67,919 research outputs found
Scaling a unitary matrix
The iterative method of Sinkhorn allows, starting from an arbitrary real
matrix with non-negative entries, to find a so-called 'scaled matrix' which is
doubly stochastic, i.e. a matrix with all entries in the interval (0, 1) and
with all line sums equal to 1. We conjecture that a similar procedure exists,
which allows, starting from an arbitrary unitary matrix, to find a scaled
matrix which is unitary and has all line sums equal to 1. The existence of such
algorithm guarantees a powerful decomposition of an arbitrary quantum circuit.Comment: A proof of the conjecture is now provided by Idel & Wolf
(http://arxiv.org/abs/1408.5728
Complex Unitary Recurrent Neural Networks using Scaled Cayley Transform
Recurrent neural networks (RNNs) have been successfully used on a wide range
of sequential data problems. A well known difficulty in using RNNs is the
\textit{vanishing or exploding gradient} problem. Recently, there have been
several different RNN architectures that try to mitigate this issue by
maintaining an orthogonal or unitary recurrent weight matrix. One such
architecture is the scaled Cayley orthogonal recurrent neural network (scoRNN)
which parameterizes the orthogonal recurrent weight matrix through a scaled
Cayley transform. This parametrization contains a diagonal scaling matrix
consisting of positive or negative one entries that can not be optimized by
gradient descent. Thus the scaling matrix is fixed before training and a
hyperparameter is introduced to tune the matrix for each particular task. In
this paper, we develop a unitary RNN architecture based on a complex scaled
Cayley transform. Unlike the real orthogonal case, the transformation uses a
diagonal scaling matrix consisting of entries on the complex unit circle which
can be optimized using gradient descent and no longer requires the tuning of a
hyperparameter. We also provide an analysis of a potential issue of the modReLU
activiation function which is used in our work and several other unitary RNNs.
In the experiments conducted, the scaled Cayley unitary recurrent neural
network (scuRNN) achieves comparable or better results than scoRNN and other
unitary RNNs without fixing the scaling matrix
Efficiency of Producing Random Unitary Matrices with Quantum Circuits
We study the scaling of the convergence of several statistical properties of
a recently introduced random unitary circuit ensemble towards their limits
given by the circular unitary ensemble (CUE). Our study includes the full
distribution of the absolute square of a matrix element, moments of that
distribution up to order eight, as well as correlators containing up to 16
matrix elements in a given column of the unitary matrices. Our numerical
scaling analysis shows that all of these quantities can be reproduced
efficiently, with a number of random gates which scales at most as with the number of qubits for a given fixed precision
. This suggests that quantities which require an exponentially large
number of gates are of more complex nature.Comment: 18 pages, 10 figure
On the Scaling Limits of Determinantal Point Processes with Kernels Induced by Sturm-Liouville Operators
By applying an idea of Borodin and Olshanski [J. Algebra 313 (2007), 40-60],
we study various scaling limits of determinantal point processes with trace
class projection kernels given by spectral projections of selfadjoint
Sturm-Liouville operators. Instead of studying the convergence of the kernels
as functions, the method directly addresses the strong convergence of the
induced integral operators. We show that, for this notion of convergence, the
Dyson, Airy, and Bessel kernels are universal in the bulk, soft-edge, and
hard-edge scaling limits. This result allows us to give a short and unified
derivation of the known formulae for the scaling limits of the classical random
matrix ensembles with unitary invariance, that is, the Gaussian unitary
ensemble (GUE), the Wishart or Laguerre unitary ensemble (LUE), and the MANOVA
(multivariate analysis of variance) or Jacobi unitary ensemble (JUE)
The block-ZXZ synthesis of an arbitrary quantum circuit
Given an arbitrary unitary matrix , a powerful matrix
decomposition can be applied, leading to four different syntheses of a
-qubit quantum circuit performing the unitary transformation. The
demonstration is based on a recent theorem by F\"uhr and Rzeszotnik,
generalizing the scaling of single-bit unitary gates () to gates with
arbitrary value of~. The synthesized circuit consists of controlled 1-qubit
gates, such as NEGATOR gates and PHASOR gates. Interestingly, the approach
reduces to a known synthesis method for classical logic circuits consisting of
controlled NOT gates, in the case that is a permutation matrix.Comment: Improved (non-sinkhorn) algorithm to obtain the proposed circui
Multicritical matrix models and the chiral phase transition
Universality of multicritical unitary matrix models is shown and a new scaling behavior is found in the microscopic region of the spectrum, which may be relevant for the low energy spectrum of the Dirac operator at the chiral phase transition
Local Relativistic Exact Decoupling
We present a systematic hierarchy of approximations for {\it local}
exact-decoupling of four-component quantum chemical Hamiltonians based on the
Dirac equation. Our ansatz reaches beyond the trivial local approximation that
is based on a unitary transformation of only the atomic block-diagonal part of
the Hamiltonian. Systematically, off-diagonal Hamiltonian matrix blocks can be
subjected to a unitary transformation to yield relativistically corrected
matrix elements. The full hierarchy is investigated with respect to the
accuracy reached for the electronic energy and molecular properties on a
balanced test molecule set that comprises molecules with heavy elements in
different bonding situations. Our atomic (local) assembly of the unitary
transformation needed for exact decoupling provides an excellent local
approximation for any relativistic exact-decoupling approach. Its order-
scaling can be further reduced to linear scaling by employing the
neighboring-atomic-blocks approximation. Therefore, it is an efficient
relativistic method perfectly well suited for relativistic calculations on
large molecules. If a large molecule contains many light atoms (typically
hydrogen atoms), the computational costs can be further reduced by employing a
well-defined non-relativistic approximation for these light atoms without
significant loss of accuracy
String Equations for the Unitary Matrix Model and the Periodic Flag Manifold
The periodic flag manifold (in the Sato Grassmannian context) description of
the modified Korteweg--de Vries hierarchy is used to analyse the translational
and scaling self--similar solutions of this hierarchy. These solutions are
characterized by the string equations appearing in the double scaling limit of
the symmetric unitary matrix model with boundary terms. The moduli space is a
double covering of the moduli space in the Sato Grassmannian for the
corresponding self--similar solutions of the Korteweg--de Vries hierarchy, i.e.
of stable 2D quantum gravity. The potential modified Korteweg--de Vries
hierarchy, which can be described in terms of a line bundle over the periodic
flag manifold, and its self--similar solutions corresponds to the symmetric
unitary matrix model. Now, the moduli space is in one--to--one correspondence
with a subset of codimension one of the moduli space in the Sato Grassmannian
corresponding to self--similar solutions of the Korteweg--de Vries hierarchy.Comment: 21 pages in LaTeX-AMSTe
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