Objective 1: Applications of Quantum Many-Body Scars
In the first part on this project, we want to understand if Quantum Many-Body Scars (QMBS), can be used to generate and manipulate quantum states with useful entanglement structures. To achieve this goal, we first developed a better understanding of the phenomenon of QMBS in Rydberg atom arrays in two- and three-dimensional lattices. This was motivated by the expectation that higher dimensional structures would be necessary to host states with useful entanglement structure. For this, we developed a novel tensor network ansatz for quantum states in higher dimensions and showed the following key properties: (1) this ansatz exactly satisfies the blockade constraint characteristic for Rydberg atom arrays, (2) it can be contracted efficiently in two and three dimensions, (3) it contains the orbits traced out by the scarred dynamics. This allowed us to predict many features of QMBS in Rydberg models two- and three-dimensional lattices. Moreover, it provided us with a parametrized family of states describing the orbits of scars. The results of this work are published in [arxiv:2311.08965].
Next, we analysed methods to control QMBS in Rydberg models. For this, we analysed a modification of the standard PXP model that describes Rydberg atom arrays and hosts QMBS on bipartite lattices. Specifically, we employed a Floquet model, where the coherent drive is alternated between two partitions of a bipartite lattice, which can be realized with a Rydberg atom array consisting of two species. This Floquet-PXP model has the advantage that there is an additional control parameter, namely the Floquet frequency. We showed that QMBS appear at any value of this control parameter, and in fact can be smoothly deformed by it. We also showed that at certain values of the control parameter, the Floquet-PXP model becomes integrable. These insights thus are important not only to understand the fundamental origin of QMBS in Rydberg atom arrays, but provide also a means to manipulate them (Objective 1.2). The results of this work are published in [arxiv:2312.16288].
Finally, in [Physical Review Letters 131, 170601 (2023) / arxiv:2305.19220] we developed a new design for a universal quantum computing unit with Rydberg atom arrays. In properly arranged atom arrays, quantum computation can emerge as a result of the collective quantum dynamics of a globally driven Rydberg array. An essential ingredient is the Floquet dynamics in PXP modes as described above. These results thus connect QMBS and applications of Rydberg atom arrays for quantum computing (Objective 1.1)
Objective 2: Quantum Dynamics of Maximum Independent Sets
The blockade mechanism gives rise to the formation of an effective low-energy Hilbert space, which coincides with the space of all maximum independent sets on certain graphs. Quantum optimization algorithms aim at steering the quantum dynamics into this low energy corner of the Hilbert space.
A key goal in this context is to understand of the performance of quantum optimization algorithms for problem classes accessible with Rydberg atom arrays. Many NP-hard optimization problems are related to the presence of spin-glass phases, where the ruggedness of the low-energy landscape is a manifestation of the problem difficulty. It signals the breakdown of simple general-purpose sampling algorithms. In [arxiv:2306.13123] we similarly relate the performance of quantum annealing algorithms to the structure of the low energy landscape and show how quantum annealing can be modified to avoid performance barriers (Objective 2.1).
A restriction of Rydberg atom arrays for quantum optimization is given by the native hardware constraint which limit applicability to unit disk graphs. In [PRX Quantum 4 (1), 010316 (2023) / arXiv:2209.03965] we showed how arbitrary optimization problems can be encoded into unit disk graphs with a polynomial overhead (Objective 2.2).
For certain atom arrangements, the space of maximum independent sets is exponentially degenerate, which can facilitate the formation of topologically ordered spin liquids, which have application for error correction (Objective 2.3). In [Physical Review Letters 129, 090401 (2022) / arxiv:2201.04034] we showed that such spin liquid states can indeed be prepared in Rydberg atom arrays via dynamical protocols. We generalized this result in [Physical Review B 106, 195155 (2022) / arxiv:2205.10387] and in [arxiv:2402.10651] to states with more complicated types of topological order.
Finally we also developed gate protocols for high-fidelity multi-qubit gates. Our protocols are robust to certain experimental imperfections [PRX Quantum 4, 020335 (2023) / arXiv:2210.08824] or minimize the execution time [arXiv:2402.12956].
Objective 3: State Characterization tools
In [Nature Communications 15, 1527 (2024) / arXiv:2209.12428] we introduced a new paradigm for quantifying topological states by combining methods of error correction with ideas of renormalization-group flow. Our approach allows for efficient and robust identification of topological order, and is applicable in the presence of incoherent noise sources, making it particularly suitable for realistic experiments.
