15 research outputs found
Transfer Matrices and Excitations with Matrix Product States
We investigate the relation between static correlation functions in the
ground state of local quantum many-body Hamiltonians and the dispersion
relations of the corresponding low energy excitations using the formalism of
tensor network states. In particular, we show that the Matrix Product State
Transfer Matrix (MPS-TM) - a central object in the computation of static
correlation functions - provides important information about the location and
magnitude of the minima of the low energy dispersion relation(s) and present
supporting numerical data for one-dimensional lattice and continuum models as
well as two-dimensional lattice models on a cylinder. We elaborate on the
peculiar structure of the MPS-TM's eigenspectrum and give several arguments for
the close relation between the structure of the low energy spectrum of the
system and the form of static correlation functions. Finally, we discuss how
the MPS-TM connects to the exact Quantum Transfer Matrix (QTM) of the model at
zero temperature. We present a renormalization group argument for obtaining
finite bond dimension approximations of MPS, which allows to reinterpret
variational MPS techniques (such as the Density Matrix Renormalization Group)
as an application of Wilson's Numerical Renormalization Group along the virtual
(imaginary time) dimension of the system.Comment: 39 pages (+8 pages appendix), 14 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)
Parallel quantum simulation of large systems on small NISQ computers
Tensor networks permit computational and entanglement resources to be concentrated in interesting regions of Hilbert space. Implemented on NISQ machines they allow simulation of quantum systems that are much larger than the computational machine itself. This is achieved by parallelising the quantum simulation. Here, we demonstrate this in the simplest case; an infinite, translationally invariant quantum spin chain. We provide Cirq and Qiskit code that translates infinite, translationally invariant matrix product state (iMPS) algorithms to finite-depth quantum circuit machines, allowing the representation, optimisation and evolution of arbitrary one-dimensional systems. The illustrative simulated output of these codes for achievable circuit sizes is given
Topological A-Type Models with Flux
We study deformations of the A-model in the presence of fluxes, by which we
mean rank-three tensors with antisymmetrized upper/lower indices, using the
AKSZ construction. Generically these are topological membrane models, and we
show that the fluxes are related to deformations of the Courant bracket which
generalize the twist by a closed 3-from , in the sense that satisfying the
AKSZ master equation implies the integrability conditions for an almost
generalized complex structure with respect to the deformed Courant bracket. In
addition, the master equation imposes conditions on the fluxes that generalize
. The membrane model can be defined on a large class of - and -structure manifolds, including geometries inspired by
supersymmetric -models with additional supersymmetries due to almost
complex (but not necessarily complex) structures in the target space.
Furthermore, we show that the model can be defined on three particular
half-flat manifolds related to the Iwasawa manifold.
When only -flux is turned on it is possible to obtain a topological string
model, which we do for the case of a Calabi-Yau with a closed 3-form turned on.
The simplest deformation from the A-model is due to the
component of a non-trivial -field. The model is generically no longer
evaluated on holomorphic maps and defines new topological invariants.
Deformations due to -flux can be more radical, completely preventing
auxiliary fields from being integrated out.Comment: 30 pages. v2: Improved Version. References added. v3: Minor changes,
published in JHE
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
Spinorial geometry and Killing spinor equations of 6-D supergravity
We solve the Killing spinor equations of 6-dimensional (1,0)-supergravity
coupled to any number of tensor, vector and scalar multiplets in all cases. The
isotropy groups of Killing spinors are Sp(1)\cdot Sp(1)\ltimes \bH (1),
U(1)\cdot Sp(1)\ltimes \bH (2), Sp(1)\ltimes \bH (3,4), , and , where in parenthesis is the number of supersymmetries
preserved in each case. If the isotropy group is non-compact, the spacetime
admits a parallel null 1-form with respect to a connection with torsion the
3-form field strength of the gravitational multiplet. The associated vector
field is Killing and the 3-form is determined in terms of the geometry of
spacetime. The Sp(1)\ltimes \bH case admits a descendant solution preserving
3 out of 4 supersymmetries due to the hyperini Killing spinor equation. If the
isotropy group is compact, the spacetime admits a natural frame constructed
from 1-form spinor bi-linears. In the and U(1) cases, the spacetime
admits 3 and 4 parallel 1-forms with respect to the connection with torsion,
respectively. The associated vector fields are Killing and under some
additional restrictions the spacetime is a principal bundle with fibre a
Lorentzian Lie group. The conditions imposed by the Killing spinor equations on
all other fields are also determined.Comment: 34 pages, Minor change
Special holonomy sigma models with boundaries
A study of (1,1) supersymmetric two-dimensional non-linear sigma models with boundary on special holonomy target spaces is presented. In particular, the consistency of the boundary conditions under the various symmetries is studied. Models both with and without torsion are discussed