5,340 research outputs found
Regular quantum graphs
We introduce the concept of regular quantum graphs and construct connected
quantum graphs with discrete symmetries. The method is based on a decomposition
of the quantum propagator in terms of permutation matrices which control the
way incoming and outgoing channels at vertex scattering processes are
connected. Symmetry properties of the quantum graph as well as its spectral
statistics depend on the particular choice of permutation matrices, also called
connectivity matrices, and can now be easily controlled. The method may find
applications in the study of quantum random walks networks and may also prove
to be useful in analysing universality in spectral statistics.Comment: 12 pages, 3 figure
Compact Neural Networks based on the Multiscale Entanglement Renormalization Ansatz
This paper demonstrates a method for tensorizing neural networks based upon
an efficient way of approximating scale invariant quantum states, the
Multi-scale Entanglement Renormalization Ansatz (MERA). We employ MERA as a
replacement for the fully connected layers in a convolutional neural network
and test this implementation on the CIFAR-10 and CIFAR-100 datasets. The
proposed method outperforms factorization using tensor trains, providing
greater compression for the same level of accuracy and greater accuracy for the
same level of compression. We demonstrate MERA layers with 14000 times fewer
parameters and a reduction in accuracy of less than 1% compared to the
equivalent fully connected layers, scaling like O(N).Comment: 8 pages, 2 figure
Estimating quantum chromatic numbers
We develop further the new versions of quantum chromatic numbers of graphs
introduced by the first and fourth authors. We prove that the problem of
computation of the commuting quantum chromatic number of a graph is solvable by
an SDP algorithm and describe an hierarchy of variants of the commuting quantum
chromatic number which converge to it. We introduce the tracial rank of a
graph, a parameter that gives a lower bound for the commuting quantum chromatic
number and parallels the projective rank, and prove that it is multiplicative.
We describe the tracial rank, the projective rank and the fractional chromatic
numbers in a unified manner that clarifies their connection with the commuting
quantum chromatic number, the quantum chromatic number and the classical
chromatic number, respectively. Finally, we present a new SDP algorithm that
yields a parameter larger than the Lov\'asz number and is yet a lower bound for
the tracial rank of the graph. We determine the precise value of the tracial
rank of an odd cycle.Comment: 34 pages; v2 has improved presentation based after referees'
comments, published versio
Lieb-Robinson bounds and the speed of light from topological order
We apply the Lieb-Robinson bounds technique to find the maximum speed of
interaction in a spin model with topological order whose low-energy effective
theory describes light [see X.-G. Wen, \prb {\bf 68}, 115413 (2003)]. The
maximum speed of interactions is found in two dimensions is bounded from above
less than times the speed of emerging light, giving a strong
indication that light is indeed the maximum speed of interactions. This result
does not rely on mean field theoretic methods. In higher spatial dimensions,
the Lieb-Robinson speed is conjectured to increase linearly with the dimension
itself. Implications for the horizon problem in cosmology are discussed.Comment: 4 pages, 1 eps figure. Bound improve
Hierarchical quantum classifiers
Quantum circuits with hierarchical structure have been used to perform binary
classification of classical data encoded in a quantum state. We demonstrate
that more expressive circuits in the same family achieve better accuracy and
can be used to classify highly entangled quantum states, for which there is no
known efficient classical method. We compare performance for several different
parameterizations on two classical machine learning datasets, Iris and MNIST,
and on a synthetic dataset of quantum states. Finally, we demonstrate that
performance is robust to noise and deploy an Iris dataset classifier on the
ibmqx4 quantum computer
Pretty good state transfer in qubit chains-The Heisenberg Hamiltonian
Pretty good state transfer in networks of qubits occurs when a continuous-time quantum walk allows the transmission of a qubit state from one node of the network to another, with fidelity arbitrarily close to 1. We prove that in a Heisenberg chain with n qubits, there is pretty good state transfer between the nodes at the jth and (n − j + 1)th positions if n is a power of 2. Moreover, this condition is also necessary for j = 1. We obtain this result by applying a theorem due to Kronecker about Diophantine approximations, together with techniques from algebraic graph theory
Universal quantum computation with unlabeled qubits
We show that an n-th root of the Walsh-Hadamard transform (obtained from the
Hadamard gate and a cyclic permutation of the qubits), together with two
diagonal matrices, namely a local qubit-flip (for a fixed but arbitrary qubit)
and a non-local phase-flip (for a fixed but arbitrary coefficient), can do
universal quantum computation on n qubits. A quantum computation, making use of
n qubits and based on these operations, is then a word of variable length, but
whose letters are always taken from an alphabet of cardinality three.
Therefore, in contrast with other universal sets, no choice of qubit lines is
needed for the application of the operations described here. A quantum
algorithm based on this set can be interpreted as a discrete diffusion of a
quantum particle on a de Bruijn graph, corrected on-the-fly by auxiliary
modifications of the phases associated to the arcs.Comment: 6 page
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