Classical data could be transformed into quantum states through quantum amplitude encodings with exponential compression, but existing protocols suffer from a sharp drop in success probability when the data is sparse or has a large dynamic range. In this work, we formulate a novel encoding algorithm which requires a minimal amount of ancillary qubits. By employing a three-step quantum circuit consisting of matching, rotation, and flipping, we boost the minimum success probability to 1 − 2−twith t ancillary qubits, independent of the data distribution. Theoretical analysis and circuit design demonstrate that this approach maintains logarithmic-scale resource consumption while being universally applicable to arbitrary datasets. This provides an efficient and robust data-loading frontend for quantum machine learning and linear algebra applications in the NISQ era.
We present a protocol for encoding
N real numbers stored in
N memory registers into the amplitudes of the quantum superposition that describes the state of
log2N 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
N. 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.
Data loading with shallow quantum circuits is a highly desirable ingredient for efficient execution of many quantum algorithms before large-scale quantum error corrections with full fault tolerance become readily available. The need for efficient data loading is especially urgent for the study of quantum machine learning. In this work, we propose a protocol that only uses a parameterized shallow quantum circuit without ancilla qubits for loading data into the amplitude of a quantum state with high fidelity. We term this data-loading method Hamiltonian-based data loading (HDL), which comprises two stages. First, the HDL algorithm identifies a Hamiltonian $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}$ whose unique ground state $|\ensuremath{\psi}\ensuremath{\rangle}$ represents the normalized data $\stackrel{P\vec}{x}$ in the form of amplitude encoding. Next, the target state is reconstructed with a parameterized quantum circuit by utilizing methods like variational quantum eigensolver to minimize energy. In this work, we provide three convincing examples to demonstrate the effectiveness of HDL for loading an
N-dimensional classical data with minimal quantum resources (
O[poly(log2N)]-depth quantum circuit without ancilla qubits). Our approach is particularly useful for quantum hardware without large-scale error corrections, and shall benefit the development of quantum machine learning algorithms.
Quantum state preparation (QSP) for a general
n-qubit state requires
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)) for a broad class of compressible data. The method first applies a classical compression step to obtain a
d-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.
Compared to general quantum states, the sparse states arise more frequently in the field of quantum computation. In this work we consider the preparation for
n-qubit sparse quantum states with
s nonzero amplitudes and propose two algorithms. The first algorithm uses
O(ns/log2n+n) gates, improving upon previous methods by
O(log2n). Moreover, the classical runtime of this algorithm is optimal. We further establish a matching lower bound for any algorithm that is not amplitude aware and employs at most
poly(n) ancillary qubits. The second algorithm is tailored for binary strings that exhibit a short Hamiltonian path. An application is encoding the input data into a state with specified Hamming weight
k in quantum machine learning, for which our algorithm constructs a circuit of size
O((kn)log2n). This surpasses previous results by
O(k/log2n) and is close to the lower bound
O((kn)). The classical runtime is also nearly optimal. Both algorithms shrink the existing gap theoretically and provide increasing advantages numerically.
Efficient data loading remains a bottleneck for near-term quantum machine learning. Existing schemes (angle, amplitude, and basis encoding) either underuse the exponential Hilbert-space capacity or require circuit depths that exceed the coherence budgets of noisy intermediate-scale quantum hardware. We introduce shot-based quantum encoding (SBQE), a data embedding strategy that distributes the hardware's native resource, shots, according to a data-dependent classical distribution over multiple initial quantum states. By treating the shot counts as a learnable degree of freedom, SBQE produces a mixed-state representation whose expectation values are linear in the classical probabilities and can therefore be composed with nonlinear activation functions. We show that SBQE is structurally equivalent to a multilayer perceptron whose weights are realized by quantum circuits, and we describe a hardware-compatible implementation protocol. Benchmarks on three image datasets, with 10 independent initializations per model, show that SBQE achieves 89.1% +- 0.9% test accuracy on Semeion (reducing error by 5.3% relative to amplitude encoding and matching a width-matched classical network), 80.95% +- 0.10% on Fashion MNIST (exceeding amplitude encoding by +2.0% and a linear multilayer perceptron by +1.3%), and 90.25% +- 0.18% on MNIST (exceeding amplitude encoding by 2.1 percentage points and the width-matched classical network by 0.3), all without any data-encoding gates.
We present an exact
n-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
k. This represents a polynomial space compression of degree
k. 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
k and identifies the gates that superpose the corresponding states with the correct amplitudes. An explicit compilation into
CNOTs and single-qubit gates is presented, with the total
CNOT-gate count of
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 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
q-Gaussian probability distribution in the non-log-concave regime with
n=6 and
k=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.