# QUANTUM COMPUTATION THE TOPOLOGICAL APPROACH

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```					QUANTUM COMPUTATION:
THE TOPOLOGICAL
APPROACH

Michael H. Freedman
Theory Group MSR

1
Classical computation based on the idea of
being able to write, erase, and read symbols.
You also need to have a few internal states
“moods” which dictate how you’ll react to what

A Turing machine formalizes this concept.
According to the thesis of Church-Turing all
REALISTIC computer architectures should
be about as efficient as each other
(POLYNOMIALLY EQUIVALENT).
2
The Turing machine lives at the heart of logic
and philosophy (the undecidability of the Halting
Problem).

But, if you think about it, even a “UNIVERSAL”
Turing machine - one capable of simulating any
other - is rather a paltry thing.

It is a victory for the plodders: as long as it can
read, write, and not misplace its records, it can
gradually accomplish almost anything! But perhaps
very SLOWLY.

3
But today the Church-Turing thesis is
in doubt. If QUANTUM MECHANICS
is correct, then the Church-Turing
thesis is almost certainly wrong!

(Another possibility is that certain
problems like FACTORING which
look hard on an ordinary computer
are actually easy.)
4
There is a new computational model “Quantum
Computation”, which is based on the ability to
“write”, “rotate”, and “read” quantum states. “The
three R’s of the 21st century.”

1. In the quantum model you “write” states in a vector
space;
2. Operate on them with “rotations” which are the
analogs of classical “gates”; and
3. “Read” the rotated state by making an observation.
What is actually observed is a frequency, say a flash
of light, corresponding to an eigenvalue of the
observable. Which eigenvalue is observed depends
probabilistically on the rotated state vector.

5
What all popular publications call “weird” is that the
state vector, living in the physical Hilbert space, does
not have to be part of some fixed standard basis (which
may have classical meaning) but rather can be any
linear combination or “superposition” of these basic
vectors.

This is only “weird” because we are big, clumsy things
and cannot usually probe, with our unaided senses, the
small scales at which superpositions are manifest. But
the mathematics is incredibly simple and not in the
least “weird”: it is just linear algebra.

(Mathematically, quantum mechanics perfect and simple, and
classical mechanics equally perfected and simple. What is
difficult still is to describe exactly how the two are related!)

6
It is not completely obvious that greater computational power
should reside in the quantum world. On the positive side, you
can prepare enormous superpositions of classical states and
try to use them to do (exponentially) many things at once. But
on the negative side, when you go to make your observation,
you have to be prepared to listen to a cacophony of replies
all chiming in at once.

There is by now an art of harnessing the interference effects
known from wave mechanics to cause the interesting bits of
the “answer” to reinforce and the annoying and useless bits
to cancel.

(One should think back to college days and remember the
double slit experiment.)

7
But this is so far pretty vague. The simplest EXAMPLE, I
know that suggests extraordinary computational power
lurking in the quantum world is a simple protocol for
querying a quantum black box with a single yes/no
question and being able to learn which object among 4
(not 2 !!!) it is “thinking of.”

While we ask one question, it is not: “Is the object A?”,
but rather a uniform superposition of four questions:
“is the object A,B,C,D?” all asked at once.

8
It is easier in symbols:
æ1 1 1     1ö    æ 1               ö
1 , 1, 1÷
ç , , ,
ç           ÷®
÷    ç- , +
ç            + + ÷
ç2 2 2
è          2÷
ø    ç 2
è         2 2 2÷  ø
æ 1               ö
1 , 1, 1÷
ç+ , -
ç            + + ÷
ç 2
è         2 2 2÷  ø
æ 1       1, 1, 1÷ö
ç+ , +
ç            -  + ÷
ç 2
è         2 2 2÷  ø
æ 1               ö
1 , 1, 1÷   according to the
ç+ , +
ç            + - ÷     answer being
ç 2
è         2 2 2÷  ø     A,B,C, or D.

9
The Black Box has communicated the hidden
object: A, B, C, or D by flipping the phase in the
1st, 2nd , 3rd, or 4th coordinate.

The punch line is that the 4x4 matrix above is
ORTHOGNONAL so we can find an observable
which – with certainty – distinguishes the four
possibilities with one observation. This is the
smallest special case of the “Grover search
algorithm.”

10
The most famous CS problem which a quantum
computer     - if built - can quickly solve is
factoring the number n (Shor). This


problem seems to scale like exp  log n
1
3

classically and like    log n 
2
in the quantum
world. So it looks like this new computational
model has considerably more power than the
old.

11
 Can we put this in perspective?
 Should we expect new computational models

I think not: Physics, like an onion, has its layers:
the classical, the quantum mechanical, and then
very far down perhaps a stringy quantum
gravity…. No one, I think, will ever make a
computer out of strings or black holes so the
quantum computer may be the final information
processing technology even if our race survives
10,000 years. It will define the boundary between
what is knowable and what is not.
12
 What exactly can we expect as an immediate
consequence of quantum computation if the
technology can be created?

With the fall of RSA and most/all other
classical encryption schemes, there would
be havoc, exciting havoc in cryptography –
probably a net minus.

Just the prospect of a future quantum
computer has already spawned a field of
“quantum security protocols” whose future
developments looks secure.

13
The Grover algorithm, that we have
already met, speeds up mindless search
(say on NP-complete problems) by a
square-root. But I would not count on
this for much: the quantum computer’s
overhead could easily eat up this
implementation which we cannot yet
know).

So what is left?

14
There is progress in quantum CS, the graph isomorphism
problem seems to hang by a thread, but even if there were
NO CS problems solved by a quantum computation we
would have a feast before us.

Feynman’s original motivation for proposing quantum
computation was the inability of classical computers to carry
out realistic simulations of the simplest quantum mechanical
systems.

Chemistry and material science (but I think not medicine)
would be revolutionized. With a quantum computer we
would soon find out how the cuprate high-temperature
superconductors work, and if the family contains the Holy
Grail: a room temperature superconductor.

15
While general NP problems would not be solved, there
is an interesting heuristic suggestion from an MIT group
(Fahri, Goldstone, et. al. …):

Create a correspondence between solutions
to a problem and the ground state of a
Hamiltonian; now take to an easy problem for
which the Hamiltonian readily attains its
ground state, then adiabatically deform to the
hard problem hoping the system remains in
its ground state.

This and many other ideas could prove quite
powerful but we will not know until be build a
quantum computer – simulating the approach
on a classical computer is exponentially
inefficient.
16
Farther down the road, it is difficult to restrain one’s
imagination. A tiny flake of material one millimeter on
a side and only a few angstroms thick could serve
as the guts of a powerful quantum computer.

Will we finally put ourselves “out of business”
by making a much more capable entity?

17
Why has AI not succeeded (an interesting
question to which I have no answer) ?

Without suggesting (ả la Penrose) that there is anything
quantum mechanical about human intelligence it seems
quite possible that quantum computers will be programmed
to appear/be intelligent.

Our ability to hold a lot in mind as “background” to decision
making might be imitated though the use of superposition.
(How this works in our - presumably classical - brains is
something I would love to know!)

18
Before we get too excited about what the NEW
WORLD will look like,

 whether it contains interesting business
opportunities – or

 even a place for humans at all,
there is a major problem to be addressed:

DECOHERENCE and the accumulation of ERRORS.

19
Decoherence is why we do not observe cats half
dead and half alive. It is the tendency of quantum
systems to become classical.

The ENVIRONMENT tends to reach into any
quantum mechanical system and MEASURE it and
reduce it to classical PROBABALISTIC
combinations - rather than the more
powerful quantum mechanical SUPERPOSITIONS.

Decoherence will always be an issue but its
significance will depend sensitively on the
proposed architecture for the quantum computer.
In particular it will depend on what “degrees of
freedom” we compute with.

20
A broad definition of quantum computation is any result you can eke
out of an experiment on a quantum mechanical system. But the
usual working definitions is the “qubit” or “quantum circuit” model.

A qubit is a two dimensional vector Space spanned by “up”
and “down”; a linearized bit, if you like: instead of a state
being entirely up or entirely down it is possible that it is in a
“superposition” a up plus b down where a and b are
complex numbers.

(“Two” is not actually important to the story – any
finite number of states gives an equivalent theory.)

21
After defining qubits, the world at large takes a bit of a wrong
turn by taking them too seriously and imagining that they,
directly, should be the guardians of quantum information. This
is too naïve.
It is imagined that if you need 10,000 qubits, well, then you
should dope a silicone wafer with 10,000 phosphorus atoms
and let their individual nuclear spins label the qubits. (Or
10,000 photon polarizations, or electron spins…etc. ). In the
conventional approach, the qubit of information is a local (in
space or in momentum space) quantum number. The
operation of the quantum computer is then imagined to be a
series of “gates” applied to the spins either individually or in
pairs.

Local degrees of freedom are fatally vulnerable to
DECOHERENCE. The same design that made it easy for
the programmer to reach into the system and gate it makes it
easy for the environment to reach in and touch those same
local degrees of freedom.
22
This conundrum has been “mathematical l y”
vanquished by the “fault tolerance” theorem which
states that once a sufficient standard of accuracy and
isolation is attained, there is a recursive strategy for
correcting errors so that one can carry out
indefinitely long computations.

As a mathematician, I admire the theorem, but as a
scientist, I regard it as nearly irrelevant since required
initial accuracy (about five decimal places) is
unrealistic. This is why NMR can factor 15 (with high
probability) but will never threaten RSA. It will never
sustain a long calculation.

23
The Topological Model

The topological approach amounts to physical
rather than “software” error correction.

Topology is the study of properties that are
retained under deformation. A physical system is
said to be in a “topological phase” if only a
change of topology can evolve the system.

Observables depend only coarsely on the
trajectories of particles; they depend on winding
numbers and their generalizations, but not on
local detail.
24
Some topology in physics is very familiar: if two identical
fermions are exchanged, the state vector is multiplied by
-1. The details of the exchange trajectory are irrelevant.

In our world, with three spatial dimensions (Do not let the string
type of exchange – up to deformation. The tw o
dimensional world is richer here: we have a clockwise
and a counter clockwise exchange, each quite different
from the other and both of infinite order in space time
(2+1 dimensions).

If many identical particles are exchanged repeatedly the
general braid can be produced.

25
Particle-antiparticle pairs are
created out of the vacuum.

birth
time

braiding

death

afterlife?
26
Around 2001, my collaborators and I * created such a
two dimensional system (mathematically) and
proved that it was a universal quantum computer.

In the last three years we ** have moved these
mathematical constructs to the brink of honest
physical descriptions: electrons feeling chemical and
Coulomb potentials and tunneling around in a two
dimensional lattice.

Several groups of chemists and physicists are
responding to these models.
* F., M. Larson, Z. Wang
** F., C. Nayak, K. Shtengel

27
Topological phases are not merely mathematical constructs.
Laughlin won the 1999 Nobel Prize in physics for his (topological)
description of Fractional Quantum Hall (FQH) fluids.

These are correlated systems of electrons trapped in a two
dimensional crystal interface within a semiconductor and subjected
to a strong transverse magnetic field. In fact, many theorists
believe that certain of the finer FQH plateaus are in states that we
now know to be “universal quantum computers.” Unfortunately they
are far too delicate (mK spectral gap) to be harnessed.

One of the first applications of the FQH effect was a measurement
of the fine structure constant to 9 decimal places: topological
phases, once created, appear to be exact (corresponding to the
mathematical fact that unitary representations of the braid group lie
only in discrete sequences.)

28
In broad terms the topological state of matter we intend to make are
mathematically isomorphic to the operator algebras of these FQH systems. We
have reasons tobelieve these ALGEBRAIC structures will be more stable.

Before we would attempt to build a quantum computer we would manufacture
first mathematically, and then materially, a little two-dimensional
universe unto itself.

It is impossible to overstate how astonishingly different the physical properties
of this little world will be from any known matter. The scientific and technological
possibilities are immense and I have no idea what most of them are.

We know certain properties of these materials in complete detail (for
mathematical reasons). These include the braiding and fusion algebra.

There are theoretical reasons to believe that such systems could actually
p r o v i d e a r o u t e t o h i g h - t e m p e r a t u r e s u p e r c o n d u c t i v i t y. T h e
complicated braiding properties of the quasiparticle excitations of these systems
do not allow them to propagate easily. However, pairs or other aggregates of
quas i parti c l es mi ght be abl e t o p ro pagat e m or e eas i l y, a nd the
formation of pairs contains the germ of superconductivity.

29
If I were allowed a metaphor here: metals with their
half-filled electronic bands are born conductors (they
sit around waiting for a potential to be applied so
that currents can flow). Semiconductors are the
perfect thing for gating currents (their conductance
depends very strongly and non-linearly on the
applied gate voltage, which acts as a switch). Our
new material, Q, will be then the natural home for
the processing of quantum information.

30
In fact, above some critical temperature quasi-particle
pairs will spontaneously arise from the ground state
(“vacuum”) and the little creature will be doing its own
unstructured quantum computation.
Perhaps like a small child idly watching a stream, its
“thoughts” will randomly be drawn this way and that:
“thinking” about nothing really, but “thinking” more deeply
than we poor classical beings could ever hope to. We will
take this dreamy, brilliant child and freeze her to a
temperature which halts these natural “thoughts” and
then (with an STM tip, pull one charged quasi-particle
around a multitude of pinned quasi-particles and so)
impose our own program on her
“mind”. This will be the quantum computer.

31
I’d like to close with a few screens showing our candidate
architecture for the quantum computing material.

Also, I have a demo (written by Dimitar Jetchev – Harvard
undergraduate) which allows the user to classically explore the
electron fluctuations mandated by the Hamiltonian operator
which governs 2-dimensional model.
1 2
The Hamiltonian       p + V describes on-site and coulomb
2m
potentials together with tunneling amplitudes for a population of
th
1
electrons 6 filling a 2-dimensional KAGOME crystal.

32
Locating Topological Phases Inside
Hubbard Type Models.

Kirill                      Chetan
Shtengel     Michael          Nayak
Freedman               33
Hubbard Model

In our model the sites (atoms) are arrayed on the
Kagome lattice

c
The colors encode differing chemical potentials ma , U , v
ab   .
c
Tunneling amplitudes tab also vary with colors.
34
We work with an equivalent triangular representation.

• In this representation particles (e.g. electrons) live on
edges.

35
Hamiltonian                Ground State Manifold Ì H 1
6
H=                         H1/6 = {all particle positions}
+
U 0 å ni2 U0 large)             {one particle per bond}
i
+
U å ni n j                  D ={dimer cover T}
i, j = 1
Now small terms:

t        Vij                mi
» e          » e   2
» e2
U        U                  U
+
å        0
tij (ci†c j + c †ci )+ å mi ni + å Vij ni n j
j
i , j 60                             i               i

j
36
Review - Perturbation Theory
H = H0 + l V
H | Y = En | Y                                       function of l
(En - H 0 )| Y = l V | Y
write | Y = P | Y + Q | Y
1         |k k|
´R
R= Q            Q= S
En - H 0    k E - e
n    k

Q | Y = l RV | Y , so
| Y = l RV | Y + P | Y             don’t like: perturbed, but can recurse
| Y = P | Y + l RV | Y + l 2 RVRV | Y + L
n|
k Vn                          k Vk ' ´ k 'Vn
|Y = | n + l S            +l 2 S                             +L
k¹ n E - e       k , k ' ¹ n (E - e )(E - e )
n    k                   n   k   n   k'
1                      42
444444444444444444444444444444444444444444        4 3
dynamic, off diag.
n | En - H 0 | Y = En - en = l        n |V |Y                           terms of projectors
diagonal terms
nVk    k Vn                          . .
of projectors E = e + l   nVn + l   2
S                     +l   3
S                +L
n   n
k¹ n    En - ek                k ,k ' ¹ n   U 2 balanced to keep
% %
mi = mj
37
We have an “occupation model” at
1/6 fill. For example, imagine that
each green atom has donated one
electron which is now free to localize
near any atom = site of Kagome (K).

Let’s look at a “game”.

38

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