Hi all,
My apologies if you receive this email twice. I will be defending
my thesis in LISE 303 (at Harvard) on April 25th at 10 am.
Refreshments will be provided. The talk will be roughly half
material Ive talked about at group meetings/seminars and half some
new results concerning the construction of unitary 2 designs.
Abstract
The existence of quantum locally generated codes is a long
standing open problem in quantum information theory. In this
thesis, we consider a bound concerning this conjecture as well as
a few constructions with codes that are 'barely non-local'. First
we formulate and analyze a bound concerning the possible
topologies of locally generated codes. It is well known that
quantum codes which can be defined on lattices (geometrically
local codes) must have sub-linear distance, and hence must be
bad quantum codes. We establish a complementary result to this
one, and show quantum codes which are strongly not embeddable
into finite dimensional lattices must also have poor distance.
Along the way we derive some results concerning "pseudorandom"
classical codes.
Given that quantum codes seem to be difficult to construct, it
seems useful to examine "bad" quantum codes for applications in
information and communication. Indeed, most of the work in quantum
error correction by researchers today is in this direction. We
construct a nearly local quantum erasure code which can achieve
the capacity of the quantum erasure channel. This code has very
poor (adversarial) distance, but still manages to correct random
erasure errors with high probability. The codes use random
Erdos-Renyi graphs to construct quantum states which are nearly
local, but also highly entangled across fixed cuts with high
probability. We derive some new results concerning classical codes
with log-sparse parity check matrices which may be of independent
interest.
Inspired by this construction, we are able to construct new
approximate unitary 2-designs or scramblers. The study of
scrambling is the study of the mixing properties of different
distributions of random unitaries. There is an inherent duality
between the study of scrambling and the study of error correction:
A good quantum code will make a good scrambler and vice versa. We
study the scrambling properties of our random Erdos-Renyi graph
state encoding circuits. We are able to show that these circuits,
when supplemented by some local Expander Graph quantum circuits,
form "weak" approximate unitary 2-designs with O(n log(n)) many
gates and depth O(log(n)). This construction, strictly speaking,
does not achieve more efficient parameters than existing
approximate 2-designs (it matches them), but might have
implementation advantages over other designs and points to a
conjecture which could yield approximate unitary designs with time
independent qubit-to-qubit coupling. This would be a very
interesting construction in the context of experimental randomized
benchmarking.
Best,
Kevin