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A Tutorial on Quantum Amplitude Embedding with Data Re-Uploading

Hyojun Ahn, Joongheon Kim

arXiv:2610.17836v1quant-phcs.AIcs.LG

TL;DR

Amplitude embedding uses few qubits for many features but restricts model outputs to quadratic forms, and the benefit of costly repeated state preparation remained unclear. The paper analyzes sequential and parallel re-uploading and measures their gains under a shared parameter budget. Re-uploading improves results on higher-order synthetic targets, while one upload remains best for quadratic targets and handwritten digits.

  • Problem

    Amplitude embedding limits outputs to quadratic forms, while which targets benefit from re-uploading—and by how much—remains unexplained despite the repeated state-preparation cost.

    Source: §I · p. 1

  • Method

    The tutorial expresses sequential and parallel re-uploading in one framework, shows each upload raises reachable polynomial order by two, and compares schemes under a shared trainable circuit family and parameter budget.

    Source: §I · p. 1, §III · p. 3, A. Experimental Setup · p. 4

  • Results

    On order-4 targets, test R2 rises from 0.689 to 0.883–0.975, and on order-6 targets from 0.337 to 0.642–0.783, with every model ahead on all ten seeds.

    Source: C. Gain on Targets of Higher Order · p. 5

  • Takeaways & Limitations

    Match upload count to task polynomial order: one upload remains best on quadratic targets and handwritten digits, and each extra upload lowers tolerated gate error.

    Source: §V · p. 6

    The results come from simulated circuits of three to twelve qubits, with nonnegative inputs and noiseless training.

    Source: §V · p. 6

Abstract

from arXiv · show

Amplitude embedding lets a variational quantum model read many classical features with few qubits, but the model output is then only a quadratic form in the normalized input. Data re-uploading, which encodes the input repeatedly between trainable blocks, can be used to enrich such a model. Because every additional upload of an amplitude embedding repeats a costly state preparation, the gain that an upload buys has to be known before the gates are spent. This tutorial thus addresses the problem of explaining what data re-uploading adds to an amplitude-embedded model and how large the gain is. We express sequential re-uploading and parallel re-uploading with two copies in one notation and show that each upload raises the reachable polynomial order by two. Under one trainable circuit family and one budget of trainable parameters, simulation results show that re-uploading raises the test coefficient of determination from 0.689 to between 0.883 and 0.975 on targets of order four and from 0.337 to between 0.642 and 0.783 on targets of order six, on all ten seeds. On a quadratic target and on handwritten digits, one upload remains the best choice, and every additional upload lowers the gate error that the classifier tolerates. These results help designers match the number of uploads to the polynomial order of the task at hand.

I. INTRODUCTION

This tutorial explains what data re-uploading adds to amplitude embedding and quantifies its performance gain. It frames the question around which targets benefit, given that each additional upload repeats a costly state preparation.

I. INTRODUCTION

  • Amplitude embedding is limited to quadratic forms, and which targets benefit from costly repeated state preparation remains unexplained.Earlier empirical comparisons changed other aspects of the model, while a study of amplitude-encoded classifiers reported dataset-dependent gains.Source: §I · p. 1
  • The paper’s contribution is a tutorial on re-uploading for amplitude embedding that explains its fundamentals and quantifies its performance gain.Source: §I · p. 1
  • Sequential and parallel re-uploading are expressed in one framework; each additional upload raises the reachable polynomial order by two and costs another state preparation.Sequential re-uploading acts as angle encoding in the state-preparation angles.Source: §I · p. 1
  • The gain is measured on synthetic targets of orders 2, 4, and 6 under one trainable circuit family, parameter budget, and training procedure.The schemes differ only in where the data enters the circuit.Source: §I · p. 1
  • On quadratic targets and handwritten digits, one upload remains the best choice, while every additional upload lowers the tolerated two-qubit gate error.These cases delimit where the paper reports an advantage from re-uploading.Source: §I · p. 1

II. DATA ENCODING IN VARIATIONAL QUANTUM MODELS

This section describes how variational quantum models encode inputs and why amplitude embedding restricts their output to a quadratic form. It contrasts that encoding with angle encoding and notes the state-preparation cost.

II. DATA ENCODING IN VARIATIONAL QUANTUM MODELS

  • A variational quantum model encodes input, applies trainable gates, measures an observable, and uses a classical optimizer to update its parameters.The circuit depends on both the classical input and trainable parameters.Source: §II · p. 2
  • Amplitude embedding normalizes the input and writes its features into quantum-state amplitudes using a state-preparation circuit.A real-input model’s output is a quadratic form in the normalized input.Source: §II · p. 2
  • Angle encoding uses one qubit per feature, whereas amplitude-embedding angles split the probability mass among amplitudes.In angle encoding, re-uploading enlarges the model’s frequency set with each repetition.Source: Figure 2 · p. 2, §II · p. 2
  • Training changes the quadratic-form matrix but not its degree, so increasing trainable-circuit depth adds no nonlinearity in the input.Source: §II · p. 2
  • Amplitude embedding uses uniformly controlled Ry rotations to split probability mass; preparing a real nonnegative vector takes 2^n − 2 CNOT gates.The encoding gate count is linear in the number of features and exponential in the number of qubits.Source: §II · pp. 2–3

III. RE-UPLOADING AN AMPLITUDE EMBEDDING

The section compares sequential and parallel ways to re-upload an amplitude embedding, showing how they expand reachable polynomial order while incurring different state-preparation and qubit costs.

  • With L sequential uploads, frequencies reach L in the state-preparation angles, enabling input polynomials through degree 2L; order-2L targets therefore need at least L uploads.Source: §III · p. 3
  • Two parallel copies produce a degree-four polynomial in the input, while k copies reach degree 2k; two copies match two sequential uploads in order at different resource cost.Source: §III · p. 3
  • Sequential re-uploading retains n qubits but multiplies encoding count and serial preparation depth; parallel copies multiply encoding count and use kn qubits.Source: §III · p. 3
  • One trainable layer of the circuits in Section IV costs only six two-qubit gates; these counts help determine whether a quantum learner fits platforms on which learned controllers operate, such as aerial and underwater vehicles.Source: §III · p. 4

IV. PERFORMANCE EVALUATION

The evaluation measures where re-uploading improves amplitude-embedded models and where its gains end, using synthetic targets, handwritten digits, and gate-error analysis.

  • The evaluation tests re-uploading on synthetic targets of varying polynomial order, handwritten digits, and under gate errors to measure where its gains end.Source: §IV · p. 4

A. Experimental Setup

The experiments compare upload schemes under a shared trainable-parameter budget across synthetic polynomial targets and handwritten-digit classification, using held-out performance measures.

A. Experimental Setup

  • All models use the same ansatz and 3nV trainable parameters, differing only in where data enters; comparisons include one upload, sequential L=2–4 uploads, and two parallel copies.Source: A. Experimental Setup · p. 4
  • The synthetic task uses nonnegative unit-sphere inputs and targets of orders 2, 4, and 6, with test R2 reported as mean and standard deviation over ten seeds.Source: A. Experimental Setup · p. 4, Figure 5 · p. 4
  • The image task uses 1,797 8 × 8 handwritten-digit images in ten classes, embeds 64 pixel intensities in six qubits, and predicts among ten outcomes measured on four qubits.Source: A. Experimental Setup · p. 4
  • Synthetic-task training runs for 2,400 full-batch Adam steps, while 600 steps suffice to fit every model's digit training set; results use ten seeds.Source: A. Experimental Setup · p. 4

B. Convergence

Re-uploading improves training loss on the order-4 target, and the reported convergence checks indicate that the one-upload model does not catch up with more steps.

B. Convergence

  • 0.312, 0.109, 0.073, and 0.064 were the training-loss plateaus for one through four uploads; two copies reached 0.022.Source: B. Convergence · p. 4
  • Within at most 1,400 of 2,400 steps, models with L=1 through L=4 were within 5% of final loss, so more steps would not let one upload catch up.Source: B. Convergence · p. 4

C. Gain on Targets of Higher Order

Re-uploading raises performance above the one-upload quadratic-form limit on higher-order targets, with gains varying by target order and upload configuration. On a quadratic target, one upload is already near-perfect and additional uploads do not improve the result.

C. Gain on Targets of Higher Order

  • 0.689 with one upload rose to 0.883, 0.916, and 0.918 with two, three, and four uploads, and 0.975 with two copies on the order-4 target.The one-upload quadratic-form limit is 0.694; each model was ahead on all ten seeds.Source: C. Gain on Targets of Higher Order · pp. 4–5
  • 0.337 with one upload rose to 0.642, 0.745, and 0.783 with two, three, and four uploads on the order-6 target.Every model was ahead of one upload on all ten seeds; two copies reached 0.771.Source: C. Gain on Targets of Higher Order · p. 5
  • 0.999 with one upload on the order-2 target exceeded sequential models at 0.975–0.977 and two copies at 0.997.The passage notes that the model classes are not nested and that realizing a pure quadratic form is harder once uploads interrupt trainable layers.Source: C. Gain on Targets of Higher Order · p. 5

D. Limits of the Gain

On handwritten digits, additional uploads reduce test accuracy despite near-perfect training accuracy, while gate errors further lower classifier accuracy. The results illustrate limits from both generalization and the hardware cost of repeated state preparations.

D. Limits of the Gain

  • 95.9% with one upload exceeded 94.8%, 92.3%, and 88.5% with two, three, and four uploads on the digits task.With 1,297 images, four uploads lost 7.4 percentage points; with 100 images, they lost 18.5 points.Source: D. Limits of the Gain · p. 5
  • At 1% two-qubit gate error, one upload retained 47.6% accuracy against 36.7% for two uploads, which add 62 CNOT gates.Four uploads fell from 87.2% without gate errors to 58.0% at 0.5% error; the two-copy model was limited by shot noise before gate errors.Source: D. Limits of the Gain · p. 5
  • Each additional upload lowers the gate error the classifier tolerates, and every model classified at least 99.9% of training images correctly.The authors attribute the digit-test deficit to generalization rather than fit; the evaluation used 1,024 shots per image and varied gate error from 0 to 2%.Source: D. Limits of the Gain · p. 5

V. CONCLUSION

The paper presents fundamentals and performance measurements for data re-uploading with amplitude embedding under a shared trainable circuit family and parameter budget. Simulations show gains on higher-order targets but favor one upload for a quadratic target and handwritten digits.

V. CONCLUSION

  • A sequential upload acts as angle encoding in the state-preparation angles, and each additional upload raises reachable polynomial order by two.A target of order 2L needs at least L uploads, each costing 2n−2 CNOT gates.Source: §V · p. 5
  • 0.689 rose to as much as 0.975 on order-4 targets, and 0.337 rose to as much as 0.783 on order-6 targets, on all ten seeds.The results were obtained with one trainable circuit family and one budget of trainable parameters.Source: §V · pp. 5–6
  • One upload remained the best choice on a quadratic target and handwritten digits, while each additional upload lowered tolerated gate error.The authors conclude that designers should match upload count to the task’s polynomial order.Source: §V · p. 6

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