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, published in the Science Advances, is expected to stimulate a more comprehensive study of tensor-network models capturing bulk-boundary correspondences.A press release can be found here.

Aug 12, 2019

Any technology requires precise benchmarking of its components, and the quantum technologies are no exception. Randomized benchmarking allows for the relatively resource economical estimation of the average gate fidelity of quantum gates from the Clifford group, assuming identical noise levels for all gates, making use of suitable sequences of randomly chosen Clifford gates. In new work in the Physical Review Letters, we report significant progress on randomized benchmarking, by showing that it can be done for individual quantum gates outside the Clifford group, even for varying noise levels per quantum gate. This is possible at little overhead of quantum resources, but at the expense of a significant classical computational cost. At the heart of our analysis is a representation-theoretic framework that we develop here which is brought into contact with classical estimation techniques based on bootstrapping and matrix pencils. We demonstrate the functioning of the scheme at hand of benchmarking tensor powers of T-gates. Apart from its practical relevance, we expect this insight to be relevant as it highlights the role of assumptions made on unknown noise processes when characterizing quantum gates at high precision.

Aug 08, 2019

Quantum communication between distant parties is based on suitable instances of shared entanglement. For efficiency reasons, in an anticipated quantum network beyond point-to-point communication, it is preferable that many parties can communicate simultaneously over the underlying infrastructure; however, bottlenecks in the network may cause delays. Sharing of multi-partite entangled states between parties offers a solution, allowing for parallel quantum communication. Specifically for the two-pair problem, the butterfly network provides the first instance of such an advantage in a bottleneck scenario. The underlying method differs from standard repeater network approaches in that it uses a graph state instead of maximally entangled pairs to achieve long-distance simultaneous communication. We will demonstrate how graph theoretic tools, and specifically local complementation, help decrease the number of required measurements compared to usual methods applied in repeater schemes. We will examine other examples of network architectures, where deploying local complementation techniques provides an advantage. We will finally consider the problem of extracting graph states for quantum communication via local Clifford operations and Pauli measurements, and discuss that while the general problem is known to be NP-complete, interestingly, for specific classes of structured resources, polynomial time algorithms can be identified. This work has just been published in the Nature Partner Journal Quantum Information.

Aug 07, 2019

Emilio Onorati, coming from ETH Zurich and now a postdoc at UCL in London, defends his PhD thesis entitled "Random processes over the unitary group: Mixing properties and applications in quantum information" with a summa cum laude distinction. Congratulations!

Jul 24, 2019

Results on the hardness of approximate sampling are seen as important stepping stones towards a convincing demonstration of the superior computational power of quantum devices. The most prominent suggestions for such experiments include boson sampling, IQP circuit sampling, and universal random circuit sampling. A key challenge for any such demonstration is to certify the correct implementation. For all these examples, and in fact for all sufficiently flat distributions, we show in new work in press in the Physical Review Letters that any non-interactive certification from classical samples and a description of the target distribution requires exponentially many uses of the device. Our proofs rely on the same property that is a central ingredient for the approximate hardness results: namely, that the sampling distributions, as random variables depending on the random unitaries defining the problem instances, have small second moments.

May 24, 2019

The entanglement of purification (EoP) is a measure of total correlation between two subsystems. New work, published in the Physical Review Letters, studies this quantity for free scalar field theory on a lattice and the transverse-field Ising model by numerical methods. In both of these models, we find that the EoP becomes a non-monotonic function of the distance between subsystems when the total number of lattice sites is small. When it is large, the EoP becomes monotonic and shows a plateau-like behavior. Moreover, we show that the original reflection symmetry which exchanges of subsystems can get broken in optimally purified systems. In the Ising model, we find this symmetry breaking in the ferromagnetic phase. We provide an interpretation of our results in terms of the interplay between classical and quantum correlations.

May 22, 2019

The von Neumann entropy is a key quantity in quantum information theory and, roughly speaking, quantifies the amount of quantum information contained in a state when many identical and independent i.i.d. copies of the state are available, in a regime that is often referred to as being asymptotic. In this work, we provide a new operational characterization of the von Neumann entropy which neither requires an i.i.d. limit nor any explicit randomness. We do so by showing that the von Neumann entropy fully characterizes single-shot state transitions in unitary quantum mechanics, as long as one has access to a catalyst - an ancillary system that can be re-used after the transition - and an environment which has the effect of dephasing in a preferred basis. Building upon these insights, we formulate and provide evidence for the catalytic entropy conjecture, which states that the above result holds true even in the absence of decoherence. If true, this would prove an intimate connection between single-shot state transitions in unitary quantum mechanics and the von Neumann entropy. Our results add significant support to recent insights that, contrary to common wisdom, the standard von Neumann entropy also characterizes single-shot situations and opens up the possibility for operational single-shot interpretations of other standard entropic quantities. In new work in press in the Physical Review Letters we discuss implications of these insights to readings of the third law of quantum thermodynamics and hint at potentially profound implications to holography.

May 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.

May 16, 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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!

May 23, 2018