Quantum Optics seminar: Zhi-Yuan Wei, Max Planck Institute for Quantum Optics

Preparation of tensor network states

Tensor network states (TNS) play a fundamental role both in quantum information processing and many-body physics. In this presentation, we study how to prepare TNS on quantum devices. First, we briefly discuss how to sequentially generate photonic matrix product states (MPS) using a microwave cavity coupled to a transmon [1], and show the sequential generation scheme can be used to generate ancilla-photon entanglement, as well as realize a photonic quantum random access memory (QRAM). Then we introduce plaquette projected entangled-pair states (P-PEPS) [2], a class of states in high-dimensional lattices that can be generated by applying sequential unitaries acting on plaquettes of overlapping regions. We show that a subclass of P-PEPS can be prepared with circuit depth T=O(L) [L is the linear size of the system]. Importantly, this class contains isometric tensor network states (isoTNS), and thus leads to an optimally scaling algorithm to create a large class of long-range correlated PEPS, which includes all string-net liquids with topological order. Finally, we address the problem of MPS preparation using local quantum circuits [3]. We first prove that faithfully preparing translation-invariant normal MPS of N sites requires a circuit depth T = Ω(log N ). We then introduce an algorithm based on the renormalization-group transformation to prepare normal MPS with an error ε in depth T = O(log(N/ε)), reaching the optimal possible scaling for a local circuit. We also show that measurement and feedback lead to an exponential speed-up of the algorithm, to T = O(log log(N/ε)). Measurements also allow one to prepare arbitrary translation-invariant MPS, including long-range (non-normal) ones, in the same depth.

[1] ZYW, Cirac, Malz, PRA 105, 022611 (2022)

[2] ZYW*, Malz*, Cirac, PRL 128, 010607 (2022)

[3] Malz*, Styliaris*, ZYW*, Cirac. PRL 132.4 (2024): 040404.

(*: equal contribution)