Robert König

Quantum Communication Theory and Quantum Computation

Technical University of Munich

TUM School of Computation, Information and Technology

Boltzmannstr. 3

85748 Garching

Tel. +49 89 289 17042


Research Webpage

I like the fundamental nature of basic questions about quantum information processing. Such problems can often be stated quite simply, typically have a natural operational motivation, and connect to interesting physics and mathematics.


Research focus: storage, manipulation, and transmission of quantum information

Our research centers around operational problems related to the storage, manipulation and transmission of quantum information. Through our work, we wish to elucidate the exact potential of quantum systems when used as an information-processing resource.

Quantum fault-tolerance and many-body physics

Many-body systems such as interacting spin lattices offer an attractive substrate for robust information processing. We are interested in explicit schemes, as well as fundamental limitations on quantum memories and fault-tolerant gates relying on collective degrees of freedom. We aim to develop tools for assessing the merits of non- standard approaches towards enhancing protocols, such as the use of Anderson localization in memories.

Finding suitable mechanisms for preparation, as well as readout of encoded information represents an interesting theoretical challenge. More generally, we focus on questions of interconvertibility of different states by a given set of operations (such as those experimentally available); these are related to the fundamental problem of classifying different phases of matter.

Selected publications

  • R. König and S. Bravyi. Classification of topologically protected gates for local stabilizer Physical Review Letters, vol. 110, no. 170503, April 2013.
  • J. Dengis, R. König and F. Pastawski. An optimal dissipative encoder for the toric code. New Journal of Physics, 16, 013023, October 2013.
  • S. Bravyi and R. König. Disorder-assisted error correction in Majorana chains. Communications in Mathematical Physics, vol. 361, pp. 641–692, October 2012.
  • R. König, G. Kuperberg and B. Reichardt. Quantum computation with Turaev-Viro codes. Annals of Physics, vol. 325, no. 12, pp. 2707–2749, December 2010.
  • Michael E. Beverland, Oliver Buerschaper, Robert Koenig, Fernando Pastawski,John Preskill, Sumit Sijher, Protected gates for topological quantum field theories, J. Math. Phys. 57, 022201 (2016)
  • Sergey Bravyi, Matthias Englbrecht, Robert Koenig, Nolan Peard, Correcting coherent errors with surface codes, arXiv:1710.02270.

Quantum information and communication

Basic tenets of classical information theory need to be reevaluated and often significantly revised in the presence of quantum effects. A major goal is to identify suitable information measures in a quantum setting, and to establish corresponding mathematical properties. This research is driven by the desire to make quantitative statements in scenarios arising in quantum communication and cryptography. More recently, our focus has been on communication over bosonic quantum channels, including cases where non-Gaussian noise is present.

Selected publications

  • R.König and G.Smith. Limits on classical communication from quantum entropy power inequalities. Nature Photonics, Vol. 7, pp. 140–146, January 2012.
  • R. König, S. Wehner and J. Wullschleger. Unconditional security from noisy quantum storage. IEEE Transactions on Information Theory, Vol. 58, no. 3, pp. 1962- 1984, March 2012.
  • R. König and S. Wehner. A strong converse for classical channel coding using entangled inputs. Phys. Rev. Lett. 103, 070504, August 2009.
  • R. König, R. Renner and C. Schaffner. The operational meaning of min- and max-entropy. IEEE Transactions on Information Theory, Vol. 55, no. 9, pp. 4337-4347, September 2009.
  • Sergey Bravyi, David Gosset, Robert Koenig, Quantum advantage with shallow circuits, arXiv:1704.00690.

Variational quantum physics and simulation

The operational viewpoint on physical systems has benefitted the development and analysis of variational methods, essentially providing a conceptual framework and a language for quantifying quantum correlations. We aim to further exploit these fruitful interactions, connecting e.g., state preparation protocols to the entanglement structure of many-body states. Of particular interest are statements about the accuracy of a given method. We aim to establish such error bounds by exploiting underlying symmetries. This involves the application of representation-theoretic methods to e.g., tensor network states.

Selected publications

  • R. König and E. Bilgin. Anyonic entanglement renormalization. Phys. Rev. B 82, 125118,September 2010.
  • R. König, B. Reichardt and G. Vidal. Exact entanglement renormalization for string-net models. Phys. Rev. B 79, 195123, May 2009.
  • M. Christandl, R. König, G. Mitchison and R. Renner. One and a half quantum de Finetti theorems.Comm. Math. Phys., 273 (2), 473–498, 2, July 2007.
  • Robert Koenig, Volkher B. Scholz,Matrix product approximations to multipoint functions in two-dimensional conformal fieltheory, Phys. Rev. Lett. 117, 121601 (2016).


Infinite-Dimensional Programmable Quantum Processors

M. Gschwendtner, A. Winter

PRX Quantum 2, 030308 (2021).

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A universal programmable quantum processor uses "program" quantum states to apply an arbitrary quantum channel to an input state. We generalize the concept of a finite-dimensional programmable quantum processor to infinite dimension assuming an energy constraint on the input and output of the target quantum channels. By proving reductions to and from finite-dimensional processors, we obtain upper and lower bounds on the program dimension required to approximately implement energy-limited quantum channels. In particular, we consider the implementation of Gaussian channels. Due to their practical relevance, we investigate the resource requirements for gauge-covariant Gaussian channels. Additionally, we give upper and lower bounds on the program dimension of a processor implementing all Gaussian unitary channels. These lower bounds rely on a direct information-theoretic argument, based on the generalization from finite to infinite dimension of a certain "replication lemma" for unitaries.

DOI: 10.1103/PRXQuantum.2.030308

On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems

I. Bardet, A. Capel, A. Lucia, D. Pérez-García, C. Rouzé

Journal of Mathematical Physics 62, 61901 (2021).

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The mixing time of Markovian dissipative evolutions of open quantum many-body systems can be bounded using optimal constants of certain quantum functional inequalities, such as the modified logarithmic Sobolev constant. For classical spin systems, the positivity of such constants follows from a mixing condition for the Gibbs measure via quasi-factorization results for the entropy. Inspired by the classical case, we present a strategy to derive the positivity of the modified logarithmic Sobolev constant associated with the dynamics of certain quantum systems from some clustering conditions on the Gibbs state of a local, commuting Hamiltonian. In particular, we show that for the heat-bath dynamics of 1D systems, the modified logarithmic Sobolev constant is positive under the assumptions of a mixing condition on the Gibbs state and a strong quasi-factorization of the relative entropy.

DOI: 10.1063/1.5142186

Energy-Constrained Discrimination of Unitaries, Quantum Speed Limits, and a Gaussian Solovay-Kitaev Theorem

S. Becker, N. Datta, L. Lami, C. Rouzé

Physical Review Letters 126, 190504 (2021).

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We investigate the energy-constrained (EC) diamond norm distance between unitary channels acting on possibly infinite-dimensional quantum systems, and establish a number of results. First, we prove that optimal EC discrimination between two unitary channels does not require the use of any entanglement. Extending a result by Acín, we also show that a finite number of parallel queries suffices to achieve zero error discrimination even in this EC setting. Second, we employ EC diamond norms to study a novel type of quantum speed limits, which apply to pairs of quantum dynamical semigroups. We expect these results to be relevant for benchmarking internal dynamics of quantum devices. Third, we establish a version of the Solovay-Kitaev theorem that applies to the group of Gaussian unitaries over a finite number of modes, with the approximation error being measured with respect to the EC diamond norm relative to the photon number Hamiltonian.

DOI: 10.1103/PhysRevLett.126.190504

Group Transference Techniques for the Estimation of the Decoherence Times and Capacities of Quantum Markov Semigroups

I. Bardet, M. Junge, N. LaRacuente, C. Rouzé, D.S. França

IEEE Transactions on Information Theory 67, 2878-2909 (2021).

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Capacities of quantum channels and decoherence times both quantify the extent to which quantum information can withstand degradation by interactions with its environment. However, calculating capacities directly is known to be intractable in general. Much recent work has focused on upper bounding certain capacities in terms of more tractable quantities such as specific norms from operator theory. In the meantime, there has also been substantial recent progress on estimating decoherence times with techniques from analysis and geometry, even though many hard questions remain open. In this article, we introduce a class of continuous-time quantum channels that we called transferred channels , which are built through representation theory from a classical Markov kernel defined on a compact group. In particular, we study two subclasses of such kernels: Hörmander systems on compact Lie-groups and Markov chains on finite groups. Examples of transferred channels include the depolarizing channel, the dephasing channel, and collective decoherence channels acting on d qubits. Some of the estimates presented are new, such as those for channels that randomly swap subsystems. We then extend tools developed in earlier work by Gao, Junge and LaRacuente to transfer estimates of the classical Markov kernel to the transferred channels and study in this way different non-commutative functional inequalities. The main contribution of this article is the application of this transference principle to the estimation of decoherence time, of private and quantum capacities, of entanglement-assisted classical capacities as well as estimation of entanglement breaking times, defined as the first time for which the channel becomes entanglement breaking. Moreover, our estimates hold for non-ergodic channels such as the collective decoherence channels, an important scenario that has been overlooked so far because of a lack of techniques.

DOI: 10.1109/TIT.2021.3065452

Convergence Rates for the Quantum Central Limit Theorem

S. Becker, N. Datta, L. Lami, C. Rouzé

Communications in Mathematical Physics 383, 223–279 (2021).

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Various quantum analogues of the central limit theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It implies that the state in any single output arm of an n-splitter, which is fed with n copies of a centred state ρ with finite second moments, converges to the Gaussian state with the same first and second moments as ρ. Here we exploit the phase space formalism to carry out a refined analysis of the rate of convergence in this quantum central limit theorem. For instance, we prove that the convergence takes place at a rate O(n−1/2) in the Hilbert–Schmidt norm whenever the third moments of ρ are finite. Trace norm or relative entropy bounds can be obtained by leveraging the energy boundedness of the state. Via analytical and numerical examples we show that our results are tight in many respects. An extension of our proof techniques to the non-i.i.d. setting is used to analyse a new model of a lossy optical fibre, where a given m-mode state enters a cascade of n beam splitters of equal transmissivities λ1/n fed with an arbitrary (but fixed) environment state. Assuming that the latter has finite third moments, and ignoring unitaries, we show that the effective channel converges in diamond norm to a simple thermal attenuator, with a rate O(n−12(m+1)). This allows us to establish bounds on the classical and quantum capacities of the cascade channel. Along the way, we derive several results that may be of independent interest. For example, we prove that any quantum characteristic function χρ is uniformly bounded by some ηρ<1 outside of any neighbourhood of the origin; also, ηρ can be made to depend only on the energy of the state ρ.

DOI: 10.1007/s00220-021-03988-1

Obstacles to Variational Quantum Optimization from Symmetry Protection

S. Bravyi, A. Kliesch, R. Koenig, E. Tang

Physical Review Letters 125, 260505 (2020).

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The quantum approximate optimization algorithm (QAOA) employs variational states generated by a parameterized quantum circuit to maximize the expected value of a Hamiltonian encoding a classical cost function. Whether or not the QAOA can outperform classical algorithms in some tasks is an actively debated question. Our work exposes fundamental limitations of the QAOA resulting from the symmetry and the locality of variational states. A surprising consequence of our results is that the classical Goemans-Williamson algorithm outperforms the QAOA for certain instances of MaxCut, at any constant level. To overcome these limitations, we propose a nonlocal version of the QAOA and give numerical evidence that it significantly outperforms the standard QAOA for frustrated Ising models.

DOI: 10.1103/PhysRevLett.125.260505

Enhanced noise resilience of the surface–Gottesman-Kitaev-Preskill code via designed bias

L. Hänggli, M. Heinze, R. König

Physical Review A 102, 52408 (2020).

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We study the code obtained by concatenating the standard single-mode Gottesman-Kitaev-Preskill (GKP) code with the surface code. We show that the noise tolerance of this surface–GKP code with respect to (Gaussian) displacement errors improves when a single-mode squeezing unitary is applied to each mode, assuming that the identification of quadratures with logical Pauli operators is suitably modified. We observe noise-tolerance thresholds of up to σ≈0.58 shift-error standard deviation when the surface code is decoded without using GKP syndrome information. In contrast, prior results by K. Fukui, A. Tomita, A. Okamoto, and K. Fujii, High-Threshold Fault-Tolerant Quantum Computation with Analog Quantum Error Correction, Phys. Rev. X 8, 021054 (2018) and C. Vuillot, H. Asasi, Y. Wang, L. P. Pryadko, and B. M. Terhal, Quantum error correction with the toric Gottesman-Kitaev-Preskill code, Phys. Rev. A 99, 032344 (2019) report a threshold between σ≈0.54 and σ≈0.55 for the standard (toric, respectively) surface–GKP code. The modified surface–GKP code effectively renders the mode-level physical noise asymmetric, biasing the logical-level noise on the GKP qubits. The code can thus benefit from the resilience of the surface code against biased noise. We use the approximate maximum likelihood decoding algorithm of S. Bravyi, M. Suchara, and A. Vargo, Efficient algorithms for maximum likelihood decoding in the surface code, Phys. Rev. A 90, 032326 (2014) to obtain our threshold estimates. Throughout, we consider an idealized scenario where measurements are noiseless and GKP states are ideal. Our paper demonstrates that Gaussian encodings of individual modes can enhance concatenated codes.

DOI: 10.1103/PhysRevA.102.052408

Quantum advantage with noisy shallow circuits

S. Bravyi, D. Gosset, R. König, M. Tomamichel

Nature Physics (2020).

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As increasingly sophisticated prototypes of quantum computers are being developed, a pressing challenge is to find computational problems that can be solved by an intermediate-scale quantum computer, but are beyond the capabilities of existing classical computers. Previous work in this direction has introduced computational problems that can be solved with certainty by quantum circuits of depth independent of the input size (so-called ‘shallow’ circuits) but cannot be solved with high probability by any shallow classical circuit. Here we show that such a separation in computational power persists even when the shallow quantum circuits are restricted to geometrically local gates in three dimensions and corrupted by noise. We also present a streamlined quantum algorithm that is shown to achieve a quantum advantage in a one-dimensional geometry. The latter may be amenable to experimental implementation with the current generation of quantum computers.

DOI: 10.1038/s41567-020-0948-z

Quantum Reverse Hypercontractivity: Its Tensorization and Application to Strong Converses

S. Beigi, N. Datta, C. Rouzé

Communications in Mathematical Physics 376, 753–794 (2020).

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In this paper we develop the theory of quantum reverse hypercontractivity inequalities and show how they can be derived from log-Sobolev inequalities. Next we prove a generalization of the Stroock–Varopoulos inequality in the non-commutative setting which allows us to derive quantum hypercontractivity and reverse hypercontractivity inequalities solely from 2-log-Sobolev and 1-log-Sobolev inequalities respectively. We then prove some tensorization-type results providing us with tools to prove hypercontractivity and reverse hypercontractivity not only for certain quantum superoperators but also for their tensor powers. Finally as an application of these results, we generalize a recent technique for proving strong converse bounds in information theory via reverse hypercontractivity inequalities to the quantum setting. We prove strong converse bounds for the problems of quantum hypothesis testing and classical-quantum channel coding based on the quantum reverse hypercontractivity inequalities that we derive.

DOI: 10.1007/s00220-020-03750-z

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