Tucker iterative quantum state preparation

TL;DR

Q-Tucker proposes a deterministic, hardware-efficient method for amplitude encoding by leveraging Tucker decomposition to factor a target quantum state into a core tensor and mode-specific unitaries, yielding circuit depth tied to entanglement structure. The approach uses an iterative process with a monotone gauge to ensure non-decreasing fidelity, supported by an oracle-guided partitioning and a stall-and-grow mechanism to guarantee convergence to exact preparation when allowed to expand to the full system. A correlation-graph heuristic biases partition choices toward high intra-block correlations and low inter-block bond dimensions, balancing computational cost with circuit depth. Numerical results on MNIST demonstrate the method can compress circuits with shallow blocks and modest-squared-depth growth, while highlighting the importance of efficient unitary synthesis for larger block sizes and the practical role of fallback strategies. Overall, Q-Tucker provides a scalable, entanglement-aware framework for quantum state preparation that adapts to hardware constraints and offers provable convergence guarantees under its iterative scheme.

Abstract

Quantum state preparation is a fundamental component of quantum algorithms, particularly in quantum machine learning and data processing, where classical data must be encoded efficiently into quantum states. Existing amplitude encoding techniques often rely on recursive bipartitions or tensor decompositions, which either lead to deep circuits or lack practical guidance for circuit construction. In this work, we introduce Tucker Iterative Quantum State Preparation (Q-Tucker), a novel method that adaptively constructs shallow, deterministic quantum circuits by exploiting the global entanglement structure of target states. Building upon the Tucker decomposition, our method factors the target quantum state into a core tensor and mode-specific operators, enabling direct decompositions across multiple subsystems.

Figures (2)

• Figure 1: Quantum circuit representation of the Tucker Iterative Process for circuit compression. The initial quantum state is decomposed into a core vector $|\text{core}\rangle$ and local hardware-efficient unitary (or isometry) factors $W_i$. At each iteration, the fidelity $F(|\text{core}_j\rangle, |0\rangle)$ is evaluated to determine whether the core can be approximated by the ground state. If the fidelity exceeds the threshold ($F \geq 1 - \epsilon$), the core is removed (indicated by the red cross). Between iterations, logical qubit swaps $\pi_j$ are applied to optimize the placement of the factors. These swaps are guided by minimizing the bond dimension between qubit partition blocks -- each corresponding to a factor $W_i$ -- which reflects the entanglement between subsystems.
• Figure 2: This study examines the relationship between iteration count, circuit depth, and fidelity loss in the Q-Tucker method. As illustrated by the scatter plot, the method achieves rapid convergence -- requiring fewer iterations -- when an acceptable fidelity loss is targeted, highlighting its potential utility.

Theorems & Definitions (5)

• lemma thmcounterlemma: monotone-gauge identity
• proof
• corollary thmcountercorollary: per-step monotonicity for any partition
• theorem thmcountertheorem: global convergence, stall-and-grow
• proof