NoRA: A Tensor Network Ansatz for Volume-Law Entangled Equilibrium States of Highly Connected Hamiltonians

Valérie Bettaque and Brian Swingle

Martin A. Fisher School of Physics, Brandeis University, Waltham, MA 02453, USA

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Abstract

Motivated by the ground state structure of quantum models with all-to-all interactions such as mean-field quantum spin glass models and the Sachdev-Ye-Kitaev (SYK) model, we propose a tensor network architecture which can accomodate volume law entanglement and a large ground state degeneracy. We call this architecture the non-local renormalization ansatz (NoRA) because it can be viewed as a generalization of MERA, DMERA, and branching MERA networks with the constraints of spatial locality removed. We argue that the architecture is potentially expressive enough to capture the entanglement and complexity of the ground space of the SYK model, thus making it a suitable variational ansatz, but we leave a detailed study of SYK to future work. We further explore the architecture in the special case in which the tensors are random Clifford gates. Here the architecture can be viewed as the encoding map of a random stabilizer code. We introduce a family of codes inspired by the SYK model which can be chosen to have constant rate and linear distance at the cost of some high weight stabilizers. We also comment on potential similarities between this code family and the approximate code formed from the SYK ground space.

Bettaque, V. (2023). Talk 110 – NoRA: A Tensor Network Ansatz for Volume-Law Entangled Equilibrium States of Highly Connected Hamiltonians. Perimeter Institute.

The more complicated quantum-mechanical systems become, the harder it is to analyze them analytically. It is therefore paramount to develop simplified models that are able to capture the essential physics and which can be simulated using either a classical or quantum computer. One class of such approximative models are multi-scale entanglement renormalization ansatz (MERA) tensor networks, which have found applications in many different branches of physics due to their ability to capture quantum phase transitions and other scale-invariant phenomena while remaining simulable using a classical computer. However, those networks rely on the system being modeled to be spatially local, which is not the case for all-to-all interacting systems like mean-field models of spin glasses and the Sachdev-Ye-Kitaev (SYK) model. In our work we therefore propose a generalization of various MERA networks, called the non-local renormalization ansatz (NoRA). The cost of this generalization is a weakening of the classical simulability, but the network can still form the basis of an efficient quantum simulation for a much wider class of models. In addition to opening up the future prospect of using NoRA to study the SYK model and its cousins, we analyzed a random version of the network in which the tensors are special Clifford gates and the network is interpreted as an encoding map of a quantum error correcting code. Finding quantum codes that are efficient at detecting and correcting errors is essential for the construction of large-scale quantum computers. We found that the NoRA network could produce quantum codes with excellent parameters, although the resulting codes are not yet practically useful

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