Complexity trapped by simplicity

Iulia Georgescu

Shawn Xingshan Cui, Michael H. Freedman, Zhenghan Wang

The second author previously discussed how classical complexity separation conjectures, we call them “axioms,” have implications in three manifold topology: polynomial length strings of operations which preserve certain Jones polynomial evaluations cannot produce exponential simplications of link diagrams. In this paper, we continue this theme, exploring now more subtle separation axioms for quantum complexity classes. Surprisingly, we now find that similar strings of operations are unable to effect even linear simplications of the diagrams.

Leonard Susskind

Gang Shu, Chen-Kuan Chou, Nathan Kurz +2 more

Trapped, laser-cooled atoms and ions produce intense fluorescence of the order 107~108 photons per second. Detection of this fluorescence enables efficient measurement of the quantum state of qubits based on trapped atoms. It is desirable to collect a large fraction of the photons to make the detection faster and more reliable. Additionally, efficient fluorescence collection can improve the speed and fidelity of remote ion entanglement and quantum gates. Refractive and reflective optics, and optical cavities have all been used to collect the trapped ion fluorescence with up to about 10% efficiency. Here we show a novel ion trap design that incorporates a metallic spherical mirror as the integral part of the trap itself, being its RF electrode. The mirror geometry enables up to 35% solid angle collection of trapped ion fluorescence. The movable central pin electrode of this trap allows precise placement of the ion at the focus of the reflector. We characterize the performance of the mirror, and measure 25% collection efficiency, likely limited by the imperfections of the mirror surface. We also study the properties of the images of single ions formed by the spherical mirror and apply aberration correction with an aspherical element placed outside the vacuum system. Owing to the simplicity of its design, this trap structure can be adapted for microfabrication and integration into more complex trap architectures.

Robert Cobb Myers, Shan-Ming Ruan

Bianca Dittrich, Erik Schnetter, Cameron Seth +1 more

Simplicity constraints play a crucial role in the construction of spin foam models, yet their effective behavior on larger scales is scarcely explored. In this article we introduce intertwiner and spin net models for the quantum group $\mathrm{SU}(2{)}_{k}\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2{)}_{k}$, which implement the simplicity constraints analogous to four-dimensional Euclidean spin foam models, namely the Barrett-Crane (BC) and the Engle-Pereira-Rovelli-Livine/Freidel-Krasnov (EPRL/FK) model. These models are numerically coarse grained via tensor network renormalization, allowing us to trace the flow of simplicity constraints to larger scales. In order to perform these simulations we have substantially adapted tensor network algorithms, which we discuss in detail as they can be of use in other contexts. The BC and the EPRL/FK model behave very differently under coarse graining: While the unique BC intertwiner model is a fixed point and therefore constitutes a two-dimensional topological phase, BC spin net models flow away from the initial simplicity constraints and converge to several different topological phases. Most of these phases correspond to decoupling spin foam vertices; however we find also a new phase in which this is not the case, and in which a nontrivial version of the simplicity constraints holds. The coarse graining flow of the BC spin net models indicates furthermore that the transitions between these phases are not of second order. The EPRL/FK model by contrast reveals a far more intricate and complex dynamics. We observe an immediate flow away from the original simplicity constraints; however, with the truncation employed here, the models generically do not converge to a fixed point. The results show that the imposition of simplicity constraints can indeed lead to interesting and also very complex dynamics. Thus we need to further develop coarse graining tools to efficiently study the large scale behavior of spin foam models, in particular for the EPRL/FK model.