Is topological quantum computing fault-tolerant?
gate
2x
1x 1x
1y M
error correction
3x 1z
If we use this definition of fault-tolerant, M
4x 2x
5x 2y M
1y
2z M
then I believe the answer is NO!
2y 3x
3
My 3y M π M
4y 3z
5y 4x M
4y M
1z 4z
2z M
3z 5x
4z 5y M
5z 5z
Theorem (von Neumann): A circuit of g gates
can be made to fail with probability ε using
But I also think this is the wrong definition
O(g log (g/ε) ) gates which fail with probability
p, assuming p<pth (vN estimated pth ~ 1%)
24. Cartoon Guide to Classical
Physical Fault-Tolerance
Energy
E = −J si sj
i,j
neighbors
total magnetization
M= si
i
+1
spin “0” as M > N0
−1
si ∈ {+1, −1} “1” as M < −N0
25. Cartoon Guide to Classical
Physical Fault-Tolerance
Simple noise model
26. Cartoon Guide to Classical
Physical Fault-Tolerance
Simple noise model
1. Pick spin at random
27. Cartoon Guide to Classical
Physical Fault-Tolerance
Simple noise model
1. Pick spin at random
2. If flipping spin would
decrease energy
a. flip spin
28. Cartoon Guide to Classical
Physical Fault-Tolerance
Simple noise model
1. Pick spin at random
2. If flipping spin would
decrease energy
a. flip spin
3. If flipping spin would
increase energy:
“bare” noise time step
a. flip spin with probability
Nr runs of 1-3
exp(−β∆E) β=T −1
29. β=T −1
Memory
1D 2D
m versus time
1.0
0.5
0.0
m
0.5
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
time
rate ~ exp(−cβJ)
m versus temperature
1.0
0.5
0.0
m
0.5
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
temperature
30. β=T −1
Memory
1D 2D
m versus time m versus time m versus time
1.0 1.0 1.0
0.5 0.5 0.5
0.0 0.0 0.0
m
m
m
0.5 0.5 0.5
1.0 1.0 1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
time time time
rate ~ exp(−cβJ) T < TC T > TC
m versus temperature
1.0 m versus temperature
1.0
0.5
0.5
0.0
m
0.0
m
0.5
0.5
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
temperature
temperature
31. Fault-Tolerance
1D 2D
m versus time m versus time
1.0
1.0
0.5 0.5
0.0
m
0.0
m
0.5 0.5
1.0
1.0
0.0 0.5 1.0 1.5 2.0 2.5
0.0 0.5 1.0 1.5 2.0 2.5
time time
Resilience to “gate” of flipping spins: self-correcting
33. Toy Model / Strawman
plaquette operator
Sp = Ze
e∈δP
vertex operator
Sv = Xe
e|v∈e
[Sp , Sv ] = 0 Sp = Sv
2 2
=I
periodic boundary Kitaev Toric Code Hamiltonian:
conditions (torus) ∆
H=− Sp + Sv
4 p v
34. A Problem?
Toric code is like 1D Ising:
m versus time
1.0 “On Thermalization in Kitaev’s 2D
0.5
model” R. Alicki, M. Fannes, and M.
0.0
Horodecki, arXiv:0810.4584
m
0.5
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
time
Decay rate for information encoded into ground state is
independent of system size, but proportional to
exp(−cβJ)
Contrast with 4D model
“On thermal stability of topological qubit in Kitaev's 4D model”
R. Alicki, (M.+P. +R.) Horodecki, arXiv:0811.0033
35. Not a Problem?
Factor a 1000 bit number:
Naive algorithm: ~109 gates Gap to temp ratio: ~20
Seems likely that exponential
suppression as function of
temp well worth working for
36. Not a Problem?
Factor a 1000 bit number:
18819881292060796383869723946165043
98071635633794173827007633564229888
59715234665485319060606504743045317
38801130339671619969232120573403187
9550656996221305168759307650257059
Naive algorithm: ~109 gates Gap to temp ratio: ~20
Seems likely that exponential
suppression as function of
temp well worth working for
37. A Problem?
What about during computing?
Creation and manipulation of
excited states require
operators like:
How to do this?
not
How do you “create” and “move” anyons without
creating errors = creating extra anyons
38. TQC: Gates
“We now introduce into the system’s Hamiltonian a scalar potential
composed of many local “traps,” each sufficient to capture exactly one
quasiparticle. These traps may be created by impurities, by small
gates, or by the potential created by tips of scanning microscopes.
The quasiparticle’s charge screens the potential introduced by the
trap and the quasiparticle-tip combination cannot be observed by
local measurements from far away. We denote the positions of these
traps by R1,...,Rk, and assume that these positions are well spaced
from each other compared to the microscopic length scales. A state
with quasiparticles at these positions can be viewed as an excited
state of the Hamiltonian without the trap potential or, alternatively, as
the ground state in the presence of the trap potential....”
C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma “Non-Abelian anyons
and topological quantum computation” Rev. Mod. Phys. 80, 1083 (2008).
39. TQC Problems?
How does one guarantee that one can trap one and
only one anyon?
Every operation the previous paragraph needs to be
demonstrated that it can be done with
EXCEPTIONALLY small error if this is to be called
physical fault-tolerance.
40. Adiabatic Dragging
Energy
Quantum Quantum
error error
correcting correcting
codespace codespace
Time
Require: degenerate through entire evolution
“Open loop” holonomic evolution
51. Variations on a Theme
Exchange interactions:
Adiabatically preparing rotated Hamiltonians:
Not all U’s allow a gap
gap: gap:
52. “Impossibility” Theorem
Is it possible to adiabatically SWAP with only one
adiabatic interpolation?
Diagonal in same basis so cannot transfer amplitude
to perform SWAP
55. Comparison
Holonomic quantum computing, but open loop
open loop holomic QC:
[D. Kult, J. Aberg, E. Sjoqvist, Phys. Rev. A 74, 022106 (2006)]
Adiabatic universal quantum computing, but with
piecewise evolution
D. Aharonov et al., in 45th Annual IEEE FOCS (2004)]
Path robustness in holonomy is subsystem dependent
compare:
[O. Oreshkov, T. A. Brun, D. A. Lidar, Phys. Rev. Lett. 102, 070502 (2009)]
Piecewise keeps gap constant:
compare spin-1 chain transmission of:
[K. Eckert, O. Romero-Isart, and A. Sanpera, New J. Phys. 9, 155 (2007)]
Details: arXiv:0905.0901
63. Deform Smooth Puncture
Turn on
While turning off
+ relevant vertex operators
Rough qubits: Ditto Rough qubit prepare +1
e.v. of encoded
64. Toolkit
1. Create small smooth qubit in +1 e.v. of Z
2. Create small rough qubit in +1 e.v. of X
3. Enlarge, move both punctures
4. Braid puncture: this give a CX target = rough
These things all commute with each other, as
of course these are Abelian anyons.
65. Universal Toolkit
1. Create small smooth qubit in +1 e.v. of Z
2. Create small rough qubit in +1 e.v. of X
3. Enlarge, move both punctures
4. Braid puncture: this give a CX target = rough
5. Measure encoded X,Z of qubits
6. Create magic states
Requires distillation.
66. Measurement
Simple, right?
But how to perform why
Hamiltonian still on?
Note: destructive Pauli
measurements are tolerant to
errors because (1)
complementary errors are large
and (2) we can remove region
being measured
67. Universal Toolkit
1. Create small smooth qubit in +1 e.v. of Z
2. Create small rough qubit in +1 e.v. of X
3. Enlarge, move both punctures
4. Braid puncture: this give a CX target = rough
5. Measure encoded X,Z of qubits
6. Create magic states
5+6 not tolerant to errors in this model
68. Summary
Adiabatically deforming between different
Hamiltonians which have stabilizer codewords as
energy eigenstates yields interesting protocols like
AGT
Using AGT as a motivator you can understand how
topological quantum computing can or cannot robustly
compute (and thus can provide rigorous error bounds.)
69. Adiabatic Gate Teleportation
Dave Bacon
University of Washington, Seattle, WA USA
[joint work with Steve Flammia (Perimeter), Alice Neels
(Washington), Andrew Landahl (Sandia Labs)]
8th Symposium on Topological Quantum Computing, Zurich 29th-31st August 2009
70. Open Questions
Like measurement based methods, but without having to
perform correction afterwards:
Can we (1) do cluster state computation in this manner?
(2) other measurement based models?
73. Self-Correcting
Calculate: free energy change to flip all spins
= +1
= −1
1D 2D
E ∝ Constant
E ∝ Perimeter
entropy wins T < TC energy wins
T > TC entropy wins
74. Von Neumann’s Model
x xy f full adder
y S f 00 1 c
01 1
1 S
NAND gate 10 1
s
11 0 a M M
xyz f 1 S
x
y 000 0 M cout
M f b
z 001 0 CS time
010 0
majority gate 011 1 suppose each gate fails
100 0 (outputs wrong answer)
101 1
110 1 independently and with
111 1 probability p
75. Von Neumann’s Conundrum
Observation: If all gates fail with probability p, then
an output bit of any circuit most fail with at least
probability p
Rest of M
output
input
Circuit
76. Von Neumann’s Conundrum
Observation: If all gates fail with probability p, then
an output bit of any circuit most fail with at least
probability p
Rest of M
output
input
Circuit
“Loophole”: replace wire by “bundle” of N wires
≤βN 0s decode as “1”
N
≥(1-β)N 0s decode as “0”
0<β<1/2
77. Von Neumann’s Solution 1
x
Simulate the majority gate redundantly: y M f
z
1x 1x
2x 1y M
x 3x 1z
4x 2x
5x 2y M
1y 2z 1f
M
2y 3x 2f
y 4y
3y
3z
3y M 4f
3f f
5y 4x 5f
1z 4z
4y M N=5
2z
z 3z 5x
4z 5y M
5z 5z
78. Von Neumann’s Solution 1
x
Simulate the majority gate redundantly: y M f
z
1x 1x
2x 1y M
x 3x 1z
4x 2x
5x 2y M
1y 2z 1f
M
2y 3x 2f
y 4y
3y
3z
3y M 4f
3f f
5y 4x 5f
1z 4z
4y M N=5
2z
z 3z 5x
4z 5y M
5z 5z
1. M gates may fail
79. Von Neumann’s Solution 1
x
Simulate the majority gate redundantly: y M f
z
1x 1x
2x 1y M
x 3x 1z
4x 2x
5x 2y M
1y 2z 1f
M
2y 3x 2f
y 4y
3y
3z
3y M 4f
3f f
5y 4x 5f
1z 4z
4y M N=5
2z
z 3z 5x
4z 5y M
5z 5z
2. incoming errors may add 1. M gates may fail
80. Von Neumann’s Solution 1
x
Simulate the majority gate redundantly: y M f
z
1x 1x
2x 1y M
x 3x 1z
4x 2x
5x 2y M
1y 2z 1f
M
2y 3x 2f
y 4y
3y
3z
3y M 4f
3f f
5y 4x 5f
1z 4z
4y M N=5
2z
z 3z 5x
4z 5y M
5z 5z
2. incoming errors may add 1. M gates may fail
81. Von Neumann’s Solution 2
“Restorative organ”:
M
Α versus Α
1.0
M 0.8
π
0.6
M
0.4
Α
M 0.2
Α
0.0
0.0 0.2 0.4 0.6 0.8 1.0
M Α
random
permutation
αN “0s” [3α2(1-α)+α3]N “0s” = α*N “0”s
![Adiabatic Gate Teleportation
& Its Applications
Dave Bacon
Department of Computer Science & Engineering
University of Washington, Seattle, WA USA
[joint work with Steve Flammia (Perimeter),
Alice Neels (Washington), Andrew Landahl (Sandia Labs)]](https://image.slidesharecdn.com/perimeter09-12738493533632-phpapp02/85/Perimeter-Institute-Talk-2009-1-320.jpg)
