Arnab Adhikary (Leibniz Universität Hannover)
Watch on YouTube: https://youtu.be/T3COUsFegqw

Quantum states picked from non-trivial symmetry protected topological (SPT) phases have computational power in measurement-based quantum computation. This power is uniform across SPT phases, and is unlocked by measurements that break the symmetry. Except at special points in the phase, all computational schemes known to date place these symmetry-breaking measurements far apart, to avoid the correlations introduced by spurious, non-universal entanglement. In this work, we investigate the opposite regime of computation where the symmetry-breaking measurements are packed densely. We show that not only does the computation still function, but in fact, under reasonable physical assumptions, this is the most resource efficient mode.

Joint work with: Wang Yang, Robert Raussendorf

Ref: Phys. Rev. Lett. 133, 160601 (2024)

Ruben Campos Delgado (Leibniz Universität Hannover)
Watch on YouTube: https://youtu.be/95E7aYm31s0

Understanding what is non-classical about quantum theory is crucial for identifying the source of the quantum speed-up. One possible candidate is contextuality. The Bell-Kochen-Specker theorem establishes the impossibility of a non-contextual hidden variable model of quantum theory, or equivalently, that quantum theory is contextual. However, non-contextuality à la Kochen-Specker is unsatisfactory for several reasons. It is strictly specific to quantum theory, it applies only to deterministic hidden variable models, and it does not extend to sharp measurements. A generalized notion of non-contextuality was proposed in 2005 by Spekkens. In this talk, after introducing the basic notions, I will discuss a necessary and sufficient condition for Spekkens non-contextuality, culminating with the result that any non-linear modification of quantum mechanics actually has the effect of making it more classical. Joint (on-going) work with M. Plávala.

Jack Davis (École Normale Supérieure Paris)
Watch on YouTube: https://youtu.be/WjUuIStE_GE

What is the origin of quantum computational advantage? Providing answers to this far-reaching question amounts to identifying the key properties, or quantum resources, that distinguish quantum computers from their classical counterparts, with direct applications to the development of quantum devices. The advent of universal quantum computers, however, relies on error-correcting codes to protect fragile logical quantum information by robustly encoding it into symmetric states of a quantum physical system. Such encodings make the task of resource identification more difficult, as what constitutes a resource from the logical and physical points of view can differ significantly. Here we introduce a general framework which allows us to correctly identify quantum resources in encoded computations, based on phase-space techniques. For a given quantum code, our construction provides a Wigner function that accounts for how the symmetries of the code space are contained within the transformations of the physical space, resulting in an object capable of describing the logical content of any physical state, both within and outside the code space. We illustrate our general construction with the Gottesman--Kitaev--Preskill encoding of qudits with odd dimension. The resulting Wigner function, which we call the Zak-Gross Wigner function, is shown to correctly identify quantum resources through its phase-space negativity. For instance, it is positive for encoded stabilizer states and negative for the bosonic vacuum. We further prove several properties, including that its negativity provides a measure of magic for the logical content of a state, and that its marginals are modular measurement distributions associated to conjugate Zak patches.

Joint work with: Nicolas Fabreand and Ulysse Chabaud

Ref: arxiv.org/abs/2407.18394

Jens Eisert (Freie Universität Berlin)
Watch on YouTube: https://youtu.be/qHOS1D_hNmY

Quantum chaos is a phenomenon in quantum many-body systems associated with various complex features, such as level repulsion in energy spectra and distinct scaling behaviors of out-of-time ordered correlation functions. In this talk, we will explore a new class of "pseudochaotic" quantum Hamiltonians that challenges traditional views on quantum chaos and its connection to computational complexity [1]. Our ensemble is computationally indistinguishable from the Gaussian unitary ensemble (GUE) of strongly-interacting Hamiltonians, which is widely regarded as a prototypical model of quantum chaos. However, despite this effective indistinguishability, our Hamiltonians lack the usual markers of chaos: they display Poissonian level statistics, low operator complexity, and weak scrambling properties. This sharp contrast between computational indistinguishability and standard chaos indicators raises important questions about the fundamental nature of quantum chaos. These findings building on methods of pseudoentanglement [2] challenge in an interesting way common views on what it means to observe complex quantum systems.

In a long outlook, we will discuss some other insights on algebraic structures in quantum computation. We will briefly discuss a full classification of Pauli Lie algebras [3] that is a compelling insight waiting for new applications to be found, and may, if time allows, discuss tools of quasi-algebraic geometry to address questions in quantum information theory [4] and sketch algebraic ideas in the context of quantum error correction [5].

[1] Simulating quantum chaos without chaos
A. Gu, Y. Quek, S. Yelin, J. Eisert, L. Leone
arXiv:2410.18196 (2024)

[2] Pseudomagic quantum states
A. Gu, L. Leone, S. Ghosh, J. Eisert, S. Yelin, Y. Quek
Phys. Rev. Lett. 132, 210602 (2024)

[3] Full classification of Pauli Lie algebras
G. Aguilar, S. Cichy, J. Eisert, L. Bittel
arXiv:2408.00081 (2024)

[4] Linear growth of quantum circuit complexity
J. Haferkamp, P. Faist, N. B. T. Kothakonda, J. Eisert, N. Yunger Halpern
Nature Phys. 18, 528 (2022)

[5] The XYZ ruby code: Making a case for a three-colored graphical calculus for quantum error correction in spacetime
J. C. Magdalena de la Fuente, J. Old, A. Townsend-Teague, M. Rispler, J. Eisert, M. Müller
PRX Quantum, in press (2025)

Markus Frembs (Leibniz Universität Hannover)
Watch on YouTube: https://youtu.be/o7mLq939GOs

Contextuality - roughly, the impossibility to assign observables definite outcomes prior to experiment - is a key distinguishing feature between classical and quantum physics. Consequently, when understood as a resource for quantum computation, contextuality is expected to hold the key to quantum advantage. Yet, despite its long recognised importance in quantum foundations and, more recently, in quantum computation, the essence of contextuality and its mathematical structure remain poorly understood.

I will give a brief introduction to the subject and present a new framework for contextuality that allows to characterise it in algebraic form. I will also discuss how it subsumes a number of other frameworks, thereby connecting various results that had previously seemed unrelated and even contradictory.

Mariami Gachechiladze (Technische Universität Darmstadt)
Watch on YouTube: https://youtu.be/rpra-FmJXH4

The rapid advancement of quantum hardware necessitates the development of reliable methods to certify their correct functioning. Existing certification techniques often fall short: they are either prohibitively expensive and rely on flawless state preparation and measurement (SPAM), or they fail to provide robust guarantees. While current SPAM-robust methods are complete, they lack soundness, meaning they do not ensure the correct implementation of quantum devices. In our recent work, we introduce quantum system quizzing, a simple yet sound certification protocol that enables the certification of entire quantum models in a black-box scenario under the dimension assumption. The protocol identifies deterministic input-output correlations of the ideal target model, which are tested during each round. This black box approach inherently eliminates SPAM errors. For single-qubit models, we derive rigorous sampling complexity guarantees. Most notably, we establish an inverse linear relationship between average gate infidelities and the number of successful protocol rounds, making the method highly practical for contemporary experimental setups. For multi-qubit quantum computers, we provide sound certification proof in the infinite statistics regime and discuss the methods to derive sample complexity results in the finite statistics regime.

Roberto Gargiulo (Forschungszentrum Jülich)
Watch on YouTube: https://youtu.be/hJ8O0vFlIOc

The computation of classical Ising partition functions, coming from statistical physics, is a natural generalization of binary optimization. This is a notoriously hard problem in general, which makes it an especially interesting task to consider in the search for practical quantum advantage in near term quantum computers. In this work we view classical Ising models (on certain graphs) as quantum imaginary time evolution, which is enabled by the use of the transfer matrix mapping. Following Onsager and Kaufman’s original solution of the 2D Ising model, we apply a Lie-theoretic approach to a more general class and consider the possibility of a similar solution. Within this class of models, we show that only the 2D Ising model may lead to an efficient classical solution. Additionally, we also consider possible quantum algorithms for the computation of partition functions and thermal averages via transfer matrices, which can be implemented either with block encodings inside larger unitaries or by approximating the state trajectories with unitary operators.

Ref: https://doi.org/10.1145/3665870.3665878

David Gross (Universität zu Köln)
Watch on YouTube: https://youtu.be/9ZK5asUnHvc

We describe the symmetry group of the stabilizer polytope for any number n of systems and any prime local dimension d. 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 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.

Otfried Gühne (Universität Siegen)
Watch on YouTube: https://youtu.be/mVh9q4UOVAE

Quantifying entanglement is difficult, especially in the multiparticle scenario. One possible quantifier is the geometric measure of entanglement. For pure states, this is essentially given by the maximal overlap of a given pure state with product states. In the mathematical literature, this is also known as the injective tensor norm. In my talk I will discuss three topics: First, I will discuss how one can find quantum states with the maximal geometric measure. Second I will present some results on subspaces where all states are highly entangled. Third, I will discuss methods to compute the geometric measure by considering multiple copies of quantum states. 

Ref: arxiv.org/abs/2210.13475

Paul Herringer (Leibniz Universität Hannover)

Watch on YouTube: https://youtu.be/9QeR9n71kAg

We present the first examples of topological phases of matter with uniform power for measurement-based quantum computation. This is possible thanks to a new framework for analyzing the computational properties of phases of matter that is more general than previous constructions, which were limited to short-range entangled phases in one dimension. We show that ground states of the toric code in an anisotropic magnetic field yield a natural, albeit non-computationally-universal, application of our framework. We then present a new model with topological order whose ground states are universal resources for MBQC. Both topological models are enriched by subsystem symmetries, and these symmetries protect their computational power. Our framework greatly expands the range of physical models that can be analyzed from the computational perspective.

Joint work with: Vir B. Bulchandani, Younes Javanmard, David T. Stephen, Robert Raussendorf

arxiv.org/abs/2410.02716

Zemin Huang (Universität zu Köln)
Watch on YouTube: https://youtu.be/eMhHHJ4_fi4

In open quantum systems, interaction with the environment yields two symmetry types: strong symmetry, where the system's symmetry charge is conserved exactly, and weak symmetry, where the symmetry charge can be exchanged with the environment but remains conserved on average. Despite that most experimental systems adopt only weak symmetries, for bosonic systems, previous studies of symmetry-protected topology have relied on additional strong symmetries, posing a major challenge to realize nontrivial topological physics. In this talk, I will show that weak symmetries alone can protect topological responses that distinguish different phases of matter. I will propose string order parameters as experimental probes, and validate these through a concrete model. Moreover, this model reveals that environmental coupling can induce a phase transition to a phase intrinsically protected by weak symmetries, with no pure-state or strong-symmetry analog.

Barbara Kraus (Technische Universität München)

We derive a cooling algorithm, which can be utilized for state preparation. We analyze its properties and show that in various scenarios the cooling algorithm outperforms the dissipative state preparation method.

Piotr Mitosek (University of Brimingham)
Watch on YouTube: https://youtu.be/PUYJnkQMg-4

The one-way model of quantum computing is used not just to implement quantum computation but also as a tool for compilation and optimisation. One-way computations are driven by successive adaptive measurements of a graph state. 'Flow structures' witness whether a computation can be performed deterministically overall, and these structures can be found in polynomial time. However, existing definitions are lengthy and complex, hindering the ability to work with flow.

We simplify these definitions by providing a new algebraic interpretation of Pauli flow: the most general flow structure. We define two matrices arising from the graph's adjacency matrix: the flow-demand matrix M and the order-demand matrix N. We show Pauli flow exists if and only if there is a right inverse C of M such that the product NC describes a directed acyclic graph. Using our algebraic interpretation, we obtain O(n^3) algorithms for finding Pauli flow, improving previous bounds. We also introduce a first lower bound for the flow-finding problem by linking it to matrix invertibility over F_2, thus showing our algorithms are optimal, barring fundamental progress in matrix multiplication algorithms. A better understanding of Pauli flow and faster flow-finding algorithms will have applications in compilation, fault tolerance, and other fields.

Ref: https://arxiv.org/abs/2410.23439, https://arxiv.org/abs/2408.06059

Hendrik Poulsen Nautrup (Universität Innsbruck)
Watch on YouTube: https://youtu.be/43gxyYVUC38

Any quantum computation consists of a sequence of unitary evolutions described by a finite set of Hamiltonians. When this set is taken to consist of only products of Pauli operators, we show that the minimal such set generating su(2^N) contains 2N+1 elements. In my talk, I provide a prescription to construct such minimal sets and an algorithm for producing sequences of rotations corresponding to any given Pauli rotation, which is shown to have optimal complexity. Going through some examples, we will see that certain sets generate su(2^N) at a faster rate than others, and I will show how this rate can be optimized by tuning the fraction of anti-commuting pairs of generators. Finally, I will present how these results can be leveraged to prove universality beyond the standard approach and I will comment on implications for measurement-based and trapped ion quantum computation as well as the construction of fault-tolerant gate sets.

Joint work with: Isaac D. Smith, Maxime Cautrès, David T. Stephen

Ref: arxiv.org/abs/2408.03294

Martin Plávala (Leibniz Universität Hannover)
Watch on Youtube: https://youtu.be/egTMM_KSmJc

Various forms of contextuality were defined and investigated in the past, leading to plethora of results on foundations of quantum theory, most importantly contextuality was shown to be a crucial resource for quantum computation. Here we take the next step in this research direction by explicitly using contextuality and the violations of contextuality inequalities to benchmark quantum devices. Namely we present a derivation and experimental implementation of a dimension-dependent contextuality inequality to certify both the quantumness and dimensionality of a given system. Existing methods for certification of the dimension of quantum system can be cheated by using larger classical systems, creating a potential loophole in these benchmarks. Our approach uses contextuality inequalities that cannot be violated by classical systems thus closing the previous loophole. We validate this framework experimentally with photons, observing violations of a CHSH-based contextuality inequality and surpassing the qutrit bound of the CGLMP4-based contextuality inequality.

Malte Pütz (Universität zu Köln)
Watch on YouTube: https://youtu.be/993OEi7EFP4

Qubit loss and coherent errors induced by weak stabilizer measurements are circuit imperfections, which imperil the formation of long-range entanglement beyond a threshold described by percolation and Nishimori criticality, respectively. Here we study the threshold behavior when the two types or errors simultaneously occur in the form of random, weak measurements by mapping out the entanglement phase diagram of surface code or GHZ-class cat states realized in shallow circuits, driven by the probability and strength of random weak measurements. The two quantum protocols are dual to each other, and both are mapped to a unified classical statistical model – a random bond Ising model (RBIM) with bond dilution. The phase boundary then connects the Nishimori critical point and percolation transition, with our numerical computation supporting an RG flow from percolation to Nishimori criticality. Our result sheds light on the entanglement criticality of monitored quantum circuits, and corroborates the “universality” of Nishimori criticality.
Our work is centered on studying entanglement structures through numerical approaches such as tensor networks and transfer matrix evolution. Positioned precisely at the intersection of quantum information and condensed matter physics, I believe our research resonates strongly with the themes of the conference and addresses several of its core objectives.

Thomas Schulte-Herbrüggen (Technische Universität München)
Watch on YouTube: https://youtu.be/yKeZoPDIl8I

What can one do with a given closed or open quantum system? Which states can one reach, or observe, or tomography, or stabilise? How can one push Markovian thermal resources to the limits? We sketch a unified Lie-theoretical frame for answering these questions rigorously. It is also meant as a step towards a dedicated quantum systems theory for the quantum engineer. Worked examples illustrate the connection between theory and experiment.

David Stephen (Quantinuum)
Watch on YouTube: https://youtu.be/LYNV7oZx1cg

The presence of string order enables 1D quantum spin chains to act as resources for long-range quantum teleportation. In this talk, I will examine the converse statement, namely whether or not string order is necessary for quantum teleportation. After discussing the intuitive connection between the two concepts, I will define a more restricted form of teleportation for which string order is indeed a necessary ingredient. Then I will present a new resource state for teleportation that goes beyond this setting, and consequently does not appear to have string order. Upon closer inspection, however, we find a more general form of string order associated with non-onsite symmetries. This analysis is facilitated by a new kind of tensor network representation, the split-index matrix product state, and we use this representation to further study the physical and computational properties of a broader class of resource states with non-onsite symmetries. In particular, we find a 1D resource state that enables universal measurement-based quantum computation, yet does not have a degenerate entanglement spectrum. These results therefore support the conjectured equivalence between string order and teleportation while also revealing how the notion of string order must be generalized.

Matthew B. Weiss (University of Massachusetts, Boston: QBism Group)
Watch on Youtube: https://youtu.be/His7JXArWtk

Can the state-space of d-dimensional quantum theory be derived from studying the behavior of a single measuring device? The answer is yes, if the measuring device corresponds to a complex-projective 3-design. In this privileged case, not only does each quantum state correspond to a probability distribution over the outcomes of a single measurement, but also the probability distributions which correspond to quantum states can be elegantly characterized as those which respect an uncertainty principle. We also give simple equations which pure-state probability distributions must satisfy, and contextualize these results by showing how 3-designs allow the structure-coefficients of the Jordan algebra of observables to be extracted from the probabilities which characterize the reference measurement itself. This leads credence to the view that quantum theory ought to be primarily understood as a set of normative constraints on probability assignments which reflect nature's lack of hidden variables, and further cements the significance of 3-designs in quantum information science.

Robert Zeier (Forschungszentrum Jülich)
Watch on YouTube: https://youtu.be/OnYQmpoPuN8

The Quantum Approximate Optimization Algorithm (QAOA) has been proposed as a method to obtain approximate solutions for combinatorial optimization tasks. In this work, we study the underlying algebraic properties of three QAOA ansätze for the maximum-cut (maxcut) problem on connected graphs, while focusing on the generated Lie algebras as well as their invariant subspaces. Specifically, we analyze the standard QAOA ansatz as well as the orbit and the multi-angle ansätze. We are able to fully characterize the Lie algebras of the multi-angle ansatz for arbitrary connected graphs, finding that they only fall into one of just six families. Besides the cycle and the path graphs, dimensions of every graph are exponentially large in the system size, meaning that multi-angle ansätze are extremely prone to exhibiting barren plateaus. Then, a similar quasi-graph-independent Lie-algebraic characterization beyond the multi-angle ansatz is impeded as the circuit exhibits additional "hidden" symmetries besides those naturally arising from a certain parity-superselection operator and all automorphisms of the considered graph. Disregarding the "hidden" symmetries, we can upper bound the dimensions of the orbit and the standard Lie algebras, and the dimensions of the associated invariant subspaces are determined via explicit character formulas. To finish, we conjecture that (for most graphs) the standard Lie algebras have only components that are either exponential or that grow, at most, polynomially with the system size. This would imply that the QAOA is either prone to barren plateaus, or classically simulable.

Joint work with Sujay Kazi, Martín Larocca, Marco Farinati, Patrick J. Coles, M. Cerezo.

Ref:  https://arxiv.org/abs/2410.05187 

Michael Zurel (Simon Fraser University Burnaby)
Watch on YouTube: https://youtu.be/SgOwwbwnpvU

The Lambda polytopes provide a geometric framework for describing and reasoning about quantum computations. They lead to the surprising result that every quantum computation can be described by a probabilistic update of a probability distribution on a finite phase space, that is, a hidden variable model and classical simulation algorithm for universal quantum computation. More recently, they have found applications in other areas of quantum computation. I will cover the basics of the Lambda polytopes including their origin, their applications, and their mathematical structure.