The AdS/CFT correspondence conjectures a holographic duality between gravity in a bulk space and a critical quantum field theory on its boundary. Tensor networks have come to provide toy models to understand such bulk-boundary correspondences, shedding light on connections between geometry and entanglement. We introduce a versatile and efficient framework for studying tensor networks, extending previous tools for Gaussian matchgate tensors in 1+1 dimensions. Using regular bulk tilings, we show that the critical Ising theory can be realized on the boundary of both flat and hyperbolic bulk lattices, obtaining highly accurate critical data. Within our framework, we also produce translation-invariant critical states by an efficiently contractible tensor network with the geometry of the multi-scale entanglement renormalization ansatz. Furthermore, we establish a link between holographic quantum error correcting codes and tensor networks. This work, going to press in the Science Advances, is expected to stimulate a more comprehensive study of tensor-network models capturing bulk-boundary correspondences.

News from Apr 20, 2019

The area law conjecture states that the entanglement entropy of a region of space in the ground state of a gapped, local Hamiltonian only grows like the surface area of the region. In new work going to press in the Physical Review Letters show that, for any quantum state that fulfills an area law, the reduced quantum state of a region of space can be unitarily compressed into a thickened surface of the region. If the interior of the region is lost after this compression, the full quantum state can be recovered to high precision by a quantum channel only acting on the thickened surface. The thickness of the boundary scales inversely proportional to the error for arbitrary spin systems and logarithmically with the error for quasi-free bosonic systems. Our results can be interpreted as a single-shot operational interpretation of the area law. The result for spin systems follows from a simple inequality showing that probability distributions with low entropy can be approximated by distributions with small support, which we believe to be of independent interest. We also discuss an emergent approximate correspondence between bulk and boundary operators and the relation of our results to tensor network states.

News from Apr 02, 2019

New work that shows how to quantum simulate topological Levin-Wen models has been published, work that brings together ideas of tensor networks with ones of mesoscopic physics and quantum simulations. The realization of topological quantum phases of matter remains a key challenge to condensed matter physics and quantum information science. In this work, we demonstrate that progress in this direction can be made by combining concepts of tensor network theory with Majorana device technology. Considering the topological double semion string-net phase as an example, we exploit the fact that the representation of topological phases by tensor networks can be significantly simpler than their description by lattice Hamiltonians. The building blocks defining the tensor network are tailored to realization via simple units of capacitively coupled Majorana bound states. In the case under consideration, this defines a remarkably simple blueprint of a synthetic double semion string-net, and one may be optimistic that the required device technology will be available soon. Our results indicate that the implementation of tensor network structures via mesoscopic quantum devices may define a powerful novel avenue to the realization of synthetic topological quantum matter in general.

News from Mar 13, 2019

New work that shows how one can simulate strongly correlated two-dimensional systems in thermal states has been published in the Physical Review Letters. Tensor network methods have become a powerful class of tools to capture strongly correlated matter, but methods to capture the experimentally ubiquitous family of models at finite temperature beyond one spatial dimension are largely lacking. We introduce a tensor network algorithm able to simulate thermal states of two-dimensional quantum lattice systems in the thermodynamic limit. The method develops instances of projected entangled pair states and projected entangled pair operators for this purpose. It is the key feature of this algorithm to resemble the cooling down of the system from an infinite temperature state until it reaches the desired finite-temperature regime. As a benchmark we study the finite-temperature phase transition of the Ising model on an infinite square lattice, for which we obtain remarkable agreement with the exact solution. We then turn to study the finite-temperature Bose-Hubbard model in the limits of two (hard-core) and three bosonic modes per site. Our technique can be used to support the experimental study of actual effectively two-dimensional materials in the laboratory, as well as to benchmark optical lattice quantum simulators with ultra-cold atoms.

News from Feb 25, 2019

A brief article on the web page of the Flagship on Quantum Technologies written by Jens Eisert summarizes what quantum simulation is all about, and what perspectives the field has.

News from Jan 10, 2019

Work on topological quantum error correction is published in Quantum, the open journal for quantum science. The color code is both an interesting example of an exactly solved topologically ordered phase of matter and also among the most promising candidate models to realize fault-tolerant quantum computation with minimal resource overhead. The contributions of this work are threefold. First of all, we build upon the abstract theory of boundaries and domain walls of topological phases of matter to comprehensively catalog the objects realizable in color codes. Together with our classification we also provide lattice representations of these objects which include three new types of boundaries as well as a generating set for all 72 color code twist defects. Our work thus provides an explicit toy model that will help to better understand the abstract theory of domain walls. Secondly, we discover a number of interesting new applications of the cataloged objects for quantum information protocols. These include improved methods for performing quantum computations by code deformation, a new four-qubit error-detecting code, as well as families of new quantum error-correcting codes we call stellated color codes, which encode logical qubits at the same distance as the next best color code, but using approximately half the number of physical qubits. To the best of our knowledge, our new topological codes have the highest encoding rate of local stabilizer codes with bounded-weight stabilizers in two dimensions. Finally, we show how the boundaries and twist defects of the color code are represented by multiple copies of other phases. Indeed, in addition to the well studied comparison between the color code and two copies of the surface code, we also compare the color code to two copies of the three-fermion model. In particular, we find that this analogy offers a very clear lens through which we can view the symmetries of the color code which gives rise to its multitude of domain walls.

News from Oct 26, 2018

The group has been successful in the first call of the Flagship initiative on quantum technologies of the European Union, with the project PASQUANS on quantum simulation. The FU Berlin's press release can be found here.

News from Oct 18, 2018

Within the last year, a number of new and unexpected applications of tensor networks have emerged in our group. This is - slightly more conventionally, the simulation of strongly correlated two-dimensional thermal systems, https://scirate.com/arxiv/1809.08258. - They also serve as design principle to create mesoscopic architectures to realize topological phases of matter and computing schemes, https://scirate.com/arxiv/1808.04529. - Maybe the least expected, they can be used in recovering non-linear classical equations of motion from data, in a tomographic mindset, https://arxiv.org/abs/1809.02448. - They can be average-case hard to contract, https://scirate.com/arxiv/1810.00738, hence identifying a first type of problem that is provably average-case hard to quantum many-body physics. - And they seem to provide new insights into an understanding of holographic models, https://scirate.com/arxiv/1711.03109, https://scirate.com/arxiv/1809.10156.

News from Oct 01, 2018

The excellence cluster MATH+ has been successful in the third round of the German excellence initiative. Jens Eisert is a PI in this network.

News from Sep 27, 2018

Quantum process tomography is aimed at learning unknown quantum processes from data. It is key to basically all applications of the quantum technologies, to build trust in the functioning of devices. However, known schemes are not sample-optimal and may suffer from state preparation and measurement (SPAM) errors. In this work, we introduce a scheme for quantum process tomography that is optimal in any desirable fashion. It makes use of (i) experimentally friendly data, it is (ii) SPAM robust and otherwise robust, (iii) exploits structure and (iv) is sample optimal. It relies heavily on new proof tools that have become available in the mathematical compressed sensing literature. This work has been published in the Physical Review Letters and selected as an Editor's choice.

News from Sep 25, 2018

Work on catalytic quantum randomness goes to press in the Physical Review X. Randomness is a defining element of mixing processes in nature and an essential ingredient to many protocols in quantum information. Specifically, we ask whether there is a gap between the power of a classical source of randomness compared to that of a quantum one. We provide a complete answer to these questions, by identifying provably optimal protocols for both classical and quantum sources of randomness, based on a dephasing construction. We find that in order to implement any noisy transition on a d-dimensional quantum system it is necessary and sufficient to have a quantum source of randomness of dimension d^1/2 or a classical one of dimension d. Interestingly, coherences provided by quantum states in a source of randomness offer a quadratic advantage. The process we construct has the additional features to be robust and catalytic, i.e., the source of randomness can be re-used. Building upon this formal framework, we illustrate that this dephasing construction can serve as a useful primitive in both equilibration and quantum information theory: We discuss applications describing the smallest measurement device, capturing the smallest equilibrating environment allowed by quantum mechanics, or forming the basis for a cryptographic private quantum channel. We complement the exact analysis with a discussion of approximate protocols based on quantum expanders deriving from discrete Weyl systems. This gives rise to equilibrating environments of remarkably small dimension. Our results highlight the curious feature of randomness that residual correlations and dimension can be traded against each other.

News from Sep 05, 2018

The European road map on quantum technologies is printed in form of an extended summary on the preprint server and the New Journal of Physics, written by an international team of researchers including Jens Eisert. For press releases, see this link.

News from Sep 01, 2018

A new exciting research network on quantum thermodynamics funded by the German Research Foundation as a Research Unit has just been installed (see the press release of the DFG). The new research field of quantum thermodynamics is enjoying significant interest recently. At the same time, demonstrations of genuine thermal machines are lacking. This research network sets out to realize such machines and demonstrate their quantum features. Partners of this German-Austrian-Israeli network are Jens Eisert (FU Berlin), spokesperson, Joachim Ankerhold (Ulm), Gershon Kurizki (Weizmann), Fred Jendrzejewski (Heidelberg), Eric Lutz (Erlangen), Ferdinand Schmidt-Kaler (Mainz) Joerg Schmiedmayer (Vienna) Kilian Singer (Kassel) Joerg Wrachtrup (Stuttgart). More information will follow soon.

News from Jul 04, 2018

Christian Krumnow, working on several aspects of tensor network states and interacting fermions, defends his PhD thesis in an impressive viva with a summa cum laude distinction. Warmest congratulations!

News from May 23, 2018

Work on anticoncentration theorems to prove speedups of quantum devices over classical computers is published in Quantum. One of the main milestones in quantum information science is to realise quantum devices that exhibit an exponential computational advantage over classical ones without being universal quantum computers, a state of affairs dubbed quantum speedup, or sometimes "quantum computational supremacy". The known schemes heavily rely on mathematical assumptions that are plausible but unproven, prominently results on anticoncentration of random prescriptions. In this work, we aim at closing the gap by proving two anticoncentration theorems and accompanying hardness results, one for circuit-based schemes, the other for quantum quench-type schemes for quantum simulations. Compared to the few other known such results, these results give rise to a number of comparably simple, physically meaningful and resource-economical schemes showing a quantum speedup in one and two spatial dimensions. At the heart of the analysis are tools of unitary designs and random circuits that allow us to conclude that universal random circuits anticoncentrate as well as an embedding of known circuit-based schemes in a 2D translation-invariant architecture. The publication in Quantum is interesting in its own right, in that it is a new open access overlay journal over the preprint server that offers a new model for publishing scientific results (Jens Eisert is a board member of Quantum).

News from May 22, 2018

Work on certifying fermionic quantum simulators is printed in the Physical Review Letters. The experimental interest in realizing quantum spin-1/2-chains has increased uninterruptedly over the last decade. In many instances, the target quantum simulation belongs to the broader class of non-interacting fermionic models, constituting an important benchmark. In spite of this class being analytically efficiently tractable, no direct certification tool has yet been reported for it. In fact, in experiments, certification has almost exclusively relied on notions of quantum state tomography scaling very unfavorably with the system size. Here, we develop experimentally-friendly fidelity witnesses for all pure fermionic Gaussian target states. Their expectation value yields a tight lower bound to the fidelity and can be measured efficiently. We derive witnesses in full generality in the Majorana-fermion representation and apply them to experimentally relevant spin-1/2 chains. Among others, we show how to efficiently certify strongly out-of-equilibrium dynamics in critical Ising chains. At the heart of the measurement scheme is a variant of importance sampling specially tailored to overlaps between covariance matrices. The method is shown to be robust against finite experimental-state infidelities.

News from May 15, 2018

The new postdoc of the group Augustine Kshetrimayum, working on tensor networks, defends his PhD thesis with summa cum laude. Congratulations!

News from Apr 20, 2018

Nathan Walk, coming from the University of Oxford, is joining the group on a prestigious Marie-Sklodowska-Curie Fellowship. Congratulations and warm welcome.

News from Apr 06, 2018

Our work on quantum simulation showing a quantum advantage while at the same time being efficiently certifiable, published in the Physical Review X, as well as work on quantum thermodynamics, published in the Nature Communications, is covered in the press. Press release on quantum simulation. Press release on quantum thermodynamics.

News from Apr 06, 2018

Work on quantum thermodynamics - and specifically on steps towards a theory of strong coupling thermodynamics - is published in the Physical Review Letters. Quantum systems strongly coupled to many-body systems equilibrate to the reduced state of a global thermal state, deviating from the local thermal state of the system as it occurs in the weak-coupling limit. Taking this insight as a starting point, we study the thermodynamics of systems strongly coupled to thermal baths. First, we provide strong-coupling corrections to the second law applicable to general systems in three of its different readings: As a statement of maximal extractable work, on heat dissipation, and bound to the Carnot efficiency. These corrections become relevant for small quantum systems and always vanish in first order in the interaction strength. We then move to the question of power of heat engines, obtaining a bound on the power enhancement due to strong coupling. Our results are exemplified on the paradigmatic situation of non-Markovian quantum Brownian motion. .

News from Mar 20, 2018