Sublinear Classical-to-Quantum Data Encoding using $n$-Toffoli Gates

TL;DR

This paper introduces a novel quantum data encoding method that achieves sublinear average circuit depth for amplitude encoding of arbitrary vectors, enhancing efficiency for near-term quantum hardware.

Contribution

The authors propose a general-purpose, sublinear-depth amplitude encoding procedure using MCX gates, based on a geometric isomorphism with hypercube graphs, suitable for near-term quantum devices.

Findings

Achieves sublinear average circuit depth in N for amplitude encoding.

Uses a probabilistic approach with success rate linked to data sparsity.

Suitable for ion trap and neutral atom quantum platforms.

Abstract

Quantum state preparation, also known as encoding or embedding, is a crucial initial step in many quantum algorithms and often constrains theoretical quantum speedup in fields such as quantum machine learning and linear equation solvers. One common strategy is amplitude encoding, which embeds a classical input vector of size N=2\textsuperscript{n} in the amplitudes of an n-qubit register. For arbitrary vectors, the circuit depth typically scales linearly with the input size N, rapidly becoming unfeasible on near-term hardware. We propose a general-purpose procedure with sublinear average depth in N, increasing the window of utility. Our amplitude encoding method encodes arbitrary complex vectors of size N=2\textsuperscript{n} at any desired binary precision using a register with n qubits plus 2 ancillas and a sublinear number of multi-controlled NOT (MCX) gates, at the cost of a probabilistic success rate proportional to the sparsity of the encoded data. The core idea of our procedure is to construct an isomorphism between target states and hypercube graphs, in which specific reflections correspond to MCX gates. This reformulates the state preparation problem in terms of permutations and \emph{binary addition}. The use of MCX gates as fundamental operations makes this approach particularly suitable for quantum platforms such as \emph{ion traps} and \emph{neutral atom devices}. This geometrical perspective paves the way for more gate-efficient algorithms suitable for near-term hardware applications.