Scientific Series

The matrix product state (MPS) ansatz makes possible computationally-efficient representations of weakly entangled many-body quantum systems with gapped Hamiltonians near their ground states, notably including massive, relativistic quantum fields on the lattice. No Wick rotation is required to apply the time evolution operator, enabling study of time-dependent Hamiltonians. Using free massive scalar field theory on the 1+1 Robertson-Walker metric as a toy example, I present early efforts to exploit this fact to model quantum fields in curved spacetime. We use the ADM formalism to write the appropriate Hamiltonian witnessed by a particular class of normal observers. Possible applications include simulations of gravitational particle production in the presence of interactions, studies of the slicing-dependence of entanglement production, and inclusion of the expectation of the stress-energy tensor as a matter source in a numerical relativity simulation.

In recent years there has been quite some effort to apply Matrix Product States (MPS) and more general Tensor Networks (TN) to lattice gauge theories. Contrary to the standard Euclidean-time Monte Carlo approach, which faces a major obstacle in the sign problem, numerical methods based on TN are free from the sign problem and allow to some extent simulating time evolution. Moreover, TN are also a suitable tool to explore proposals for potential future quantum simulators for lattice gauge theories.

In this talk I am going to present some examples where these possibilities allow novel insight into lattice gauge theories. After briefly introducing MPS, I will mainly focus on two models: The first part of the talk is going to be about the Schwinger model. I will show how MPS can help to explore proposals for potential future quantum simulators for this model by studying their spectral properties and simulating adiabatic preparation protocols for the interacting vacuum.

Furthermore, I will show an explicit example where TN allow to overcome the Monte Carlo sign problem in a lattice calculation by studying the zero-temperature phase structure for the two-flavor case at non-zero chemical potential with MPS.

In the second part, I am focusing on a non-Abelian gauge model, namely a 1+1 dimensional SU(2) lattice gauge theory. Using MPS, the phenomenon of string breaking in this theory can be studied in real time, thus allowing to gain new insight into this process. Moreover, I will show how the gauge field can be integrated out for systems with open boundary conditions and how to obtain a formulation which allows to address the model more efficiently with MPS.

Quantum tomography is an important tool for characterizing the parameters of unknown states, measurements, and gates.  Standard quantum tomography is the practice of estimating these parameters with known measurements, states, or both, respectively.  In recent years, it has become important to address the issue of working with systems where the ``devices'' used to prepare states and make measurements both have significant errors.  Of particular concern to me is whether such state-preparation and measurement errors are correlated with each other.  In this talk, I will share a solution to assessing such correlations with an object called a partial determinant.  Further, I will show how this technique suggests a perspective for such correlated quantum states and observables (over the space of device settings) is analogous to the non-holonomic perspectives of thermodynamic heat and work (over the macroscopic state space.)

In this talk I will give a short introduction into Projected Entangled-Pair States (PEPS), and their infinite variant iPEPS, a class of tensor network Ansatz targeted at the simulation of 2D strongly correlated systems. I will present work on two recent

projects: the first will be an application of the iPEPS algorithm to a Kitaev-Heisenberg model, a model which through-out recent years has received a lot of attention due to its potential connection to the physics of a subclass of the so-called Iridate compounds. The second will be work related to the development of the iPEPS method to specifically target cylindrical geometries. Here I will present some preliminary results where we apply the methods to the Heisenberg and Fermi-Hubbard models and evaluate their performance in comparison to infinite Matrix Product States. As a final part of my talk I will, depending on time, elaborate somewhat on potential future topics including (but not restricted to):  the main challenges of iPEPS simulations from a numerical perspective and what pre-steps we have experimented with to tackle these, the possibility of applying recent proposals for finite-temperature calculations within the PEPS framework to frustrated spin systems and the use of Tensor Network Renormalization for the study of RG flows.

Seminal work of Steve Lack showed that universal algebraic theories (PROPs) may be composed to produce more sophisticated theories.  I’ll apply this method to construct an axiomatic version of the theory of a pair of complementary observables starting from the theory of monoids.  How far can we get with this?  Quite far!  We’ll get a large chunk of finite dimensional quantum theory this way —but the fact that quantum systems have non-trivial dynamics means that it’s (always) possible to present the resulting theory as a composite PROP in Lack’s sense.  If time permits, I’ll also discuss how this approach can serve as a way of constructing toy models with specific properties.
TRIGGER WARNING: category theory, blasphemy.

Unlike entanglement entropy and mutual information which may mix both classical and quantum correlations, entanglement negativity received extensive interest recently, for its merit of measuring the pure quantum entanglement in the system. In this talk, I will introduce the entanglement negativity in 2+1 dimensional topologically ordered phases. For a bipartitioned or tripartitioned spatial manifold, we show how the universal part of entanglement negativity depends on the presence of quasiparticles and the choice of ground states. Besides interpreting recent results in exactly solvable lattice models, we give new results on non-Abelian topologically ordered phases.

1. The notion of wall-crossing structure (as defined by Maxim Kontsevich and myself in arXiv: 1303.3253)
provides the  universal framework for description of different types of wall-crossing formulas (e.g. Cecotti-Vafa in 2d or KSWCF in 4d). It also gives
a language and tools for proving algebraicity and analyticity of arising generating series (e.g. for BPS invariants).

2. Holomorphic Floer theory is the Floer theory of a pair of complex Lagrangian subvarieties  of a complex symplectic manifold (maybe infinite-dimensional).
This geometry underlies several important topics, both in mathematics and physics. Those  include  questions about analytic continuation of exponential integrals
(e.g. Feynman integrals),
deformation quantization of holomorphic symplectic manifolds (and related Riemann-Hilbert correspondences), Geometric Langlands correspondence, etc.

3. It was known for a long time that many a priori divergent series (like e.g. formal WKB series for solutions of equations with a small parameter) become analytic or meromorphic functions after taking their Laplace transform
(Borel resummation). This property was called  the resurgence property of the divergent series. The relation of the resurgence phenomenon to simplest wall-crossing formulas was realized in the early 90's in the work of French mathematicians (Ecale, Voros, Pham , Malgrange and  others).

I plan to discuss how the combination of 1 and 2 can be applied to 3 in a very general situation.
Main idea goes back to our theory of Donaldson-Thomas invariants (arXiv: 0811.2435).
Namely, analyticity of the formal series  follows from existence of a global analytic object which is glued from the local ones by means of the formal series.
This analytification of  an a priori formal variety is a by-product of the growth estimates on the data of the underlying wall-crossing structure.

In case of the Holomorphic Floer theory (which underlies e.g. the Stokes phenomenon for the WKB solutions of PDE or difference equations) one needs an estimate on the number of pseudo-holomorphic discs
with the boundary on the union of our Lagrangian subvarieties. In practice it often appears as an estimate on the number of gradient lines between two critical points
of the action or potential.

Inflation is proposed as a means of explaining why the Universe is currently so homogeneous on larger scales, solving both the horizon and flatness problems in early universe cosmology. However, if inflation itself requires homogeneous conditions to get started, then inflation is not a solution to the horizon problem. Most work up until now has focussed on a dynamical systems approach to classifying the stability of inflationary models, but recently Numerical Relativity (NR) has been used to simulate the actual evolution of the inflaton field, leading to new insights. I will describe a recent work (https://arxiv.org/abs/1608.04408) in which we used NR to consider the robustness of generic small and large field inflationary models to initial inhomogeneities in the inflaton field and the extrinsic curvature of the metric.

To build a fully functioning quantum computer, it is necessary to encode quantum information to protect it from noise. Topological codes, such as the color code, naturally protect against local errors and represent our best hope for storing quantum information. Moreover, a quantum computer must also be capable of processing this information. Since the color code has many computationally valuable transversal logical gates, it is a promising candidate for a future quantum computer architecture.

In the talk, I will provide an overview of the color code. First, I will establish a connection between the color code and a well-studied model - the toric code. Then, I will explain how one can implement a universal gate set with the subsystem and the stabilizer color codes in three dimensions using techniques of code switching and gauge fixing. Next, I will discuss the problem of decoding the color code. Finally, I will explain how one can find the optimal error correction threshold by analyzing phase transitions in certain statistical-mechanical models.

The talk is based on http://arxiv.org/abs/1410.0069, http://arxiv.org/abs/1503.02065 and recent works with M. Beverland, F. Brandao, N. Delfosse, J. Preskill and K. Svore.

In order to create ansatz wave functions for models that realize topological or symmetry protected topological phases, it is crucial to understand the entanglement properties of the ground state and how they can be incorporated into the structure of the wave function.

In this first part of this talk, I will discuss entanglement properties of models of topological crystalline insulators and spin liquids and show how to incorporate topological order, symmetry fractionalization, and lattice symmetry protected topological order into tensor network wave functions.

In the second part of this talk, I will discuss intrinsically fermionic topological phases and an exactly solvable model we built to elucidate the structure of the ground state wave functions in these phases.

References:

https://arxiv.org/abs/1507.00348

https://arxiv.org/abs/1605.06125

Condensed matter realizations of Majorana zero modes constitute potential building blocks of a topological quantum computer and thus have recently been the subject of intense theoretical and experimental investigation.  In the first part of this talk, I will introduce a new scheme for preparation, manipulation, and readout of these zero modes in semiconducting wires coated with mesoscopic superconducting islands.  This approach synthesizes recent materials growth breakthroughs with tools long successfully deployed in quantum-dot research, notably gate-tunable island couplings, charge-sensing readout, and charge pumping.  Guided by these capabilities, we map out numerous milestones that progressively bridge the gap between Majorana zero-mode detection and long-term quantum computing applications.  These include (1) detecting non-Abelian anyon ‘fusion rules’ in two complementary schemes, one based on charge sensing, the other using a novel Majorana-mediated charge pump, (2) validation of a prototype topological qubit, (3) braiding to demonstrate non-Abelian statistics, and (4) observing the elusive topological phase transition accompanying the onset of Majorana modes.  With the exception of braiding, these proposed experiments require only a single wire with as few as two islands, a setup already available in the laboratory.  In the second part of the talk, I will introduce a new class of 2D microscopic models---termed ‘Majorana-dimer models’---which generalize well-known quantum dimer models by dressing the bosonic dimers with pairs of Majorana modes.  These models host a novel interacting topological phase of matter which has the same bulk anyonic content as the chiral Ising theory, albeit with a fully gapped edge.  These seemingly contradictory statements can be reconciled by noting that our phase is inherently fermionic:  it can be understood as the product of an Ising phase with a topological p-ip superconductor.  Potential physical realizations of this exotic state via a lattice of strongly interacting Majorana modes will be discussed.

The past few years have seen a surge of interest in six-dimensional superconformal field theories (6D SCFTs).  Notably, 6D SCFTs have recently been classified using F-theory, which relates these theories to elliptically-fibered Calabi-Yau manifolds.  Classes of 6D SCFTs have remarkable connections to structures in group theory and therefore provide a physical link between two seemingly-unrelated mathematical objects.  In this talk, we describe this link and speculate on its implications for future studies of 6D SCFTs.