End-to-end quantum algorithms for tensor problems

TL;DR

This work develops an end-to-end quantum algorithm for tensor problems, notably tensor PCA and planted kXOR, using a native qubit-based Kikuchi encoding to enable explicit circuit constructions and non-asymptotic resource estimates. The approach leverages guided ground-state energy estimation with a specially prepared guiding state and quantum signal processing, achieving superquadratic to quartic speedups in key regimes, and extends to sparse tensor PCA and tensor completion as well as asymmetric tensors. Core contributions include concrete state-preparation and phase-estimation circuits, substantial reductions in constant overheads via a novel guiding-state technique, and rigorous recovery proofs for sparse and asymmetric settings, backed by detailed resource estimates (e.g., ~900 logical qubits and ~10^15 non-Clifford gates for representative instances). The results position tensor problems as practical candidates for quantum advantage on fault-tolerant hardware, while also clarifying the interplay with improved classical algorithms and highlighting the potential for further compiler and algorithmic improvements.

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.

Figures (11)

• Figure 1: Overview of the block-encoding construction, inspired by quantum walks berry2009black.
• Figure 2: Performance of the Kikuchi method for symmetric and asymmetric tensor completion tasks $(k=4)$. The setting is detailed in Section \ref{['sec:planted-kxor']}. Reported points are the average of 30 independent trials. For the symmetric case, we choose $n=20$. For the asymmetric case, we choose $n=7$; due to symmetric embedding, this amounts to 28-dimensional tensor. In both cases, we use the Kikuchi method with $\ell=6$.
• Figure 3: The circuit for $l=2$ ($k=2^l$) in the \ref{['alg:circuit_dicke_state']} which generates a circuit in \ref{['eq:condition_dicke_state']}.
• Figure 4: The circuit description of $O_H$. The circuit gives the full oracle $O_H$ by combining the two oracles $O_A$ and $O_E$.
• Figure 5: Circuit description of $P_i$. The circuit decomposes each $P_i$ into Toffoli and Incrementor gates, with the Incrementor itself further decomposable into Toffoli gates. For the qubits of $U_i$ in register $A$, we need to apply all possible 4-body Toffoli gates; there are six such gates, and each requires two Toffoli gates to implement. The Incrementor acts on seven qubits, which requires a total of 21 Toffoli gates for its implementation. Therefore, the total number of Toffoli gates needed is $21 + 2 \times 2 \times 6 = 45$. The Toffoli depth of the Incrementor is $D$, and the depth for the Toffoli gates acting between $A$ and the ancilla ($anc$) can be achieved with a depth of $k(k-1)/2 + 1$. This yields a total circuit depth of $D + 7 = 13$.

Theorems & Definitions (39)

• Proposition 2.1
• proof
• Proposition 2.2: Signed tensor entries
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Proposition 3.1: Prop.2.15 in schmidhuber2024quartic
• Proposition 3.2: Prop.2.16 in schmidhuber2024quartic