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Vol.20
No.5&6,
May 1, 2020
Research Articles:
Two-outcome synchronous
correlation sets and
Connes'
embedding problem
(pp361-374)
Travis B. Russell
We show that
Connes'
embedding problem is equivalent to the weak
Tsirelson
problem in the setting of two-outcome synchronous correlation sets. We
further show that the extreme points of two-outcome synchronous
correlation sets can be realized using a certain class of universal
C*-algebras.
We examine these algebras in the three-experiment case and verify that
the strong and weak
Tsirelson
problems have affirmative answers in that setting.
Persistent homology analysis
of multiqubit entanglement
(pp375-399)
Ricardo Mengoni, Alessandra Di Pierro, Leleh Memarzadeh, and
Stefano Mancini
We introduce a
homology-based
technique for the { classification of
multiqubit
state vectors with genuine entanglement}. In our approach, we associate
state vectors to data sets by introducing a metric-like measure in terms
of bipartite entanglement, and investigate the persistence of
homologies
at different scales. This leads to a novel classification of
multiqubit entanglement. The
relative occurrence frequency of various classes of entangled states is
also shown.
Algebraic units, anti-unitary
symmetries, and a small catalogue of SICs
(pp400-417)
Ingemar
Bengtsson
In complex vector spaces
maximal sets of
equiangular
lines, known as
SICs,
are related to real quadratic number fields in a dimension dependent
way. If the dimension is of the form
$n^2+3$
the base field has a fundamental unit of negative norm, and there exists
a SIC with anti-unitary symmetry. We give eight examples of exact
solutions of this kind, for which we have
endeavoured
to make them as simple as we can---as a belated reply to the referee of
an earlier publication, who claimed that our exact solution in dimension
28 was too complicated to be fit to print. An interesting feature of the
simplified solutions is that the components of the
fiducial
vectors largely consist of algebraic units.
Realization of Quantum Oracles using
Symmetries of Boolean Functions
(pp418-448)
Peng Gao, Yiwei Li, Marek Perkowski, and Xiaoyu Song
Designing a quantum oracle is an important step in practical realization
of Grover algorithm, therefore it is useful to create methodologies to
design oracles. Lattice diagrams are regular two-dimensional structures
that can be directly mapped onto a quantum circuit. We present a quantum
oracle design methodology based on lattices. The oracles are designed
with a proposed method using generalized Boolean symmetric functions
realized with lattice diagrams. We also present a decomposition-based
algorithm that transforms non-symmetric functions into symmetric or
partially symmetric functions. Our method, which combines logic
minimization, logic decomposition, and mapping, has lower quantum cost
with fewer ancilla
qubits. Overall, we obtain encouraging synthesis results superior to
previously published data.
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