No file available
To read the file of this research,
you can request a copy directly from the authors.
Quantum max-flow in the bridge graph
Preprints and early-stage research may not have been peer reviewed yet.
Abstract
The quantum max-flow quantifies the maximal possible entanglement between two regions of a tensor network state for a fixed graph and fixed bond dimensions. In this work, we calculate the quantum max-flow exactly in the case of the bridge graph. The result is achieved by drawing connections to the theory of prehomogenous tensor and the representation theory of quivers. Further, we highlight relations to invariant theory and to algebraic statistics.
ResearchGate has not been able to resolve any citations for this publication.
In this paper, we study sample size thresholds for maximum likelihood estimation for tensor normal models. Given the model parameters and the number of samples, we determine whether, almost surely, (1) the likelihood function is bounded from above, (2) maximum likelihood estimates (MLEs) exist, and (3) MLEs exist uniquely. We obtain a complete answer for both real and complex models. One consequence of our results is that almost sure boundedness of the log-likelihood function guarantees almost sure existence of an MLE. Our techniques are based on invariant theory and castling transforms.
Tensor product state (TPS) based methods are powerful tools to efficiently simulate quantum many-body systems in and out of equilibrium. In particular, the one-dimensional matrix-product (MPS) formalism is by now an established tool in condensed matter theory and quantum chemistry. In these lecture notes, we combine a compact review of basic TPS concepts with the introduction of a versatile tensor library for Python (TeNPy) [1]. As concrete examples, we consider the MPS based time-evolving block decimation and the density matrix renormalization group algorithm. Moreover, we provide a practical guide on how to implement abelian symmetries (e.g., a particle number conservation) to accelerate tensor operations.
Tensor networks are efficient representations of high-dimensional tensors which have been very successful for physics and mathematics applications. We demonstrate how algorithms for optimizing such networks can be adapted to supervised learning tasks by using matrix product states (tensor trains) to parameterize models for classifying images. For the MNIST data set we obtain less than 1% test set classification error. We discuss how the tensor network form imparts additional structure to the learned model and suggest a possible generative interpretation.
The quantum max-flow min-cut conjecture relates the rank of a tensor network to the minimum cut in the case that all tensors in the network are identical\cite{mfmc1}. This conjecture was shown to be false in Ref. \onlinecite{mfmc2} by an explicit counter-example. Here, we show that the conjecture is almost true, in that the ratio of the quantum max-flow to the quantum min-cut converges to 1 as the dimension N of the degrees of freedom on the edges of the network tends to infinity. The proof is based on estimating moments of the singular values of the network. We introduce a generalization of "rainbow diagrams"\cite{rainbow} to tensor networks to estimate the dominant diagrams. A direct comparison of second and fourth moments lower bounds the ratio of the quantum max-flow to the quantum min-cut by a constant. To show the tighter bound that the ratio tends to 1, we consider higher moments. In addition, we show that the limiting moments as agree with that in a different ensemble where tensors in the network are chosen independently, this is used to show that the distributions of singular values in the two different ensembles weakly converge to the same limiting distribution. We present also a numerical study of one particular tensor network, which shows a surprising dependence of the rank deficit on and suggests further conjecture on the limiting behavior of the rank.
This is a partly non-technical introduction to selected topics on tensor
network methods, based on several lectures and introductory seminars given on
the subject. It should be a good place for newcomers to get familiarized with
some of the key ideas in the field, specially regarding the numerics. After a
very general introduction we motivate the concept of tensor network and provide
several examples. We then move on to explain some basics about Matrix Product
States (MPS) and Projected Entangled Pair States (PEPS). Selected details on
some of the associated numerical methods for 1d and 2d quantum lattice systems
are also discussed.
A representation of a quiver is given by a collection of matrices. Semi-invariants of quivers can be constructed by taking admissible partial polarizations of the determinant of matrices containing sums of matrix components of the representation and the identity matrix as blocks. We prove that these determinantal semi-invariants span the space of all semi-invariants for any quiver and any infinite base field. In the course of the proof we show that one can reduce the study of generating semi-invariants to the case when the quiver has no oriented paths of length greater than one.
Physical interactions in quantum many-body systems are typically local:
Individual constituents interact mainly with their few nearest neighbors. This
locality of interactions is inherited by a decay of correlation functions, but
also reflected by scaling laws of a quite profound quantity: The entanglement
entropy of ground states. This entropy of the reduced state of a subregion
often merely grows like the boundary area of the subregion, and not like its
volume, in sharp contrast with an expected extensive behavior. Such "area laws"
for the entanglement entropy and related quantities have received considerable
attention in recent years. They emerge in several seemingly unrelated fields,
in the context of black hole physics, quantum information science, and quantum
many-body physics where they have important implications on the numerical
simulation of lattice models. In this Colloquium we review the current status
of area laws in these fields. Center stage is taken by rigorous results on
lattice models in one and higher spatial dimensions. The differences and
similarities between bosonic and fermionic models are stressed, area laws are
related to the velocity of information propagation, and disordered systems,
non-equilibrium situations, classical correlation concepts, and topological
entanglement entropies are discussed. A significant proportion of the article
is devoted to the quantitative connection between the entanglement content of
states and the possibility of their efficient numerical simulation. We discuss
matrix-product states, higher-dimensional analogues, and states from
entanglement renormalization and conclude by highlighting the implications of
area laws on quantifying the effective degrees of freedom that need to be
considered in simulations.
Associated to a closed, oriented surface S is the complex vector space with basis the set of all compact, oriented 3-manifolds which it bounds. Gluing along S defines a Hermitian pairing on this space with values in the complex vector space with basis all closed, oriented 3-manifolds. The main result in this paper is that this pairing is positive, i.e. that the result of pairing a nonzero vector with itself is nonzero. This has bearing on the question of what kinds of topological information can be extracted in principle from unitary (2+1)-dimensional TQFTs.
The proof involves the construction of a suitable complexity function c on all closed 3-manifolds, satisfying a gluing axiom which we call the topological Cauchy-Schwarz inequality, namely that c(AB) ≤max(c(AA),c(BB)) for all A,B which bound S, with equality if and only if A=B.
The complexity function c involves input from many aspects of 3-manifold topology, and in the process of establishing its key properties we obtain a number of results of independent interest. For example, we show that when two finite-volume hyperbolic 3-manifolds are glued along an incompressible acylindrical surface, the resulting hyperbolic 3-manifold has minimal volume only when the gluing can be done along a totally geodesic surface; this generalizes a similar theorem for closed hyperbolic 3-manifolds due to Agol-Storm-Thurston.
The variety of uniform matrix product states arises both in algebraic geometry as a natural generalization of the Veronese variety, and in quantum many-body physics as a model for a translation-invariant system of sites placed on a ring. Using methods from linear algebra, representation theory, and invariant theory of matrices, we study the linear span of this variety.
We study the problem of generating independent samples from the output distribution of Google’s Sycamore quantum circuits with a target fidelity, which is believed to be beyond the reach of classical supercomputers and has been used to demonstrate quantum supremacy. We propose a method to classically solve this problem by contracting the corresponding tensor network just once, and is massively more efficient than existing methods in generating a large number of uncorrelated samples with a target fidelity. For the Sycamore quantum supremacy circuit with 53 qubits and 20 cycles, we have generated 1×106 uncorrelated bitstrings s which are sampled from a distribution P^(s)=|ψ^(s)|2, where the approximate state ψ^ has fidelity F≈0.0037. The whole computation has cost about 15 h on a computational cluster with 512 GPUs. The obtained 1×106 samples, the contraction code and contraction order are made public. If our algorithm could be implemented with high efficiency on a modern supercomputer with ExaFLOPS performance, we estimate that ideally, the simulation would cost a few dozens of seconds, which is faster than Google’s quantum hardware.
We propose a tensor network approach to compute amplitudes and probabilities for a large number of correlated bitstrings in the final state of a quantum circuit. As an application, we study Google's Sycamore circuits, which are believed to be beyond the reach of classical supercomputers and have been used to demonstrate quantum supremacy. By employing a small computational cluster containing 60 graphical processing units (GPUs), we compute exact amplitudes and probabilities of 2×10^{6} correlated bitstrings with some entries fixed (which span a subspace of the output probability distribution) for the Sycamore circuit with 53 qubits and 20 cycles. The obtained results verify the Porter-Thomas distribution of the large and deep quantum circuits of Google, provide datasets and benchmarks for developing approximate simulation methods, and can be used for spoofing the linear cross entropy benchmark of quantum supremacy. Then we extend the proposed big-batch method to a full-amplitude simulation approach that is more efficient than the existing Schrödinger method on shallow circuits and the Schrödinger-Feynman method in general, enabling us to obtain the state vector of Google's simplifiable circuit with n=43 qubits and m=14 cycles using only one GPU. We also manage to obtain the state vector for Google's simplifiable circuits with n=50 qubits and m=14 cycles using a small GPU cluster, breaking the previous record on the number of qubits in full-amplitude simulations. Our method is general in computing bitstring probabilities for a broad class of quantum circuits and can find applications in the verification of quantum computers. We anticipate that our method will pave the way for combining tensor network-based classical computations and near-term quantum computations for solving challenging problems in the real world.
Tensor network states form a variational ansatz class widely used, both analytically and numerically, in the study of quantum many-body systems. It is known that if the underlying graph contains a cycle, e.g., as in projected entangled pair states, then the set of tensor network states of given bond dimension is not closed. Its closure is the tensor network variety. Recent work has shown that states on the boundary of this variety can yield more efficient representations for states of physical interest, but it remained unclear how to systematically find and optimize over such representations. We address this issue by defining an ansatz class of states that includes states at the boundary of the tensor network variety of given bond dimension. We show how to optimize over this class in order to find ground states of local Hamiltonians by only slightly modifying standard algorithms and code for tensor networks. We apply this method to different models and observe favorable energies and runtimes when compared with standard tensor network methods.
Tensor networks, a model that originated from quantum physics, has been gradually generalized as efficient models in machine learning in recent years. However, in order to achieve exact contraction, only treelike tensor networks such as the matrix product states and tree tensor networks have been considered, even for modeling two-dimensional data such as images. In this work, we construct supervised learning models for images using the projected entangled pair states (PEPS), a two-dimensional tensor network having a similar structure prior to natural images. Our approach first performs a feature map, which transforms the image data to a product state on a grid, then contracts the product state to a PEPS with trainable parameters to predict image labels. The tensor elements of PEPS are trained by minimizing differences between training labels and predicted labels. The proposed model is evaluated on image classifications using the Modified National Institute of Standards and Technology database (MNIST) and the Fashion-MNIST datasets. We show that our model is significantly superior to existing models using treelike tensor networks. Moreover, using the same input features, our method performs as well as the multilayer perceptron classifier, but with much fewer parameters and is more stable. Our results shed light on potential applications of two-dimensional tensor network models in machine learning.
Originally developed in the context of condensed-matter physics and based on renormalization group ideas, tensor networks have been revived thanks to quantum information theory and the progress in understanding the role of entanglement in quantum many-body systems. Moreover, tensor network states have turned out to play a key role in other scientific disciplines. In this context, here I provide an overview of the basic concepts and key developments in the field. I briefly discuss the most important tensor network structures and algorithms, together with an outline of advances related to global and gauge symmetries, fermions, topological order, classification of phases, entanglement Hamiltonians, holografic duality, artificial intelligence, the 2D Hubbard model, 2D quantum antiferromagnets, conformal field theory, quantum chemistry, disordered systems and many-body localization.
Consider a linear space L of complex D-dimensional linear operators and assume that some power Lk of L is the whole set End(CD). Perez-Garcia, Verstraete, Wolf, and Cirac conjectured that the sequence L
1
, L
2
, . .. stablilizes after O(D
2
) terms; we prove that this happens after O(D
2
log D) terms, improving the previously known bound of O(D
4
).
In this note we discuss the geometry of matrix product states with periodic boundary conditions and provide three infinite sequences of examples where the quantum max-flow is strictly less than the quantum min-cut. In the first we fix the underlying graph to be a 4-cycle and verify a prediction of Hastings that inequality occurs for infinitely many bond dimensions. In the second we generalize this result to a 2d-cycle. In the third we show that the 2d-cycle with periodic boundary conditions gives inequality for all d when all bond dimensions equal two, namely a gap of at least 2^{d-2} between the quantum max-flow and the quantum min-cut.
We propose a new class of tensor network state as a model for the AdS/CFT correspondence and holography. This class is demonstrated to retain key features of the multi-scale entanglement renormalization ansatz (MERA), in that they describe quantum states with algebraic correlation functions, have free variational parameters, and are efficiently contractible. Yet, unlike MERA, they are built according to a uniform tiling of hyperbolic space, without inherent directionality or preferred locations in the holographic bulk, and thus circumvent key arguments made against the MERA as a model for AdS/CFT. Novel holographic features of this tensor network class are examined, such as an equivalence between the causal cones C(R) and the entanglement wedges E(R) of connected boundary regions R.
We refine the classification of prehomogeneous vector spaces, provided by Sato-Kimura, in the case of tensor spaces, presenting a quick way to check whether a given tensor space is prehomogeneous or not.
These notes are intended as an introduction to the theory of prehomogeneous spaces, under the perspective of projective geometry. This is motivated by the fact that in the classification of irreducible prehomogeneous spaces (up to castling transforms) that was obtained by Sato and Kimura, most cases are of parabolic type. This means that they are related to certain homogeneous spaces of simple algebraic groups, and reflect their geometry. In particular we explain how the Tits-Freudenthal magic square can be understood in this setting. © 2013, Rendiconti del Seminario Matematico. All rights reserved.
We study the left-right action of on m-tuples of matrices with entries in an
infinite field K. We show that invariants of degree define the null
cone. Consequently, invariants of degree generate the ring of
invariants if . We also prove that for ,
invariants of degree at least are required to
define the null cone. We generalize our results to matrix invariants of
m-tuples of matrices, and to rings of semi-invariants for
quivers. For the proofs, we use new techniques such as the regularity lemma by
Ivanyos, Qiao and Subrahmanyam, and the concavity property of the tensor
blow-ups of matrix spaces. We will discuss several applications to algebraic
complexity theory, such as a deterministic polynomial time algorithm for
non-commutative rational identity testing, and the existence of small
division-free formulas for non-commutative polynomials.
We describe how to efficiently construct the quantum chemical Hamiltonian
operator in matrix product form. We present its implementation as a density
matrix renormalization group (DMRG) algorithm for quantum chemical applications
in a purely matrix product based framework. Existing implementations of DMRG
for quantum chemistry are based on the traditional formulation of the method,
which was developed from a viewpoint of Hilbert space decimation and attained a
higher performance compared to straightforward implementations of matrix
product based DMRG. The latter variationally optimizes a class of ansatz states
known as matrix product states (MPS), where operators are correspondingly
represented as matrix product operators (MPO). The MPO construction scheme
presented here eliminates the previous performance disadvantages while
retaining the additional flexibility provided by a matrix product approach; for
example, the specification of expectation values becomes an input parameter. In
this way, MPOs for different symmetries - abelian and non-abelian - and
different relativistic and non-relativistic models may be solved by an
otherwise unmodified program.
We present how the surface/state correspondence, conjectured in
arXiv:1503.03542, works in the setup of AdS3/CFT2 by generalizing the
formulation of cMERA. The boundary states in conformal field theories play a
crucial role in our formulation and the bulk diffeomorphism is naturally taken
into account. We give an identification of bulk local operators which
reproduces correct scalar field solutions on AdS3. We also calculate the
information metric for a locally excited state and show that it is given by
that of 2d hyperbolic manifold, which is argued to describe the time slice of
AdS3.
We propose a family of exactly solvable toy models for the AdS/CFT
correspondence based on a novel construction of quantum error-correcting codes
with a tensor network structure. Our building block is a special type of tensor
with maximal entanglement along any bipartition, which gives rise to an exact
isometry from bulk operators to boundary operators. The entire tensor network
is a quantum error-correcting code, where the bulk and boundary degrees of
freedom may be identified as logical and physical degrees of freedom
respectively. These models capture key features of entanglement in the AdS/CFT
correspondence; in particular, the Ryu-Takayanagi formula and the negativity of
tripartite information are obeyed exactly in many cases. That bulk logical
operators can be represented on multiple boundary regions mimics the
Rindler-wedge reconstruction of boundary operators from bulk operators,
realizing explicitly the quantum error-correcting features of AdS/CFT recently
proposed by Almheiri et. al in arXiv:1411.7041.
Introduction. The problem discussed in this paper was formulated by T. Harris as follows:
“Consider a rail network connecting two cities by way of a number of intermediate cities, where each link of the network has a number assigned to it representing its capacity. Assuming a steady state condition, find a maximal flow from one given city to the other.”
Let Q be a quiver without oriented cycles and let α be a dimension vector such that Gl(α) has an open orbit on the representation space R(α). We find representations {Si} and corresponding polynomials {Psi α} on R(α), which generate the semi-invariants and are algebraically independent.
Two theorems on the canonical Kronecker form of a perturbed matrix pencil and the characterization of the closure of the set of all matrix pencils with a fixed Kronecker canonical form are given.
We describe the null-cone of the representation of G on M
p
, where either G = SL(W) × SL(V) and M = Hom(V,W) (linear maps), or G = SL(V) and M is one of the representations S
2(V
*) (symmetric bilinear forms), Λ2(V
*) (skew bilinear forms), or V* ÄV*V^* \otimes V^* (arbitrary bilinear forms). Here V and W are vector spaces over an algebraically closed field K of characteristic zero and M
p
is the direct sum of p of copies of M. More specifically, we explicitly determine the irreducible components of the null-cone on M
p
. Results of Kraft and Wallach predict that their number stabilises at a certain value of p, and we determine this value. We also answer the question of when the null-cone in M
p
is defined by the polarisations of the invariants on M; typically, this is only the case if either dim V or p is small. A fundamental tool in our proofs is the Hilbert–Mumford criterion for nilpotency (also known as unstability).
The representation of dimension vector α of the quiver Q can be parametrised by a vector space R(Q, α) on which an algebraic group Gl(α) acts so that the set of orbits is bijective with the set of isomorphism classes of representations of the quiver. We describe the semi-invariant polynomial functions on this vector space in terms of the category of representations. More precisely, we associate to a suitable map between projective representations a semi-invariant polynomial function that describes when this map is inverted on the representation and we show that these semi-invariant polynomial functions form a spanning set of all semi-invariant polynomial functions in characteristics 0. If the quiver has no oriented cycles, we may replace consideration of inverting maps between projective representations by consideration of representations that are left perpendicular to some representation of dimension vector α. These left perpendicular representations are just the cokernels of the maps between projective representations that we consider.
The density matrix renormalization group is a method that is useful for describing molecules that have strongly correlated electrons. Here we provide a pedagogical overview of the basic challenges of strong correlation, how the density matrix renormalization group works, a survey of its existing applications to molecular problems, and some thoughts on the future of the method.
I show how recent progress in real space renormalization group methods can be
used to define a generalized notion of holography inspired by holographic
dualities in quantum gravity. The generalization is based upon organizing
information in a quantum state in terms of scale and defining a higher
dimensional geometry from this structure. While states with a finite
correlation length typically give simple geometries, the state at a quantum
critical point gives a discrete version of anti de Sitter space. Some finite
temperature quantum states include black hole-like objects. The gross features
of equal time correlation functions are also reproduced in this geometric
framework. The relationship between this framework and better understood
versions of holography is discussed.
This note discusses the problem of maximizing the rate of flow from one terminal to another, through a network which consists of a number of branches, each of which has a limited capacity. The main result is a theorem: The maximum possible flow from left to right through a network is equal to the minimum value among all simple cut-sets. This theorem is applied to solve a more general problem, in which a number of input nodes and a number of output nodes are used.
We prove an area law for the entanglement entropy in gapped one dimensional quantum systems. The bound on the entropy grows surprisingly rapidly with the correlation length; we discuss this in terms of properties of quantum expanders and present a conjecture on completely positive maps which may provide an alternate way of arriving at an area law. We also show that, for gapped, local systems, the bound on Von Neumann entropy implies a bound on R\'{e}nyi entropy for sufficiently large and implies the ability to approximate the ground state by a matrix product state. Comment: 9 pages, 1 figure; typo fixed in Eq. 41; typo fixed in Eq. 4
An Introduction to Quiver Representations
- H Derksen
- J Weyman
H. Derksen and J. Weyman. An Introduction to Quiver Representations, volume 184
of Graduate Studies in Mathematics. American Mathematical Society, Providence, 2017.
doi:10.1090/gsm/184.
Representation Theory: a First Course
- W Fulton
- J Harris
W. Fulton and J. Harris. Representation Theory: a First Course, volume 129 of Graduate Texts
in Mathematics. Springer, New York, 1991. doi:10.1007/978-1-4612-0979-9.