Department of Physics • University of Cologne

We generalize the notion of quantum state designs to infinite-dimensional spaces. We first prove that, under the definition of continuous-variable (CV) state t-designs from Comm. Math. Phys. 326, 755 (2014), no state designs exist for t≥2. Similarly, we prove that no CV unitary t-designs exist for t≥2. We propose an alternative definition for CV state designs, which we call rigged t-designs, and provide explicit constructions for t=2. As an application of rigged designs, we develop a design-based shadow-tomography protocol for CV states. Using energy-constrained versions of rigged designs, we define an average fidelity for CV quantum channels and relate this fidelity to the CV entanglement fidelity. As an additional result of independent interest, we establish a connection between torus 2-designs and complete sets of mutually unbiased bases.

In its many variants, randomized benchmarking (RB) is a broadly used technique for assessing the quality of gate implementations on quantum computers. A detailed theoretical understanding and general guarantees exist for the functioning and interpretation of RB protocols if the gates under scrutiny are drawn *uniformly* at random from a compact group. In contrast, many practically attractive and scalable RB protocols implement random quantum circuits with local gates randomly drawn from some gate-set. Despite their abundance in practice, for those *non-uniform* RB protocols, general guarantees under experimentally plausible assumptions are missing. In this work, we derive such guarantees for a large class of RB protocols for random circuits that we refer to as *filtered RB*. Prominent examples include linear cross-entropy benchmarking, character benchmarking, Pauli-noise tomography and variants of simultaneous RB. Building upon recent results for random circuits, we show that many relevant filtered RB schemes can be realized with random quantum circuits in linear depth, and we provide explicit small constants for common instances. We further derive general sample complexity bounds for filtered RB. We show filtered RB to be sample-efficient for several relevant groups, including protocols addressing higher-order cross-talk. Our theory for non-uniform filtered RB is, in principle, flexible enough to design new protocols for non-universal and analog quantum simulators. On the technical side, we develop a general perturbative description of noise in random circuits in terms of harmonic analysis that can also be used to analyze the noise-robustness of random circuit protocols beyond RB.

Huang, Kueng, Preskill introduced the learning task now known as “classical shadows”: given few copies of an unknown state ρ, construct a classical description of the state from independent measurements that can be used to predict certain properties of the state. Specifically, they show Θ(B/epsilon^2) samples of ρ suffice to approximate the expectation value Tr(Oρ) of any Hermitian observable O to within additive error epsilon provided Tr(O^2) ≤ B and the eigenvalues of O are contained in [-1,1]. We consider the task of constructing classical shadows with joint measurements and pure unknown states. We show Θ(√B/epsilon + 1/epsilon^2) copies are necessary and sufficient to construct classical shadows in this setting. https://uni-koeln.zoom.us/j/99938061817?pwd=bk5hemQzbmY4ZVd6dFFhTnlsb0lEUT09

Extracting important information from a quantum system as efficiently and tractably as possible is an important subroutine in most quantum technologies. We present an efficient method for constructing an approximate classical description of a quantum state using very few measurements of the state. This description, called a classical shadow, can be used to predict many different properties. The required number of measurements is independent of the system size and saturates information-theoretic lower bounds [arXiv:2002.08953]. These quantum-to-classical converters pave the way for new synergies between (near-term) quantum computing and classical machine learning [arXiv:2106.12627] in the context of quantum many-body physics. Conversely, instances where classical shadows fail constitute promising candidates for new types of quantum advantage (quantum-enhanced learning) [arXiv:2112.00778].

A central result in the foundations of quantum mechanics is the Kochen-Specker theorem. In short, it states that quantum mechanics cannot be reconciled with classical models that are noncontextual for ideal measurements. This phenomenon is also known as quantum contextuality. The first explicit derivation by Kochen and Specker was rather complex, but considerable simplifications have been achieved thereafter. We propose a systematic approach to find minimal Hardy-type and Greenberger-Horne-Zeilinger-type (GHZ-type) proofs of the Kochen-Specker theorem, these are characterized by the fact that the predictions of classical models are opposite to the predictions of quantum mechanics. Based on our results, we show that the Kochen-Specker set with 18 vectors from Cabello et al. [A. Cabello et al., Phys. Lett. A 212, 183 (1996)] is the minimal set for any dimension, verifying a longstanding conjecture by Peres. In this talk, we will give an introduction to quantum contextuality, Kochen-Specker theorem and our proof of Peres conjecture. Zoom: https://uni-koeln.zoom.us/j/97462167683?pwd=RFE3VzhsWkIzUkxWUXJCS1dIN3Q0dz09

Bell tests are the prime example of device-independent experiments where hence the specific design of the experimental devices does not influence the conclusions of the experiment. However, using statistical analysis of data from Bell tests one can show, for example, that an explanation of the data requires superluminal communication. Is this a breakdown of special relativity? In this talk I will discuss why one should not jump to the latter conclusion and how data analysis could be used to eventually improve experimental setups---if desired by the research community. https://uni-koeln.zoom.us/j/6233837377

Bell inequalities are central tools for studying nonlocal correlations and their applications in quantum information processing. Identifying inequalities for many particles or measurements is, however, difficult due to the computational complexity of characterizing the set of local correlations. We develop a method to characterize Bell inequalities under constraints, which may be given by symmetry or other linear conditions. This allows to search systematically for generalizations of given Bell inequalities to more parties. As an example, we find all possible generalizations of the two-particle inequality by Froissart [Il Nuovo Cimento B64 , 241 (1981)], also known as I3322 inequality, to three particles. For the simplest of these inequalities, we study their quantum mechanical properties and demonstrate that they are relevant, in the sense that they detect nonlocality of quantum states, for which all two-setting inequalities fail to do so.

In (quantum) information theory and statistics it is very useful to know inference relations between random variables. These relations are generally difficult to find from the data of the random variables, for example because there are random variables that cannot be observed. However, in the relatively new study of causal inference through causal structures, we aim to find such inference relations. Causal structures are Directed Acyclic Graphs that depict correlations and causation between random variables. By analysing the entropy inequalities that are implied by such a structure, one can exclude certain causal relations by comparing the given data to the inequalities. In this talk, I will introduce a non-standard entropy inequality, known as a Zhang-Yeung inequality. We will see that there also exists a partially quantum version of this inequality. I will explain the effects of this inequality, or rather its unexpected lack thereof, on certain quantum causal structures. To conclude, I will pose a question that is still unanswered and for which I'm open to new ideas.

We compare the BFGS optimizer, ADAM and Natural Gradient Descent (NatGrad) in the context of Variational Quantum Eigensolvers (VQEs). We systematically analyze their performance on the QAOA ansatz for the Transverse Field Ising Model as well as on overparametrized circuits with the ability to break the symmetry of the Hamiltonian. The BFGS algorithm is frequently unable to find a global minimum for systems beyond about 20 spins and ADAM easily gets trapped in local minima. On the other hand, NatGrad shows stable performance on all considered system sizes, albeit at a significantly higher cost per epoch. In sharp contrast to most classical gradient based learning, the performance of all optimizers is found to decrease upon seemingly benign symmetry-preserving and more complex symmetry-breaking overparametrization of the ansatz class, with BFGS and ADAM failing more often and more severely than NatGrad. Additional tests for the Heisenberg XXZ model corroborate the accuracy problems of BFGS in high dimensions, but they reveal some shortcomings of NatGrad as well. Our results suggest that great care needs to be taken in the choice of gradient based optimizers and the parametrization for VQEs. This is joint work with Michael (Kastoryano) and Christian (Gogolin).

In this study, we seek to provide a new way of studying classical fluctuation theorems. Fully quantum fluctuation theorems employ machinery that involves explicit modeling of all degrees of freedom in a quantum setting- the system that delivers the work, the heat bath, the system that imposes the control, as well as the energy reservoir or battery that fuels the dynamics. We intend to study the effects of introducing this explicit modeling in a purely classical phase space setting. To meet this end we construct the required mathematical machinery in classical phase space and derive four versions of fluctuation theorems in terms of evolution induced on the energy reservoir by the system. One of the aims is to eliminate the implicit assumptions counted upon in standard approaches and study the effects of explicitly modeling batteries and external controls.

This thesis explores one of the most prominent objects in quantum computing and fault-tolerance, the Clifford group, from a representation theory perspective. In particular, we look at the structure of the t tensor power oscillator representation of the symplectic group, intimately connected to the Clifford group, for multi-qudit systems -with odd local dimensions- described by the phase space formalism. The motivation of our work stems from the characterization of tensor power representations of the Clifford group. These representations appear in many applications of Quantum Information, from randomized benchmarking to quantum state distinction, as well as in quantum state tomography. Working with qudits, we are able to parametrize a Clifford unitary with the metaplectic representation of the symplectic group, which provides us with the capacity to investigate such representations in order to gain valuable information about the Clifford group. In parallel, the Clifford group and its related oscillator representation of the symplectic group are also important in representation theory, particularly in the study of the Howe duality, which involves a well-known notion in mathematical community, the so called theta correspondence. These representations are well understood in the case t < n in terms of self-orthogonal Calderbank-Shor-Steane (CSS) quantum codes and a correspondence termed "eta correspondence". The regime t > n was left unexplored. This thesis is an effort to investigate this regime. The objective is to decompose the t tensor power oscillator representation into irreducible ones, a goal that entailed two main levels. In the first, we decompose the representation into Sp irreducible ones, and the second, we restrict it to a special Abelian subgroup of the symplectic group, and we perform a further decomposition of the restricted oscillator representation into the subgroup’s irreducible ones (known as the N-spectrum in the literature). The analysis was carried out by implementing analytical formulas into the GAP System for Computational Discrete Algebra along with Sage Math and the proof techniques are based on mathematical tools from number theory as well as representation and character theory. The work done in this master thesis is a step forward in the characterization of the oscillator representation of the symplectic group, in an attempt to give relevant insights about the Clifford group.

Networks based on entangled quantum systems enable interesting applications in quantum information processing and the understanding of the resulting quantum correlations is essential for advancing the technology. We show that the theory of quantum coherence provides powerful tools for analyzing this problem. For that, we demonstrate that a recently proposed approach to network correlations based on covariance matrices can be improved and analytically evaluated for the most important cases.

Since the beginning of Information Theory, people have been interested in the limitations of communication and information. These limitations are completely characterized for two or three parties by Shannon inequalities: Strong subadditivity and monotonicity. However, with four or more parties it turns out there are infinitely many (!) relevant non-Shannon type inequalities, called Zhang-Yeung inequalities. The origin of these inequalities is still poorly understood. In the quantum case, Shannon inequalities are replaced by Von Neumann inequalities, but the question is whether there are indeed also non Von Neumann inequalities. I will talk about such inequalities for specific quantum states and their implications for (quantum) causal structures. These causal structures are a relatively recent development in the area of causal inference. The use of quantum Zhang-Yeung type inequalities might further specify the difference between certain quantum causal structures.

Unsharp quantum measurements provide a resource in scenarios where one faces the trade-off between information gain and disturbance. In this work we propose a scheme for semi-device-independent self-testing of unsharp measurements and show that all two-outcome qubit measurements can be self-tested in a robust way. Along with the main result we introduce a method to bound correlations in sequential measurement scenarios using semi-definite programming. This method can also be applied to refine security analysis of the semi-device-independent one-way quantum key distribution. We also present new information gain – disturbance relations for pairs of dichotomic measurements.

Sampling from quantum Gibbs states has proven to be a task of major importance in quantum computation. This problem is not trivial even for quantum computers since the exponential of a quantum many-body Hamiltonian requires too much computational power. On the one hand, different methods to prepare quantum Gibbs states have already been proposed. On the other hand, as far as we know, no certification protocol has been proposed so far. Here, we propose methods to certify one-dimensional Gibbs state preparations. The main idea behind the methods is to build a fidelity witness for the purification of the target Gibbs state, which is known as the thermofield double state. The procedure to build the parent Hamiltonian relies on a useful locality bound for imaginary-time evolutions, in the same way Lieb-Robinson bounds are used for real-time ones. The method allows a highly efficient estimation of the fidelity lower-bound through block quantum tomography.

Fully quantum fluctuation relations incorporate coherence and quantum correlations, but typically require explicit modeling of components that more traditional approaches (both classical and quantum) only treat implicitly. Such components include the system that delivers the work, as well as the system that imposes the external control. In this project, we strive to mimic the general machinery of fully quantum fluctuation relations in a purely classical phase space setting, and within this construction derive classical counterparts to various fully quantum fluctuation relations. One goal is to understand the effects of introducing explicit batteries and external controls in the classical scenario, and how this relates to standard classical approaches. A further aim is to enable a comparison between fully quantum fluctuation relations and classical fluctuation relations on a more equal footing.

In this talk, I will review a recent paper (https://arxiv.org/abs/1912.02207) giving a somewhat interesting perspective on Grover's algorithm in terms of billiard balls.

The study of polynomials that are positive on certain sets has a rich history, going back to Hilbert's seventeenth problem. Here we will look at multivariate polynomials (and more generally, tensor contractions) that have matrices as their variables. We present a family of maps that are positive semi-definite on the positive cone. This extends the well-known concept of positive maps as used in entanglement theory to the multilinear case. We will present connections to polynomial identity rings and central polynomials, concepts that found recent use in quantum information in the context of MPS bond dimension witnesses and remote time manipulation.

We give a quantum speedup for solving the canonical semidefinite programming relaxation for binary quadratic optimization. The class of relaxations for combinatorial optimization has so far eluded quantum speedups. Our methods combine ideas from quantum Gibbs sampling and matrix exponent updates. A de-quantization of the algorithm also leads to a faster classical solver. For generic instances, our quantum solver gives a nearly quadratic speedup over state-of-the-art algorithms. We also provide an efficient randomized rounding procedure that converts approximately optimal SDP solutions into constant factor approximations of the original quadratic optimization problem.

The postulates of quantum mechanics can be divided into two parts: the dynamical part which defines the pure states, their dynamics and composition rule and the probabilistic part which defines the structure of measurements and the probability rule. In this talk I will show that the dynamical part of quantum theory fully determines the probabilistic part, within the operational framework. First I will show that for single systems there are infinitely many probabilistic structures compatible with the dynamical structure, and that these can be put into correspondence with certain representations of the unitary group. Following this I will outline the operational conditions which arise when considering multiple systems, and how they constrain the possible probabilistic structures. I will then outline how the condition of associativity (of systems) singles out the measurement postulates of quantum mechanics amongst all possible alternatives. In a final part I will discuss how this result relates to existing work, such as Gleason’s theorem and Zurek’s derivation of the Born rule from envariance.

In this work, we consider optimal state discrimination for a quantum system that interacts with an environment, i.e., states evolve under a quantum channel. We show the conditions on a quantum channel and an ensemble of states such that a measurement for optimal state discrimination is preserved. In particular, we show that when an ensemble of states with equal a priori probabilities is given, an optimal measurement can be preserved over any quantum channel by applying local operations and classical communication, that is, by manipulating the quantum states before and after the channel application. Examples are provided for illustration. Our results can be readily applied to quantum communication protocols over various types of noise.

Semidefinite programs have found applications in many fields of physics and mathematics. They are well suited to the computation of upper bounds on the maxima of commutative or noncommutative polynomial problems. They also match nearly one to one the objects studied in quantum information theory. However, SDPs have richer structure, and are computationally more intensive than linear programs. Symmetry reduction is a well-known tool to reduce the computational requirements of solving those problems. However, there is currently no turn-key solution to exploit symmetries coming from representation of arbitrary groups. In this talk, we discuss numerical techniques to decompose finite representations of arbitrary finite groups, and a recent Octave/MATLAB implementation.

In spite of being more than 5 decades old, Bell's theorem is still at the center of quantum foundations, also playing a crucial role in the device-independent framework for quantum information processing. In this talk we will discuss some fresh results about Bell non-local correlations. First we introduce a natural and geometrical way to quantify non-locality. Based on that, we will see interesting concentration phenomena taking place in the set of Bell correlations. Finally, we will discuss something we have called quantum voids: regions on the faces of the non-signalling set (those compatible with special relativity) where all non-local correlations are of a post-quantum nature. By the end, and if time permits, we will see how deep learning techniques provide a novel tool to characterize and discover new kinds of non-local correlations.

Port-based teleportation (PBT) is a variant of the well-known task of quantum teleportation where the receiver Bob, instead of having to apply a non-trivial correction unitary, merely has to pick up the right quantum system at a “port” specified by the classical message he received from Alice. While much more resource-demanding than standard teleportation, PBT has applications to instantaneous non-local computation and can be used to attack position-based quantum cryptography. A perfect PBT protocol would violate the linearity of quantum theory, so there is a trade-off between error and entanglement consumption (or the number of ports) which can be analyzed using representation theory of the symmetric and unitary groups. In particular the resource state has a “purified" Schur-Weyl duality symmetry. I will give an introduction to the task of PBT and its symmetries. Subsequently, I will present how the asymptotics of existing formulas for the optimal performance for a given number of ports can be derived. To achieve this, a connection between representation theory and the random matrix ensemble GUE_0, and a surprising appearance of the discrete Laplacian are exploited.

The circuit model of quantum computation defines its logical depth or “computational time” in terms of temporal gate sequences, but the measurement-based model could allow totally dif- ferent time ordering and parallelization of logical gates. We introduce a deterministic scheme of universal measurement-based computation, using only Pauli measurements on multi-qubit hyper- graph states generated by the Controlled-Controlled-Z (CCZ) gates. In contrast to the cluster-state scheme where the Clifford gates are parallelizable, our scheme enjoys massive parallelization of CCZ and SWAP gates, so that the computational depth grows with the number of global applications of Hadamard gates, or, in other words, with the number of changing computational bases. An exponentially-short depth implementation of an N-times Controlled-Z gate illustrates a novel trade- off between space and time complexity.