Geometric optimization problems in quantum computation and discrete mathematics: Stabilizer states and lattices

This thesis consists of two parts: Part I deals with properties of stabilizer states and their convex hull, the stabilizer polytope. Stabilizer states, Pauli measurements and Clifford unitaries are the three building blocks of the stabilizer formalism whose computational power is limited by the Gottesman- Knill theorem. This model is usually enriched by a magic state to get a universal model for quantum computation, referred to as quantum computation with magic states (QCM). The first part of this thesis will investigate the role of stabilizer states within QCM from three different angles. The first considered quantity is the stabilizer extent, which provides a tool to measure the non-stabilizerness or magic of a quantum state. It assigns a quantity to each state roughly measuring how many stabilizer states are required to approximate the state. It has been shown that the extent is multiplicative under taking tensor products when the considered state is a product state whose components are composed of maximally three qubits. In Chapter 2, we will prove that this property does not hold in general, more precisely, that the stabilizer extent is strictly submultiplicative. We obtain this result as a consequence of rather general properties of stabilizer states. Informally our result implies that one should not expect a dictionary to be multiplicative under taking tensor products whenever the dictionary size grows subexponentially in the dimension. In Chapter 3, we consider QCM from a resource theoretic perspective. The resource theory of magic is based on two types of quantum channels, completely stabilizer preserving maps and stabilizer operations. Both classes have the property that they cannot generate additional magic resources. We will show that these two classes of quantum channels do not coincide, specifically, that stabilizer operations are a strict subset of the set of completely stabilizer preserving channels. This might have the consequence that certain tasks which are usually realized by stabilizer operations could in principle be performed better by completely stabilizer preserving maps. In Chapter 4, the last one of Part I, we consider QCM via the polar dual stabilizer polytope (also called the Lambda-polytope). This polytope is a superset of the quantum state space and every quantum state can be written as a convex combination of its vertices. A way to classically simulate quantum computing with magic states is based on simulating Pauli measurements and Clifford unitaries on the vertices of the  Lambda-polytope. The complexity of classical simulation with respect to the polytope   is determined by classically simulating the updates of vertices under Clifford unitaries and Pauli measurements. However, a complete description of this polytope as a convex hull of its vertices is only known in low dimensions (for up to two qubits or one qudit when odd dimensional systems are considered). We make progress on this question by characterizing a certain class of operators that live on the boundary of the  Lambda-polytope when the underlying dimension is an odd prime. This class encompasses for instance Wigner operators, which have been shown to be vertices of  Lambda. We conjecture that this class contains even more vertices of  Lambda. Eventually, we will shortly sketch why applying Clifford unitaries and Pauli measurements to this class of operators can be efficiently classically simulated. Part II of this thesis deals with lattices. Lattices are discrete subgroups of the Euclidean space. They occur in various different areas of mathematics, physics and computer science. We will investigate two types of optimization problems related to lattices. In Chapter 6 we are concerned with optimization within the space of lattices. That is, we want to compare the Gaussian potential energy of different lattices. To make the energy of lattices comparable we focus on lattices with point density one. In particular, we focus on even unimodular lattices and show that, up to dimension 24, they are all critical for the Gaussian potential energy. Furthermore, we find that all n-dimensional even unimodular lattices with n   24 are local minima or saddle points. In contrast in dimension 32, there are even unimodular lattices which are local maxima and others which are not even critical. In Chapter 7 we consider flat tori R^n/L, where L is an n-dimensional lattice. A flat torus comes with a metric and our goal is to approximate this metric with a Hilbert space metric. To achieve this, we derive an infinite-dimensional semidefinite optimization program that computes the least distortion embedding of the metric space R^n/L into a Hilbert space. This program allows us to make several interesting statements about the nature of least distortion embeddings of flat tori. In particular, we give a simple proof for a lower bound which gives a constant factor improvement over the previously best lower bound on the minimal distortion of an embedding of an n-dimensional flat torus. Furthermore, we show that there is always an optimal embedding into a finite-dimensional Hilbert space. Finally, we construct optimal least distortion embeddings for the standard torus R^n/Z^n and all 2-dimensional flat tori

Changes of endocannabinoid plasma levels following type I trauma

The endocannabinoid system has emerged as one of the most important facilitators of stress adaptation in the body. So far, little to no research took place concerning the endocannabinoid response to stress in humans. In this study, we investigated the influence of a type I trauma on endocannabinoid plasma levels in humans over a period of 6 months, compared with non-traumatized controls. We measured endocannabinoid plasma levels (using high performance liquid chromatography-tandem mass spectrometry) as well as hippocampal and amygdala volumes in fourteen participants who had experienced a psychic trauma and developed an acute stress disorder. Fourteen healthy non-traumatized age- and gender-matched controls were studied in comparison over a 6 months period. Data were collected at three points of time: within 48 hours after the traumatic event, after one month and after six months. At each point of time a psychiatric interview was conducted and the Clinician Administered PTSD Scale (CAPS), HAMD and BDI were rated. When 2-Arachidonoylglycerol (2-AG) levels were compared between traumatized subjects and non-traumatized controls, there was a statistical significant difference one month after trauma (p=0.046). Regarding endocannabinoid levels over the course of time, 2-AG decreased in the trauma group between one month and six months after the initial trauma. This finding was in line with the hypothesis that endocannabinoids act on-demand as protective mechanism, being released after a stressful stimulus to inhibit the development of posttraumatic sequelae. In traumatized subjects, anandamide levels after six months showed a strong negative correlation with hippocampal volumes at this time point. A statistically significant negative correlation between left amygdala volume at six months and anandamide level at this time point in traumatized subjects was found. In non-traumatized controls, remarkably, there were positive correlations of 2-AG levels and amygdala volumes one month after trauma, which were not detected at the other time points. In contrast, endocannabinoid levels were neither correlated with any demographic and clinical variable, nor with volumes of cingulate regions. In conclusion, the results of our study point to a delayed and possibly regulatory response of the endocannabinoid system after a single psychic trauma over the course of six months in humans. Moreover, the data point to a genuine relation of peripheral endocannabinoid levels possibly reflecting central endocannabinoid activity and neuroplasticity in brain key regions involved in the generation of traumatic memories, i.e. hippocampus and amygdale

Efficient classical simulation of quantum computation beyond Wigner positivity

We present the generalization of the CNC formalism, based on closed and noncontextual sets of Pauli observables, to the setting of odd-prime-dimensional qudits. By introducing new CNC-type phase space point operators, we construct a quasiprobability representation for quantum computation which is covariant with respect to the Clifford group and positivity preserving under Pauli measurements, and whose nonnegative sector strictly contains the subtheory of quantum theory described by nonnegative Wigner functions. This allows for a broader class of magic state quantum circuits to be efficiently classically simulated than those covered by the stabilizer formalism and Wigner function methods.Comment: 25 pages, 4 figure, 2 algorithm

The axiomatic and the operational approaches to resource theories of magic do not coincide

Stabiliser operations occupy a prominent role in the theory of fault-tolerant quantum computing. They are defined operationally: by the use of Clifford gates, Pauli measurements and classical control. Within the stabiliser formalism, these operations can be efficiently simulated on a classical computer, a result which is known as the Gottesman-Knill theorem. However, an additional supply of magic states is enough to promote them to a universal, fault-tolerant model for quantum computing. To quantify the needed resources in terms of magic states, a resource theory of magic has been developed during the last years. Stabiliser operations (SO) are considered free within this theory, however they are not the most general class of free operations. From an axiomatic point of view, these are the completely stabiliser-preserving (CSP) channels, defined as those that preserve the convex hull of stabiliser states. It has been an open problem to decide whether these two definitions lead to the same class of operations. In this work, we answer this question in the negative, by constructing an explicit counter-example. This indicates that recently proposed stabiliser-based simulation techniques of CSP maps are strictly more powerful than Gottesman-Knill-like methods. The result is analogous to a well-known fact in entanglement theory, namely that there is a gap between the class of local operations and classical communication (LOCC) and the class of separable channels. Along the way, we develop a number of auxiliary techniques which allow us to better characterise the set of CSP channels.Comment: 10 pages + 12 pages appendix, 2 figures. Fixed grammar mistake in title. Added one and two-qubit case. Corrected wrong equatio

Wigner's Theorem for stabilizer states and quantum designs

We describe the symmetry group of the stabilizer polytope for any number $n$ of systems and any prime local dimension $d$. In the qubit case, the symmetry group coincides with the linear and anti-linear Clifford operations. In the case of qudits, the structure is somewhat richer: for $n=1$, it is a wreath product of permutations of bases and permutations of the elements within each basis. For $n>1$, the symmetries are given by affine symplectic similitudes. These are the affine maps that preserve the symplectic form of the underlying discrete phase space up to a non-zero multiplier. We phrase these results with respect to a number of a priori different notions of "symmetry'', including Kadison symmetries (bijections that are compatible with convex combinations), Wigner symmetries (bijections that preserve inner products), and symmetries realized by an action on Hilbert space. Going beyond stabilizer states, we extend an observation of Heinrich and Gross (Ref. [25]) and show that the symmetries of fairly general sets of Hermitian operators are constrained by certain moments. In particular: the symmetries of a set that behaves like a 3-design preserve Jordan products and are therefore realized by conjugation with unitaries or anti-unitaries. (The structure constants of the Jordan algebra are encoded in an order-three tensor, which we connect to the third moments of a design). This generalizes Kadison's formulation of the classic Wigner Theorem on quantum mechanical symmetries.Comment: 21 pages, v2: minor notation changes and references adde

Hidden variable model for quantum computation with magic states on qudits of any dimension

It was recently shown that a hidden variable model can be constructed for universal quantum computation with magic states on qubits. Here we show that this result can be extended, and a hidden variable model can be defined for quantum computation with magic states on qudits with any Hilbert space dimension. This model leads to a classical simulation algorithm for universal quantum computation.Comment: 31 pages. v3=version accepted for publication in Quantu

Beurteilung des Kurzzeiteffektes transkutaner elektrischer Nervenstimulation (TENS) in der Therapie schmerzhafter Kraniomandibulärer Dysfunktionen

Ziel dieser randomisierten, placebokontrollierten, klinischen Studie war die Untersuchung des Kurzzeiteffektes transkutaner elektrischer Nervenstimulation (TENS) im Burst-Modus in der Therapie schmerzhafter Kraniomandibulärer Dysfunktionen (CMD). Untersucht wurden die Parameter Schmerzreduktion, Verbesserung der Unterkiefermobilität und das subjetive Erfolgsgefühl der Patienten/innen (n=15 pro Gruppe). Im Placebovergleich führte eine 30-minütige TENS-Behandlung im Burst-Modus (Einstellungsparameter: Asymmetrischer Bi-Phasen-Rechteckimpuls/Impulsdauer 80 Mikrosekunden/9 Impulse pro Burst mit einer Frequenz von 100 Hz/2 Bursts pro Sekunde/Individuell regelbare Impulsamplitude nach dem subjektiven Erfolgsempfinden/Maximale Spannung 9 V) zu einer statistisch hochsignifikanten unmittelbaren Schmerzreduktion (alpha=0,000), aber nicht zu einer kompletten Schmerzfreiheit. Die Dauer der Schmerzreduktion war kurzzeitig (1 h) und trat bei 20 Prozent der Patienten/innen nicht auf. Ferner konnte im Placebovergleich eine statistisch signifikante Steigerung der maximalen Mundöffnung ohne Schmerz (alpha=0,002) sowie der Protrusion (alpha=0,040) beobachtet werden. Wie lange diese Steigerung anhielt, lässt sich mit dieser Studie nicht beantworten. Die subjektive Patientenzufriedenheit war im Placebvergleich 24 h nach der TENS im Burst-Modus-Behandlung signifikant höher (alpha=0,046). TENS im Burst-Modus mit den in der Studie untersuchten Stimulationsparameter kann als eine adjuvante therapeutische Maßnahme in der symptomatischen Therapie schmerzhafter Kraniomandibulärer Dysfunktionen empfohlen werden

Coordinated population activity underlying texture discrimination in rat barrel cortex

Rodents can robustly distinguish fine differences in texture using their whiskers, a capacity that depends on neuronal activity in primary somatosensory \u201cbarrel\u201d cortex. Here we explore how texture was collectively encoded by populations of three to seven neuronal clusters simultaneously recorded from barrel cortex while a rat performed a discrimination task. Each cluster corresponded to the single-unit or multiunit activity recorded at an individual electrode. To learn how the firing of different clusters combines to represent texture, we computed population activity vectors across moving time windows and extracted the signal available in the optimal linear combination of clusters. We quantified this signal using receiver operating characteristic analysis and compared it to that available in single clusters. Texture encoding was heterogeneous across neuronal clusters, and only a minority of clusters carried signals strong enough to support stimulus discrimination on their own. However, jointly recorded groups of clusters were always able to support texture discrimination at a statistically significant level, even in sessions where no individual cluster represented the stimulus. The discriminative capacity of neuronal activity was degraded when error trials were included in the data, compared to only correct trials, suggesting a link between the neuronal activity and the animal's performance. These analyses indicate that small groups of barrel cortex neurons can robustly represent texture identity through synergistic interactions, and suggest that neurons downstream to barrel cortex could extract texture identity on single trials through simple linear combination of barrel cortex responses