Robbie King

Matthew Ding, Robbie King, Bobak T. Kiani et al.

Finding the ground state of strongly-interacting fermionic systems is often the prerequisite for fully understanding both quantum chemistry and condensed matter systems. The Sachdev--Ye--Kitaev (SYK) model is a representative example of such a system; it is particularly interesting not only due to the existence of efficient quantum algorithms preparing approximations to the ground state such as Hastings--O'Donnell (STOC 2022), but also known no-go results for many classical ansatzes in preparing low-energy states. However, this quantum-classical separation is known to \emph{not} persist when the SYK model is sufficiently sparsified, i.e., when terms in the model are discarded with probability $1-p$, where $p=Θ(1/n^3)$ and $n$ is the system size. This raises the question of how robust the quantum and classical complexities of the SYK model are to sparsification. In this work we initiate the study of the sparse SYK model where $p \in [Θ(1/n^3),1]$ and show there indeed exists a certain robustness of sparsification. We prove that with high probability, Gaussian states achieve only a $Θ(1/\sqrt{n})$-factor approximation to the true ground state energy of sparse SYK for all $p\geqΩ(\log n/n^2)$, and that Gaussian states cannot achieve constant-factor approximations unless $p \leq O(\log^2 n/n^3)$. Additionally, we prove that the quantum algorithm of Hastings--O'Donnell still achieves a constant-factor approximation to the ground state energy when $p\geqΩ(\log n/n)$. Combined, these show a provable separation between classical algorithms outputting Gaussian states and efficient quantum algorithms for the goal of finding approximate sparse SYK ground states whenever $p \geq Ω(\log n/n)$, extending the analogous $p=1$ result of Hastings--O'Donnell.

Casper Gyurik, Alexander Schmidhuber, Robbie King et al.

Topological data analysis (TDA) aims to extract noise-robust features from a data set by examining the number and persistence of holes in its topology. We show that a computational problem closely related to a core task in TDA -- determining whether a given hole persists across different length scales -- is $\mathsf{BQP}_1$-hard and contained in $\mathsf{BQP}$. This result implies an exponential quantum speedup for this problem under standard complexity-theoretic assumptions. Our approach relies on encoding the persistence of a hole in a variant of the guided sparse Hamiltonian problem, where the guiding state is constructed from a harmonic representative of the hole.

Amira Abbas, Robbie King, Hsin-Yuan Huang et al.

The success of modern deep learning hinges on the ability to train neural networks at scale. Through clever reuse of intermediate information, backpropagation facilitates training through gradient computation at a total cost roughly proportional to running the function, rather than incurring an additional factor proportional to the number of parameters - which can now be in the trillions. Naively, one expects that quantum measurement collapse entirely rules out the reuse of quantum information as in backpropagation. But recent developments in shadow tomography, which assumes access to multiple copies of a quantum state, have challenged that notion. Here, we investigate whether parameterized quantum models can train as efficiently as classical neural networks. We show that achieving backpropagation scaling is impossible without access to multiple copies of a state. With this added ability, we introduce an algorithm with foundations in shadow tomography that matches backpropagation scaling in quantum resources while reducing classical auxiliary computational costs to open problems in shadow tomography. These results highlight the nuance of reusing quantum information for practical purposes and clarify the unique difficulties in training large quantum models, which could alter the course of quantum machine learning.