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

quant-phOct 15, 2026

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

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

Organizations: Department of Computer Science, LMU Munich, Germany · Aqarios GmbH, Munich, Germany

Abstract

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 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.

Figures & tables

Figure 1: (a) Chain length at

E=4

: the number

L+1

of blocks, all of which enter every term of every Fourier coefficient (Lemma 4.1 ). (b) Phase-locking. At each input, every parameter of the serial circuit moves the state along the one vector

|\psi^{\perp}\rangle

(Lemma 4.2 ); on two qubits, in the ancilla and the parallel circuit alike, it moves it along three orthogonal vectors (Remark 4.3 ). Symbols as in Lemma 4.2 and Remark 4.3 .

Figure 2: Share of converged runs (final loss

\leq 10^{-6}

) against the encoding budget

E

, for flat targets and the three optimisers. Bars: 95% Wilson intervals; 100 runs per point.

Figure 3: Poisson-kernel targets, Adam at matched parameter counts. (a) Share of runs that reach the least-squares optimum; open diamonds: Levenberg–Marquardt on the same runs of the serial circuit (both two-qubit circuits reach the optimum in every run under it). Bars: 95% Wilson intervals; 100 runs per point. (b) Held-out error

1-R^{2}

: median over runs, bars from the lower to the upper quartile; open diamonds: median of the serial circuit under Levenberg–Marquardt.

Figure 4: Real-world series, Adam at matched parameter counts; panels as in Figure 3 , 100 runs per point.

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

$N$ , $L$	number of qubits; number of encoding layers
$E$	encoding budget, the number of encoding gates ( $NL$ if all qubits are encoded; $L$ for the ancilla circuit)
$\boldsymbol{\theta}$ , $P$	Lie-algebra coordinates of all gates; their number
$S(x)$	encoding layer, $R_{X}(x)$ on the encoded qubits
$W^{(\ell)}$	block of trainable gates, $\ell=0,\dots,L$
$M$	observable, $Z$ on the last qubit
$J$	coefficient Jacobian $\partial\tilde{\mathbf{c}}/\partial\boldsymbol{\theta}$

Appendix

Table 1: Notation of the paper.

Figure 5: (a) Median

\log_{10}\kappa_{J}

at initialisation against

E

, flat targets; the dotted line marks double precision. (b) Median

\log_{10}\kappa_{J}

(solid) and median

\log_{10}

loss (dashed) along gradient descent, flat targets,

E=16

		loss reaches $10^{-4}$					loss does not reach $10^{-4}$
threshold	circuit	above at start	$\kappa_{J}$ first	same interval	loss first	never above	above later	never above
$10^{-2}$	serial	4	189	175	3	35	1	593
	ancilla	12	432	330	10	38	2	176
	parallel	8	447	305	14	36	0	190
$10^{-3}$	serial	36	330	38	0	2	50	544
	ancilla	142	588	90	0	2	21	157

Appendix

Table 2: Gradient descent on the exact loss: order of the events “

\kappa_{J}

exceeds the threshold” (seen at checkpoints) and “loss

\leq 10^{-4}

” (seen at every step); numbers of runs, pooled over target families and

E

(1,000 runs per circuit). Categories as defined in the text.

circuit	qubits	encoded	$(m_{a})$	per layer	upper bound
serial ∗	1	1	(1,1)	2	2
ancilla †	2	1	(2,2)	6	8
parallel ∗	2	2	(1,2,1)	10	10

Appendix

Table 3: Tangent directions added by every layer after the first (exact numerical rank at random gates) for the three circuits of this paper, against the upper bound

d^{2}-\sum_{a}m_{a}^{2}

of Appendix B , with

m_{a}

the multiplicities of the eigenvalues of the encoding generator. Rows marked ∗ are proved, the row marked † is proved as a lower bound (Proposition 4.4 ).

targets	$E$	failed	stalled	any	two	share $\geq 0.99$	$\log_{10}$ gradient	failed	converged	initial	$n_{\rm init}$
				$\sigma_{i}\leq 10^{-8}\sigma_{1}$ at the end point				$\log_{10}\sigma_{\min}/\sigma_{1}$
flat	16	4	4	3	3	3	$-6.0$	$-9.4$	$-1.5$	$-11.6$	2
flat	24	10	10	9	9	9	$-7.4$	$-11.2$	$-1.8$	$-16.2$	8
flat	32	17	17	15	14	15	$-6.8$	$-10.9$	$-2.0$	$-17.0$	16
$1/k$	12	1	1	1	1	1	$-10.9$	$-13.0$	$-1.9$	$-7.9$	0.5
$1/k$	16	4	4	3	3	3	$-8.6$	$-10.7$	$-2.2$	$-10.8$	2

Appendix

Table 4: Serial Levenberg–Marquardt runs with final loss above

10^{-6}

(the two-qubit circuits have none). Stalled: the loss fell by less than

1\%

over the last quarter of the iterations. Middle block: runs with at least one and with exactly two singular values

\sigma_{i}\leq 10^{-8}\sigma_{1}

at the end point, and runs in which the corresponding left-singular directions carry at least

0.99

of the squared residual. Then medians of the

\log_{10}

relative gradient at the failed end points and of

\log_{10}\sigma_{\min}/\sigma_{1}

at the failed end points, the converged end points and the initial points of the configuration.

n_{\rm init}

: median number of singular values

\leq 10^{-8}\sigma_{1}

at the initial points.

		runs at the optimum			ancilla $-$ serial	median test $R^{2}$		held-out error
target	$E$	serial	ancilla	par. 2	difference [95%]	serial	ancilla	serial/ancilla
flat	8	0.95	1.00	1.00	+0.05 [+0.00, +0.11]	1.0000	1.0000	—
flat	12	0.87	1.00	1.00	+0.13 [+0.07, +0.21]	1.0000	1.0000	—
flat	16	0.84	1.00	1.00	+0.16 [+0.09, +0.24]	1.0000	1.0000	—
flat	24	0.60	1.00	1.00	+0.40 [+0.30, +0.50]	1.0000	1.0000	—
flat	32	0.36	1.00	1.00	+0.64 [+0.54, +0.73]	0.9783	1.0000	—

Appendix

Table 5: Synthetic targets, full-batch Adam, 100 runs per cell. Runs at the optimum: share with excess training loss at most

10^{-6}

of the variance. Ancilla

-

serial: difference of these shares with a 95% Newcombe interval. Held-out error: median and 10%/90% quantiles of

(1-R^{2}_{\rm serial})/(1-R^{2}_{\rm ancilla})

over runs on the same target; for flat targets the ancilla error is at the double-precision floor and the ratio is not shown.

		runs at the optimum			median test $R^{2}$		held-out error	serial runs
series	$E$	serial	ancilla	par. 2	serial	ancilla	serial/ancilla [10%, 90%]	worse
gasprice	24	0.04	1.00	1.00	0.9569	0.9591	1.05 [1.00, 1.15]	0.92
gasprice	32	0.00	0.98	0.96	0.9682	0.9786	1.49 [1.30, 1.83]	1.00
ibmclose	24	0.01	1.00	1.00	0.9607	0.9641	1.09 [1.04, 1.14]	0.97
ibmclose	32	0.00	1.00	1.00	0.9629	0.9638	1.03 [0.97, 1.09]	0.70
nottem	24	0.09	1.00	1.00	0.8995	0.9004	1.01 [0.97, 1.02]	0.71

Appendix

Table 6: Real-world series, full-batch Adam, 100 initialisations per architecture on the same data. Runs at the optimum as in Table 5 . Held-out error ratio paired by run index; last column: share of serial runs with higher held-out error than the paired ancilla run.

		Adam			Levenberg–Marquardt
target	$E$	at optimum	error ratio [10%, 90%]	$>5\%$ worse	at optimum	error ratio [10%, 90%]	$>5\%$ worse
Poisson kernel	8	0.77	1.00 [1.00, 1.19]	0.17	0.99	1.00 [1.00, 1.00]	0.01
Poisson kernel	12	0.42	1.01 [1.00, 1.91]	0.43	0.97	1.00 [1.00, 1.00]	0.03
Poisson kernel	16	0.17	1.29 [1.00, 2.33]	0.71	0.94	1.00 [1.00, 1.00]	0.06
Poisson kernel	24	0.03	1.57 [1.02, 3.70]	0.83	0.79	1.00 [1.00, 1.30]	0.19
Poisson kernel	32	0.01	1.61 [1.01, 3.48]	0.83	0.72	1.00 [1.00, 1.16]	0.18

Appendix

Table 7: Held-out error of the serial circuit relative to the control under Adam and under Levenberg–Marquardt (same data and initial gates): share of serial runs at the optimum; ratio of the held-out errors

(1-R^{2})

, serial over ancilla, paired by run index (for the real series the data are the same and the pairing is arbitrary), median and 10% and 90% quantiles; share of serial runs more than 5% worse than the paired control run. 100 runs per cell.

		Adam			Levenberg–Marquardt			ancilla $-$ serial
target	$E$	serial	ancilla	par. 2	serial	ancilla	par. 2	Levenberg–Marquardt [95%]
flat	8	0.95	1.00	1.00	0.98	1.00	1.00	+0.02 [-0.02, +0.07]
flat	12	0.87	1.00	1.00	1.00	1.00	1.00	+0.00 [-0.04, +0.04]
flat	16	0.84	1.00	1.00	0.96	1.00	1.00	+0.04 [-0.00, +0.10]
flat	24	0.60	1.00	1.00	0.94	1.00	1.00	+0.06 [+0.01, +0.12]
flat	32	0.36	1.00	1.00	0.82	1.00	1.00	+0.18 [+0.11, +0.27]

Appendix

Table 8: Training on data, Levenberg–Marquardt from the same data and initial gates as Adam: share of runs at the optimum (excess training loss at most

10^{-6}

) under both optimisers, 100 runs per cell, and the difference ancilla minus serial under Levenberg–Marquardt with a 95% Newcombe interval.

series (Rdatasets)	length	$R^{2}_{16}$	$R^{2}_{24}$	$R^{2}_{32}$	$R^{2}_{32}-R^{2}_{16}$	$\sup\|f_{32}\|$
AER/DJFranses	770	$0.762$	$0.822$	$0.858$	$+0.096$	0.76
AER/DutchSales	425	$0.271$	$0.168$	$0.020$	$-0.251$	0.46
AER/MarkDollar	518	$-0.057$	$-0.089$	$-0.283$	$-0.226$	0.31
AER/MarkPound	1974	$-0.137$	$-0.245$	$-0.525$	$-0.388$	0.15
CardioDataSets/cv_mortality_ts	508	$0.600$	$0.594$	$0.544$	$-0.056$	0.75
DAAG/mdbAVtJtoD	867	$-0.347$	$-0.596$	$0.670$	$+1.017$	0.51

Appendix

Table 9: Screening of real-world series (Appendix A ): median held-out

R^{2}

of the least-squares fit of degree

E

over five random splits, the gain from

E=16

32

, and the supremum of the degree-32 fit. Duplicates and series with missing values omitted.

		peak rate (tuned $\to$ bracketed)			runs at the optimum, serial $\|$ ancilla
target	$E$	serial	ancilla	par. 2	tuned	bracketed
flat	8	0.03 ∗	0.01 ∗	0.01 ∗	0.95 $\|$ 1.00	0.95 $\|$ 1.00
flat	12	0.01 ∗	0.01 ∗	0.01 ∗	0.87 $\|$ 1.00	0.87 $\|$ 1.00
flat	16	0.01 ∗	0.01 ∗	0.01 ∗	0.84 $\|$ 1.00	0.84 $\|$ 1.00
flat	24	0.03 ∗	0.01 ∗	0.01 ∗	0.60 $\|$ 1.00	0.60 $\|$ 1.00
flat	32	0.03	0.01 ∗	0.01 ∗	0.36 $\|$ 1.00	0.36 $\|$ 1.00

Appendix

Table 10: Training on data, learning-rate bracketing (Appendix A ): peak rate before and after the extension of the grid ( ∗ : best tuning score at the floor, not extended; † : near-best rates still include an end of the extended grid), and share of runs at the optimum for the serial and the ancilla circuit under the tuned and under the bracketed rates.

		runs at the optimum, serial $\|$ ancilla			ancilla $-$ serial [95%]
target	$E$	1 $\times$	4 $\times$ held	4 $\times$ stretched	4 $\times$ held
flat	12	23/30 $\|$ 30/30	24/30 $\|$ 30/30	21/30 $\|$ 30/30	+0.20 [+0.05, +0.37]
flat	24	22/30 $\|$ 30/30	15/30 $\|$ 30/30	15/30 $\|$ 30/30	+0.50 [+0.30, +0.67]
Poisson kernel	12	14/30 $\|$ 30/30	20/30 $\|$ 30/30	21/30 $\|$ 30/30	+0.33 [+0.15, +0.51]
Poisson kernel	24	1/30 $\|$ 17/30	0/30 $\|$ 28/30	3/30 $\|$ 30/30	+0.93 [+0.75, +0.98]
nottem	24	6/30 $\|$ 30/30	1/30 $\|$ 30/30	3/30 $\|$ 30/30	+0.97 [+0.79, +0.99]

Appendix

Table 11: Training on data, step budget (Appendix A ): runs at the optimum out of 30, serial

|

ancilla, for the main experiment (

1\times

, 20,000 steps) and for 80,000 steps with the additional steps at the peak rate (

4\times

held) or with the decay spread over the whole run (

4\times

stretched); last column: ancilla minus serial under the held schedule, with a 95% Newcombe interval. Same data and initialisations in all three columns.

Figure 6: Flat targets in the setup of Section 6 (Adam at matched parameter counts): share of runs that reach the least-squares optimum; open diamonds: Levenberg–Marquardt on the same runs of the serial circuit (both two-qubit circuits reach the optimum in every run under it). Bars: 95% Wilson intervals; 100 runs per point.

Figure 7: Share of converged runs (final loss

\leq 10^{-6}

) against the encoding budget

E

, for

1/k

targets and the three optimisers. Bars: 95% Wilson intervals; 100 runs per point.

		gradient descent		Levenberg–Marquardt		Adam
targets	$E$	runs	difference [95%]	runs	difference [95%]	runs	difference [95%]
flat	8	93 $\|$ 100	$+0.07$ [ $+0.02$ , $+0.14$ ]	100 $\|$ 100	$+0.00$ [ $-0.04$ , $+0.04$ ]	100 $\|$ 100	$+0.00$ [ $-0.04$ , $+0.04$ ]
	12	85 $\|$ 100	$+0.15$ [ $+0.08$ , $+0.23$ ]	100 $\|$ 100	$+0.00$ [ $-0.04$ , $+0.04$ ]	95 $\|$ 100	$+0.05$ [ $+0.00$ , $+0.11$ ]
	16	65 $\|$ 100	$+0.35$ [ $+0.26$ , $+0.45$ ]	96 $\|$ 100	$+0.04$ [ $-0.00$ , $+0.10$ ]	83 $\|$ 100	$+0.17$ [ $+0.10$ , $+0.26$ ]
	24	24 $\|$ 99	$+0.75$ [ $+0.65$ , $+0.82$ ]	90 $\|$ 100	$+0.10$ [ $+0.04$ , $+0.17$ ]	66 $\|$ 100	$+0.34$ [ $+0.25$ , $+0.44$ ]
	32	3 $\|$ 99	$+0.96$ [ $+0.89$ , $+0.98$ ]	83 $\|$ 100	$+0.17$ [ $+0.10$ , $+0.26$ ]	53 $\|$ 100	$+0.47$ [ $+0.37$ , $+0.57$ ]

Appendix

Table 12: Exact loss: runs out of 100 with final loss

\leq 10^{-6}

, serial

|

ancilla, and the difference of the shares (ancilla minus serial) with a 95% Newcombe interval, per target family, encoding budget and optimiser. ∗ : neither circuit converges in half of the runs.

optimiser	targets	threshold	$E=8$	$E=12$	$E=16$	$E=24$	$E=32$
GD	flat	$10^{-4}$	93 $\|$ 100 $\|$ 100	86 $\|$ 100 $\|$ 100	65 $\|$ 100 $\|$ 100	27 $\|$ 100 $\|$ 100	3 $\|$ 100 $\|$ 99
		$10^{-6}$	93 $\|$ 100 $\|$ 100	85 $\|$ 100 $\|$ 100	65 $\|$ 100 $\|$ 100	24 $\|$ 99 $\|$ 99	3 $\|$ 99 $\|$ 97
		$10^{-8}$	92 $\|$ 100 $\|$ 100	83 $\|$ 100 $\|$ 100	63 $\|$ 100 $\|$ 100	21 $\|$ 99 $\|$ 99	1 $\|$ 98 $\|$ 97
		$10^{-10}$	92 $\|$ 100 $\|$ 100	83 $\|$ 100 $\|$ 100	62 $\|$ 100 $\|$ 100	19 $\|$ 99 $\|$ 99	1 $\|$ 98 $\|$ 95
	$1/k$	$10^{-4}$	78 $\|$ 100 $\|$ 100	43 $\|$ 100 $\|$ 100	11 $\|$ 93 $\|$ 87	0 $\|$ 29 $\|$ 24	0 $\|$ 0 $\|$ 0
		$10^{-6}$	71 $\|$ 100 $\|$ 100	32 $\|$ 99 $\|$ 100	5 $\|$ 77 $\|$ 72	0 $\|$ 16 $\|$ 15	0 $\|$ 0 $\|$ 0

Appendix

Table 13: Exact loss: runs out of 100 with final loss at or below the threshold, serial

|

ancilla

|

parallel. The threshold of the main text is

10^{-6}

. GD: gradient descent; LM: Levenberg–Marquardt. For Adam,

10^{-10}

lies within the range at which converged runs end (Appendix A ).

		serial				ancilla
targets	$E$	7,500	14,500	27,000	50,000	7,500	14,500	27,000	50,000
flat	8	74	86	90	93	100	100	100	100
	12	51	74	80	85	100	100	100	100
	16	16	34	55	65	97	100	100	100
	24	0	3	6	24	51	92	98	99
	32	0	0	0	3	1	46	94	99

Appendix

Table 14: Gradient descent on the exact loss: runs out of 100 with loss

\leq 10^{-6}

after 7,500, 14,500, 27,000 and all 50,000 steps. These are four of the checkpoints at which the loss was recorded (Appendix A ), each about twice the one before.

		tuned		bracketed		common
targets	$E$	rate ( $10^{-3}$ )	runs	rate ( $10^{-3}$ )	runs	runs
flat	8	3 $\|$ 1 $\|$ 1	100 $\|$ 100 $\|$ 100	3 $\|$ 0.1 † $\|$ 0.1 †	100 $\|$ 100 $\|$ 100	99 $\|$ 100 $\|$ 100
	12	1 $\|$ 1 $\|$ 1	95 $\|$ 100 $\|$ 100	0.1 † $\|$ 0.1 † $\|$ 0.1 †	82 $\|$ 100 $\|$ 100	95 $\|$ 100 $\|$ 100
	16	1 $\|$ 1 $\|$ 1	83 $\|$ 100 $\|$ 100	0.3 $\|$ 0.3 $\|$ 0.1 †	81 $\|$ 100 $\|$ 100	83 $\|$ 100 $\|$ 100
	24	1 $\|$ 1 $\|$ 1	66 $\|$ 100 $\|$ 100	0.3 $\|$ 0.1 † $\|$ 0.1 †	66 $\|$ 100 $\|$ 100	66 $\|$ 100 $\|$ 100
	32	3 $\|$ 1 $\|$ 1	53 $\|$ 100 $\|$ 100	3 $\|$ 0.1 † $\|$ 0.1 †	53 $\|$ 100 $\|$ 100	51 $\|$ 100 $\|$ 100

Appendix

Table 15: Adam on the exact loss, serial

|

ancilla

|

parallel: peak learning rate and runs out of 100 with final loss

\leq 10^{-6}

under the tuned rate (main text), under the rate selected after the extension of the grid ( † : near-best rates still include an end of the extended grid), and under the common rate

10^{-3}

Quantum Circuit Trainability

Quantum circuit trainability studies how circuit structure, initialization, noise, and objectives affect optimization of parameterized quantum models.

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Architecture Shape Governs QNN Trainability: Jacobian Null Space Growth and Parameter Efficiency

Michael Poppel, David Bucher, Maximilian Zorn, Markus Baumann+4

Department of Computer Science, LMU Munich, Germany · Aqarios GmbH, Munich, Germany

Variational Quantum Circuits Quantum Circuit Trainability Quantum Neural Networks

Open paper AskPDF arXiv

Variational quantum circuits with angle encoding implement truncated Fourier series, and architectures arranging

N

qubits with

L

encoding layers each -- sharing encoding budget

E = NL

-- generate identical frequency spectra, identical frequency redundancy, and require the same minimum parameter count for coefficient control. Despite this equivalence, trainability varies substantially with architecture shape

(N,L)

at fixed

E

. We identify structural rank deficiency of the coefficient matching Jacobian

J

as the mechanism responsible. For serial single-qubit architectures, we prove

\mathrm{rank}(J) \leq 2L+1

regardless of parameter count

P

, with

\dim(\ker J) \geq P-(2L+1)

growing without bound -- a phenomenon we term structural gradient starvation: a growing fraction of parameters become structurally decoupled from the loss as

P

increases at fixed

L

. Parallel architectures avoid this via independent phase trajectories, ensuring

σ_{\min}(J^{(\mathrm{par})}) > 0

generically for

P \leq 2E+1

, so no parameter lies in

\ker J

. For practitioners, we further show that the two natural routes to increasing parameter count have fundamentally different effects: adding feature map (FM) layers monotonically strengthens the Jacobian QFIM eigenvalue spectrum and achieves

R^2 \geq 0.95

with

1.6

2.2\times

fewer parameters than adding trainable blocks across all tested architectures, while trainable blocks improve training only through the classical interpolation mechanism with no quantum-specific benefit.

optimiser	targets	threshold	$E=8$	$E=12$	$E=16$	$E=24$	$E=32$
GD	flat	$10^{-4}$	93 $\|$ 100 $\|$ 100	86 $\|$ 100 $\|$ 100	65 $\|$ 100 $\|$ 100	27 $\|$ 100 $\|$ 100	3 $\|$ 100 $\|$ 99
		$10^{-6}$	93 $\|$ 100 $\|$ 100	85 $\|$ 100 $\|$ 100	65 $\|$ 100 $\|$ 100	24 $\|$ 99 $\|$ 99	3 $\|$ 99 $\|$ 97
		$10^{-8}$	92 $\|$ 100 $\|$ 100	83 $\|$ 100 $\|$ 100	63 $\|$ 100 $\|$ 100	21 $\|$ 99 $\|$ 99	1 $\|$ 98 $\|$ 97
		$10^{-10}$	92 $\|$ 100 $\|$ 100	83 $\|$ 100 $\|$ 100	62 $\|$ 100 $\|$ 100	19 $\|$ 99 $\|$ 99	1 $\|$ 98 $\|$ 95
	$1/k$	$10^{-4}$	78 $\|$ 100 $\|$ 100	43 $\|$ 100 $\|$ 100	11 $\|$ 93 $\|$ 87	0 $\|$ 29 $\|$ 24	0 $\|$ 0 $\|$ 0
		$10^{-6}$	71 $\|$ 100 $\|$ 100	32 $\|$ 99 $\|$ 100	5 $\|$ 77 $\|$ 72	0 $\|$ 16 $\|$ 15	0 $\|$ 0 $\|$ 0

		tuned		bracketed		common
targets	$E$	rate ( $10^{-3}$ )	runs	rate ( $10^{-3}$ )	runs	runs
flat	8	3 $\|$ 1 $\|$ 1	100 $\|$ 100 $\|$ 100	3 $\|$ 0.1 † $\|$ 0.1 †	100 $\|$ 100 $\|$ 100	99 $\|$ 100 $\|$ 100
	12	1 $\|$ 1 $\|$ 1	95 $\|$ 100 $\|$ 100	0.1 † $\|$ 0.1 † $\|$ 0.1 †	82 $\|$ 100 $\|$ 100	95 $\|$ 100 $\|$ 100
	16	1 $\|$ 1 $\|$ 1	83 $\|$ 100 $\|$ 100	0.3 $\|$ 0.3 $\|$ 0.1 †	81 $\|$ 100 $\|$ 100	83 $\|$ 100 $\|$ 100
	24	1 $\|$ 1 $\|$ 1	66 $\|$ 100 $\|$ 100	0.3 $\|$ 0.1 † $\|$ 0.1 †	66 $\|$ 100 $\|$ 100	66 $\|$ 100 $\|$ 100
	32	3 $\|$ 1 $\|$ 1	53 $\|$ 100 $\|$ 100	3 $\|$ 0.1 † $\|$ 0.1 †	53 $\|$ 100 $\|$ 100	51 $\|$ 100 $\|$ 100

Abstract

Quantum Circuit Trainability

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