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Same Spectrum, Harder First-Order Training: Phase-Locking in Single- versus Multi-Qubit Re-Uploading Circuits

Michael Poppel, David Bucher, Maximilian Zorn, Markus Baumann, Sebastian Wölckert, Claudia Linnhoff-Popien, Philipp Altmann, Jonas Stein

arXiv:2610.21778v1quant-phcs.LG

TL;DR

For one-dimensional inputs, a single-qubit serial circuit matches the functions representable by a multi-qubit circuit with the same encoding budget, but its training reliability is lower. The paper compares their tangent structures and measures training outcomes against a two-qubit control.

  • Problem

    Although width can be exchanged for depth without losing expressivity for one-dimensional inputs, the training consequences of that exchange remain to be measured.

    Source: §1 · pp. 1–2

  • Method

    The paper proves structural properties of the serial circuit and compares its first-order training with two-qubit controls sharing its encoding budget and model class.

    Source: §1 · p. 1, §3 · p. 3

  • Results

    The serial circuit reaches its optimum less often, with the gap growing with E on exact flat-target losses; on real-world series, its held-out error is up to about 1.5 times the control under Adam.

    Source: §1 · p. 2

  • Takeaways & Limitations

    For one-dimensional inputs, width-for-depth preserves expressivity but costs first-order training reliability; a second qubit avoids that cost at the price of two-qubit gates.

    Source: §7 · p. 10

    Results are limited to one-dimensional inputs, Haar-random initial gates, noise-free double-precision statevector simulation, fixed step budgets, and one learning-rate schedule.

    Source: §7 · p. 9

Abstract

from arXiv · show

Angle-encoded variational quantum circuits implement truncated Fourier series. For one-dimensional inputs, a single-qubit re-uploading (serial) circuit represents every function that a multi-qubit circuit with the same encoding budget $E$ can; we find that this exchange of width for depth is not free in first-order training. Because its state lives on the Bloch sphere, all parameters of the serial circuit move the state, at each input, along one complex direction, which we call \emph{phase-locking}. We prove that this costs no Fourier direction but leaves no direction to spare: however many parameters it has, the serial circuit has at most $2E+2$ tangent directions, one more than its $2E+1$ Fourier coefficients need, while a two-qubit control with the same encoding budget and model class generically has at least $6E+6$. The cost in training is measured against this control, as the number of runs out of 100 that reach the optimum within a fixed step budget. On an exact loss with flat-spectrum targets the gap grows with $E$; at $E=32$ the serial circuit converges in 3, 53 and 83 runs under gradient descent, Adam and Levenberg--Marquardt, the control in 99, 100 and 100. On three real-world series, at matched parameter counts, the serial circuit reaches the least-squares optimum in at most 9 runs under Adam and in 32 to 93 under Levenberg--Marquardt, the control in at least 98 and in all 100; its held-out error is up to about $1.5$ times that of the control under Adam, but no larger in the typical run under Levenberg--Marquardt.

1. Introduction

For one-dimensional inputs, a single-qubit serial circuit can represent every function in the same encoding-budget model class, but its structure differs from two-qubit controls. The paper measures the training cost of that difference without claiming it causes the observed outcomes.

1. Introduction

  • For one-dimensional inputs, the serial circuit realizes every real trigonometric polynomial of degree at most E with values in [−1, 1], so width can be exchanged for depth without loss of expressivity.The accessible spectrum depends only on encoding budget E.Source: §1 · p. 1
  • At each input, all serial-circuit parameters move the state along one complex direction, leaving at most one more tangent direction than the 2E + 1 Fourier coefficients require.The two-qubit circuits compared have several times as many tangent directions.Source: §1 · p. 1
  • The controlled comparison uses a two-qubit ancilla circuit with the serial circuit’s encoding budget, chain length, and model class, but three complex tangent components rather than one.A two-qubit parallel circuit has three components as well but half the chain length.Source: §1 · p. 1
  • The paper proves the tangent-count and phase-locking results and full row rank of the coefficient Jacobian, with the proofs also formalised in Lean 4 using cited results as hypotheses.Training differences are measured empirically, and the authors claim no causal link between structure and training outcomes.Source: §1 · p. 2
  • On exact flat-target losses, the serial circuit reaches its optimum less often than the control, with a gap that grows with E; on real-world series its held-out error is up to about 1.5 times the control under Adam.Under Levenberg–Marquardt, the typical run predicts as well as the control.Source: §1 · p. 2

2. Related work

Related work covers Fourier expressivity, quantum signal processing, and approaches to trainability. The paper distinguishes its fixed-width training comparison from barren-plateau studies and from trainable-encoding optimization traps.

2. Related work

  • Prior work studies how encoding budget determines accessible spectra, how trainable gates determine realized coefficients, and where gates must be placed for coefficients to be reachable [Wiedmann et al., 2025; Oh et al., 2026].The cited literature also studies coefficient statistics over random parameters.Source: §2 · p. 2
  • Single-qubit repeated-encoding circuits are universal for univariate functions, while quantum signal processing work studies phase-finding and reports local-minimum outcomes from random phases [P´erez-Salinas et al., 2021; Yu et al., 2022; Motlagh & Wiebe, 2024; Dong et al., 2021].This paper instead starts from Haar-random gates and compares two-qubit circuits with the same spectrum.Source: §2 · p. 2
  • For random-phase QSP, prior work gives phase-derivative and tangent-kernel statistics; this paper’s envelope is the expectation-value counterpart and is used at the Fourier-coefficient level.The cited QSP result gives |µ(x)|2 = 1 − f(x)2 for the envelope.Source: §2 · p. 2
  • The paper distinguishes its fixed-width training cost from barren plateaus, defined as gradients vanishing exponentially with qubit count, and relates its tangent counts across inputs to fixed-input quantum Fisher information rank [McClean et al., 2018; Cerezo et al., 2021].Its encodings are fixed, unlike the trainable encoding weights implicated in “Fourier locking”.Source: §2 · p. 2

3. Background

The background defines the angle-encoded circuit and its Fourier representation, then sets out the Jacobians and circuit pair used in the comparison. The serial and ancilla circuits share a model class and generically retain all coefficient directions.

3. Background

  • An angle-encoded VQC interleaves encoding layers with L + 1 trainable-gate blocks and outputs the expectation of a non-identity Pauli word.Each encoding layer applies a Pauli rotation by the input angle to each encoded qubit.Source: §3 · p. 2
  • The serial circuit uses one qubit and E layers; the ancilla control encodes on one qubit and reads out another, while the parallel circuit encodes both qubits with half as many layers.The ancilla circuit matches the serial circuit’s encoding budget, block count, and model class, but has a four-dimensional state space and 15 parameters per gate rather than 3.Source: §3 · p. 3
  • With Pauli-angle encoding, the output is a truncated Fourier series with spectrum {−E, …, E}, where E is the total number of encoding gates, and a real function is specified by 2E + 1 real coefficients.The paper fixes RX encoding, Z readout, and the |0⟩ initial state without changing the model class or relevant training statements.Source: §3 · pp. 2–3
  • The coefficient Jacobian differentiates the real Fourier-coefficient vector with respect to parameters, while the data Jacobian differentiates outputs at specified inputs.The data Jacobian is the real Fourier design matrix of the inputs multiplied by the coefficient Jacobian.Source: §3 · p. 3
  • The serial and ancilla circuits have the same model class, the degree-at-most-E real trigonometric polynomials bounded by 1, and their coefficient Jacobians have rank 2E + 1 for almost every gate choice.The parallel circuit serves as a chain-length reference; the paper does not prove the same model-class or rank results for it.Source: §3 · p. 3
  • Figure 1 contrasts the serial circuit’s single state-motion direction at each input with the three orthogonal directions of two-qubit circuits.Its other panel depicts chain length as the number of blocks, L + 1.Source: Figure 1 · p. 3

4. Tangent structure of single- and multi-qubit circuits

The section distinguishes serial and multi-qubit circuits through chain length and parameter-motion directions, then counts how those directions accumulate across inputs. The serial circuit has no spare tangent direction, unlike the two-qubit circuits.

4. Tangent structure of single- and multi-qubit circuits

  • The section establishes structural counts, while whether those differences affect training is measured in later sections rather than derived here.For a serial circuit, the tangent count is the rank of the matrix H; the count is the real-span dimension of the tangent-coefficient functions across parameters and inputs.Source: §4 · p. 3, §4.3 · p. 5

4.1. Polynomial structure and chain length

  • At fixed encoding budget E, the serial circuit has E + 1 blocks, the parallel circuit has E/N + 1, and the ancilla circuit removes this chain-length difference.Chain length counts the blocks in each coefficient-expansion term; it is a property of the expansion, not a claim that every coefficient depends on every block.Source: §4.1 · p. 4

4.2. Phase-locking: one complex direction for all parameters

  • In the serial circuit, every parameter’s state derivative orthogonal to the state is a complex multiple of one vector, and all parameters share the same envelope µ(x).The parameter-specific h_j is determined by the sub-circuit up to the gate containing that parameter and has degree at most the number of preceding encoding layers; µ has degree at most L.Source: §4.2 · p. 4

4.3. Tangent count and spare directions

  • For generic gates, the serial circuit has at most 2(E + 1) tangent directions, while the ancilla circuit has at least 6(E + 1) and the parallel circuit at most 5E + 6.The serial circuit’s 2E + 2 directions leave no spare direction beyond those needed to change its 2E + 1 Fourier coefficients and its readout rotation; the two-qubit circuits have at least 4E − 2 spare directions for the ancilla and 3E − 2 for the parallel circuit.Source: §4.3 · p. 5

5. First-order training on an exact loss

The exact-loss experiments compare serial and two-qubit circuits on shared targets and initial gates, then test whether convergence differences persist across optimisers. The serial circuit converges less often, with the gap especially pronounced at larger encoding budgets and under Adam.

5. First-order training on an exact loss

  • The experiment uses exact losses for random degree-E trigonometric targets, so the represented serial and ancilla targets have zero optimal loss.Each configuration has 100 runs; under Adam the comparison is also repeated at matched parameter counts.Source: §5 · p. 6

5.1. Gradient descent

  • At E = 32 on flat targets, the serial circuit converges in 3 of 100 runs under gradient descent, versus 99 of 100 for the ancilla circuit.Across flat targets, the convergence gap grows with E; at E = 16 the counts are 65 of 100 and 100 of 100, respectively.Source: §5.1 · p. 6
  • Gradient-descent convergence counts are limited by the step budget: at E = 24 on flat targets, 18 of 24 successful runs converge only after step 27,000.The remaining runs may be slow or stuck; the counts alone do not distinguish these possibilities.Source: §5.1 · p. 6

5.2. Conditioning and Levenberg–Marquardt

  • Initial conditioning does not account for the convergence difference: configurations from different circuits with matched initial κJ still differ, and within a configuration initial κJ does not predict which runs converge.κJ never rises above 10−4 in 465 of 1,000 serial runs, compared with 127 ancilla runs; the authors call this a description, not a mechanism.Source: §5.2 · p. 7
  • Levenberg–Marquardt converges in 923 of 1,000 serial runs and all 2,000 two-qubit runs, narrowing but not eliminating the gap.At E ≥24, the serial circuit converges in 73 to 90 of 100 runs while the ancilla circuit converges in all runs; the difference is significant in those four configurations.Source: §5.2 · p. 7

5.3. Adam

  • Under Adam on 1/k targets at E = 24 and 32, the serial circuit converges in 23 and 9 of 100 runs, versus 100 and 97 for the ancilla circuit.With parameter counts matched on flat targets at E = 32, the serial circuit reaches the least-squares optimum in 36 of 100 runs, while the control does so in every run.Source: §5.3 · pp. 7–8

6. Training on data

On Poisson-kernel targets and three real-world series, the serial circuit reaches the least-squares optimum less often than the two-qubit controls under Adam and Levenberg–Marquardt. The training shortfall can carry over to held-out error, though its size depends on the series and optimiser.

Setup

  • The data experiments use 200 training and 100 test points on [−π, π), matched parameter counts, tuned full-batch Adam, and same-initial-gate Levenberg–Marquardt comparisons.Targets include Poisson-kernel sums not representable at degree E and three real-world series; the real-data task fits random points rather than forecasting.Source: §6 · p. 8

Poisson-kernel targets

  • At E = 32 on Poisson-kernel targets, serial reaches the optimum in 1 of 100 runs under Adam versus 24 for the ancilla control; under Levenberg–Marquardt, it reaches it in 72 versus 100 for both two-qubit circuits.The serial shortfall is significant at each E under both comparisons.Source: §6 · p. 8, Figure 3 · p. 9

Real-world data

  • The three real-world series are Nottingham monthly temperatures, weekly US gasoline prices, and daily IBM closing prices; the latter two were screened for degree-32 fit and held-out variance explained.The temperatures were selected before the screen; the other two series were selected using a prespecified screen.Source: §6 · p. 8
  • At most 9 of 100 serial runs reach the optimum on the three real-world series under Adam, versus at least 96 for the two-qubit circuits; under Levenberg–Marquardt, serial reaches it in 32–93 runs versus 100 for both controls.The serial-control difference is significant in all six Levenberg–Marquardt configurations.Source: §6 · p. 8, Figure 4 · p. 9
  • Under Levenberg–Marquardt, the serial circuit’s median held-out-error ratio is between 1.00 and 1.01 on every series and every Poisson-kernel configuration, indicating performance comparable to the control.At E = 32, 30 of 100 serial runs on ibmclose and 61 of 100 on the temperatures have lower held-out error than the control.Source: §6 · p. 9

7. Discussion and conclusion

For one-dimensional inputs, the serial circuit preserves expressivity but has fewer tangent directions than the two-qubit circuits, and first-order optimisers find its optimum less reliably. The paper reports this training cost without establishing phase-locking as its cause.

Discussion

  • The serial and ancilla circuits represent the same functions without losing Fourier directions, yet first-order methods reach the serial optimum less often, with the gap growing with E; Levenberg–Marquardt narrows but does not close it at large E or on real-world series.The initial conditioning and poor starting points do not explain the observed difference.Source: §7 · p. 9
  • The serial circuit and ancilla control represent the same functions without structural loss of Fourier directions, yet tangent vectors have one complex component per input versus three, yielding 2E + 2 versus at least 6E + 6 directions.Source: §7 · p. 9
  • The authors do not establish that phase-locking causes the training cost, because its effect cannot be separated from the two-dimensional state-space dimension.Along gradient descent, loss falls after the Jacobian conditioning recovers in nearly every observable run, and recovery occurs less often for the serial circuit.Source: §7 · p. 9

Limitations

  • The study is limited to one-dimensional inputs, Haar-random initial gates, noise-free double-precision statevector simulation, fixed step budgets, and one learning-rate schedule.From E = 24, the simulation no longer resolves the serial circuit’s initial conditioning.Source: §7 · p. 9

Conclusion

  • The two-qubit circuits avoid the measured training cost at the price of two-qubit gates, while Levenberg–Marquardt removes it from the typical run but requires a linear solve per step.Whether the extra two-qubit directions enable conditioning to recover along the training path remains open.Source: §7 · p. 10

Overview of the appendices and the code

The appendices organize the paper’s protocol, proofs, formalisation, and further results. The repository provides code and artifacts for reproducing the figures and tables.

Appendices

Code

  • The repository contains simulation code, result files, scripts to regenerate figures and tables, and a Lean 4/Mathlib formalisation of the paper’s theoretical results.Figures and tables are regenerated from result files alone; those scripts do not run circuits.Source: Overview of the appendices and the code · p. 13

A. Experimental details

The experimental details specify circuit architectures, parameterisations, targets, optimisers, evaluation measures, and reproducibility checks. The data experiments use matched parameter counts and fixed train/test protocols, while the exact-loss experiments compare convergence and conditioning.

Architectures

  • The serial circuit uses one qubit and E encoding layers, the ancilla circuit encodes on one of two qubits for E layers, and the parallel circuit encodes on both for E/2 layers.Gates are independently Haar-initialised; two-qubit readout is on qubit 1.Source: A. Experimental details · p. 13

Parameterisation

  • Each gate is stored as an SU(2N) matrix and updated in Pauli-generator coordinates, which provide every gate tangent direction at the same scale.These coordinates yield Riemannian steepest descent; optimiser trajectories depend on the coordinates, and training with angle parameterisations was not studied.Source: A. Experimental details · p. 13

Targets and data

  • Exact-loss targets are degree-E flat-spectrum and 1/k random trigonometric polynomials; data experiments use Poisson-kernel targets and real series with specified random train/test splits.The exact loss is evaluated on 2E + 1 equispaced points, whose grid average equals the circle integral exactly.Source: A. Experimental details · p. 14

Conditioning and endpoints

  • Conditioning is tracked through Jacobian singular values at initialisation and gradient-descent checkpoints, while Levenberg–Marquardt endpoints are examined for residual and small-singular-direction behavior.Singular values below roughly 10^-15 are unresolved at double precision; the appendix’s stalled-run criterion is less than 1% loss reduction over the final quarter of iterations.Source: A. Experimental details · p. 14

Optimisers and budgets

  • For data, serial and ancilla blocks each have 15 parameters and parallel blocks have 30 at half chain length; Adam runs 20,000 steps and Levenberg–Marquardt runs 2,000 iterations.Adam’s rate is tuned on separate targets; Levenberg–Marquardt uses the same data and initial gates.Source: A. Experimental details · p. 15

Real-series screen

  • The real-series screen was fixed before running circuits and selected series with degree-32 held-out R2 of at least 0.95 plus specified improvement and fit criteria.The 0.95 threshold was set after seeing the Nottingham temperatures, which were part of the design before screening.Source: A. Experimental details · p. 15

Measures

  • A run reaches the data optimum when excess training loss over the degree-E least-squares fit, normalised by training variance, is at most 10^-6; held-out prediction uses 1 − R2.Held-out error is also compared as the serial-over-ancilla ratio for matched target runs.Source: A. Experimental details · p. 15

Protocol checks

  • The step-budget check tests seven configurations using 80,000 steps instead of 20,000, while the learning-rate check tests whether selected rates are constrained by the grid edge.The learning-rate rule is applied equally across circuits; a changed selected rate triggers repeated evaluation runs.Source: A. Experimental details · p. 15

B. Proofs

The proofs show that serial re-uploading preserves the full Fourier model class while phase-locking constrains its tangent directions; they establish generic tangent counts and connect full-rank training to least-squares fitting.

Training geometry

  • On 2E+1 equispaced inputs, the Fourier design matrix is invertible, so data-output Jacobian rank is linked to coefficient-Jacobian rank.The design-matrix columns are orthogonal on these inputs.Source: B. Proofs · p. 16

Expressivity and coefficient rank

  • Every real trigonometric polynomial of degree at most E bounded by 1 is realised by the serial circuit, and the ancilla circuit has the same model class.The serial construction uses Fejér–Riesz factorisation; the ancilla construction embeds the serial state.Source: B. Proofs · pp. 16–17

Phase-locking

  • Phase-locking makes every serial-circuit parameter derivative share one complex envelope, yet the coefficient Jacobian still has full row rank almost everywhere.The proof uses the one-dimensional orthogonal complement of a qubit state and the resulting shared complex factor in its output derivatives.Source: B. Proofs · pp. 17, 19

Tangent directions

  • For one encoded qubit, tangent directions grow by 2(d−1) per layer; the two-qubit parallel circuit has generic tangent dimension 6+10L, or 5E+6.The proof establishes lower bounds at identity gates and extends them generically using polynomial rank conditions.Source: B. Proofs · pp. 18–19
  • For generic gates, serial (d=2, L=E) has 2E+2 tangent directions, ancilla (d=4, L=E) has at least 6E+6, and two-qubit parallel (L=E/2) has 5E+6.Source: B. Proofs · pp. 18–19

Spare directions

  • The serial circuit has no spare tangent directions beyond readout rotations at full row rank, while the ancilla and parallel circuits have at least 4E−2 and 3E−2 spare directions.These counts subtract the coefficient-Jacobian rank and the dimensions attributable to readout rotations.Source: B. Proofs · p. 19

Optimiser steps

  • Under Riemannian gradient descent, stacking t gates in a block scales its first-order step by t, but this argument does not extend to Adam.Adam’s coordinate-wise normalisation is not invariant under the orthogonal coordinate maps used in the proof.Source: B. Proofs · pp. 19–20

Training geometry

  • With a full-row-rank coefficient Jacobian, a stationary point of squared data loss is a least-squares fit over degree-at-most-E real trigonometric polynomials.If the least-squares fit lies in the model class but the loss is above optimum, a stationary point must instead have a rank-deficient Jacobian.Source: B. Proofs · p. 20

C. Lean formalisation

The paper formalises its Sections 3–4 and Appendix B results in Lean 4, with cited mathematical results supplied as explicit hypotheses. The formalisation also records which statements and supporting arguments remain outside its scope.

Formalisation scope

  • Every lemma, proposition, remark, and corollary in Sections 3–4 and Appendix B is stated and proved in Lean 4, with no `sorry` and only three reported axioms.The proofs are kernel-checked; the cited results enter as explicit hypotheses rather than axioms.Source: C. Lean formalisation · p. 20

Formalisation limits

  • The formalisation does not cover optimiser-run correspondence, several frame-change arguments, or whether two-qubit readout rotations preserve the output beyond first order.The project also lacks a coefficient Jacobian for the parallel circuit; its first-order unchanged-output directions are bounded by the span of output derivatives.Source: C. Lean formalisation · p. 20

D. Additional results

Additional checks confirm that the serial circuit reaches the optimum less often than the two-qubit controls at matched parameter counts. Initial conditioning does not explain the gap, and conditioning changes during training before the loss reaches its threshold in most observed runs.

D. Additional results

  • The κJ bound is one-sided: good conditioning guarantees progress, but poor conditioning does not rule it out.Source: D. Additional results · p. 20
  • Despite worse serial initial conditioning, circuits with similar median κJ still differed in convergence, while run-level initial κJ did not predict outcomes (AUC 0.40–0.65).At E=8, 12, and 16, serial medians were 10^-4.5, 10^-7.1, and 10^-11.6; values at E=24 and 32 were below about 10^-15, at double-precision resolution.Source: D. Additional results · p. 21
  • In 544/551 ancilla, 586/598 parallel, and 322/330 serial runs, κJ crossed 10^-4 before loss reached 10^-4; the authors describe this ordering, not a mechanism.κJ never exceeded 10^-4 in 465/1,000 serial runs, versus 127/1,000 ancilla runs; at the higher threshold 10^-2, event ordering was less clear.Source: D. Additional results · p. 21
  • At E=8 on flat targets, 77 serial versus 54 ancilla runs fell below 10^-10, although all 200 runs were below 10^-6; Adam rate bracketing changed 23 of 27 selections.The paper notes that 10^-10 cuts through converged runs and that near-best rates often still included an end of the extended grid.Source: D. Additional results · p. 21
  • Across seven data configurations, the serial circuit reached the optimum less often under both step-budget schedules; under the held schedule the gap ranged from 0.20 to 0.97, with every interval excluding zero.The gap depended on the budget and did not change in only one direction across the reported checks.Source: D. Additional results · p. 22, Table 11 · p. 26
  • At matched counts, serial Adam reached optimum in 95, 87, 84, 60, and 36/100 runs at E=8–32, versus every run for both two-qubit circuits.Levenberg–Marquardt yielded 98, 100, 96, 94, and 82 serial runs at E=8–32, versus all runs for both controls.Source: D. Additional results · p. 22, Figure 6 · p. 24
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