783 research outputs found
Shaded Tangles for the Design and Verification of Quantum Programs (Extended Abstract)
We give a scheme for interpreting shaded tangles as quantum programs, with
the property that isotopic tangles yield equivalent programs. We analyze many
known quantum programs in this way -- including entanglement manipulation and
error correction -- and in each case present a fully-topological formal
verification, yielding in several cases substantial new insight into how the
program works. We also use our methods to identify several new or generalized
procedures.Comment: In Proceedings QPL 2017, arXiv:1802.0973
Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?
We study the commutation relations within the Pauli groups built on all
decompositions of a given Hilbert space dimension , containing a square,
into its factors. Illustrative low dimensional examples are the quartit ()
and two-qubit () systems, the octit (), qubit/quartit () and three-qubit () systems, and so on. In the single qudit case,
e.g. , one defines a bijection between the maximal
commuting sets [with the sum of divisors of ] of Pauli
observables and the maximal submodules of the modular ring ,
that arrange into the projective line and a independent set
of size [with the Dedekind psi function]. In the
multiple qudit case, e.g. , the Pauli graphs rely on
symplectic polar spaces such as the generalized quadrangles GQ(2,2) (if
) and GQ(3,3) (if ). More precisely, in dimension ( a
prime) of the Hilbert space, the observables of the Pauli group (modulo the
center) are seen as the elements of the -dimensional vector space over the
field . In this space, one makes use of the commutator to define
a symplectic polar space of cardinality , that
encodes the maximal commuting sets of the Pauli group by its totally isotropic
subspaces. Building blocks of are punctured polar spaces (i.e. a
observable and all maximum cliques passing to it are removed) of size given by
the Dedekind psi function . For multiple qudit mixtures (e.g.
qubit/quartit, qubit/octit and so on), one finds multiple copies of polar
spaces, ponctured polar spaces, hypercube geometries and other intricate
structures. Such structures play a role in the science of quantum information.Comment: 18 pages, version submiited to J. Phys. A: Math. Theo
About the Dedekind psi function in Pauli graphs
We study the commutation structure within the Pauli groups built on all
decompositions of a given Hilbert space dimension , containing a square,
into its factors. The simplest illustrative examples are the quartit ()
and two-qubit () systems. It is shown how the sum of divisor function
and the Dedekind psi function enter
into the theory for counting the number of maximal commuting sets of the qudit
system. In the case of a multiple qudit system (with and a prime),
the arithmetical functions and count the
cardinality of the symplectic polar space that endows the
commutation structure and its punctured counterpart, respectively. Symmetry
properties of the Pauli graphs attached to these structures are investigated in
detail and several illustrative examples are provided.Comment: Proceedings of Quantum Optics V, Cozumel to appear in Revista
Mexicana de Fisic
Layered Quantum Key Distribution
We introduce a family of QKD protocols for distributing shared random keys
within a network of users. The advantage of these protocols is that any
possible key structure needed within the network, including broadcast keys
shared among subsets of users, can be implemented by using a particular
multi-partite high-dimensional quantum state. This approach is more efficient
in the number of quantum channel uses than conventional quantum key
distribution using bipartite links. Additionally, multi-partite
high-dimensional quantum states are becoming readily available in quantum
photonic labs, making the proposed protocols implementable using current
technology.Comment: 11 pages, 5 figures. In version 2 we extended section 4 about the
dimension-rate trade-off and corrected minor error
Depicting qudit quantum mechanics and mutually unbiased qudit theories
We generalize the ZX calculus to quantum systems of dimension higher than
two. The resulting calculus is sound and universal for quantum mechanics. We
define the notion of a mutually unbiased qudit theory and study two particular
instances of these theories in detail: qudit stabilizer quantum mechanics and
Spekkens-Schreiber toy theory for dits. The calculus allows us to analyze the
structure of qudit stabilizer quantum mechanics and provides a geometrical
picture of qudit stabilizer theory using D-toruses, which generalizes the Bloch
sphere picture for qubit stabilizer quantum mechanics. We also use our
framework to describe generalizations of Spekkens toy theory to higher
dimensional systems. This gives a novel proof that qudit stabilizer quantum
mechanics and Spekkens-Schreiber toy theory for dits are operationally
equivalent in three dimensions. The qudit pictorial calculus is a useful tool
to study quantum foundations, understand the relationship between qubit and
qudit quantum mechanics, and provide a novel, high level description of quantum
information protocols.Comment: In Proceedings QPL 2014, arXiv:1412.810
Discrete Wigner Function Derivation of the Aaronson-Gottesman Tableau Algorithm
The Gottesman-Knill theorem established that stabilizer states and operations
can be efficiently simulated classically. For qudits with dimension three and
greater, stabilizer states and Clifford operations have been found to
correspond to positive discrete Wigner functions and dynamics. We present a
discrete Wigner function-based simulation algorithm for odd- qudits that has
the same time and space complexity as the Aaronson-Gottesman algorithm. We show
that the efficiency of both algorithms is due to the harmonic evolution in the
symplectic structure of discrete phase space. The differences between the
Wigner function algorithm and Aaronson-Gottesman are likely due only to the
fact that the Weyl-Heisenberg group is not in for and that qubits
have state-independent contextuality. This may provide a guide for extending
the discrete Wigner function approach to qubits
Anyonic entanglement renormalization
We introduce a family of variational ansatz states for chains of anyons which
optimally exploits the structure of the anyonic Hilbert space. This ansatz is
the natural analog of the multi-scale entanglement renormalization ansatz for
spin chains. In particular, it has the same interpretation as a coarse-graining
procedure and is expected to accurately describe critical systems with
algebraically decaying correlations. We numerically investigate the validity of
this ansatz using the anyonic golden chain and its relatives as a testbed. This
demonstrates the power of entanglement renormalization in a setting with
non-abelian exchange statistics, extending previous work on qudits, bosons and
fermions in two dimensions.Comment: 19 pages, 10 figures, v2: extended, updated to match published
versio
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