Mariami Gachechiladze

PhD student
Room: B-111
Phone: +49 271 740 3716
marigachi@physik.uni-siegen.dePreprints
See also arxiv
Mariami Gachechiladze, Otfried Gühne and Akimasa Miyake
Changing the circuit-depth complexity of measurement-based quantum
computation with hypergraph states
arXiv:1805.12093
The circuit model of quantum computation defines its logical depth or
"computational time" in terms of temporal gate sequences, but the
measurement-based model could allow totally different time ordering and
parallelization of logical gates. We introduce a deterministic scheme of
universal measurement-based computation, using only Pauli measurements on
multi-qubit hypergraph states generated by the Controlled-Controlled-Z (CCZ)
gates. In contrast to the cluster-state scheme where the Clifford gates are
parallelizable, our scheme enjoys massive parallelization of CCZ and SWAP
gates, so that the computational depth grows with the number of global
applications of Hadamard gates, or, in other words, with the number of changing
computational bases. An exponentially-short depth implementation of an N-times
Controlled-Z gate illustrates a novel trade-off between space and time
complexity.
Publications
M. Hebenstreit, M. Gachechiladze, O. Gühne and B. Kraus
The Entanglement Hierarchy 2 x m x n Systems
Phys. Rev. A 97,
032330
(2018),
arXiv:1710.00981
We consider three-partite pure states in the Hilbert space $\mathbb{C}^2
\otimes \mathbb{C}^m \otimes \mathbb{C}^n$ and investigate to which states a
given state can be locally transformed with a non-vanishing probability.
Whenever the initial and final states are elements of the same Hilbert space,
the problem can be solved via the characterization of the entanglement classes
which are determined via stochastic operations and classical communication
(SLOCC). In general, there are infinitely many SLOCC classes. However, when
considering transformations from higher- to lower-dimensional Hilbert spaces,
an additional hierarchy among the classes can be found. This hierarchy of SLOCC
classes coarse grains SLOCC classes which can be reached from a common resource
state of higher dimension. We first show that a generic set of states in
$\mathbb{C}^2 \otimes \mathbb{C}^m \otimes \mathbb{C}^n$ for $n=m$ is the union
of infinitely many SLOCC classes, which can be parameterized by $m-3$
parameters. However, for $n \neq m$ there exists a single SLOCC class which is
generic. Using this result, we then show that there is a full-measure set of
states in $\mathbb{C}^2 \otimes \mathbb{C}^m \otimes \mathbb{C}^n$ such that
any state within this set can be transformed locally to a full measure set of
states in any lower-dimensional Hilbert space. We also investigate resource
states, which can be transformed to any state (not excluding any zero-measure
set) in the smaller-dimensional Hilbert space. We explicitly derive a state in
$\mathbb{C}^2 \otimes \mathbb{C}^m \otimes \mathbb{C}^{2m-2}$ which is the
optimal common resource of all states in $\mathbb{C}^2 \otimes \mathbb{C}^m
\otimes \mathbb{C}^m$. We also show that for any $n < 2m$ it is impossible to
reach all states in $\mathbb{C}^2 \otimes \mathbb{C}^m \otimes
\mathbb{C}^{\tilde{n}}$ whenever $\tilde{n}>m$.
Mariami Gachechiladze, Nikolai Wyderka and Otfried Gühne
The structure of ultrafine entanglement witnesses
J. Phys. A: Math. Theor. 51,
365307
(2018),
arXiv:1805.06404
An entanglement witness is an observable with the property that a negative
expectation value signals the presence of entanglement. The question arises how
a witness can be improved if the expectation value of a second observable is
known, and methods for doing this have recently been discussed as so-called
ultrafine entanglement witnesses. We present several results on the
characterization of entanglement given the expectation values of two
observables. First, we explain that this problem can naturally be tackled with
the method of the Legendre transformation, leading even to a quantification of
entanglement. Second, we present necessary and sufficient conditions that two
product observables are able to detect entanglement. Finally, we explain some
fallacies in the original construction of ultrafine entanglement witnesses [F.
Shahandeh et al., Phys. Rev. Lett. 118, 110502 (2017)].
Mariami Gachechiladze and Otfried Gühne
Completing the proof of "Generic quantum nonlocality"
Phys. Lett. A 381,
1281
(2017),
arXiv:1607.02948
In a paper by Popescu and Rohrlich [Phys. Lett. A 166, 293 (1992)] a proof
has been presented showing that any pure entangled multiparticle quantum state
violates some Bell inequality. We point out a gap in this proof, but we also
give a construction to close this gap. It turns out that with some extra effort
all the results from the aforementioned publication can be proven. Our
construction shows how two-particle entanglement can be generated via
performing local projections on a multiparticle state.
Mariami Gachechiladze, Nikoloz Tsimakuridze and Otfried Gühne
Graphical description of unitary transformations on hypergraph states
J. Phys. A: Math. Theor. 50,
19LT01
(2017),
arXiv:1612.01447
Hypergraph states form a family of multiparticle quantum states that
generalizes cluster states and graph states. We study the action and graphical
representation of nonlocal unitary transformations between hypergraph states.
This leads to a generalization of local complementation and graphical rules for
various gates, such as the CNOT gate and the Toffoli gate. As an application,
we show that already for five qubits local Pauli operations are not sufficient
to check local equivalence of hypergraph states. Furthermore, we use our rules
to construct entanglement witnesses for three-uniform hypergraph states.
Mariami Gachechiladze, Costantino Budroni and Otfried Gühne
Extreme violation of local realism in quantum hypergraph states
Phys. Rev. Lett. 116,
070401
(2016),
arXiv:1507.03570
Hypergraph states form a family of multiparticle quantum states that
generalizes the well-known concept of Greenberger-Horne-Zeilinger states,
cluster states, and more broadly graph states. We study the nonlocal properties
of quantum hypergraph states. We demonstrate that the correlations in
hypergraph states can be used to derive various types of nonlocality proofs,
including Hardy-type arguments and Bell inequalities for genuine multiparticle
nonlocality. Moreover, we show that hypergraph states allow for an
exponentially increasing violation of local realism which is robust against
loss of particles. Our results suggest that certain classes of hypergraph
states are novel resources for quantum metrology and measurement-based quantum
computation.
Master Thesis