Low-depth measurement-based deterministic quantum state preparation

TL;DR

This paper tackles the challenge of loading arbitrary classical data into quantum states via amplitude encoding under a low-depth, deterministic paradigm. It builds on the divide-and-conquer encoding by introducing a mid-circuit disentangling measurement that removes ancilla qubits while preserving amplitudes, enabling $O(n)$-depth circuits with $O(2^n)$ ancillas. The authors demonstrate the method on dense and W-state examples, and extend the framework to sparse states, mid-circuit qubit resets, and hybrid combine-and-conquer schemes. The work provides practical, measurement-based techniques suited to near-term quantum hardware and offers code resources for implementation. Overall, the approach achieves a favorable time-space trade-off relative to previous methods and broadens the repertoire for amplitude-encoded state preparation on quantum devices.

Abstract

We present a low-depth amplitude encoding method for arbitrary quantum state preparation. Building on the foundation of an existing divide-and-conquer algorithm, we propose a method to disentangle the ancillary qubits from the final state. Our method is measurement-based but deterministic, and offers an alternative approach to existing state preparation algorithms. It has circuit depth O(n), which is known to be optimal, and O(2^n) ancilla qubits, which is close to optimal. We illustrate our method through detailed worked examples of both a ``dense'' state and a W-state. We discuss extensions to the algorithm resetting qubits mid-circuit, and construct hybrid algorithms with varying space and circuit depth complexities.

Figures (11)

• Figure 1: Recursive binary tree structure with weights $\omega_f^j$ picked up at each level for each state
• Figure 2: Recursive binary tree structure [solid-teal] of a general input vector $\vec{x}$ of size $N = 8$ [dashed-blue].
• Figure 3: Quantum circuit realising the general state preparation using the time encoding method for an $N=8$ real vector input.
• Figure 4: Quantum circuit realising the divide-and-conquer algorithm, including disentanglement stages for an N=8 input.
• Figure 5: A section of the divide-and-conquer circuit showing how the moveable CSWAPs can be shifted inside the circuit to be performed in parallel with the rigid CSWAPs.