QUANTUM COMPUTATION: THE TOPOLOGICAL APPROACH

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QUANTUM COMPUTATION THE TOPOLOGICAL APPROACH

• Michael H. Freedman
• Theory Group MSR

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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 youll react to what you just read.
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).
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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.
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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.)
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There is a new computational model Quantum
Computation, which is based on the ability to
write, rotate, and read quantum states.
The three Rs of the 21st century.

• In the quantum model you write states in a
  vector space
• Operate on them with rotations which are the
  analogs of classical gates and
• 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.

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• 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!)

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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.)
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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.
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It is easier in symbols
according to the answer being A,B,C, or D.
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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.
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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
classically and like in the
quantum world. So it looks like this new
computational model has considerably more power
than the old.
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• Can we put this in perspective?
• Should we expect new computational models to
  emerge every few decades?

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.
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• 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.
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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 computers
overhead could easily eat up this advantage
(depending on details of implementation which we
cannot yet know).
So what is left?
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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. Feynmans 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.
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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.
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Farther down the road, it is difficult to
restrain ones 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?
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Why has AI not succeeded (an interesting question
to which I have no answer) ?
Without suggesting (? 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!)
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• 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.
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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.
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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 up plus down where
and are complex numbers.
(Two is not actually important to the story
any finite number of states gives an equivalent
theory.)
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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
spinsetc. ). 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.
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This conundrum has been mathematically
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.
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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.
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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 theorists unsettle you about this
point!) , there is really only a single type of
exchange up to deformation. The two 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 (21 dimensions). If many identical
particles are exchanged repeatedly the general
braid can be produced.
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Particle-antiparticle pairs are
created out of the vacuum.
birth braiding
time
death
afterlife?
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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
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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.)
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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 provide
a route to high-temperature superconductivity.
The

complicated
braiding properties of the quasiparticle
excitations of these systems do not allow them to
propagate easily. However, pairs or other
aggregates of quasiparticles might be able to
propagate more easily, and the formation of
pairs contains the germ of superconductivity.
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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.
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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.
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Id 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.
The Hamiltonian describes
on-site and coulomb potentials together with
tunneling amplitudes for a population of
electrons filling a 2-dimensional KAGOME
crystal.
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Locating Topological Phases Inside Hubbard Type
Models.
Kirill Shtengel
Chetan Nayak
MichaelFreedman
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Hubbard Model
In our model the sites (atoms) are arrayed on
the Kagome lattice
The colors encode differing chemical potentials
. Tunneling amplitudes tab
also vary with colors.
c
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We work with an equivalent triangular
representation.

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

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Hamiltonian Ground State Manifold H H1/6
all particle positions (U0 large) one
particle per bond D dimer cover
T Now small terms
j
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Review - Perturbation Theory
function of l
dont like perturbed, but can recurse
dynamic, off diag. terms of projectors
.
.
diagonal terms of projectors
balanced to keep
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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).Lets look at a
game.