Quantum Uploading in Quantum Computing

• Quantum uploading is a framework for injecting classical or quantum data into computational degrees of freedom in quantum circuits via methods like re-uploading, amplitude encoding, and state transfer.
• It enables repeated insertion of input information at multiple circuit layers, bolstering model expressivity while balancing circuit depth and trainability.
• Empirical and theoretical studies show that different uploading methods affect performance and noise resilience, with hardware implementations achieving up to 95% accuracy on classification tasks.

Quantum uploading denotes a family of procedures that inject information into quantum computational degrees of freedom. In the literature assembled here, the term is used for at least three distinct operations: repeated embedding of classical data inside variational quantum circuits, explicit preparation of amplitude-encoded states whose amplitudes equal sampled function values, and transfer of an unknown state from ordinary physical space into the virtual correlation space of a tensor-network computer (Pérez-Salinas et al., 2019, Guseynov et al., 2024, Morimae, 2010). The most active contemporary usage is data re-uploading, in which the same input is encoded at multiple circuit depths rather than only once at the input layer, but recent work shows that this architectural idea must be distinguished from amplitude loading, quantum-input processing, and virtual-space state transfer.

1. Competing technical meanings

The sources considered here assign “uploading” to technically different tasks. This suggests that the term functions more as an umbrella label for information injection into quantum models than as a single protocol.

Sense of uploading	Representative operation	Representative source
Classical data re-uploading	Repeatedly apply $U(x)$ or $E(x)$ between trainable layers	(Pérez-Salinas et al., 2019)
Amplitude/state-preparation uploading	Prepare $\ket{\psi}$ with amplitudes equal to sampled $f(x_k)$	(Guseynov et al., 2024)
Virtual-space or quantum-state uploading	Transfer an unknown state into correlation space or interact with fresh copies of $\rho$	(Morimae, 2010, Cha et al., 23 Sep 2025)

Within variational quantum machine learning, uploading usually means feature embedding. Pérez-Salinas et al. formulate the basic single-qubit classifier as

$$\mathcal{U}(\vec{\phi}, \vec{x}) \equiv U(\vec{\phi}_N)U(\vec x)\ldots U(\vec{\phi}_1)U(\vec x),$$

so the same classical input is inserted repeatedly between trainable unitaries (Pérez-Salinas et al., 2019). Later work broadens this idea to multi-qubit, qudit, QCNN, Hamiltonian-embedding, and pulse-native settings without changing the central architectural motif: data are injected repeatedly into an evolving quantum state rather than only once.

A separate line of work uses uploading in the amplitude-encoding sense. There the goal is not classification or feature maps but preparation of a state whose computational-basis amplitudes equal discretized function values. For a polynomial

$f(x)=\sum_{i=0}^Q \sigma_i x^i,$

the target state is

$$\ket{\psi}^n= \frac{\sum_{k=0}^{2^n-1}\sum_{i=0}^Q \sigma_i x_k^i \ket{k}^n} {\sqrt{\sum_{\kappa=0}^{2^n-1}\left|\sum_{j=0}^Q \sigma_j x_\kappa^j\right|^2}},$$

which is an explicit state-preparation problem rather than a variational feature-embedding problem (Guseynov et al., 2024).

An older and conceptually distinct meaning appears in computational tensor networks. There, uploading is the transfer of an unknown state from real physical space into the virtual correlation space where tensor-network computation is performed (Morimae, 2010). More recent work extends re-uploading to genuinely quantum inputs by making an ancilla interact sequentially with fresh copies of an input state $\rho$, thereby realizing a cascade of $\rho$-dependent completely positive and trace-preserving maps (Cha et al., 23 Sep 2025).

2. Classical-data re-uploading as layered quantum feature construction

In the foundational single-qubit formulation, repeated uploading compensates for the limited local state space of one qubit. Pérez-Salinas et al. define a processing layer by

$E(x)$0

and also introduce the practically important merged form

$E(x)$1

where the data enter linearly in the gate angles while observables depend nonlinearly on the resulting trigonometric compositions (Pérez-Salinas et al., 2019). The same work states that “a single rotation cannot capture any non-trivial separation of patterns in the original data,” and motivates re-uploading as the quantum analogue of repeatedly feeding the input into hidden units of a neural network.

This idea generalizes in several directions. A single qudit can be used in place of qubits, with

$E(x)$2

and with upload blocks built from noncommuting generators such as $E(x)$3, $E(x)$4, and the squeezing operator $E(x)$5 (Wach et al., 2023). The qudit setting adds a label-geometry effect: when class labels, basis states, and encoding operators are aligned, performance can improve substantially; when they are not aligned, the same ladder structure becomes a bias rather than an advantage.

Several papers then specialize re-uploading to structured architectures. In layered-uploading QCNNs, a standard single-upload sequence

$E(x)$6

is modified to

$E(x)$7

so fresh uploads occur after pooling has reduced the active register (Barrué et al., 2024). The architectural point is explicit in the supplied material: more features can be injected without increasing the number of qubits, at the price of greater sequential depth.

A closely related but distinct strategy is incremental data-uploading. Instead of re-uploading the entire input repeatedly, the input is partitioned and the pieces are spread across the circuit, with one variational layer inserted between successive partial uploads (Periyasamy et al., 2022). In the reported MNIST and Fashion-MNIST experiments, interleaving more partial uploads improved accuracy monotonically, with IDU$E(x)$8 outperforming both DRU and IDU$E(x)$9. For MNIST, the reported test accuracies were $\ket{\psi}$0 for DRU, $\ket{\psi}$1 for IDU$\ket{\psi}$2, and $\ket{\psi}$3 for IDU$\ket{\psi}$4 (Periyasamy et al., 2022).

Image-specific re-uploading can also be built around Hamiltonian embeddings. In that case the image $\ket{\psi}$5 is first Hermitianized,

$\ket{\psi}$6

and the upload unitary becomes

$\ket{\psi}$7

The overall classifier is

$\ket{\psi}$8

so the same image is reintroduced at every layer through a matrix-level time-evolution operator rather than a feature-to-angle map (Wang et al., 2024).

3. Uploading as explicit amplitude state preparation

Amplitude-uploading work addresses a different bottleneck: how to load structured classical functions into amplitudes without assuming QRAM or arbitrary-state-preparation oracles. The explicit construction of (Guseynov et al., 2024) starts from a discretized interval

$\ket{\psi}$9

and targets the state

$f(x_k)$0

specialized first to polynomial $f(x_k)$1.

The core reduction writes a complex polynomial as

$f(x_k)$2

where $f(x_k)$3 are odd, $f(x_k)$4 are even, all four are real, and $f(x_k)$5 on $f(x_k)$6 (Guseynov et al., 2024). This places the problem in a QSVT/QSP-compatible form. A linear-amplitude state $f(x_k)$7 is first prepared, converted into a diagonal block-encoding $f(x_k)$8 of the coordinate values, and then transformed into polynomial oracles $f(x_k)$9 by alternating phase modulation. The four components are finally assembled through a linear combination of unitaries.

The resource claims are explicit. The paper states and derives $\rho$0 scaling for degree-$\rho$1 polynomials, $\rho$2 pure ancillas, and success probability $\rho$3 (Guseynov et al., 2024). For piece-wise polynomial functions, the scaling becomes

$\rho$4

with a comparator $\rho$5 used to select the appropriate segment. The same framework extends, by polynomial or piece-wise polynomial approximation, to continuous and piece-wise continuous targets.

This line of work differs sharply from variational data re-uploading. Re-uploading enriches a hypothesis class by repeatedly inserting a classical input into trainable dynamics; amplitude uploading constructs a specific quantum state whose amplitudes are prescribed by a structured classical function. The two share the language of uploading, but not the objective, circuit semantics, or complexity analysis.

4. Expressivity, universality, and limits

Re-uploading models are often introduced through universality claims, but the literature considered here makes clear that universality is only one part of the story. Pérez-Salinas et al. motivate a universal-classifier picture for single-qubit re-uploading (Pérez-Salinas et al., 2019). Qudit results confirm that additional uploads increase accessible functional complexity, and the one-dimensional regression example

$\rho$6

is fit exactly by a qutrit with two layers but not one (Wach et al., 2023). More recently, an experimental photonic implementation of the original separated re-uploading architecture proves a sharper learning-theoretic statement: the separated model is universal, while in the analyzed one-dimensional setting its finite-depth VC dimension is

$\rho$7

whereas the compressed scheme has infinite VC dimension in the worst case (Mauser et al., 7 Jul 2025).

At the same time, expressivity is spectrally structured. For scalar-input QRU,

$\rho$8

and (Barthe et al., 2023) proves that high-frequency components vanish on average. The paper introduces absorption witnesses to compare gradient variances of re-uploading models with those of related data-less PQCs, and shows that the average Fourier weight concentrates in a Gaussian envelope whose width scales as $\rho$9, not $$\mathcal{U}(\vec{\phi}, \vec{x}) \equiv U(\vec{\phi}_N)U(\vec x)\ldots U(\vec{\phi}_1)U(\vec x),$$0. A plausible implication is that re-uploading architectures possess a built-in low-frequency bias: repeated uploads enlarge formal frequency support, but typical high-frequency mass remains suppressed.

Predictive performance imposes a second limit. For $$\mathcal{U}(\vec{\phi}, \vec{x}) \equiv U(\vec{\phi}_N)U(\vec x)\ldots U(\vec{\phi}_1)U(\vec x),$$1-qubit circuits processing $$\mathcal{U}(\vec{\phi}, \vec{x}) \equiv U(\vec{\phi}_N)U(\vec x)\ldots U(\vec{\phi}_1)U(\vec x),$$2 Gaussian features through $$\mathcal{U}(\vec{\phi}, \vec{x}) \equiv U(\vec{\phi}_N)U(\vec x)\ldots U(\vec{\phi}_1)U(\vec x),$$3 encoding layers, (Wang et al., 24 May 2025) proves

$$\mathcal{U}(\vec{\phi}, \vec{x}) \equiv U(\vec{\phi}_N)U(\vec x)\ldots U(\vec{\phi}_1)U(\vec x),$$4

with $$\mathcal{U}(\vec{\phi}, \vec{x}) \equiv U(\vec{\phi}_N)U(\vec x)\ldots U(\vec{\phi}_1)U(\vec x),$$5. If

$$\mathcal{U}(\vec{\phi}, \vec{x}) \equiv U(\vec{\phi}_N)U(\vec x)\ldots U(\vec{\phi}_1)U(\vec x),$$6

then for any repetition count $$\mathcal{U}(\vec{\phi}, \vec{x}) \equiv U(\vec{\phi}_N)U(\vec x)\ldots U(\vec{\phi}_1)U(\vec x),$$7 and any observable with eigenvalues in $$\mathcal{U}(\vec{\phi}, \vec{x}) \equiv U(\vec{\phi}_N)U(\vec x)\ldots U(\vec{\phi}_1)U(\vec x),$$8,

$$\mathcal{U}(\vec{\phi}, \vec{x}) \equiv U(\vec{\phi}_N)U(\vec x)\ldots U(\vec{\phi}_1)U(\vec x),$$9

For balanced binary classification this yields

$f(x)=\sum_{i=0}^Q \sigma_i x^i,$0

so deep, narrow re-uploading on high-dimensional data can collapse predictive performance toward random guessing on unseen inputs (Wang et al., 24 May 2025). This shows that expressivity and trainability do not guarantee useful prediction under distributional averaging.

A third theoretical strand studies the cost of removing tunable upload frequencies. In single-qubit re-uploading with fixed upload primitive $f(x)=\sum_{i=0}^Q \sigma_i x^i,$1, (Liu et al., 24 Jun 2026) proves that tunable upload circuits can still be approximated with depth

$f(x)=\sum_{i=0}^Q \sigma_i x^i,$2

up to a target-dependent constant overhead, while mismatch-class targets obey logarithmic lower bounds

$f(x)=\sum_{i=0}^Q \sigma_i x^i,$3

The conceptual conclusion is explicit in that work: expressive power removed from tunable frequencies can be transferred into circuit depth, but the tradeoff is quantitative rather than free.

5. Hardware realizations and application domains

The re-uploading paradigm has moved well beyond abstract circuit design. On a four-qubit superconducting transmon simulator, a hybrid model with data re-uploading achieved classification accuracy around $f(x)=\sum_{i=0}^Q \sigma_i x^i,$4 on simple tasks and around $f(x)=\sum_{i=0}^Q \sigma_i x^i,$5 on handwritten decimal digits, with a 15-layer MNIST architecture using 244 trainable parameters and one-vs-others classification (Tolstobrov et al., 2023). The same paper emphasizes that, for image recognition, “the information about every pixel is also written several times.”

Image-focused Hamiltonian re-uploading has also been benchmarked against QCNNs. On the reported tasks, test accuracies were $f(x)=\sum_{i=0}^Q \sigma_i x^i,$6 vs $f(x)=\sum_{i=0}^Q \sigma_i x^i,$7 on CT images, $f(x)=\sum_{i=0}^Q \sigma_i x^i,$8 vs $f(x)=\sum_{i=0}^Q \sigma_i x^i,$9 on sklearn digits, $$\ket{\psi}^n= \frac{\sum_{k=0}^{2^n-1}\sum_{i=0}^Q \sigma_i x_k^i \ket{k}^n} {\sqrt{\sum_{\kappa=0}^{2^n-1}\left|\sum_{j=0}^Q \sigma_j x_\kappa^j\right|^2}},$$0 vs $$\ket{\psi}^n= \frac{\sum_{k=0}^{2^n-1}\sum_{i=0}^Q \sigma_i x_k^i \ket{k}^n} {\sqrt{\sum_{\kappa=0}^{2^n-1}\left|\sum_{j=0}^Q \sigma_j x_\kappa^j\right|^2}},$$1 on MNIST, and $$\ket{\psi}^n= \frac{\sum_{k=0}^{2^n-1}\sum_{i=0}^Q \sigma_i x_k^i \ket{k}^n} {\sqrt{\sum_{\kappa=0}^{2^n-1}\left|\sum_{j=0}^Q \sigma_j x_\kappa^j\right|^2}},$$2 vs $$\ket{\psi}^n= \frac{\sum_{k=0}^{2^n-1}\sum_{i=0}^Q \sigma_i x_k^i \ket{k}^n} {\sqrt{\sum_{\kappa=0}^{2^n-1}\left|\sum_{j=0}^Q \sigma_j x_\kappa^j\right|^2}},$$3 on FashionMNIST, with a doubled-depth FashionMNIST model reaching $$\ket{\psi}^n= \frac{\sum_{k=0}^{2^n-1}\sum_{i=0}^Q \sigma_i x_k^i \ket{k}^n} {\sqrt{\sum_{\kappa=0}^{2^n-1}\left|\sum_{j=0}^Q \sigma_j x_\kappa^j\right|^2}},$$4 (Wang et al., 2024). The same source states that the proposed model outperforms the QCNN baseline by “up to over $$\ket{\psi}^n= \frac{\sum_{k=0}^{2^n-1}\sum_{i=0}^Q \sigma_i x_k^i \ket{k}^n} {\sqrt{\sum_{\kappa=0}^{2^n-1}\left|\sum_{j=0}^Q \sigma_j x_\kappa^j\right|^2}},$$5 on MNIST test set.”

Time-series work extends re-uploading beyond static classification. In traffic forecasting on Athens loop-detector data, a hybrid architecture using an LSTM autoencoder and a quantum layer with $$\ket{\psi}^n= \frac{\sum_{k=0}^{2^n-1}\sum_{i=0}^Q \sigma_i x_k^i \ket{k}^n} {\sqrt{\sum_{\kappa=0}^{2^n-1}\left|\sum_{j=0}^Q \sigma_j x_\kappa^j\right|^2}},$$6 qubits and $$\ket{\psi}^n= \frac{\sum_{k=0}^{2^n-1}\sum_{i=0}^Q \sigma_i x_k^i \ket{k}^n} {\sqrt{\sum_{\kappa=0}^{2^n-1}\left|\sum_{j=0}^Q \sigma_j x_\kappa^j\right|^2}},$$7 re-uploading blocks is reported to outperform the corresponding classical recurrent baseline for 6 qubits or more, while in a purely feed-forward replacement scenario classical and hybrid scores become compatible from 10 qubits onward (Schetakis et al., 22 Jan 2025). That paper explicitly interprets re-uploading as a quantum analogue of recurrence or a memory cell.

Pulse-native reformulations push the trainable part of re-uploading down to the control layer. In (Acedo et al., 11 Dec 2025), the single-qubit variational block

$$\ket{\psi}^n= \frac{\sum_{k=0}^{2^n-1}\sum_{i=0}^Q \sigma_i x_k^i \ket{k}^n} {\sqrt{\sum_{\kappa=0}^{2^n-1}\left|\sum_{j=0}^Q \sigma_j x_\kappa^j\right|^2}},$$8

replaces gate-based $$\ket{\psi}^n= \frac{\sum_{k=0}^{2^n-1}\sum_{i=0}^Q \sigma_i x_k^i \ket{k}^n} {\sqrt{\sum_{\kappa=0}^{2^n-1}\left|\sum_{j=0}^Q \sigma_j x_\kappa^j\right|^2}},$$9 layers while keeping the repeated encoding blocks. In noisy transmon simulations calibrated to IBM Brisbane parameters, the gate-based model remained near $\rho$0 test accuracy across depths, whereas the pulse-based model reached about $\rho$1; under increasing depolarizing probability, the pulse-based model maintained roughly $\rho$2 test accuracy up to about $\rho$3 (Acedo et al., 11 Dec 2025). The data embedding itself remains gate-based in that work, so the present result is best read as pulse-native variational re-uploading rather than fully pulse-native feature uploading.

Photonic hardware adds another implementation axis. The original separated re-uploading architecture has been realized on an integrated photonic processor, with perfect classification of circles and moons at three layers, and a theoretical analysis tying the architecture to finite VC dimension and learnability (Mauser et al., 7 Jul 2025). This experimental result reinforces a broader methodological point: the placement and factorization of uploads are not merely stylistic circuit choices but affect statistical complexity, optimization geometry, and noise sensitivity.

6. Quantum-state uploading beyond classical feature maps

The most direct extension from classical to quantum data replaces feature angles by repeated physical interaction with the input state itself. In (Cha et al., 23 Sep 2025), a signal qubit $\rho$4 interacts sequentially with fresh copies of a quantum input $\rho$5 on register $\rho$6, according to

$\rho$7

This realizes a discrete cascade of $\rho$8-dependent CPTP maps, formally analogous to a collision model in open-system dynamics. The paper proves that this architecture can approximate any bounded continuous real-valued function of a finite-dimensional quantum input using only one ancilla qubit and single-qubit measurements, with qubit reuse reducing the physical requirement from $\rho$9 qubits to $\rho$0 qubits for $\rho$1 layers and $\rho$2-qubit inputs (Cha et al., 23 Sep 2025).

An older but conceptually related meaning appears in tensor-network computation. There, uploading refers to moving an unknown state from a physical qubit into the virtual correlation space of an MPS resource. Starting from

$\rho$3

and projecting qubits $\rho$4 and $\rho$5 onto

$\rho$6

the post-measurement state becomes

$\rho$7

which is a teleportation-like upload into correlation space (Morimae, 2010). The same paper first shows that cloning between real physical space and correlation space is impossible, so upload must consume rather than duplicate the original state. It also shows that the teleportation-like method is operationally preferable to simply inverting the previously known downloading procedure.

Taken together, these results indicate that “quantum uploading” now spans three non-equivalent but structurally related tasks: repeated insertion of classical data into a variational dynamics, explicit loading of structured classical amplitudes into quantum states, and coherent transfer or repeated interaction of genuinely quantum inputs. A plausible synthesis is that the unifying theme is not a particular gate set or model family, but the controlled creation of information-bearing quantum degrees of freedom from data sources—classical, functional, or quantum—under hardware and trainability constraints.