Abstract
This dissertation is a collection of three papers written by the author and her advisor, which concern the quantum information (and more specifically quantum resource theory) properties of the Sachdev-Ye-Kitaev (SYK) model and other (fermionic) many-body systems.
In the first paper, we propose the Non-local Renormalization Ansatz (NoRA) tensor network as a way to prepare states with volume-law entanglement and large ground-state degeneracy, as expected for mean-field models such as spin glasses and SYK. Using entanglement and complexity considerations, we argue that this ansatz is indeed capable of approximately capturing the ground state behavior of SYK in particular. Since the network is not contractible on classical computers, we instead explore the special case of it being defined in terms of random Cliffords. Treating the resulting ensemble of isometries as the embeddings for a certain class of random error-correcting stabilizer codes, we are able to show that these codes have constant rates and linear distance, while also having large (but not fully saturated) stabilizer weights. We point out connections between this class of codes and approximate codes derived from the actual SYK ground space.
The second paper formally introduces and studies Clifford operators defined in terms of Majorana fermions, and how they relate to the known concept of fermionic stabilizer codes. Unlike their Pauli equivalents, these objects are required to obey fermion parity superselection rules, meaning they have to commute with the total parity of the system. We prove that the subgroup of Majorana Cliffords that preserve said parity can be represented by orthogonal matrices over $\mathbb{F}_2$, and also show how they can be efficiently tracked and sampled from on classical computers, in analogy to the Gottesmann-Knill theorem. These statements are then used to show how one can construct/sample any Majorana-based stabilizer. Lastly, we prove that the subgroup of parity-preserving Cliffords still form a 3-design, albeit when constrained to sector of the Hilbert space with fixed parity.
In the third and final paper, we study the decomposition of the SYK Gibbs state in terms of a complete basis of observables given by strings of Majorana fermions. By evaluating moments of the corresponding expectation values in the disorder-averaged large-$N$ limit using a path integral saddle point approximation, we show that for $q > 2$ they behave like real Gaussian random variables with zero mean, which is due to them depending erratically on the choice of couplings in the SYK Hamiltonian. We also observe that the integrable nature of the Gaussian $q = 2$ case causes the associated moments to obey different (non-Gaussian) statistics. Using the fact that measures of non-stabilizerness (magic) such as the robustness of magic and stabilizer Rényi entropy can be expressed/bounded in terms of a sum over all such moments, we are able to compute them for the SYK thermal state using a saddle point analysis. Furthermore, we are able to relate these results to calculations in the holographic Jackiw-Teichelboim (JT) gravity dual at low temperatures, where the same moments can be retrieved from a wormhole geometry that is stabilized by the insertion of a heavy operator dual to the Majorana string.
We conclude this dissertation by pointing out future directions for which the results in these publications might prove useful, in particular with regard to the relationship between randomness, wormholes, and closed universes, and what it can tell us about a potential holographic description of (nonlocal) quantum magic.