Diploma thesis
In 2005 I completed my Diploma thesis in the quantum information group at the University of Potsdam. Questions on linear optics including how to exploit different degrees of freedom were of central interest.
PhD
Between October 2005 and October 2008 I was a PhD student in the Quantum Information Theory group at the Institute for Mathematical Sciences, Imperial College London, sponsored by Microsoft Research through the European PhD Scholarship. The Physics behind the effects I was interested in are described by Quantum Optics. More precisely, my PhD project is concerned with the capabilities of linear optics as possible hardware to realize quantum information processing devices.
That means on the one hand to find methods to describe and solve the problem of how to implement given unitaries on dualrail encoding with beam splitters. Particularly interesting is, of course, to find the optimal solutions to these problem, i.e., the solutions that work with the highest probability of success within the set of rules defined in the beginning.
A more relaxed version of this task is concerned with the generation of special quantum states. Given a certain input (e.g., single photons), how to produce a given state.
On the other hand, given certain quantum gates that are implemented in linear optics (and therefore only work probabilistically), how to apply them to given resources (e.g., EPR pairs) to generate other states, such that the resource requirement is the least possible. Actually, this task is an entirely classical problem.
Further, questions for fundamental limits inherent in linear optics were addressed.
The final version of my PhD thesis can be found here, a PDF suitable for eBookreaders is available here.
Research Interest
At the moment I am a postdoctoral researcher at the University of Potsdam.
Besides various questions in linear optics I am investigating optomechanical systems. These are, for example,
tiny mirrors which are probed with classical or quantum states of light. Signatures of quantum behaviour of this
mesoscopic mechanical system and effects of the environment can be identified.
A BibTeX file of my publications can be found here.
Publications

Konrad Kieling.
Linear Optics Methods in Quantum Information ProcessingQuantum computing is a promising attempt for the next step in the evolution of computation technnology. In this thesis, methods that can be used for implementing quantum circuits with optics are studied. This is particularly done within the framework of linear optics. The linear optics toolbox is developed and the common problem of whether the access to multiple degrees of freedom of photons may enhance linear optics' efficiency is addressed. Except for some special smallscale computation problems, exploiting multiple degrees of freedom does not seem to result in an efficiency gain. Consequences of the restriction to linear elements include the probabilistic nature of elementary quantum gates, giving rise to several schemes of computation that shift this drawback to the step of resource preparation. Procedures of how to generate these resource states are investigated for some gate model based schemes as well as for the cluster model. Further, the question of how to use the allowed tools most e ciently is investigated. Different strategies for cluster state preparation, one of the most efficient strategies among them, are proposed and numerically simulated.,
University of Potsdam,
2005
.

Konrad Kieling and David Gross and Jens Eisert.
Minimal resources for linear optical oneway computingWe address the question of how many maximally entangled pairs are needed in order to build up cluster states for quantum computing using the toolbox of linear optics. As the needed gates in dualrail encoding are necessarily probabilistic with known optimal success probability, this question amounts to finding the optimal strategy for building up cluster states, from the perspective of classical control. We develop a notion of classical strategies, and present rigorous statements on the ultimate maximal and minimal use of resources of the globally optimal strategy. We find that this strategy  being also the most robust with respect to decoherence  gives rise to an advantage of already more than an order of magnitude in the number of maximally entangled pairs when building chains with an expected length of L=40, compared to other legitimate strategies. For twodimensional cluster states, we present a first scheme achieving the optimal quadratic asymptotic scaling. This analysis shows that the choice of appropriate classical control leads to a very significant reduction in resource consumption.,
J. Opt. Soc. Am. B 24 (184),
2007
[pdf]
.

David Gross and Konrad Kieling and Jens Eisert.
Potential and limits to clusterstate quantum computing using probabilistic gatesWe establish bounds to the necessary resource consumption when building up cluster states for oneway computing using probabilistic gates. Emphasis is put on state preparation with linear optical gates, as the probabilistic character is unavoidable here. We identify rigorous general bounds to the necessary consumption of initially available maximally entangled pairs when building up onedimensional cluster states with individually acting linear optical quantum gates, entangled pairs, and vacuum modes. As the known linear optics gates have a limited maximum success probability, as we show, this amounts to finding the optimal classical strategy of fusing pieces of linear cluster states. A formal notion of classical configurations and strategies is introduced for probabilistic nonfaulty gates. We study the asymptotic performance of strategies that can be simply described, and prove ultimate bounds to the performance of the globally optimal strategy. The arguments employ methods of random walks and convex optimization. This optimal strategy is also the one that requires the shortest storage time, and necessitates the fewest invocations of probabilistic gates. For twodimensional cluster states, we find, for any elementary success probability, an essentially deterministic preparation of a cluster state with quadratic, hence optimal, asymptotic scaling in the use of entangled pairs. We also identify a percolation effect in state preparation, in that from a threshold probability on, almost all preparations will be either successful or fail. We outline the implications on linear optical architectures and faulttolerant computations.,
Phys. Rev. A 74 (042343),
2006
[pdf]
.

Konrad Kieling and Terry Rudolph and Jens Eisert.
Percolation, renormalization, and quantum computing with nondeterministic gatesWe apply a notion of static renormalization to the preparation of entangled states for quantum computing, exploiting ideas from percolation theory. Such a strategy yields a novel way to cope with the randomness of nondeterministic quantum gates. This is most relevant in the context of optical architectures, where probabilistic gates are common, and cold atoms in optical lattices, where hole defects occur. We demonstrate how to efficiently construct cluster states without the need for rerouting, thereby avoiding a massive amount of conditional dynamics; we furthermore show that except for a single layer of gates during the preparation, all subsequent operations can be shifted to the final adapted singlequbit measurements. Remarkably, cluster state preparation is achieved using essentially the same scaling in resources as if deterministic gates were available.,
Phys. Rev. Lett. 99 (130501),
2007
[pdf]
.

N. M. VanMeter and P. Lougovski and D. B. Uskov and K. Kieling and J. Eisert and Jonathan P. Dowling.
General linearoptical quantum state generation scheme: Applications to maximally pathentangled statesWe introduce schemes for linearoptical quantum state generation. A quantum state generator is a device that prepares a desired quantum state using product inputs from photon sources, linearoptical networks, and postselection using photon counters. We show that this device can be concisely described in terms of polynomial equations and unitary constraints. We illustrate the power of this language by applying the Grobnerbasis technique along with the notion of vacuum extensions to solve the problem of how to construct a quantum state generator analytically for any desired state, and use methods of convex optimization to identify bounds to success probabilities. In particular, we disprove a conjecture concerning the preparation of the maximally pathentangled n,0)+0,n) (NOON) state by providing a counterexample using these methods, and we derive a new upper bound on the resources required for NOONstate generation.,
Phys. Rev. A 76 (063808),
2007
[pdf]
.

Konrad Kieling and David Gross and Jens Eisert.
Cluster state preparation using gates operating at arbitrary success probabilitiesSeveral physical architectures allow for the sequential preparation of cluster states for measurementbased quantum computing using probabilistic quantum gates. In such an approach, the order in which partial resources are combined to form the final cluster state turns out to be crucially important. We determine the influence of this classical decision process on the expected size of the final cluster. Extending earlier work, we consider different quantum gates operating at various probabilites of success. For finite resources, we employ a computer algebra system to obtain the provably optimal classical control strategy and derive symbolic results for the expected final size of the cluster. Surprisingly, two substantially different regimes can be identified: When the success probability of the elementary gates is high, the influence of the classical control strategy is found to vanish. In that case, other figures of merit become more relevant. For small probabilities of success, the choice of an appropriate strategy is crucial.,
New J. Phys. 9 (200),
2007
[pdf]
.

Konrad Kieling and Jens Eisert.
Percolation in quantum computation and communication in “Quantum and Semiclassical Percolation and Breakdown in Disordered Solids”,
Lecture Notes in Physics 762,
2007
[pdf]
.

Konrad Kieling and Jeremy O'Brien and Jens Eisert.
On photonic controlled phase gatesAs primitives for entanglement generation, controlled phase gates take a central role in quantum computing. Especially in ideas realizing instances of quantum computation in linear optical gate arrays a closer look can be rewarding. In such architectures, all effective nonlinearities are induced by measurements: Hence the probability of success is a crucial parameter of such quantum gates. In this work, we discuss this question for controlled phase gates that implement an arbitrary phase with one and two control qubits. Within the class of postselected gates in dualrail encoding with vacuum ancillas we identify the optimal success probabilities. We construct networks that allow for an implementation by means of todays experimental capabilities in detail. The methods employed here appear specifically useful with the advent of integrated linear optical circuits, providing stable interferometers on monolithic structures.,
New Journal of Physics 12 (013003),
2010
[pdf]
.

Matthias Ohliger and Konrad Kieling and Jens Eisert.
Limitations of quantum computing with Gaussian cluster statesAbstract: We discuss the potential and limitations of Gaussian cluster states for measurementbased quantum computing. Using a framework of Gaussian projected entangled pair states (GPEPS), we show that no matter what Gaussian local measurements are performed on systems distributed on a general graph, transport and processing of quantum information is not possible beyond a certain influence region, except for exponentially suppressed corrections. We also demonstrate that even under arbitrary nonGaussian local measurements, slabs of Gaussian cluster states of a finite width cannot carry logical quantum information, even if sophisticated encodings of qubits in continuousvariable (CV) systems are allowed for. This is proven by suitably contracting tensor networks representing infinitedimensional quantum systems. The result can be seen as sharpening the requirements for quantum error correction and fault tolerance for Gaussian cluster states, and points towards the necessity of nonGaussian resource states for measurementbased quantum computing. The results can equally be viewed as referring to Gaussian quantum repeater networks.,
2010,
arXiv:1004.0081
[pdf]
[ps].

Andrea Mari and Konrad Kieling and Bo Melholt Nielsen and Eugene S. Polzik and Jens Eisert.
Directly estimating nonclassicalityWe establish a method of directly measuring and estimating nonclassicality  operationally defined in terms of the distinguishability of a given state from one with a positive Wigner function. It allows to certify nonclassicality, based on possibly much fewer measurement settings than necessary for obtaining complete tomographic knowledge, and is at the same time equipped with a full certificate. We find that even from measuring two conjugate variables alone, one can infer the nonclassicality of quantum mechanical modes. This method also provides a practical tool to eventually certify such features in mechanical degrees of freedom in optomechanics. The proof of the result is based on Bochner's theorem characterizing classical and quantum characteristic functions and on semidefinite programming. In this joint theoreticalexperimental work we present data from experimental optical Fock state preparation, demonstrating the functioning of the approach.,
2010,
arXiv:1005.1665
[pdf]
[ps].

Karel Lemr and Antonin Cernoch and Jan Soubusta and Konrad Kieling and Jens Eisert and Miloslav Dusek.
Experimental implementation of the optimal linearoptical controlled phase gateWe report on the first experimental realization of optimal linearoptical controlled phase gates for arbitrary phases. The realized scheme is entirely flexible in that the phase shift can be tuned to any given value. All such controlled phase gates are optimal in the sense that they operate at the maximum possible success probabilities that are achievable within the framework of any postselected linearoptical implementation. The quantum gate is implemented using bulk optical elements and polarization encoding of qubit states. We have experimentally explored the remarkable observation that the optimum success probability is not monotone in the phase.,
2010,
arXiv:1007.4797
[pdf]
[ps].
Posters

Minimal resources for linear optical oneway computingWe address the question of how many maximally entangled pairs are needed in order to build up cluster states for quantum computing using the
toolbox of linear optics. As the needed gates in dualrail encoding are necessarily probabilistic with known optimal success probability, this question
amounts to ﬁnding the optimal strategy for building up cluster states, from the perspective of classical control. We develop a notion of classical
strategies, and present rigorous statements on the ultimate maximal and minimal use of resources of the globally optimal strategy. We ﬁnd that
this strategy – being also the most robust with respect to decoherence – gives rise to an advantage of already more than an order of magnitude
in the number of maximally entangled pairs when building chains with an expected length of L = 40, compared to other legitimate strategies. For
twodimensional cluster states, we present a ﬁrst scheme achieving the optimal quadratic asymptotic scaling. This analysis shows that the choice of
appropriate classical control leads to a ver y signiﬁcant reduction in resource consumption.,
2007
[pdf]
.

Percolation, renormalization, and quantum computing with nondeterministic gatesWe apply a notion of static renormalization to the preparation of cluster states for quantum computing, exploiting ideas from percolation theory. Such a strategy yields a novel way to cope with the randomness of nondeterministic quantum gates. This is most relevant in the context of linear optical architectures, where probabilistic gates are inevitable. We demonstrate how to efficiently construct cluster states without the need for rerouting, thereby avoiding a massive amount of feedforward and conditional dynamics, and furthermore show that except for a single layer of fusion measurements during the preparation, all further measurements can be shifted to the final adapted single qubit measurements. Remarkably, the cluster state preparation is achieved using essentially the same scaling in resources as if deterministic gates were available.,
2007
[pdf]
.

Linear optics toolbox for integrated opticsConsidering optical modes and restricting the allowed operations to be energy conserving, one ends up in the framework of linear optics. By further ﬁxing an encoding of a
computational Hilbert space into the modes’ Fock space, these devices can be employed for quantum information processing. For a given task a variety of ﬁgures of merit – such
as the probability of success or the number of auxiliary photons – can be studied. We introduce methods to assess those quantities in the singlephoton (discrete variables) regime
and give construction prescriptions for integrated optics networks implementing controlled phasegates, state transformations and measurements in an optimal fashion.,
2010
[pdf]
.

Directly measuring the Wigner function negativityComparing the Wigner function of harmonic oscillators with classical phasespace distributions, it becomes
evident that the minimum distance of a given state with negative Wigner function to one with a positive Wigner
function (classical) is a natural measure of nonclassicality. By combining Bochners theorem with semideﬁnite
programming we present a method that can strictly bound this quantity by using a ﬁnite number of measurements
of just two quadratures. That is to say, nonclassicality is directly measured, using much less than tomographic
knowledge, and at the same time with fully certiﬁed error bars.
To demonstrate the potential of our method we show an application to data from an experiment preparing optical
modes in nonclassical states, giving rise to a tight bound on the negativity.,
2010
[pdf]
.
Talks

Linear optics state preparationAs the linear optical toolbox does not allow for interactions between single photons, these nonlinearities are simulated by projective measurements. This results in a probabilistic nature of any quantum computational scheme based entirely on linear optics. As already shown by KLM arbitrarily high success probabilities (ie. near deterministic operation of the gates) are possible in linear optics. The tradeo one has to accept are daunting overheads of optical devices and the need for expensive resource state. When it comes to experimental realization of quantum gates the question of optimal operation with limited resources arises. This includes linear optical Bell state discrimination, nonlinear signshift gates (building blocks for KLM s CNOT) and the CNOT gate itself. Recent results are discussed and simple tools and methods useful for adressing the question of optimal success will be presented. Especially linear optical implementations of the controlled NOT gate will be of interest.,
IQING 4 (Paris),
2005
.

Optimal strategies for fusing optical cluster statesWe investigate and solve the problem of how to optimally build up cluster states for quantum computing with linear optical means using fusion gates, entirely from the perspective of classical control. We develop a notion of classical strategies and identify the optimal one to prepare linear cluster states with probabilisitic gates. We find that this globally optimal strategy gives rise to an advantage of more than an order of magnitude in the number of maximally entangled pairs already when building chains with an expected length of L = 100, compared to another strategy that could equally well be thought as being optimal. For twodimensional cluster states, we present a proof of global optimality of the asymptotic behavior in the resources. This analysis shows that the choice of appropriate classical control may lead to a very significant reduction in resource consumption.,
DPG Spring Meeting (Frankfurt),
2006
.

Optimal strategies for fusing cluster states with probabilistic gatesWe investigate and solve the problem of how to optimally build up cluster states for quantum computing with probabilistic gates entirely from the perspective of classical control. As in the context of linear optics nonunit success probabilities occur quite naturally, we will focus especially on this architecture. We develop a notion of classical strategies and identify the optimal one to prepare linear cluster states. For twodimensional cluster states we present a scheme that shows optimal behavior in the asymptotic scaling in the resources. Further, a method will be presented to achieve this scaling even without the need for expensive feedforward during the preparation step.,
Baton Rouge,
2006
.

Cluster state preparation with probabilistic gatesWe investigate and solve the problem of how to optimally build up cluster states for quantum computing with probabilistic gates entirely from the perspective of classical control. As in the context of linear optics nonunit success probabilities occur quite naturally, we will focus especially on this architecture. We develop a notion of classical strategies and identify the optimal one to prepare linear cluster states. For twodimensional cluster states we present a scheme that shows optimal behavior in the asymptotic scaling in the resources. Further, a method will be presented to achieve this scaling even without the need for expensive feedforward during the preparation step, thus decreasing the amount of required quantum memory.,
QUOXIC (Oxford),
2006
.

Percolation, renormalization, and quantum computing with nondeterministic gatesWe apply a notion of static renormalization to the preparation of cluster states for quantum computing, exploiting ideas from percolation theory. Such a strategy yields a novel way to cope with the randomness of nondeterministic quantum gates. This is most relevant in the context of linear optical architectures, where probabilistic gates are inevitable. We demonstrate how to efficiently construct cluster states without the need for rerouting, thereby avoiding a massive amount of feedforward and conditional dynamics, and furthermore show that except for a single layer of fusion measurements during the preparation, all further measurements can be shifted to the final adapted single qubit measurements. Remarkably, the cluster state preparation is achieved using essentially the same scaling in resources as if deterministic gates were available.,
International Workshop on MBQC (Oxford),
2007
.

Percolation, renormalization, and quantum computing with nondeterministic gatesWe apply a notion of static renormalization to the preparation of cluster states for quantum computing, exploiting ideas from percolation theory. Such a strategy yields a novel way to cope with the randomness of nondeterministic quantum gates. This is most relevant in the context of linear optical architectures, where probabilistic gates are inevitable. We demonstrate how to efficiently construct cluster states without the need for rerouting, thereby avoiding a massive amount of feedforward and conditional dynamics, and furthermore show that except for a single layer of fusion measurements during the preparation, all further measurements can be shifted to the final adapted single qubit measurements. Remarkably, the cluster state preparation is achieved using essentially the same scaling in resources as if deterministic gates were available. Further, techniques to reduce the size of the required resource states will be presented.,
DPG Spring Meeting (Düsseldorf),
2007
.

Percolation, renormalization, and quantum computing with nondeterministic gatesWe apply a notion of static renormalization to the preparation of cluster states for quantum computing, exploiting ideas from percolation theory. Such a strategy yields a novel way to cope with the randomness of nondeterministic quantum gates. This is most relevant in the context of linear optical architectures, where probabilistic gates are inevitable. We demonstrate how to efficiently construct cluster states without the need for rerouting, thereby avoiding a massive amount of feedforward and conditional dynamics, and furthermore show that except for a single layer of fusion measurements during the preparation, all further measurements can be shifted to the final adapted single qubit measurements. Remarkably, the cluster state preparation is achieved using essentially the same scaling in resources as if deterministic gates were available. Further, techniques to reduce the size of the required resource states will be presented.,
IQING 5 (Innsbruck),
2007
.

Percolation, renormalization, and quantum computing with nondeterministic gatesWe apply a notion of static renormalization to the preparation of entangled states for quantum computing, exploiting ideas from percolation theory. Such a strategy yields a novel way to cope with the randomness of nondeterministic quantum gates. When the gate probabilities exceed some threshold of the underlying lattice, a phase transition gives rise to an asymptotically certain emergence of the desired cluster states. We demonstrate how to efficiently construct cluster states without the need for rerouting, thereby avoiding a massive amount of feedforward and conditional dynamics. Remarkably, the cluster state preparation is achieved using essentially the same scaling in resources as if deterministic gates were available, or arbitrary conditional dynamics would have been allowed for.,
First European Young Scientists Conference on Quantum Information (Vienna),
2007
.

Linear optics unleashedLinear optical architectures for quantum information pro cessing are based on singlephoton sources, photonnumber preserving optical elements, and photon (number resolving) detectors. Due to the allowed set of tools being narrowed down, speciﬁc problems can often be cast into a mathematical framework that allows for the assessment of possible state manipulation, measurement and preparation. The measurement based nature of optical gates and the issue of encoding qubits in a way to easily access them exp erimentally leads to a rich structure of problems. In this talk, we will discuss issues of resource consumption in optical state preparation, optimal success probabilities, and prescriptions of how to build linear optical quantum gates and measurement devices.,
DPG Spring Meeting (Darmstadt),
2008
.

Percolation meets linear optical quantum computingWe apply a notion of static renormalization to the preparation of entangled states for quantum computing, exploiting ideas from percolation theory. Such a strategy yields a novel way to cope with the randomness of nondeterministic quantum gates. This is most relevant in
the context of optical architectures, where probabilistic gates are common, and cold atoms in optical lattices, where hole defects occur. We demonstrate how to efficiently construct cluster states without the need for rerouting, thereby avoiding a massive amount of conditional dynamics; we furthermore show that except for a single layer of gates during the preparation, all subsequent operations can be shifted to the final adapted singlequbit measurements. Remarkably, cluster state preparation is achieved using essentially the same scaling in resources as if deterministic gates were available.,
SPIE Optics and Photonics (San Diego),
2008
.

Unleashing linear optics  explicit construction of small networks,
QQQ Meeting (PaulDrudeInstitute Berlin),
2008
.

New perspectives in optomechanics,
MINOS Project Meeting (IQOQI Vienna),
2008
.

Measuring nonMarkovian quantum noise in optomechanical systemsOptomechanical systems offer the perspective of driving mechanical modes to close to the quantum ground state by a suitable radiation pressure coupling to the light field of a cavity. In this work, we study the influence of complex thermal baths to which the mechanical mode is coupled, and discuss effects of nonOhmic damping. Complementary to efforts of cooling down the mirror to observe quantum mechanical behaviour, a new perspective of using such systems will be presented. We will discuss ideas of using the mechanical mirror at a finite temperature as an ultrasensitive device to probe properties of complex baths
 which are unaccessible so far. This is done without making any possibly unjustified assumptions: Using the device as a black box in systems identification, one can think of certifiably and quantitatively probing properties of decohering environments.,
DPG Spring Meeting (Hamburg),
2009
.
Public Talks
