End-to-end quantum algorithms for tensor problems

Abstract

We present a comprehensive end-to-end quantum algorithm for tensor problems, including tensor PCA and planted kXOR, that achieves potential superquadratic quantum speedups over classical methods. We build upon prior works by Hastings~(\textit{Quantum}, 2020) and Schmidhuber~\textit{et al.}~(\textit{Phys.~Rev.~X.}, 2025), we address key limitations by introducing a native qubit-based encoding for the Kikuchi method, enabling explicit quantum circuit constructions and non-asymptotic resource estimation. Our approach substantially reduces constant overheads through a novel guiding state preparation technique as well as circuit optimizations, reducing the threshold for a quantum advantage. We further extend the algorithmic framework to support recovery in sparse tensor PCA and tensor completion, and generalize detection to asymmetric tensors, demonstrating that the quantum advantage persists in these broader settings. Detailed resource estimates show that 900 logical qubits, $\sim 10^{15}$ gates and $\sim 10^{12}$ gate depth suffice for a problem that classically requires $\sim 10^{23}$ FLOPs. The gate count and depth for the same problem without the improvements presented in this paper would be at least $10^{19}$ and $10^{18}$ respectively. These advances position tensor problems as a candidate for quantum advantage whose resource requirements benefit significantly from algorithmic and compilation improvements; the magnitude of the improvements suggest that further enhancements are possible, which would make the algorithm viable for upcoming fault-tolerant quantum hardware.