This paper systematically develops the concept of entanglement threads that characterize the entanglement structure of holographic duality. Behind this framework lies a simple philosophy: holographic quantum entanglement can be visualized using threadlike objects. Inspired by the fact that tensor network models can be deformed into a quantum circuit form with flow-conserving features, we abstract the concept of entanglement threads. These entanglement threads can be understood as a preset ensemble of wires in a holographic quantum circuit, and we propose that they characterize the underlying partially ordered structure of holographic quantum entanglement. Combining the concepts of entanglement threads and kinematic space, a elegant circuit interpretation for the holographic complexity is provided. We also clarify the connection and distinction between entanglement threads and the previously proposed concept of bit threads.
This paper presents a novel theoretical framework, Dynamic Holographic Resonance (DHR), exploring the potential of holographic principles to accelerate the development of highly efficient quantum computation. We propose a model wherein the holographic principle is treated as a continuously evolving, dynamic field, capable of optimizing quantum circuit designs and algorithms through iterative refinement. The core mechanism leverages this dynamic field to identify and exploit subtle correlations inherent in quantum systems, ultimately aiming to surpass the limitations of current quantum computing approaches. We investigate the interplay between the holographic principle, circuit optimization, and the inherent properties of quantum entanglement, suggesting a pathway toward achieving exponential speedups in quantum computations. This work contributes to a deeper understanding of the fundamental role of holographic data in guiding the design of quantum systems.
We introduce ``holographic shadows,'' a new class of randomized measurement schemes for classical shadow tomography that achieves the optimal scaling of sample complexity for learning geometrically local Pauli operators at any lengthscale, without the need for fine-tuning protocol parameters such as circuit depth or measurement rate. Our approach utilizes hierarchical quantum circuits, such as tree quantum circuits or holographic random tensor networks. Measurements within the holographic bulk correspond to measurements at different scales on the boundary (i.e., the physical system of interests), facilitating efficient quantum state estimation across observable at all scales. Considering the task of estimating stringlike Pauli observables supported on contiguous intervals of
k sites in a one-dimensional system, our method achieves an optimal sample complexity scaling of $\ensuremath{\sim}{d}^{k}\mathrm{poly}(k)$, with
d the local Hilbert space dimension. We present a holographic minimal cut framework to demonstrate the universality of this sample complexity scaling and validate it with numerical simulations, illustrating the efficacy of holographic shadows in enhancing quantum state learning capabilities.