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arXiv:2505.06054 · IEEE International Conference on Quantum Computing and Engineering (QCE)

Sublinear Classical-to-Quantum Data Encoding Using nnn-Toffoli Gates

Vittorio Pagni, Gary Schmiedinghoff, Kevin Lively, Michael Epping, Michael Felderer

Published Aug 30, 2025• 1 citation
arXivHTMLPDF

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$\mathrm{N}=2^{\mathrm{n}}$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$\mathrm{N}=2^{n}$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 binary addition. The use of MCX gates as fundamental operations makes this approach particularly suitable for quantum platforms such as ion traps and neutral atom devices. This geometrical perspective paves the way for more gate-efficient algorithms suitable for nearterm hardware applications.

DOI
10.1109/qce65121.2025.00034

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Chapman & Hall/CRC eBooks•Nov 28, 2023•0 citations
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 sizeN=2n\mathrm{N}=2^{\mathrm{n}}N=2nin 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 sizeN=2n\mathrm{N}=2^{n}N=2nat any desired binary precision using a register withnnnqubits 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 binary addition. The use of MCX gates as fundamental operations makes this approach particularly suitable for quantum platforms such as ion traps and neutral atom devices. This geometrical perspective paves the way for more gate-efficient algorithms suitable for nearterm hardware applications.
We present a protocol for encoding NNN real numbers stored in NNN memory registers into the amplitudes of the quantum superposition that describes the state of log2N{log}_{2}Nlog2​N qubits. This task is one of the main steps in quantum machine learning algorithms applied to classical data. The protocol combines partial CNOT gate rotations with probabilistic projection onto the desired state. The number of additional ancilla qubits used during the implementation of the protocol, as well as the number of quantum gates, scales linearly with the number of qubits in the processing register and hence logarithmically with NNN. The average time needed to successfully perform the encoding scales logarithmically with the number of qubits, in addition to being inversely proportional to the acceptable error in the encoded amplitudes. It also depends on the structure of the dataset in such a way that the protocol is most efficient for nonsparse data.
Quantum state preparation (QSP) for a general nnn-qubit state requires O(2n)O(2^n)O(2n) CNOT gates and circuit depth, making exact amplitude encoding (EAE) impractical for near-term quantum hardware. We introduce an ancilla-free hybrid classical-quantum strategy that reduces this cost to O(poly(n))O(poly(n))O(poly(n)) for a broad class of compressible data. The method first applies a classical compression step to obtain a ddd-sparse representation of the input, loads this sparse vector using a sparse-state preparation routine, and then reconstructs the target state through a polynomial-depth quantum inverse transform. We evaluate the framework on synthetic benchmark signals and real biomedical time series using Fourier and Haar transforms, demonstrating substantial reductions in CNOT counts and circuit depth compared to EAE, together with competitive performance relative to the Fourier Series Loader (FSL). The quantum simulation results show that combining classical data compression with quantum decompression provides a scalable framework for efficient QSP, reducing quantum overhead without requiring variational training or ancillary registers.
Quantum processing of classical data fundamentally relies on efficiently mapping classical datasets onto the amplitudes of quantum states. Preparing large amplitude-encoded states, however, remains a major bottleneck in quantum computing. In this paper, we introduce a novel state-preparation framework we call Q-MERGE, that addresses this bottleneck by partitioning a large target state into MMM nnn-qubit segments, preparing these segments independently and in parallel, and then coherently combining them into a single amplitude-encoded state using SELECT-SWAP operations and measurement. Q-MERGE is agnostic to the segment-level preparation method, allowing existing techniques to be applied to smaller subproblems while providing a tunable trade-off between circuit depth and qubit count. With mid-circuit measurement and preparation-register reuse, the required ancilla qubits can be reduced from O(Mn) to O(n). Applied to a real-world 32x102432 x 102432x1024 ultrasound dataset, Q-MERGE achieves an infidelity of 3.5x10−93.5 x 10^{-9}3.5x10−9, compared with 6.2x10−36.2 x 10^{-3}6.2x10−3 for direct preparation using the same underlying method, a more than six orders of magnitude improvement at equal CNOT circuit depth. We demonstrate its feasibility experimentally with a 53 qubit circuit comprised of 954 1-qubit gates and 478 2-qubit gates, on the Quantinuum System Model H2 trapped-ion quantum computer. The prepared state is validated using shadow-overlap tomography. Numerical analysis of Haar-random states indicates consistent success probability for up to M=107M=10^7M=107 segments, supporting the scalability of Q-MERGE for encoding massive classical datasets.
We present an exact nnn-qubit computational-basis amplitude encoder of real- or complex-valued data vectors of $d=(\genfrac{}{}{0}{}{n}{k})$ components into a subspace of fixed Hamming weight kkk. This represents a polynomial space compression of degree kkk. The circuit is optimal in that it expresses an arbitrary data vector using only $d\ensuremath{-}1$ (controlled) reconfigurable beam splitter (RBS) gates. It is constructed by an efficient classical algorithm that sequentially generates all bitstrings of weight kkk and identifies the gates that superpose the corresponding states with the correct amplitudes. An explicit compilation into CNOTs\mathrm{CNOT}\mathrm{s}CNOTs and single-qubit gates is presented, with the total CNOT\mathrm{CNOT}CNOT-gate count of O(kd)\mathcal{O}(kd)O(kd) provided in analytical form. In addition, we show how to load data in the binary basis by sequentially stacking encoders of different Hamming weights using O(dlog(d))CNOT\mathcal{O}(d\mathrm{log}(d))\phantom{\rule{0.2em}{0ex}}\mathrm{CNOT}O(dlog(d))CNOT gates. Moreover, using generalized RBS gates that mix states of different Hamming weights, we extend the construction to efficiently encode arbitrary sparse vectors. We perform a proof-of-principle demonstration of our scheme on a commercial trapped-ion quantum computer. We successfully upload a qqq-Gaussian probability distribution in the non-log-concave regime with n=6n=6n=6 and k=2k=2k=2. We also illustrate how the effect of hardware noise can be alleviated by quantum error mitigation. Numerically, we show how our encoder can improve the performance of variational quantum algorithms for problems that include particle-preserving symmetries. Our results constitute a versatile framework for quantum data compression with various potential applications in fields such as quantum chemistry, quantum machine learning, and constrained combinatorial optimizations.
Quantum state preparation is the process of producing a target quantum state that will be used as the input to a quantum circuit. Many quantum algorithms require an input state where amplitudes, phases, or basis probabilities encode problem data, and the cost of preparing this state can be significant in gate count and circuit depth. This review summarizes common goals, assumptions, and methods for quantum state preparation, with emphasis on preparing states from classical vectors, probability distributions, and feature data used in quantum machine learning. We organize approaches by the information they load and by the resources they require, including gate count, circuit depth, qubit overhead, and classical preprocessing. It also compares exact and approximate preparation procedures, and discusses how precision targets affect cost. The review highlights links between families of methods, typical sources of resource estimates, and criteria that help match a preparation method to a task and hardware constraints.
Amplitude encoding encodes a real or complexed value vector of the length one into the amplitudes of a quantum state. We describe the top-down strategy and indicate the algorithm step by step. Then we describe the combining states strategy. Instead of representing the binary tree by multi-control rotation gates, we can use controlled SWAP operators with simple rotation gates. The resulting circuits depth is less than top-down divide strategy, however, we require the same number of qubits as the number of rotation gates and the qubits are entangled after the operation.