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
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