Apparent Compression, Real Stability: The Intrinsic Dimension of Learning a Quantum Wavefunction

Lu Wei; Yufeng Wang; Chenfeng Cao; Haibin Ling

cs.LGSep 27, 2026

Apparent Compression, Real Stability: The Intrinsic Dimension of Learning a Quantum Wavefunction

Authors: Lu Wei, Yufeng Wang, Chenfeng Cao, Haibin Ling

Organizations: Data Science Department, Stony Brook University · Department of Computer Science, Stony Brook University · Dahlem Center for Complex Quantum Systems, Freie Universität Berlin · Westlake University

Abstract

How many directions in weight space does training need? The intrinsic dimension answers this with the smallest number of random directions in which training still reaches a target accuracy, and small values have motivated parameter-efficient methods such as LoRA. We measure it for variational Monte Carlo (VMC), which trains a neural network to represent the ground state of a quantum many-body system. VMC is a demanding test, because the network generates its own training samples and every gradient is noisy, and a revealing one, because the exact answer is known and every run can be scored. We train only a small latent vector that a frozen random map turns into the network's weights, with no change to the standard natural-gradient optimizer. We find that a small dimension can be misleading, while the stability it brings is real. On a magnet with a hard sign pattern, a network that cannot represent signs reaches its best energy in 8 of 28,642 directions, but only because no such network can go lower; once signs are learnable, neither the signs nor the magnitudes are cheap. The dimension rises across a quantum phase transition, so it tracks how difficult a state is at far less compute than fitting a scaling law, yet it never falls below a floor set by the random subspace itself, even where the ground state is nearly trivial. Training in the subspace, in contrast, never diverged in our experiments, whereas full-parameter training with the same settings did, and a control with matched solvers attributes the difference to the reduced dimension.

Figures & tables

Figure 1: Method overview. (a) The measurement. Training is confined to a random

d

-dimensional slice of weight space through the initialization

\theta_{0}

: the map

g

and

\theta_{0}

are frozen, so only the latent

z\in\mathbb{R}^{d}

moves and the slice is fixed before the first step. Its orientation is drawn at random, and a different seed draws a different slice, which is why the seed selects the basin a run ends in (Sec. 4.1 ). (b) What a measured dimension has to be read against, on the

4\times 4

frustrated magnet. A network with non-negative amplitudes reaches

\varepsilon_{\mathrm{rel}}=0.5313

in 8 of its 28,642 directions and 0.5271 with every direction trainable, so

3{,}500\times

the directions buy 0.8% of the error: both sit at the positive-cone bound, the lowest energy any non-negative wavefunction can reach. Letting the model represent signs is worth

12\times

, and supplying the exact sign a further

9\times

(Secs. 4.1 , 4.2 ).

Configuration	$d_{\mathrm{eff}}$	min $\varepsilon$	med. $\varepsilon$
TC1 (full)	28,642	0.5271	0.5271
TC2 (linear)	8	0.5287	0.5313
TC2 (linear)	32	0.5279	0.6477
TC2 (linear)	128	0.5273	0.5274
TC2 (linear)	512	0.5272	0.5272
TC3 (tanh map)	8	1.4401	1.4617

Table 2: Each row gives the minimum and median best

\varepsilon_{\mathrm{rel}}

over seeds for one configuration at one

d

, within the first 880 steps.

Figure 2: Sign/amplitude dissection in one sign-capable model (

4\times 4

J_{2}/J_{1}=0.5

; best

\varepsilon_{\mathrm{rel}}

within 1,500 steps; lines are medians of 3 seeds, small dots are seeds). (a) Amplitude swept with the sign block complete (orange), beside the exact-sign control (blue; the star is the full network, which completed 1,380–1,460 steps) and the positive-cone bound, below which no sign-free configuration of Table 2 can go (dotted). (b) Sign swept with the amplitude fixed at 1,024 dimensions; dashed: the complete sign block.

Figure 3:

d^{*}(h)

\tau=10^{-2}

for TFIM chains at

L=16

and

L=24

(deep protocol, Table 6 ). The demand for trainable dimensions rises

{\sim}4\times

across the quantum phase transition at both sizes. Deep in the paramagnet (

L=16

, broken axis) it relaxes only to 64 dimensions, although at

h/J=10

a product state without parameters is within

\varepsilon_{\mathrm{rel}}=2.5\times 10^{-3}

of the ground-state energy.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Every seed of the

6\times 6

sign scan. Best relative error per seed against

J_{2}/J_{1}

for the full-parameter reference TC1c (5 seeds per coupling) and the saturated subspace TC2c at

d=3072

(2 seeds), scored against the exact energies of Appendix C . At four couplings the median TC1c seed sits in the bad basin (shaded), and its seeds spread over more than an order of magnitude; the two subspace seeds stay within

3.3\times

of each other. The subspace runs logged no divergence event, the reference 41 events and 3 aborts (Sec. 4.4 ).

$J_{2}/J_{1}$	$E_{0}$ (exact)	$E_{0}/N$	$E_{\mathrm{DMRG}}$	$(E_{\mathrm{DMRG}}-E_{0})/\|E_{0}\|$ ( $10^{-4}$ )
0.40	$-19.0708202$	$-0.5297450$	$-19.0604506$	5.4
0.45	$-18.5636661$	$-0.5156574$	$-18.5496834$	7.5
0.50	$-18.1371475$	$-0.5038097$	$-18.1179292$	10.6
0.55	$-17.8263972$	$-0.4951777$	$-17.8000652$	14.8
0.60	$-17.7565894$	$-0.4932386$	$-17.7403613$	9.2
0.65	$-18.2371629$	$-0.5065879$	$-18.2368670$	0.16

Appendix

Table 3: Exact ground-state energies of the

6\times 6

J_{1}

–

J_{2}

torus beside the DMRG references (

\chi=4096

Figure 5: The plot shows best

\varepsilon_{\mathrm{rel}}

within the first 880 steps against trainable dimension

d

for the four configurations of Table 2 , at

4\times 4

and

J_{2}/J_{1}=0.5

. Small dots are single seeds, and the lines are medians over 3 seeds. The linear subspace (TC2) reaches the positive-cone bound shared by every sign-free configuration already at

d=8

. The nonlinear map (TC3) lies above the linear curve at every matched

d

and is slowest at

d=8

$J_{2}/J_{1}$	0.40	0.45	0.50	0.55	0.60	0.65	0.70
$4\times 4$ , $d=128$	0.256	0.278	0.274	0.266	0.225	0.203	0.207
$4\times 4$ , $d=512$	0.083	0.094	0.103	0.103	0.075	0.055	0.051
$4\times 4$ , $d=2048$	0.043	0.057	0.056	0.051	0.049	0.037	0.031
$6\times 6$ , $d=3072$	0.061	0.071	0.088	0.156	0.079	0.056	0.048

Appendix

Table 4: Error of the sign-capable model vs. frustration at fixed subspace dimension.

4\times 4

rows: median best

\varepsilon

within 1,380 steps (phase-weighted allocation, 2 seeds).

6\times 6

row: median

\varepsilon

at fixed

d=3072

, the saturated allocation at that size.

Figure 6: Error of the sign-capable model against frustration

J_{2}/J_{1}

at fixed subspace dimension, on the

4\times 4

J_{1}

–

J_{2}

lattice (Sec. 4.2 , Table 4 ). The plot shows the median best

\varepsilon_{\mathrm{rel}}

within 1,380 steps over 2 seeds, with phase-weighted allocation at

d\in\{128,512,2048\}

. At

d=128

and

d=512

every coupling with

J_{2}/J_{1}\leq 0.55

has a higher error than every coupling with

J_{2}/J_{1}\geq 0.60

. At

d=2048

the two ranges overlap.

Sign	Ampl.	median	Exact sign
dims	dims	$\varepsilon$	(ratio)
1,633	2	0.415	0.511 ( $0.81\times$ )
1,633	8	0.284	0.367 ( $0.77\times$ )
1,633	32	0.207	0.202 ( $1.02\times$ )
1,633	128	0.136	0.094 ( $1.44\times$ )
1,633	415	0.053	0.057 ( $0.94\times$ )

Appendix

Table 5: Sign/amplitude dissection in one sign-capable model (

4\times 4

J_{2}/J_{1}=0.5

; median best

\varepsilon_{\mathrm{rel}}

within 1,500 steps, 3 seeds). Top: sign block complete, amplitude swept, beside the exact-sign control at the same amplitude dimension (interpolated log-log at 415 and 1,024). Bottom: amplitude fixed at 1,024 dimensions, sign swept, as a ratio to the complete block. From 415 amplitude dimensions on the learned sign matches the exact one. With the amplitude this well resolved, a quarter of the sign block already leaves the median error

1.24\times

higher.

Figure 7: The sign/amplitude grid behind Table 5 and Fig. 2 , for one sign-capable model at $4\times 4$ and $J_{2}/J_{1}=0.5$ . Each cell shows the median best $\varepsilon_{\mathrm{rel}}$ within 1,500 steps over 3 seeds, and the white contours lie at $\varepsilon_{\mathrm{rel}}=0.05$ , $0.1$ , $0.2$ and $0.4$ . In each amplitude column a red outline marks the smallest sign block whose median is within $1.1\times$ of that of the complete block. With 2–128 amplitude dimensions a 402-dimensional sign block is enough. From 415 amplitude dimensions on, the sign needs its whole block.

$h/J$	0.6	0.8	0.9	0.95	1.0	1.05	1.1	1.2	1.3	1.4	1.6	2.0	4	10
$L=16$	32	64	64	64	64 †	128	128 †	256 †	128 †	128	128	128	64 †	64
$L=24$	32	—	64	—	64	64 †	128	128	128	128	—	—	—	—

Appendix

Table 6: The staircase at two sizes, as

d^{*}(h)

per cell at

\tau=10^{-2}

under the deep protocol (plotted as Fig. 3 in the main text). The edge of the step falls inside the critical window at both sizes, and the demand relaxes to 64 deep in the paramagnet. A dash marks a column that was not run at that size. A dagger (

\dagger

) marks a value of

d^{*}

decided by a cell whose median lies within

\pm 10\%

of the threshold.

Figure 8: Control for the independence of the

d^{*}(h)

staircase from the step budget (Sec. 4.3 ). The base protocol (TFIM,

L=16

, fixed 2000-step budget) is compared with the deep protocol (6000 steps,

3\times

, with

d

extended to 256 in the critical region). At

\tau=10^{-2}

, all 7 cells that the two protocols share are unchanged under the

3\times

budget, and 0 cells shrank. The two cells that are censored under the base protocol resolve within the deep grid (

h/J=1.1\to 128

1.2\to 256

). At this threshold the staircase is therefore not limited by the budget. At tighter thresholds the control fails. At

3\times 10^{-3}

the outcome is mixed, and at

10^{-3}

the one comparable cell moved

128\to 64

. This is why we restrict claims about

d^{*}

\tau=10^{-2}

Figure 9: Median $\varepsilon_{\mathrm{rel}}$ against trainable dimension $d$ , with one curve per transverse field $h$ (TFIM, $L=16$ , deep protocol, Sec. 4.3 ). The $d^{*}(h)$ staircase of Fig. 3 is a level cut of this smooth error surface $\varepsilon_{\mathrm{med}}(h,d)$ at the threshold $\tau=10^{-2}$ . Across the scanned window $h/J\leq 2$ , larger fields beyond the transition need a systematically larger $d$ to reach the same accuracy.

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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.

Abstract

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