Quantum circuits for topological state generation

Jiannis K. Pachos

Yichen Huang, Xie Chen

Topological quantum states cannot be created from product states with local quantum circuits of constant depth and are in this sense more entangled than topologically trivial states, but how entangled are they? Here we quantify the entanglement in one-dimensional topological states by showing that local quantum circuits of linear depth are necessary to generate them from product states. We establish this linear lower bound for both bosonic and fermionic one-dimensional topological phases and use symmetric circuits for phases with symmetry. We also show that the linear lower bound can be saturated by explicitly constructing circuits generating these topological states. The same results hold for local quantum circuits connecting topological states in different phases.

Jincheng Zhang

Quantum computing holds immense potential for revolutionary advancements across numerous fields, from drug discovery to materials science. However, the current limitations of quantum algorithms are largely constrained by the inherent fragility of quantum states and the difficulty in creating complex, dynamically adaptable circuits. This paper explores a novel approach to quantum computing based on non-linear topological structures, proposing a framework that utilizes dynamic topology generation to create quantum circuits with complex and self-adaptive structures. We introduce a new method for constructing quantum circuits, leveraging the principles of topology to induce intricate and responsive topological phases within the circuits. The core mechanism focuses on building circuits that inherently evolve in response to external stimuli or internal quantum states, enabling the generation of intricate and complex quantum states. This innovation addresses a key challenge in quantum computing – the lack of inherent adaptability – and offers a promising path toward the development of more robust and versatile quantum algorithms.

Tapio Simula

Giuseppe Mussardo

Xie Chen, Michael Hermele, David T. Stephen

In previous work, it was shown that non-trivial gapped states can be generated from a product state using a sequential quantum circuit. Explicit circuit constructions were given for a variety of gapped states at exactly solvable fixed points. In this paper, we show that a similar generation procedure can be established for chiral topological states as well, despite the fact that they lack a zero-correlation-length exactly solvable form. Instead of sequentially applying local unitary gates, we sequentially evolve the Hamiltonian by changing local terms in one subregion and then the next. The Hamiltonian remains gapped throughout the process, giving rise to an adiabatic evolution mapping the ground state from a product state to a chiral topological state. We demonstrate such a sequential adiabatic generation process for free fermion chiral states like the Chern Insulator and the p+ip p + i p superconductor. Moreover, we show that coupling a quantum state to a discrete gauge group can be achieved through a sequential quantum circuit, thereby generating interacting chiral topological states from the free fermion ones.