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Whilst tomography has dominated the theory behind reconstructing/approximating quantum objects, such as states or channels, conducting full tomography is often not necessary in practice. If one is interested in learning properties of a quantum system, side-stepping the exponential lower bounds of tomography is then possible. In this talk, we will introduce various learning models for approximating quantum objects, survey the literature of quantum learning theory and explore instances where learning can be fully time- and sample efficient.

Particle physics underpins our understanding of the world at a fundamental level by describing the interplay of matter and forces through gauge theories. Yet, despite their unmatched success, the intrinsic quantum mechanical nature of gauge theories makes important problem classes notoriously difficult to address with classical computational techniques. A promising way to overcome these roadblocks is offered by quantum computers, which are based on the same laws that make the classical computations so difficult. Here, we present a quantum computation of the properties of the basic building block of two-dimensional lattice quantum electrodynamics, involving both gauge fields and matter. This computation is made possible by the use of a trapped-ion qudit quantum processor, where quantum information is encoded in d different states per ion, rather than in two states as in qubits. Qudits are ideally suited for describing gauge fields, which are naturally high-dimensional, leading to a dramatic reduction in the quantum register size and circuit complexity. Using a variational quantum eigensolver we find the ground state of the model and observe the interplay between virtual pair creation and quantized magnetic field effects. The qudit approach further allows us to seamlessly observe the effect of different gauge field truncations by controlling the qudit dimension. Our results open the door for hardware-efficient quantum simulations with qudits in near-term quantum devices.

BosonSampling is one of the leading candidate models for a demonstration of quantum computational advantage. However, there are still important gaps between our best theoretical results and what can be implemented realistically in the laboratory. One of the largest gaps concerns the scaling between the number of modes (m) and number of photons (n) in the experiment. The original proposal by Aaronson and Arkhipov, as well as all subsequent improvements, required m to scale as n^2, whereas most state-of-the-art typically operate in a regime where m is linear in n. In this talk, I will describe how our recent work bridges this gap by providing evidence that BosonSampling remains hard even for m as low as 2n. I will review the template for proofs of computational advantage used in BosonSampling and other proposals, and discuss how we solved the new challenges that appear in this regime.

It is known that measurement-based quantum computations (MBQCs) which compute a non-linear Boolean function with sufficiently high probability of success are contextual, i.e., they cannot be described by a non-contextual hidden variable model. It is also known that contexuality has descriptions in terms of cohomology [1,2]. And so it seems in range to obtain a cohomological description of MBQC. And yet, the two connections mentioned above are not easily strung together. In a previous work [3], the cohomological description for MBQC was provided for the temporally flat case. Here we present the extension to the general temporally ordered case. [1] S. Abramsky, R. Barbosa, S. Mansfield, The Cohomology of Non-Locality and Contextuality, EPTCS 95, 2012, pp. 1-14 [2] C. Okay, S. Roberts, S.D. Bartlett, R. Raussendorf, Topological proofs of contextuality in quantum mechanics, Quant. Inf. Comp. 17, 1135-1166 (2017). [3] R. Raussendorf, Cohomological framework for contextual quantum computations, Quant. Inf. Comp. 19, 1141-1170 (2019) This is jount work with Polina Feldmann and Cihan Okay

Demonstrating quantum advantages in near term quantum devices is a notoriously difficult task. Ongoing efforts try to overcome different limitations of quantum devices without fault-tolerance, such as their limited system size or obstacles towards verification of the outcome of the computation. Proposals that exhibit more reliable quantum advantages for classically hard-to-simulate verifiable problems lack, at the same time, practical applicability. In this talk we will review different approaches to demonstrate quantum advantages inspired from many-body quantum physics. The first of them use entangled quantum resources such as cluster states, which are useful to demonstrate verifiable quantum advantages based on sampling problems (Theory proposal Phys. Rev. X 8, 021010, 2018 and recent experimental demonstration arXiv preprint arXiv:2307.14424). The second probe measurement of many-body quantities such as dynamical structure factors in quantum simulation setups (Proceedings of the National Academy of Sciences 117 (42), 26123-26134).

Nonlocal games are a foundational tool for understanding entanglement and constructing quantum protocols in settings with multiple spatially separated quantum devices. However, the spatial separation between devices can be difficult to enforce in practice. To this end, Kalai et al. (STOC '23) initiated the study of compiled nonlocal games. The KLVY compilation procedure transforms any k-prover nonlocal into a game with a classical verifier and a single cryptographically limited quantum prover. Kalai et al. showed that their compilation procedure is sound against classical provers and complete for entangled provers. Natarajan and Zhang (FOCS '23) showed that the compiled two-prover CHSH game is sound against quantum provers. I will discuss recent work, showing that the compiler is sound for any two-player XOR game. I will also discuss challenges and open questions in extending results from nonlocal games to the compiled setting.

Information-theoretic insights have proven fruitful in many areas of quantum physics. But can the fundamental dynamics of quantum systems be derived from purely information-theoretic principles, without resorting to Hilbert space structures such as unitary evolution and self-adjoint observables? Here we provide a model where the dynamics originates from a condition of informational non-equilibrium, the deviation of the system’s state from a reference state associated to a field of identically prepared systems. Combining this idea with three basic information-theoretic principles, we derive a notion of energy that captures the main features of energy in quantum theory: it is observable, bounded from below, invariant under time-evolution, in one-to-one correspondence with the generator of the dynamics, and quantitatively related to the speed of state changes. Our results provide an information-theoretic reconstruction of the Mandelstam-Tamm bound on the speed of quantum evolutions, establishing a bridge between dynamical and information-theoretic notions.

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With the recent construction of quantum low-density parity-check (LDPC) codes with optimal asymptotic parameters, finding methods to perform low-overhead computation using those constructions has become a central problem of quantum error-correction. In particular, triorthogonal codes---which admit transversal non-Clifford operations---are of particular interest, but few examples of these codes are presently known. In our work, we introduce a new family of codes, the quantum rainbow codes, a generalization of pin codes and color codes, that can be constructed from any chain complex. When applied to the hypergraph product of three complexes, we show that those codes can implement transversal non-Clifford gates and have improved parameters compared to pin codes. Considering expander graphs with large girth as the input complexes, we can for instance obtain families of triorthogonal codes with parameters [[n,Θ(n^{2/3}),Θ(log(n))]].

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In this talk the role of information theory in the description of physical evolutions will be discussed. After defining information quantifiers, their contractivity with respect to physical dynamics will be explained, a requirement which simply encodes the intuition that noisy transformations should lose information. The interplay between the two concepts will be exemplified for Markovian evolutions, showing how Markovianity can be defined in purely information theoretic terms. Extending on this result, we prove our main theorem: that all physical maps can be defined solely in terms of a particular metric on the space of density matrices, the Fisher information. This result should be understood in the context of reconstruction of quantum mechanics, proving once again the key role of information in shaping our description of the world.

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