Rethink the Role of Deep Learning towards Large-scale Quantum Systems
Authors: Yusheng Zhao
Chi Zhang
Yuxuan Du
Venue: ICML 2025 poster (2025)
Keywords: learning
quantum system
ground state property estimation
Primary area: applications->chemistry_physics_and_earth_sciences

Abstract: Characterizing the ground state properties of quantum systems is fundamental to capturing their behavior but computationally challenging. Recent advances in AI have introduced novel approaches, with diverse machine learning (ML) and deep learning (DL) models proposed for this purpose. However, the necessity and specific role of DL models in these tasks remain unclear, as prior studies often employ varied or impractical quantum resources to construct datasets, resulting in unfair comparisons. To address this, we systematically benchmark DL models against traditional ML approaches across three families of Hamiltonian, scaling up to $127$ qubits in three crucial ground-state learning tasks while enforcing equivalent quantum resource usage. Our results reveal that ML models often achieve performance comparable to or even exceeding that of DL approaches across all tasks. Furthermore, a randomization test demonstrates that measurement input features have minimal impact on DL models' prediction performance. These findings challenge the necessity of current DL models in many quantum system learning scenarios and provide valuable insights into their effective utilization.

Rethink the Role of Deep Learning towards Large-scale Quantum Systems
Yusheng Zhao
Chi Zhang'
Yuxuan Du 2
Abstract
2024). Despite the significance, the curse of dimension- ality renders classical simulations of large-scale quantum systems infeasible for practical applications (Ceperley & Alder, 1986; White, 1992; Becca & Sorella, 2017). Quan- tum computers offer a promising alternative; however, their current limitations, including unavoidable quantum noise and restricted quantum resources, constrain their immediate utility (Preskill, 2018). These challenges underscore the pressing need for innovative strategies to quantify GSP and classify quantum phases in large-qubit systems efficiently.
Characterizing the ground state properties of quan- tum systems is fundamental capturing their be- havior but computationally challenging. Recent advances in Al have introduced novel approaches, with diverse machine learning (ML) and deep learning (DL) models proposed for this purpose. However, the necessity and specific role of DL models in these tasks remain unclear, as prior stud- ies often employ varied or impractical quantum resources to construct datasets, resulting in unfair comparisons. To address this, we systematically benchmark DL models against traditional ML ap- proaches across three families of Hamiltonian, scaling up to 127 qubits in three crucial ground- state learning tasks while enforeing equivalent quantum resource usage. Our results reveal that ML models often achieve performance compara- ble to even exceeding that of DL approaches across all tasks. Furthermore, a randomization test demonstrates that measurement input features have minimal impact on DL models' prediction performance. These findings challenge the ne- cessity of current DL models in many quantum system learning scenarios and provide valuable insights into their effective utilization.
Artificial intelligence has recently emerged as a transforma- tive tool in scientific discovery, augmenting and accelerating research across diverse domains. Within quantum physics and quantum computing, this synergy has given rise to bur- geoning field, dubbed quantum system learning (QSL) (Car- leo & Troyer, 2017; Gebhart et al., 2023), where classical learners extract essential information from quantum systems to enable efficient predictions about related systems (see Appendix A for details). A seminar work to this field was made by Huang et al. (Huang et al., 2022), who proved that classical algorithms incapable of leveraging data cannot achieve comparable performance guarantees in the context of GSP estimation (GSPE) and quantum phase classifica- tion (QPC). Building on this, several machine learning (MI.) models (Lewis et al., 2024; Cho & Kim, 2024; Wanner et al., 2024) with provable efficiency have been devised to tackle different classes of quantum systems.
1. Introduction
The remarkable success of deep learning (DL) over ML in many fields such computer vision and language process- ing has inspired researchers to explore advanced DL frame- works for tackling GSPE and QPC tasks (Wang et al., 2022b; Tran et al., 2022: Zhang & Di Ventra, 2023; Tang et al., 2024a; Wu et al., 2024; Rouzé & França, 2024). Despite re- cent advancements, most DL-based methods incur substan- tially higher computational and memory costs compared to their ML-based counterparts. Moreover, the claimed heuris- tic advantages of DL models often ignore their dependence quantum resources, resulting an unfair comparison with ML approaches. More specifically, the success of many DL protocols frequently relies on leveraging expensive, or even unlimited, quantum measurements to generate labels, whereas ML models rely on a limited number of measure- ments. This disparity in quantum resource requirements creates an inherently inequitable basis for comparison. With
The efficient estimation of ground state property (GSP) and the classification of phases of matters are among the most fundamental problems in unraveling the myth of the quantum world, with far-reaching implications in quantum physics (Carleo & Troyer, 2017; Torlai al., 2018; Geb- hart et al., 2023), chemistry (Aspuru-Guzik et al., 2005; Kandala et al., 2017: Cao et al., 2019; Bauer et al., 2020), and materials discovery (Bauer et al., 2020; Clinton et al.,
Universitty of Science & Technology of China, Hefei, China NNnyangg Technological University, Singapore. Correspondence to: Chi Zhang <chizhang @ uust.edu.cn?. Yuxuan Du <duyux- uan 23@ gmail.com>.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
the searcity of quantum resources likely persist in the foreseeable future, a pivotal question emerges:
and the ground state energy of H((x) refer to (1) and X1 respectively. Throughout the study, interchangeably use (((()) and (15) to specify the ground state.
Do we truly need deep learning in QSL with restrictive quantum resources?
Fundamental properties of ground states. A crucial way to understand quantum systems involves quantifying their ground state properties. This process relies on quantum measurements, which extract classical information from the ground state (v)). Mathematically, when applying an ob- servable O (i.e., Hermitian matrix with the size 2 2 to ( the expectation values of measurement outcomes is tr (pO), where = | {eb is the density matrix formalism of (eb) with 1-) (-1 being the outer product operation. By selecting different O, key properties of quantum systems, e.g. correlation functions, entanglement entropies, and the classification of phases, can be acquired.
To address the above question, here we systematically revisit the role of DL in QSL, especially for GSPE and QPC tasks across various families of quantum systems. By maintaining the same total number of queries (i.e., measurements), we first confirm the scaling law of current ML and DL. models. We next uncover a counterintuitive phenomenon, where existing ML models often outperform current DL models in QSL tasks. We last design randomization tests to examine the role of measurements as data features in GSPE and QPC tasks. The results reveal an opposite function of measure- ment outcomes. Specifically, measurement outcomes are largely redundant as input representations for GSPE tasks but significantly improve performance for QPC tasks. These results offer concrete guidance for future model design to advance QSL, Part of source code of dataset generation are open-sourced in this Github Repository.
Correlation functions. Correlation functions reveal the un- derlying interactions and symmetries of the system. The two-point correlation functions of ((b) is matrix C' with the size N x N, whose entry Ci₃ refers to the expectation value of the observable O = (X,X,, + Y,Y, + 2,2,)/3 with
Ciy tr(X;X,p) + tr (Y,),))) tr (z(22)
(2)
2. Preliminaries
Here we briefly recap the basic concepts of quantum sys- tems. Refer to Appendix A for the omitted details.
where P.P, represents N --qubit Pauli string such that Pauli operator P E {X, Y, ) act on the i-th and j-th qubits, and the identity operator I acts on all other qubits.
The computation of fundamental properties of ground states in quantum systems has far-reaching impacts on physics, materials science, and chemistry (Bauer et al., 2020; Geb- hart et al., 2023; Clinton et al., 2024). In such systems, the interactions among particles are mathematically described by a Hamiltonian, which encapsulates the system's 's energy and dynamics. For an N N-qubit system, the Hamiltonian can generally be expressed as H(a)= = Ei= a,G, E 2 x where E TRA refers to the Pauli expansion coefficients, G E {1,X, Y, Y,,, is the i-th Pauli-string Vi E [ = [ 0 is the identity matrix, and X = [9]1, Y [ 1..... Z = I are Pauli operators along X, Y, Z axes.
Entanglement entropies. Entanglement entropies quantify the strength of quantum correlations. For the 2-order Rényi entanglement entropy of the specified subsystem A (i.e., a subset of qubits among N-qubits), it takes the form as
S2(PA) = log (r())
where PA (rA(()) (1)) (0)) is the reduced density matrix.
Quantum phase. The quantum phase of ground states re- veals the underlying symmetries and critical phenomena of the explored quantum system. Conceptually, two ground states (x)) and 1y/) belong to the same phase if they share identical relevant order parameters or topological invariants. The choice of observables O used to quantify these phases is inherently problem-dependens and varies based on the specific properties under study.
The mathematical formulation of Hamiltonian indicates that it is Hermitian operator. Consequently, it can always be expressed through its digen-decompossiion
N H(a) Σ Ai x2*x
(1)
Finite measurements and classical shadow. Due to the probabilistic nature of quantum mechanies, the exact value of a desired property, i.e., tr(pO), requires infinite measure- ments on identically prepared quantum states p. In practice, only a finite number of measurements M can be performed, leading to statistical uncertainty. Accordingly, the estimated mean value of an observable O is given by 1 where o, represents the i-th measurement outcome, and the variance of the estimate scales as var(O) ~ 1/M.
Suppose that the eigenvalues are arranged as an ascending order, i.e,, A S A2 I ...2ZN. Then the eigenvalue A, rep- resents the i-th energy level and the eigenvector (ab) is the corresponding energy state. Here we follow the conven- tions to use Dirac notations to represent vectors (Nielsen & Chuang, 2010), where (b))) E IR² is the column vector and ( is the conjugate transpose of (&%).. The ground state
M ΣM' %
2

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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
A computationally and memory-efficient approach for es- timating the expectation values of many local observables is classical shadow (Huang et al., 2020). The fundamental principle of classical shadow lies in the 'measure first and ask questions later' strategy. For an unknown N-qubit state p, the Pauli-based classical shadow repeats the following procedure T times. At the 1-th time, the state is first operated with a unitary Ug- = UL. @ .... UU, @ UN. randomly sampled from the predefined unitary ensemble single-qubit Clifford gates and then each qubit is measured under the 7. basis to obtain an N-bit string denoted by b, E {0, 13 The shadow formalism of the j-th qubit is P3. = 3 j.t (bbba))) (bz, UU, I, Vj E [N]. Suppose the observable is O- = Pi @ ...P... @ Pv. The shadow estimation of tr(pO) is I E 1 tr(pseP)).
where N; (I 222/2 at qubit i and x: = (52, Ra, a, A). For ease of notations, we denote the ground states of Hна (2), HTFMM and (Ry)(z) as (x))) [(TFIM)())) and [ (z)), and their density matrices as pim (3)), PIFIM((:), and (Rya(ar), respectively.
2 is the occupation number operator
3.2. QSL Tasks and benchmark learning models
The primary objective of the classical ML and DL models under investigation is to extract meaningful knowledge from the training dataset, enabling accurate predictions of target properties for a given class of Hamiltonians.
Dataset construction with unified resources. Unlike prior studies, we impose restrictions on the quantum resources available for constructing the training dataset. This ap- proach ensures a fair comparison between classical ML and DL models while adhering to practical constraints. More specifically, let D = {(x(), ( represent the training dataset, where (x)) ~ IDx is the i-th classical input param- eter of the explored class of Hamiltonians sampled from prior data distribution Dx, and v(()) is the corresponding measurement information (e.g.. classical shadows) collected from the ground state [ę)())) with M snapshots.
3. Problem setup
To fairly evaluate the necessity of DL in QSL, we reformu- late the QSPE and QSP tasks to ensure equivalent quantum resource usage. This section begins by introducing the explored families of Hamiltonian, followed by a detailed description of the MI and DL. approaches employed for these tasks, and an explanation of the unified resource cost in dataset construction.
We adopt the total number of measurements as the measure of the quantum resource cost. Specifically, when construct- ing the dataset D, we restrict the total amount of the query complexity of the quantum systems, i.e., M, ensuring it does not exceed specified threshold. This restriction stems from the fact that given the scarcity of available quantum re- sources in the foresecable future, it is essential to minimize n. x M required to train learning models (see Appendix A.4).
3.1. Explored familiis of Hamiltonians
We consider three families of Hamiltonians, i... Heisenberg models (HB), transverse-felld Ising models (TFIM) (Dutta et al., 2015), and Rydberg atom models (Bernien et al., 2017; Browaeys & Lahaye, 2020). These Hamiltonians were chosen because they are widely studied in the field of QSL (Luo et al., 2022; An et al., 2023; Gebhart et al., 2023; Du et al., 2023; An et al., 2024), owing to their versatility and practical significance.
With access to the constructed training dataset D, ML and DL models utilize it to infer hypothesis function h(x z), which is then employed to predict a target property of unseen inputs. Here, z represents optional auxiliary information, the details of which will be clarified later. A standard met- ric to measure the prediction performance of ML and DL models is the expected risk (a.k.a,, prediction error), i.e.,
Following concepts introduced in Sec. 2, the mathematical expression of an N-qubit Heisenberg models is
Him (ar) = Σ Jj + Y.Y + 222),
(4)
R(h) = E. [lh(2, 2) - -xx))
where x = }Ji.} contains all coupling strength parameters between adjacent qubits i and j. for Vi.j E [N]. For an N-qubit ) TFIM, its formula is
where f(x) is the target property with the input xc.
Tasks. We apply ML and DL models to learn QSL tasks from D. The first two tasks belong to QSPE, where f f(x) corresponds to Cipy in Eq. (2) and S2 in Eq. (3) for |WHB ())) and [VTFII())) The last task belongs to QPC, which classifies the ordered phases of War)((r)) Specifically, there are three phases of [URyd Define two observ- >
N HTFM = Σ J,Z,Z,+1
Σ N2
(5)
where x = (J.} U { contains the coupling strengths {J.) of all qubit pairs and the transverse field {h₂]. For an N-qubit Rydberg atom model, it takes the form as
ables as O, ZV 2k=o - (T Z-e+1 with * (2,3).
The state [x)) is classified into Z2-order phase if (P)Rya(x)O,)) > (rox(ttt( phase if tr (z)))()))0)) > maxftr((Ryall (z) 02).) and the disordered phase if neither condition is satisfied.
0.7]: Zg-order
SURS a0Ii - 1
N N.N,
Ω X, A,N, (6)
(x=) =
Σ (
2
3

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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
The primary distinction between current ML and DL mod- els in achieving the above task lies in their implementation approaches. For clarity, we next introduce their implemen- tations separately.
mizing the MSE loss & = - (bkkmmal (0; -) g(o)z with Kernel E {DK, REFK, NTK For self-consistenny, we also employ these three kernels to complete the QPC task. The only modification is the loss function, which is the
cross entropy (CE) loss C = - +
n Ei= E (() log(pe") ())) g()) for the class E [3].
ML models. A common characteristic of ML models is the use of tailored explicit feature maps to extract information from the training dataset D without involving the auxiliary information in Eq. (7). In most cases, their mathematical form can be unified as
where (()) is the one-hot vector of and pa refers to the predicted score with the class c.
Tree models. Tree ensemble models are representative ML classifiers, which empirically outperform DL in certain dataset types, such as tabular data (Grinsztajn et al., 2022). For this reason, we choose four of them for the QPC task: RandomForest (RF), BraddientooostingTTee (GBT), Light- GBM (LGBM), and XGBoost (XGB). These methods share a common feature map, i.c., (xx) = [x](x)..... Φ (T(x)), where (x)) = [I( E Li),.. I(x E L with I(.) be- ing the indicator function and L' denoting the j-th leaf of tree t. The lcarning model ( w²) is obtained via minimizing CE loss L as with the kernel method with Tree E {RF, GBT, LGBM, XGBoost].
h(æ; u)) (wx, ((x)),
(8)
where E Id refers model parameters and <i.) E Rd denotes the problem-informet feature map.
Here, we focus on three classes of advanced ML mod- els commonly employed in QSL tasks. All these models utilize a supervised learning framework to derive the hy- pothesis h(æ;; w) from a pre-processed dataset D. That is, in alignment with the supervised learning framework, the collected measurement information ((6) is used to acquire the estimated label of the target property, i.e., D + D {x("), g(), ). where g( refers to the estimation of the target property (() (e.g., Cij, Sg, or the ordered phase class).
DL models. Unlike ML, a key feature of DL is their use of neural networks to implicitly extract knowledge from D. Current DL models are primarily categorized into two paradigms: supervised learning and self-supervised learn- ing (SSL). We next separately outline their mechanisms.
Remark. Although the labels in D vary depending the specified QSI. tasks, the construction of D is performed entirely classically using the original dataset 2> without any quantum resources. In addition, when M - oo, the labels become exact g(0) = (z). Vi E [n].
Supervised learning. Current DL models in this category can be classified into two classes based on the use of auxil- iary information z in Eq. (7). In the first class, DL models exclude = and are expressed as h(x;; u)) = gNN (a:; w, arc), where are specifies the neural architecture. In the second class, DL models incorporate the measured outcomes v of p (z) as z, with the form la w) = (x, z; w. are).
Existing ML models differ in the design of (o-) and the use of D to complete the training, as recapped below.
Linear Regressor (LR). When the feature map ((x) is de- signed according to the geometry property of the given Hamiltonian, ((x;; w) can be implemented using Lasso or Ridge regression to achieve provable guarantees for many QSPE tasks (Lewis et al., 2024; Wanner et al., 2024). In addition, prior studies have proposed the use of random Fourier features an alternative to these feature maps of x)) to achieve similar performance. We follow this convention to implement the linear regressor hLL (f) ww), which is ob- tained by minimizing the mean square error (MSE) loss func- tion with L, regularization: L- = 1 Ei= [ w)
For both classes, their implementation and optimization are similar to ML models. The dataset D is preprocessed to D, which is used to optimizing these DL models via the MSE or CE loss. Here we consider two deep neural networks broadly used in QSL tasks: multilayer Perceptron (MLP) and convolutional neural networks (CNN). For clarity, DL models with auxiliary information are denoted as MLP-A and CNN-A while MLP and CNN refer to the first subclass.
Self-supervise learning. The key feature of SSL models is their two-stage optimization process, consisting of a pre- training stage and a fine-tuning stage (Devlin, 2018). The aim of pre-training is capture general representations of quantum systems via unlabeled data. In contrast, the aim of fine-tuning is to adapt the pre-trained representations to specific QSL tasks, using a few labeled data.
g 1|3 + ? with A 2 R, being hyperparameter, LR E [Lasso, Ri ] and * E (1,2}.
Kernel Methods, For Hamiltonians that satisfy the smooth- ness property, Huang et al. (2022) proved that the truncated Dirichlet kernel (DK) can achieve provable efficiency in many QSPE tasks. For comparison, they employed radial ba- sis function kernel (RBFK) K) and neural tangent kernel (NTK) as references. We follow the same convention of using these three kernels to tackle the explored two GSPE tasks. Accord- ing to the dual form of linear regression, these kernel meth- ods, denoted by hiKemet(;;; w), can be optimized by mini-
The SSL framework utilizes two datasets: a pretrain dataset (8) 1 Aue D pre = (x)()) uprre Ji=1 and a supervised finetune (SFT) dataset Da = ((x(1, U N= to complete the optimiza-
(t) tion, where the definitions of npre. nstt Uprc. and u align
4

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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
with the definitions of and M in D. To ensure consistent quantum resource usage with other learning models, the datasets must satisfy "pr X Mpre + Aist x Msn = x M.
HB-Lasso
HB-Ridge TFIM-Ridge
HB-LLMAQPE-F
TFIM-Lasso
TFIM-LLM4QPES
Correlation (n 100)
Enkropy (n 100)
0.04
0.25
During pretraining, a neural network 9pre (xr (2; 2; w, are),
R04SE
0.03
RMSB
0.17
which utilizes measurement outcomes { as ancillary information {=()), ). is optimized by sequentially predicting measurement outcomes for individual qubits. The objective to minimize the KL-divergence loss. During finetuning, the optimized 9pre serves as backbone and is paired with trainable head. This head is updated using a labeled dataset Dan which is preprocessed from Dst on the classical side, by minimizing either MSE loss or CE loss.
0.02
0.09
0.01
64
128
256
512
0.01
64
120
256
512
M
0.04
Correlation (M -M-38 512)
0.25
Entropy (M -M 512)
RMSE
0.03
RMSF
0.17
0.02
0.09
We employ two advanced SSL models as benchmarks: Shadow Generator (SG) (Wang et al., 2022b) and LLM4QPE (Tang et al., 2024a). For clarity, SG is an only autoregressive model without supervised finetuning: LIM4QPE-F refers to SSL models trained without pretrain- ing, while LLM4QPE--1 -T denotes those trained with both pretraining and fine-tuning.
0.01
20
10
80
100
0.01
20
10
60
80
100
Figure 1: Scaling behavior of learning models when applied to GSPE tasks with 127-qubit (ынн) and (TTFIM). The up- per (lower) two subplots explore the scaling behavior of learning models for M (for n) when applied to predicting e(C) and e(52). while keeping n 100 (M = 512). The notation "a-b" means that the explored task is a and the employed model is b.
4. Experiment and results
We now conduct systematic experiments two GSPE tasks and one QPC task introduced in Sec. 3 to investigate the role of ML and DL models under consistent quantum resource constraints. Refer to Appendix C for the omitted details.
and entropy of 127-qubit (ынв) and (TFFMM). For each task, the training size and the snapshots are varied as n E {20, 40, 60, 80, 100} and M E {64, 128, 256, 512). Each learning model uses the same amount of quantum resources to complete the training. The hyperparameter À of two ML models is fixed to be 10 The feature dimension of these two models is the same, which is d = 2 x 1272 for Iymn) and d = x (127 - 1) for [TTFIM). LLM40PE-E with about 18. 1M parameters is trained under its default setting.
4.1. Results of ground state property estimations
For GSPE tasks, the dataset construction of Hus and (TFM((#) is as follows. For Ham (x+) in Eq. (4), we set Jig = 369
1- with varying a E (1, uniformly. For HIFM (x) in Eq. (5), Jj is sampled from [0, 2] uniformly. The number of training examples n and snapshots M varies depending on the tasks and will be detailed later. Each task is repeated five times under each setting to collect statistical results.
The achieved results in Fig. I confirm the scaling law of the employed ML and DL models, more training examples n and more snapshots M enable a better prediction per- formance. In addition, the prediction performance linearly (logarithmically) depends on the training size n (number of shots M), highlighting the dominant role of training size. For the same task, the two ML models, Lassc and Ri dge, often achieve similar performance and consistently outperform the DL model. For instance, when = 100 and M 512, the quantity E(C) for Lasso, Ridge and LLM4CPE-E when applied to (НнB) is 0.011, 0.012, and 0.017, respectively. See Appendix C.2 for details.
We employ test error as a surrogate to evaluate the expected risk R(h) in Eq. (7) by fixing The 200 test examples. For clarity, we only present the averaged results for the explored QSPE tasks in the main text. Specifically, the given dataset D is used to train different learning models to predict Cij for all qubit pairs (i,j), with the corresponding test errors (1, recorded. The root mean squared error (RMSE)
on all cases yields (C)) = ( 1 Ex, (CC,)) Similarly, Nz
in entropy estimation in Eq. (3), the subsystem A refers to all neighboring qubit pairs, i.c., i-th and (i 1)-th qubits. record the test errors for all pairs (c(S2.)): h and compute RMSE across these N - settings, i.c., e(S²) = ( ENT 1/22
Does DL really outperforms ML in QSPE? We inves- tigate the performance of all learning models presented in Sec. 3 when applied predict correlations of [ыHB). where the qubit count has four varied settings, i.e., N E (48, 64, 100, 127]. For LIM4QPE-T, we set Upre = 100 and Mpre = 1024. To ensure fairness, the cost of quantum resources in these models satisfy n x M = n&n x MaR + npre x Mpre, where nott E (20, 60, 100) and M = Man- We additionally adopt classical shadow (C5) as the baseline.
Following the same routine as e(C),
Scaling behavior of QSL models. To explore the scaling behavior of different QSL models on GSPE tasks, we apply Lasso, Ridge and LLM4QPE-E to learn the correlation
5

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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
Table I: The RMSE result on correlation prediction of (WB)) with varied N and Natt M is fixed to 64. The best results are highlighted in boldface while the second-hest results are distinguished in underlined. The results are averaged over 5 independent runs with different random seeds. The standard deviation is very small (< 10 10" such that we omit it.
N
48
N = 63
N = 100
127 N
Methods
Run
20
Pse
60
Run
100
Run
20
0.n 100
P.n.
20
Tha 60
100
20
-00
100
0.21113
0.21257
0.21399
0.21447
MLF-4 layers
0,03422
0:03625
0.03524
0.066663
0.04435
0,03833
0.07533
0,05952
0.06010
0.07971
0.09173
0.016608
- layere
0.06484
0.04899
0.03456
0.00621
0.03008
0.03100
0.06436
0.03425
0.02808
0.07441
0.03196
0.05221
SG
0.22127
0.20645
0.20033
0,21988
0.21354
0.203391
0.21981
0.22062
0.21066
0,23070
0.20901
0.20575
LIMAOPI-T T
0,05189
0.03368
0,03197
0.06111
0.03364
0.02863
0.05050
0:032227
0.02726
0.05079
0.03184
0.02631
DK
0.03336
0.06008
0.071-17
0.06277
0.05977
0.07179
0.05146
0.05489
0.06180
0.05057
0.06358
0.06331
RBFK
0,05452
0.04176
0.04101
0.04726
0.03829
0.03922
0.04096
0.03299
0.01252
0.03850
0.03115
0.03086
NIK
0.06828
0:01074
0.03903
0.066822
0.01107
0.01391
0.05960
0:03311
0.02871
0.05676
0
0.02706
Tasso 0.04221
0.026:36
0.02488
0.04856
0.02791
0.02326
0.04219
0.02602
0.02646
0.04137
0.03222
0.02083
Ridge 0,04247
0.02881
0.02475
0.04216
0,02816
0.02
0.04191
0.02711
0.02251
0.041
0.02620
0.02161
Tab. 1 demonstrates the achieved results. The overall results indicate that while all learning models exhibit the expected scaling behavior, ML models consistently outperform DL models under fair quantum resource constraints. Among DL models, LIM4QPE-T achieves the best performance, highlighting the effectiveness of SSL. Furthermore, higher Nsst with fewer pre generally leads to lower prediction er- rors compared to the reverse setting. Additionally, unlike other learning models, SG only attains performance compa- rable to classical shadows, underscoring the importance of high-quality labeled data in training. We conduct additional benchmarks on TFIM for predicting entanglement entropy and observe similar results (see Appendix C.3).
For the supervised learning framework, use Lasso, Ri dge, RBFK, and CNN as benchmark models. To assess the impact of measurement outcomes e, we compare model performance with and without using v as auxiliary informa- tion. When æ is concatenated with v, the feature dimension of both ML models is fixed to be d = 127 x (M + 254). Otherwise, it is 2 1272. As for CNN, input dimension is 127 x (M + 127) and 127?. respectively. For the SSL frame- work, we employ LLM4 and design randomization rest. In this test, the ground truth measurement outcomes v in both Dre and Dan are replaced with randomized coun- terparts v', where each element e'y is uniformly sampled from [0, 5] as an integer value.
The achieved results are exhibited in Fig. 2. The left panel illustrates the results of supervised learning models. A key observation is that ML models are more sensitive to whether measurement information is included. For example, when n = 60, cCC) for Ridge-A A (or ANN is 0.066 (or 0.032) while Ri Ridge (or CNN) is 0.032 (or 0.031). The right panel shows the performance when using real and randomized measurement outcomes as input features, indicating that characterizing real outcomes embeddings shows evi- dent performance improvement of LLM4QPE-T in GSPE tasks. Refer to Appendix C.4 for additional experiments.
LIMIOPET
Lasso
RHFK
SC
64
Grand truth Random
0.10
Leseo-A
Ridge
RBFKA
0.04
122
BidgeA
CNN-A
RMMSTE
256
0.05
0,00
0.50
50
100
40
60
80
Tratning stze
Tratning size
Figure 2: Role of measurements as input representations on GSPE tasks. RMSE e(C) when applied to the employed learning models to predict correlations of 127-qubij (е/н).. Left panel: Performance supervised learning models with varied training size n and fixed snapshots M = 64. Right panel: Performance of based model I.IMM QPE-T with varied training size and snapshots M. The notation "a-A" (or "a") refers that the learning model a uses (or does not use) as auxiliary information.
Is bigger often better? To evaluate how DL model size impacts performance in GSPE tasks, we benchmark multiple MLPs with different number of parameters when predicting correlation of 127-qubit (ын)) with M = 512 and n 100. For completeness, we also exploit LLM4QPE-F (with
~ 18. 1M parameters), as well as Lasso (with ~ 0. parameters), as two baselines. The quantum resource cost to train all models is kept to be the same.
Are measurement outcomes redundant? We conduct two independent experiments to investigate the role of mea- surement outcomes as input features. The first focuses supervised learning models, and the second evaluates the SSL-based model using a randomization test. For both ex- periments, the task is to predict correlations of the 127-q (HB)), with quantum resource costs of each learning model kept consistent with the scaling behavior experiments.
To explore the role of the model size, the employed MLPs are composed of only one hidden layer whose dimension varies (16, 32, 64, 128). In this way, the model size spans from 0.5M to 4.1M. All models are trained in 1000 epochs with
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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
== 0.001
1
3854
RP
RF-A
Ground truth
Random
Lasso (0.1M)
LIMIQEE (18.1M)
1.00
1.00
6 10
Test ascuracy
Test ancuracy
x 10
0.95
0.85
x
64
xx
0.90
0.70
0.5M
1M
2.1M
64
128
0.85
20
40
60
80
100
0.55
20
40
60
80
100
Training size
Training size
Figure 3: Performance of MLPS with varied model sizes on predicting correlation of 127-qubit (ruB). The x-axis represents the number of parameters in MLPs with 'M' being the abbreviation of million. The notation X refers to regularization weights used in MLPs, and "A (6 M)" refers to the performance of model A (i.e., Lasso, and LLM4QPE) with b million parameters.
Figure 5: Role of measurements on QPC tasks. The left and right panels separately exhibit the performance of RF and LLM4 QPE-T T when applied to learn 31-qubit (&nya) with varied training size and snapshots M.
ing behavior, we explore the performance of RF, XGB, and LLM4QPE-E when they are applied to predict the class of (aya((:)) The qubit count, training size, and the snapshots vary with N E {16, 25, 31 n E {20, 40, 60, S0, 100), and M E {64, 128, 256), respectively. For fair comparison, the quantum resource cost is fixed across all models. In addi- tion, all two ML models undergo hyperparameter optimiza- tion using Optuna (Akiba et al., 2019), while LLM4QPE-F are well trained to ensure optimal performance.
an early stopping strategy. We consider (z-regularization weights as hyperparameters.
The achieved results are exhibited in Fig. 3. When À is small, increasing the model size results in degraded per- formance, due to the overfitting. In contrast, when À be- comes large, MLPS could match the performance of the optimal LLM4QPE-F, despite having only 1/36 the param- eters. These findings suggest that optimizing performance depends more on balancing model size and hyperparameters than simply increasing model size.
Fig. 4 verifies the scaling law of the employed three learning models, as the growth and M enables a better test accu- racy, Moreover, a common feature of both subplots is that the test accuracy linearly (logarithmically) increases with the training size (the number of snapshots M), implying that fraining size is more crucial for accuracy improvement. For the same task, the two ML models, RF and XGB, per- form similarly and are consistently superior to LLM4OPE-F. For instance, when n = 100 and M = 256, the accuracy for RF, XGB and LLM40PE-F when applied to (&nya) is 4, 95.05%, and 91.90%, respectively.
4.2. Results of quantum phases classification
For QPC task presented in Sec. 3, the training dataset construction of (æya(x) is followed by the configuration of (Wang et al., 2022b; Tang et al., 2024a), i.c., Rs/a E [1, 2.95.. A [02, 3.r). and Ω 10T. The distribution of the labels exhibits long-tail distribution, hence, we pre- process the original dataset to achieve label balance. The training size n and snapshots M varies depending on the tasks and will detailed later. As with Sec. 4, we use test accuracy to evaluate the classification performance, where nee 1600 test examples are used for all cases.
Does DL really outperform classical ML? We evaluate the performance of all classifiers presented in Sec. 3.2 when applied to classifying phases of 31 (x))) with M E {64, 128, 256]. The resource constraint is the same as those Sec. 4.1. For LLm4 -I, we set ppre 100 and Mpre 512, and the cost of quantum resources in all classifiers satisfy x M nut x Mat Mpr x Mpres where Aatt {20, 60, 100} and M Mat.
XGBoost
Random Forest
LLM4QPE-F
1.0
1.0
Test accuracy
0.9
Test accuracy
0.9
0.8
0.8
The achieved results are summarized in Tab. 2. Although the overall results exhibit the scaling behavior, classical ML models perform on par with or better than DL models in most cases. Moreover, in contrast with QSPE tasks, LLM4CPE-T is inferior to simpler DL models such as MLP and CNN. This result reflects the weakness of current SSL models in manipulating QPC tasks. Another observation is that L.M4QPPETT is more sensitive to changes of the training size naft. different from its performance on QSPE tasks.
0.7
0.7
0.6
64
128 M
256
0.6
20
40
60
80
100
0
Figure 4: Scaling behavior of ML and DL models when applied to QPC tasks with (Rya). The left (right) subplots explore the scaling behavior of learning models for M (m) when applied to the QPC task, while keeping 10 (M = 256). The shadow region refers to the standard deviation.
Scaling behavior of QSL. models. To investigate the scal-
Are measurement outcomes redundant? We conduct two
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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
Table 2: The test accuracy (%) of quantum phase classification on 3 31-qubit (d/yya) with varied M and Ruft. The best results are highlighted in boldface while the second-best results are distinguished in underlined.
M
M Puu
M 256
Methods
Run
20
100
Puac = 20
Fun 100
Pla = 20
Pun, 60
Pun = 100
MTP
92.43
92.93
92.79
94.36
92.92
94.50
94.36
92.93
94.50
CNN
92.42
92.64
92.50
94.93
93.36
92.79
94.93
93.36
92.79
-P
79.64
86,50
93.64
73.00
81.07
89.64
70.07
78.57
90.57
83.57
90.57
90.21
87.71
91.86
90.43
87.71
91.86
90.43
RBFK
86.79
92.93
93.29
89.64
93.57
93.79
89. 89.61
93.57
93.79
NTT
91.64
89.37
93.64
92.43
92.64
94.36
92.43
92.64
94.36
RF
57.71
91.93
92.14
93.43
91.86
93.57
88.36
91.86
91.21
T.GRM
92.93
91.14
95.66
88.43
92.43
94.57
88.43
92.50
93.50
GBT
93.21
92.43
95.14
93.93
92.21
95.57
93.93
92.36
94.93
XGB
94.21
92.36
94.43
89.79
92.07
94.43
89.79
91.36
94.43
classifiers to explore the role of measurement outcomes. Particularly, RF and LLM4 are employed to predict the phases of 31-qubit y with M E (64, 128, 256} and n E {20, 40, 60, 80, 100}. The experiment setups are almost similar to the one used in QSPE. In the randomization test, the randomized input v' are uniformly sampled from {0, 13. due Pauli-Z measurements applied the state.
been developed to learn Hamiltonians under various con- ditions, with improved sample and runtime complexities (Lewis et al.. 2024; Wanner et al., 2024). More recently, ex- perimental studies have validated the practical effecttvvnness of this approach (Cho & Kim, 2024).
The second category of learning- pearning-based approaches employs deep learning (DL) models to study quantum systems. Un- like ML models, which focus on optimizing computational complexities across various quantum systems, this research direction emphasizes the development of novel learning paradigms and neural architectures to achieve heuristic im- provements, particularly in prediction accuracy. In recent years, a variety of DL models have been introduced to predict specific or multiple properties of quantum systems within supervised and self-supervised learning frameworks (Zhu et al., 2022; Wang et al., 2022b; Zhang & Di Ventra, 2023; Tang et al., 2024a; Wu et al., 2024).
The results are exhibited in Fig. 5. Different from the results in QSPE tasks, using measurement outcomes as auxiliary information can indeed boost the test accuracy of both clas- sical ML and DL models on QPC tasks. For example, When n = 100 and M = 256, the test accuracy of RE increases from 13.57% to 98%. For LLM4QPE-T, there an ev- ident performance gap when the learning model is trained with ground-truth and randomized measurement outcomes, where the former is 85.36% and the latter is 66.93% when 100 and M = 64.
While our work also explores the capabilities of learning models in understanding quantum systems, it distinguishes itself from prior literature by focusing specifically on the role of DL. in this context. Through systematic and extensive experiments, we fill this knowledge gap, uncovering the potential and the limitations of current DL models.
5. Related works
Prior approaches to understanding quantum systems can be categorized into simulation-based (White, 1992), quan- tum tomography
and learning- based methods (Huang et al., 2022). This work primarily focuses the learning- based paradigm. Below, we briefly summarize prior studies on learning the ground state properties of Hamiltonians and elucidate their connection to our approach. Additional details are provided in Appendix A.3.
6. Conclusion
In this study, we revisit current QSL models by unifying quantum resource usage and systematically verifying the scaling laws of learning models applied to GSPE and QPC tasks standard families of Hamiltonians. Our results exhibit that increased training data and quantum resources generally enhance prediction performance. Counterintu- itively, we find that ML models often outperform DL mod- els in these tasks, raising questions about the necessity of existing DL models in QSL. Furthermore, randomization test highlights the redundancy of measurement features in the benchmark tasks. All of these findings offer concrete guidance for model design.
The first category of learning-basee approaches leverages conventional ML models to predict interested properties in quantum systems. A key feature of these methods is their typically provable guarantees, where the computational com- plexities scale at most polynomially with system size. A seminal contribution in this area the work by (Huang et al., 2022), which demonstrated the efficiency of kernel- based methods in learning ground state properties and predicting different phases of smooth Hamiltonians. Building on this foundation, a series of provably efficient MI. models have
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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
Two future research directions are identifying novel QSL DL. pro-
Z., Fedele, F,, al. Artificial intelligence for quantum computing. arXiv preprint arXiv:2411.0913/ 2024.
tasks where DL models excel and developing tocols tailored for such scenarios. For the first direction, we will explore the potential of DL models on more challeng- ing quantum many-body systems and large-scale molecular systems (McArdle et al., 2020; Bauer et al., 2020). Addi- tionally, we aim to evaluate the capabilities of ML and DL models designed for digital quantum computers and their associated applications (Tian et al., 2023; Alexeev et al., 2024; Nation et al., 2025; Du et al., 2025b). For the second direction, we will exploit the advanced neural architectures and training strategies to develop novel DL models for QSL.
Amazon. Amazon braket pricing, 2024. URL https:
/ //aws. amazon. wtt.ccoo///a.eee wemm/baaakt/rrrciiggg
An, Z., Wu, J., Yang, M., Zhou, D., and Zeng. B. Uni- fied quantum state tomography and hamiltonian learn- ing using transformer models: language language-translatio translatoon like approach for quantum systems. arXiv preprint 2r04v:2304.122 /2010, 2023.
An, Z., Wu, J., Yang, M., Zhou, D., and Zeng, B. Unified quantum state tomography and hamiltonian learning: A banguage-translation-ii approach for quantum systems. Physical Review Applied, 21(1):014037, 2024.
Impact Statement
Quantum system learning (QSL) holds transformative poten- tial for advancing quantum technologies. Our work revisits prior QSI. frameworks, offering new insights into their scal- ability, efficiency, and practical applicability under unified quantum resource constraints. However, as QSL becomes more practical, societal implications must be carefully con- sidered. The increased accessibility of QSL could accelerate technological progress but also exacerbate existing inequali- ties, such disparities in access quantum resources and expertise. Moreover, the integration of advanced AI models into quantum technologies raises ethical concerns, including potential misuse and unintended societal consequences. We emphasize the need for interdisciplinary dialogue and the development of responsible policy frameworks to ensure that the advancements in QSL lead equitable and positive outcomes for society.
A.. Dutoi, A. D., Love, P. J., and Head- Gordon, M. Simulated quantum computation of molecu- lar energies. Science, 309(5741):1044-770) 2005.
Bauer, B., Bravyi, S., Motta, M., and Chan, G. K.-L. Quan- tum algorithms for quantum chemistry and quantum ma- terials science. Chemical reviews, 120(22): 12855-17777 2020.
Becca, F. and Sorella, S. Quantum Monte Carlo approaches for correlated systems. Cambridge University Press, 2017.
Bernien, H., Schwartz, S., Keesling, A., Levine, H., Om- ran, A., Pichler, H., Choi, S., Zibrov, A. S., Endres, M.. Greiner, M., al. Probing many-body dynamics on 51-atom quantum simulator. Nature, 551(768)))5797-5588 2017.
Acknowledgments
Y.Z. and C.Z. are supported by the NSFC under Grant No.62072426 and No.61871362. Y.D. is funded by the SUG project.
Browaeys, A. and Lahaye, T. Many-body physics with individually controlled rydberg atoms. Nature Physics, 16(2):132-142, 2020.
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Wu, Y.DD,, Zhu, Y., Wang, Y., and Chiribella, G. Learning quantum properties from short-range correlations using multi-task networks. Nature Communications, 15(1): 8796, 2024.
Tian, J., Sun, X., Du, Y., Zhao, S., Liu, Q., Zhang, K., Yi. W., Huang, W., Wang, C., Wu, X., et al. Recent advances for quantum neural networks in generative learning. IEEE Transactions on Pattern Analysis and Machine Intelli- gence, 45(10):12321-12344 2023.
Zhang. X., Luo, M., Wen, Z., Feng, Q., Pang, S., Luo, W., and Zhou, X. Direct fidelity estimation of quantum states using machine learning. Physical Review Letters, 127 (13):130503, 2021.
Zhang, Y.H.. and Di Ventra, M. Transformer quantum state: A multipurpose model for quantum many-body problems. Physical Review B, 107(():07514))) 2023.
Torlai, G. and Fishman, M. PastaQ: A package for sim- ulation, tomography and analysis of quantum comput- ers, 2020. URL https: aastaQ.j11.
com//Torlai/
Zhou, Y. and Liu, Q. Performance analysis of multi-shot shadow estimation. Quantum, 7:1044, 2023.
Torlai, G., Mazzola, G., Carrasquilla, J., Troyer, M., Melko, R., and Carleo, G. Neural-network quantum state tomog- raphy. Nature physics, 14(5):447-450, 2018.
Zhu, Y., Wu, Y.-D,, Bai, G., Wang, D.-S., Wang, Y., and Chiribella, G. Flexible learning of quantum states with generative query neural networks. Nature communica- tions, 13(1):6222, 2022.
Tran, V. T., Lewis, L., Huang, H.-Y., Kofler, J., Kueng, R., Hochreiter, S., and Lehner, S. Using shadows to learn ground state properties of quantum hamiltonians. In Machine Learning and Physical Sciences Workshop at the 36th Conference on Neural Information Processing Systems (NeurIPS), 2022.
Vadali, A., Kshirsagar, R., Shyamsundar, P., and Perdue, G. N. Quantum circuit fidelity estimation using machine learning. Quantum Machine Intelligence, 6(1):1, 2024.
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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
A. Preliminaries of quantum system learning
In this section, we supplement the contents of Sec. In addition, we introduce related work and quantum resource in detail.
A.1. Basics of quantum computation
N -uquiit state. The atom unit in quantum system learning is the quantum bit, namely qubit. The single-qubit state can be expressed as a linear combination of two normalized, orthogonal complex vectors in the Hilbert space C2, In Dirac notation (Nielsen & Chuang, 2010), a single-qubit qubit state is defined as
(Q) co (0) + [1]
co C1
E C2,
where (0) = [1, and (1) = [0, 1] specify two unit bases and the coefficients co, Cj yield /col + 1e12 Similarly, an Nqqubit state is defined as a unit vector in C²* i.e.,
(
Gj ley),
where lez) E I== is the computational basis whose j-th entry is 1 and other entries are 0, and Σ |cy1 with ej E C.
Density matrix. Besides Dirac notation, the density matrix can be used to describe more general qubit states evolve under the open system. More specifically, the density matrix of the -ubit state (et) takes the form as
p (b) (
C2 X2"
where (16) = |!5)1 refers to the complex conjugate transpose of lttt).
For set of N-qubit states (pj, 1o with Pj > 0, -== Pj = 1, and (yy E C² for j E [m]. its density matrix formalism is
m
p
PiPi
with Pj = 1 (1yg and tr(p) =
Quantum measurement. The quantum measurement refers the procedure of extracting classical information from the quantum state. is mathematically specified by a Hermitian matrix H called the observable. Applying the observable I to the quantum state (ıb) yields random variable whose expectation value is ( H (+)) Additionally, the measurement outcome corresponds to one of the eigenvalues of the observable H, and the probability of obtaining specific eigenvalue X is determined by the projection of (ي²) onto the corresponding eigenstate of H. For a pure state = (w) (w), the expectation value can equivalently be expressed as tr( p). In the case mixed states represented by a density matrix p. the expectation value generalizes to tr(Hp) = E3 (, pÿ), reflecting the statistical mixture of quantum states.
A.2. Classical shadow
Classical shadows provide a computationally and memory-efficient method for storing quantum states on classical computers, enabling the estimation of expectation values for local observables (Huang et al., 2020). The core principle of classical shadows the 'measure first and ask questions later' strategy. Here we recap the implementation of Pauli-based classical shadows, as a widely used approach in the context of QSL. Interested readers can refer to tutorials and surveys (Huang, 2022), or recent progress (Huang et al., 2021; Nguyen et al., 2022; Zhou & Liu, 2023; Nakaji et al., 2023; Ippoliti, 2024; Rouzé & França, 2024) for more comprehensive details.
The procedure of based classical shadow for an unknown N N-qubit state p is repeating the following procedure M times. At each time, the state p is first operated with unitary U randomly sampled from the single-qubit Clifford (CI) group CI(2) and then each qubit is measured under the Z basis to obtain an -itit string denoted by b {0, 1)N, In this scenario, the unitary at the t-th time takes the form as U. U2 @ US ... à UN, U CI(2) with uniform weights.
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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
As proved in Ref. (Huang al., 2020), the classical shadow with respect to the 1-th snapshot for VZ E [M] is
P. =
(9)
Then, the estimated state of p by Pauli-based classical shadows is
M Σ M Pt.
PM
(10)
with Eipas] =
Note that Pauli-based classical shadows, i.e., (Ut,,, bis it are memory and computation-efficine Namely, O(NM) bits are sufficient store PM and ((N computational time is enough load (T the classical memory. Next, when the locality of the observable O E; O₂ is well bounded, the shadow estimation of the expectation value, i.e., tr(pas O,), can be performed in ((M) time after PM is loaded into the classical memory (Huang et al., 2020).
A.3. Related work
In this subsection, delve into prior literature relevant to this study, including works omitted in the main text. For clarity, we systematically outline the connections and differences between our work and each category.
Learning-free model. Previous studies on learning-fiee models in QSL can be classified into two subcategories based on whether they involve interaction with quantum systems.
The first subcategory consists of classical simulators, including state-vector simulators and tensor-network simulators, each with its own strengths and limitations. State-vector simulators (Aaronson & Gottesman, 2004; Smelyanskiy et al., 2016; Li al., 2021) can handle arbitrary quantum states but limited small qubit counts, typically no more than 50, Tensor-network simulators (White, 1992; Perez-Garcia et al., 2006; Sandvik & Vidal, 2007: Corboz, 2016; Cirac et al., 2021), on the other hand, are capable of simulating large-qubit states but are most effective for states with low entanglement (Orás, 2019).
The second category consists of measurement based protocols, which require direct access to the quantum system. These protocols assume that the ground state of the explored Hamiltonian has already been prepared using a quantum analog simulator or a quantum digital computer through specific algorithms. The primary focus of this category is on effectively extracting the target information from these prepared quantum states. A representative example in this context is classical shadow protocol (Huang et al., 2020; 2021; Nguyen et al., 2022; Zhou & Liu, 2023; Ippoliti, 2024).
As this study focuses primarily on learning based approaches, we do not perform a systematic benchmark against learning- free models. However, these learning-free models are complementary to learning-basee methods. For instance, classical shadows are utilized in this work to generate training examples.
ML models for QSL, Existing ML models in the field of QSL encompass kernel methods (Huang al., 2022), 4 Ey-regularized regression (Lasso) (Lewis al., 2024), and fz-regularized regression (Ridge) (Wanner et al., 2024). These MI models leverage structured representations of Hamiltonian parameters and associated measurement outcomes to predict target properties or phases, For quantum states prepared by quantum computers, recent study proposed a truncated trigonometric monomial kernel to efficiently predict different linear properties (Du et al., 2025a). As highlighted in the main text, key feature of ML models is provably efficient, which has been validated empirically on real experimental quantum data (Cho Kim, 2024).
DL models for QSL. Significant efforts have been devoted to utilizing deep learning (DL) techniques enhance the understanding of large-scale quantum systems. This research line be categorized into three subclasses: using DL to reconstruct quantum states, to predict the properties of quantum states prepared by quantum circuits, and to predict the properties of ground states for family of Hamiltonian. The primary focus of this study lies in the last subclass. For clarity, we briefly summarize each subclass below.
Quantum state reconstruction. This subclass primarily focuses on using various DL models to reconstruct the explored quantum state. The first approach involves employing DI. models to generate the explicit form of the density matrix,
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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
with the input being the classical description of the quantum state under investigation. Typical examples of this approach include (Lanyon al., 2017; Ahmed et al., 2021; Cha et al., 2021; Ma et al., 2023). Another approach involves using generative models to implicitly reconstruct the given quantum state, where the outputs of the neural networks replicate the measurement outcomes of the quantum state. Typical examples of this approach include auto-regressive models such as Boltzmann machine (Carleo & Troyer, 2017). Recurrent Neural Networks (RNNs) (Carrasquilla et al., 2019; Hibat-Allah et al., 2020; Carrasquilla & Torlai, 2021; Li et al., 2024) and Transformers (Carrasquilla et al., 2021; Cha et al., 2021; Wang et al., 2022b; Zhang & Di Ventra, 2023; Viteritti al., 2023; Sprague & Czischek, 2024; Fitzek et al., 2024)
Properties prediction of quantum states prepared by quantum computers. Initial studies have been carried out to use DL models predict the interesting properties of the quantum states prepared by quantum computers. For example, different DL models have been proposed to complete nonlocality detection (Deng, 2018; Kriváchy et al., 2020), estimation of quantum state entanglement (Ma & Yung, 2018; Gray et al., 2018; Koutný et al., 2023; Rieger et al., 2024), entropy estimation (Shin et al., 2024; Goldfeld et al., 2024), and fidelity prediction (Zhang et al., 2021; Wang et al., 2022a; Du et al., 2023; Wu et al., 2023; Vadali et al., 2024; Qian et al., 2024; Qin et al., 2024).
Properties prediction of ground states, As mentioned in the main text, the main focus of this research line is developing different neural architectures to predict the properties of ground states under supervised, semi-supervised, or self-supervised learning paradigms. In the supervised learning framework, typical examples include (Wu al., 2024; Wanner et al., 2024). In the semi-supervised framework, there have (Tang et al., 2024b). In the self-supervised framework, there have (Zhu et al., 2022; Wang et al., 2022b; Tang et al., 2024a).
Our work distinguishes itself from prior studies by focusing specifically on the role of DL in quantum system learning. Rather than designing new network architectures or learning paradigms, we conduct systematic analysis of the performance of existing DL models compared ML models. To the best of our knowledge, this is the first empirical investigation that explores how classical ML outperforms DL and identifies scenarios where DL excels in QSL. This study provides a comprchensive comparison across various Hamiltonians of different sizes, including large-scale systems.
A.4. Quantum resource cost
In this subsection, we present evidence to justify our choice of using the number of measurements the metric for quantum resource usage. In particular, we restrict the total number of measurements in the dataset, i.e., n x M. when conducting the training. This decision is primarily based on two considerations: (i) in line with the conventions of QSL,, learners typically rely on single-copy measurements, and (ii) conducting measurements is resource-intensive on near-term quantum cloud computing platforms. By aligning our analysis with this constraint, we ensure a fair and practical assessment of quantum resource efficiency, bridging the gap between theoretical learning paradigms and real-world economic considcrations.
For illustrative, we summarize pricing data from leading quantum cloud providers (Microsoft, 2024; IBM, 2024; Amazon, 2024), highlighting cost disparities, shown in Tab. 3 and Tab. 4. An observation is that with the increased number of measurements, the required cost becomes extremely expensive even unaffordable. For example, when 10000 and M = 1000, the cost of IonQ-Forte is about 80, 000 USD based on Tab. 3.
Table 3: Price (USD) for every shot execution of different quantum machines, until December 2024. A shot is a single execution of a quantum algorithm on a QPU, such as the time evolution of Hamiltonian on QuEra-Aquila,
Table 4: Price (USD) for every hour usage of different quan- tum machines, until December 2024.
Quantum machine
System size
Price (USD) / hour
Quantum machine
system size
Price (USD) / shot
IQM-Garnet
20
3000.00
IonQ-Forte
36
7000.00
IQM-Garnet
20
0.00145
IonQ-Aria
25
7000.00
AoraaAria
25
0.03000
QuEra-Aquila
256
2500.00
IonQ-Forte
36
0.08000
IBM-QPU
2 127
5760.00
Rigetti Ankaa
84
0.00090
PASQAL Fresnel
100
3000.00
QuEra-Aquila 256
0.01000
Rigetti-Ankaa-3
82
4680.00
we next highlight the runtime cost associated with collecting the training dataset. Suppose the wall-clock time required by
IonQ-Forte to perform a single-shot measurement is second. When n = 10000 and M = 10000 the dataset construction requests around 115 days, which is unacceptable in practice.
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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
B. More details of quantum system learning models
This section provides the detailed information of quantum system learning (QSL) models corresponding to the supplements of Sec. 3.2 in the main text. We present overview our QSL pipeline (Appendix B.1), classical ML-based (Appendix B.2) and DL-based QSL models (Appendix B.3) separately.
B.1. Overview
The overall QSL pipeline is illustrated in Fig. 6. The quantum system evolves under a specified Hamiltonian, and classical information is extracted by collecting measurement outcomes, subject external quantum resource constraints. In the classical machine learning (ML) process, QSL features are derived from quantum system parameters using customized feature maps, and the resulting features are input into trainable ML models. In the deep learning (DL) process, neural networks synchronously serve as both feature maps and trainable learners, forming an end-to-end learning paradigm. Under resource constraints, supervised labels are approximated from measurement outcomes using learning-fred estimators, such as classical shadows. We benchmark both classical ML and DL learning processes, focusing in particular on the scaling behavior of model performance with respect to key parameters of resources, as well the impact of real versus randomized measurements as representations of QSL models.
Quantum system
Learning process in classical computer
Comparison
Systeh porameters
QPC
Learning process
mAR
observable
GOU
Classical ML
Hamiltonion
Quantum system
Pecome
Scaling behavior
Measurement
operotons
query
GSPE
Training
Measurament shots M Model size
Role of measurement
(jm@) mG
Gp
Randomized
Sel-sgaeriied
Superwsei
Resource constraints
Researcher
Classical ML
DL process
Clossical ML model
DL model
Figure 6: Overview of our methodology on quantum system learning (QSL). The left: The quantum data is collected from the quantum system that evolved by specific Hamiltonian under external quantum resource constraints. The center: We organize learning patterns, namely classical ML and DL, explicitly or implicitly characterize the features of collected quantum data towards GSPE and QPC tasks. The right: We benchmark the performance of classical ML and DL, discussing the scaling behavior and necessary features (especially on the measurement information) of QSL models.
B.2. ML models
The mechanism of applying ML models to complete QSL is demonstrated in Fig. 6. In particular, system parameters including Hamiltonian parameters (e.g., coupling strength) and system conditions (e.g., the interaction range of |/Rya))) are characterized as input features, as well as approximate labels that obtained from measurement outcomes via classical shadow, together form supervised dataset. The classical ML model learns from the dataset and is applied to GSPE and QPC tasks.
Dirichlet Kernel (DK) (Huang et al., 2022)
The Dirichlet kernel of order is defined as:
Nox (xx x²) Σ cos( k x²)),
(11)
where & is coefficients vector. The feature map (xx) = 1 Σn
( (p(x(0)) with cutoff A = 3. As explained in the main text, we follow the same routine in Ref. (Huang et al., 2022), we use random Fourier features as the surrogate of DK. The mathematical formulation of random Fourier features takes the form as x axd to / 2 cos cos(war + b)
Σ
kEz
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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
where wi are random frequencies sampled from the normal distribution and b, are random biases sampled uniformly from [0, 2m)] and we consider it new features as training data for SVM or KR with linear kemnl..
Radial basis function kernel (RBFK) (Huang et al., 2022) RBF kernel (Buhmann, 2000), also known as the Gaussian kernel, is widely used kernel function for handling non-linear data by mapping the input space into a higher-dimensional feature space where the data becomes more separable. The RBF kernel between two feature vectors æ and x' is defined as:
x 112 (RBPK(2, x') = exp (-1lae
|| a
=exp
(12)
27
where || x - x' 1|a the squared Euclidean distance between two vectors and 722 Σ. ET
|| x 0
The feature map
2nd
((x) = Ei= KRBFK (x, _= (o(x((())))) In practical, we explicitly use SVM or KR with Gaussian kernel to implement RBFK. We choose hyperparameter y2 from a list of discrete candidate values.
Neural Tangent Kernel (NTK) (Huang et al., 2022) The Neural Tangent Kernel (NTK) (Jacot et al., 2018) is theoretical framework that describes the behavior of infinitely wide neural networks during training via gradient descent. For a neural network fx; f) with parameters 0, the NTK is defined as the kernel function:
KNTK (æz, æ²) Ee-p( ((Vol(( (Vo (ax; 0), vol(a';0)))
(13)
where (o 0) is the gradient of the network's output and Egup represents the expectation over the distribution of ( " where KNTK is the NTK of one specific neural network and K is the corresponding kernel matrix. In experiments, the employed neural network are 2 layers, each of which is with hidden dimension of 32 and an activation function ReI.U, implemented by library neural.tangent (Novak et al., 2020; 2022). NTK is used to map features onto higher dimension feature space, which is considered as features of training data for SVM or KR with linear kernel.
initial parameter 0. The feature map ((x) x Σ Σi=
KNTK (2x, x x0))
< d-regularized regression (Lasso) (Lewis et al., 2024) Lasso is linear regression technique that incorporates & regularization. The feature map ((x) is groblem-informed and designed according to Hamiltonian geometry property. The observable corresponding to GSPs decomposes as
Σ app
where P is geometrically local Pauli observable. ((x)='.P of Lasso takes the form as T[x: E Te), where T[:] is the indicator function, x' is vector in the discrete input feature space, and Tw' includes the coordinates close to æ' in each coordinate set close to P. In experiments, ((x) is implemented by random Fourier features, with a rigorously theoretical guarantee (Lewis et al., 2024). We practically map x nxd to 2 /o/ccos(W) (ossWw + b), and concatenate with æ as new feature [z, /2/dcos(// Wx x b)], where W; is sampled from a normal distribution and b, is uniformly sampled [o, 2T).
lz-regularized regression (Ridge) (Wanner et al., 2024) Ridge is a linear regression technique that incorporates & regularization to address multicollinearity and overfitting in regression models. The feature map ((())rr takes the form as pa V | E Tl, where æ' is a vector the discrete input feature space, and T includes the coordinates close to x' in each coordinate set close to P. In experiments, is mapped via random Fourier features, the same Lasso did.
B.3. DL models
The mechanism of applying DL models to complete QSL is demonstrated in Fig. 6. In particular, there are two learning patterns, one of which is SSL, DL model is train on masked measurement information, enables itself learning prior knowledge from measurement (c.g., classically generate measurement strings according to specific system parameters in the inference phase). The other is supervised learning pattern, which is similar to ML, encodes system parameters and (optional) measurement information into input features, as well as approximate labels that obtained from measurement outcomes via classical shadows, together form supervised dataset. The supervised DL. model learns from the dataset and
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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
XL
Quantum phoses
Ground state properties estimation
clossification
Embeddings
Layer normalization
Softmax
layer
Decoder
Forwaforward
network
FC layer
Aggregation
Conditional embedding
Token embedding
Pasitional embeddlig
Masked
multi-heads
Decoder
Decoder
attention
Hamiitonian
Measunement
parameters
outcomes
Supervised training somples
1. Embedding
2. SelSSefffuueee supervsuee pretrain
3. Supervised finetune
Figure 7: Model architecture of the advanced SSL-based learning model like LLM4QPE. Part I: Three embeddings are concatenated as one, two which are embedded from Hamiltonian parameters (e-g.. coupling strength) and measurement outcomes; Part Pretrain model is a I--ayer transformer-based Decoder, taking embeddings as input and output the same size tensors; Part 3: The decoder used for supervised finetuning, incorporated with other specific components for adaptation to downstream GSPE and QPC tasks.
applied to GSPE and QPC tasks. The SSL-based model is finetuned on supervised dataset and then applied to both tasks, or directly estimate properties via classical shadow on its own generated measurement information.
As illustrated in Fig. 6, the DL learning process categorized into two patterns, one of which is the self-supervised learning (SSL.) paradigm, learning the the prior knowledge from unlabeled quantum data (e.g., measurement information), such as shadow generator (SG) and the pretrain phase of LLM4QPE, the other is supervised learning models, such as MLP, CNN, and the finetune phase of LIM4 OPE. For deeper MLP and CNN, We incorporate well-known techniques such as dropout, residual connections, and appropriate & regularization to ensure effective training.
Table Model architecture MI.P- layer for QSI. models. Di is up to input feature dimension. Dout is up to task dimension. By default, The following w width 128.
Table 6: Model architecture of CNN-1 Tayer for QSI, models. Di is up to input feature dimension. Dout is up to task dimension. We consider 32 convolutional kernels in the model.
Layer
Settings
Layer
Settings
Fully Connected
Din x width ReLU
Convolutional
5 x kernel; stride 1; padding
Activation
Activation
ReLU
Dropout
p p= 0.5
Dropout
p 0.5
Fully Connected
width x Dout
Fully Connected
(32 x Día) x 128 ReL.U
Activation
Dropout
p = 0.5
Fully Connected
128 x Dout
Muti-layer Perceptron -bbasd model We implement MLP -based QSL models with several fully connected (FC) hidden layers, each of which is implemented as detailed in Tab. 5. We tune the model size via varying the width of hidden layers as the depth of the network is fixed.
Convolutional Neural Network (CNN)-based model We implement CNN-hased QSL models with several convolutional layers, each of which is implemented as detailed in Tab. 6.
Transformer-basse model We primarily follow the work named LLM4QPE (Tang et al., 2024a), which is the advanced language model-like QSL paradigm, We refer the readers to their paper for model details. LLM4 QPE is mainly composed of the pre-processing embeddings and decoder-only architecture with trainable parameters (See Fig. 7). The training process includes a self-supervised pre-training phase and a supervised fine-tuning (SFT) phase. Specifically, we embed the Hamiltonian system parameters (e.g., coupling strength) and measurement outcomes as well as positional embeddings. We configure the model with heads, 8 layers (i.e., = 8) with a 128-dimensional FFN, and FC hidden layer of dimension 128 for both GSPE and QPC tasks. The maximum dimensions of embeddings are 512 and 256 for GSPE and QPC tasks,
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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
respectively.
The other transformer-basee model we consider in this paper is shadow generator (SG) (Wang et al., 2022b), which is open-sourced LPennyLaneAI, 2022), and we refer its original model settings.
C. Additional experiment results
This section provides additional experiments and results to complement the main text, offering more comprehensive analysis of ML and DL models QSL tasks. Appendix C.1 details the dataset construction process, including the simulation tools and parameter settings used generate the data. The subsequent three subsections present supplementary results topics omitted from the main text, including scaling behavior (Appendix C.2), performance comparison between MLL and DL models under fair resource constraints (Appendix C.3), and the role of measurement outcomes as input features through a randomization test (Appendix C.4). Additionally, we investigate some new topics that are not addressed in the main text. Appendix C.5 examines the metrics and conditions under which DL models can outperform ML models. We explore whether deeper neural networks (Appendix C.6) and infinite measurement shots (Appendix C.7) could influence the results. Appendix C.8 and C.9 further describe the redundancy current measurement embedding strategy and the necessity for design. Besides, Appendix C.10 provides visualization case 48-qubit system.
C.1. Dataset generation tools
We utilize different simulation tools to generate the datasets and obtain exact results for the two GSPE tasks and one QPC task introduced in Sec. 3. Below, provide detailed explanation of the tools employed.
The training dataset ID for two GSPE tasks (i.e., correlation prediction and entanglement entropy prediction) of both (чTним) and WHe) is constructed by using PastaQ il (Torlai & Fishman, 2020), which can effectively generate classical shadows in the labit-qubit scenario.
To evaluate the performance of different QSL models, we also need the corresponding ground-truth results. These results obtained by using ITensors jil and ITensorMPS j1. The following are two pieces of codes approximate estimation of two-point correlation Cij in Eq. (2) and the entanglement entropy S2 in Eq. (3), respectively.
function compute_
xxcorr humppptte__aatt_.oorreaaaiioonn :MPS)
(psi), "Sx" "Sx")
yycorr corrclattion_matiiicccattt (cmpppp (psi), "Sy", "Sy")
zzcorr correlation_matzzae (complex (psi), "S2", "S2")
corr = (xxcorr + yycorr + 22corr) ./ 3.0
dig = sqrt. (corg)))
for i-1: (en:h))
c corr corrli,;] dig[i]
corrl;,il corr [; digli]
10
end;
return vec (real (corr))
end;
13
exast_correlation
compute_ccaet_corrcllaiiioo (psi)
function compute_cxact_renytte
a::Int: b:sInt)
sitcs = sitcinds (psi)
8assert 1 <= a b <= (en)th
@assert b
orthogenalize! (psi,a)
psidag-prime (dag (psi), linkindd (psi))
rho_A = prime (psila), linkinds (psi, a-11) prime (psidag [al, sites (a))
r prime(psilbl, linkinds (psi, b)) prime (psidag [b), sites b
D1, sigen(rho A)
10
return -log2 (sum (real (D1) (d)
" end;
exact_entrope [omppptte.eeeteeeeyyyooooooooeee (psi, a+1) for in 1:N-1];
The training dataset D for the QPC task (i.c., phase classification) of (vkya) and the corresponding ground- -ruth results are collected by using the source code provided in Ref. CennyLanneA (2022).
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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
C.2. Scaling behavior of QSL models
In this subsection, we provide more experiment results related to the scaling behavior of ML and DL models when applied to solve the two QSPE tasks and one QPC task introduced in Sec. 3.
HB-Lasso
TFsM-Lasso
HB-Ridge
TFIM-Ridge
HB-LLM4QPE
TFIM-LLM4QPE-F
M 64
128
M
256
M
512
0.04
0.04
0.04
RMSE
0.04
RMSE
RMSE
0.03
RMSF
0.03
0.03
0.02
0.02
0.02
0.02
0.01
20
40
60
80
100
0.01
20
40
60
80
100
0.01
20
40
60
80
100
0.01
20
40
60
80
100
Training size n
Training size n
Training size
Training size n
(a) The subplots explore the scaling behavior of learning models for when applied to predicting c(C).
M
0.25
0.25
0.25
0.25
RMSE
0.17
RMSE
0.17
RMSE
0.17
RMSE
0.17
0.09
0.09
0.09
0.09
0.01
20
40
60
80
100
0.01
20
40
60
80
100
0.01
20
40
60
80
100
0.01
20
40
60
80
100
Training size
Training size
Training size
Training size
(b) The subplots explore the scaling behavior of learning models for when applied predicting c(S2).
Correlation (n= 20)
0.04
Correlation (n= 40)
Correlation (n= 60)
0.04
Correlation (n 80)
Correlation (n= 100)
RMSE
0.04
RMSE
0.03
RMSE
0.03
RMSC
0.03
RMSE
0.03
0.02
0.02
0.02
0.02
0.02
0.01
64
128
256
512
0.01
64
128
256
512
0.01
64
128
256
512
0.01
64
128
256
512
0.01
64
128
256
512
M
M
M
M
M
(c) The subplots explore the scaling behavior of learning models for M when applied to predicting <CC).
0.25
n- 20
0.25
A
40
0.25
0.25
n-
80
0.25
2- 100
RMSR
0.17
ROASE:
10.17
RMSE
0.17
RMSE
0.17
RMSE
0.17
0.09
0.09
0.09
0.09
0.09
0.01 64
128
256
512
0.01
64
128
256
512
0.01
64
128
256
512
0.01
64
128
256
512
0.01
64
126
256
512
M
M
(d) The subplots explore the scaling behavior of learning models for M when applied predicting e e(S2).
Figure 8: Scaling behavior of learning models when applied to GSPE tasks with 127-qubit (vH)) and (<TRIи)). All notations follow the same meaning those introduced in Fig. 1.
Additional results with respect to two GSPE tasks. Recall that in the main text, we use two-point correlation prediction and entanglement entropy prediction as benchmark tasks to evaluate the performance of ML and DL. models. Here, we present additional results for these tasks, obtained by applying ML and DL models under identical parameter settings as described in the main text. The only difference lies in using more refined training sizes and snapshots M to collect the relevant results.
The achieved results related to the correlation prediction are shown in Fig. 8(a) and Fig. 8(c). For each subplot in Fig. 8(a), we fix the snapshots to be M E {64, 128, 256, 512) and vary the training size with n E {20,40 40, 60, 80, 100). Conversely, for each subplot in Fig. 8(c), we fix the training size to be n E (20, 40, 60, 80, 100} and vary the snapshots with M E (64, 128, 256, 512). All of these results validate the scaling behavior of ML. and DL. models, echoing the statement in
20

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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
the main text, i.e., results shown in Fig
The results for entanglement entropy prediction are presented in Fig. 8(b) and Fig. 8(d), with parameter settings identical to those used for the correlation tasks. While the achieved results also indicate the consistent scaling law in Sec. 4. an interesting phenomenon is that when M is small (e.g.. M = 64), the performance of ML and DL models is unsatisfied, no matter how the training size n scales. RMSE < ((2) of Lasso, Ridge, and L.M4QPE-F maintain around averaged 0.165, 0.164, and 0. 0.176, respectively, on 127-qubit (UTFIM), without notable scaling.
Additional results with respect to the QPC task. We present the complete results of the exploration of scaling behavior on the QPC task, i.e., the phase classification of 127-qubit (vRyd) with the same settings as those in the main text. As shown assertion the scaling behavior of QSL models in QPC tasks,
in Fig. (a) and Fig. 9(b,, the achieved results confirm mentioned in Sec. 4.2: the test accuracy increases with the training size and measurement shots M. According to the increasing trend in the measure of test accuracy, the training size n plays a more significant role in performance improvement compared to the number of measurements M. Besides, Tree models RF and XGB are more robust than LLM4QPE-F to fewer shots M and samples n. For example, when n = 20 and M = 64, the test accuracy of RF and XGB are 90.05% and 92.55%, respectively, while the test accuracy of LLM4QPE-F is only 60.17%.
XGBoust
Random Forest
LLMNQPE-F
1.0
1.0
256
0.9
0..9
Test accura
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
Test
0.5
0.6
40
60
80
100
0.5
20
40
60
80
100
(a) The scaling bchavior of learning models of the varied training size n.
XGBoost
Random Forest
LLM4QPE-F
1.0
n n-20
1.0
1.0
n- 60
1.0
1.0
0 A=100
0.9
Test aoc coeuracy
Test accuracy
Test accuracy
0.9
0.8
0.7
0.8
0.8
0.8
0.8
Test
0.6
0.7
0.7
0.7
0.77
64
0.6
128
256
0.6
64
128
256
0.6
256
0.6
64
128
256
M
M
(b) The scaling behavior of learning models for M.
Figure 9: Scaling behavior of ML and DL models when applied to QPC tasks with 127-qubit (1Rgyd). The shadow region refers to the standard deviation. All notations follow the same meaning those introduced in Fig. 4.
C.3. Does DL really outperform classical ML?
In this subsection, we present additional experiments and results comparing classical ML and DL models GSPE tasks introduced in Sec. 3. These findings complement the results shown in Tab. I in the main text, which only exhibits the results of QSL models when applied to predict correlations of ((/н))
For completeness, here we apply QSL models to predict two-point correlations Cij of (FTFIM) and to predict entanglement entropy S₂ of (ынн) and (ртттм) All learning models presented in Sec. 3.2 are considered, where the parameter settings of these models are the same as those introduced in the main text.
Tab. 7 summarizes the achieved RMSE c(C) when applying ML and DL models predict correlation of |/TFII)) As with the conclusion in the main text, ML models outperform DL models in this task. In particular, Ridge and Lasso are superior to all other employed models. In addition, LLM4 QP ø-T T is still better than other DI. models. In addition, given a few shots per sample (M = 64), the autoregressive model SG only attains a similar performance with CS. This observation further validates the importance of fine-tuning.
Tab. and Tab. separately summarize the comparison benchmark results of predicting entanglement entropy Sa of (WHB)
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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
Table 7: The RMSE result on correlation prediction of (РTRIм) with varied N and Natt. M is fixed to 64. The best results are highlighted in boldface while the second-best results are distinguished in underlined. The results are averaged over 5 independent runs with different random seeds.
N-
N- 63
N 100
N 127
Methods
Pur 20
Ovn 60
Pun 1010
Paate 201
Platt 60
Plat: 100
Nich 20
Fiet 60
Plat: 100
Flat: 20
Ren
80
Rett 100
CS
0.20924
0:20360
0.21092
0,21180
MLF-1 loyer
0.15304
0.17621
0.18264
0.14469
0.17311
0.18283
0.135144
0.15827
0.17499
0.12615
0.14982
0,16912
layer
0.12003
0,10860
0.09798
0.10801
0.09005
0.07831
0.03357
0.01538
0.0:100
0.08093
0
0.03056
SG
0.20766
0.21469
0,21312
0 08202188
0. 0,20179
0.21969
0.21574
0,21629
0.21887
0.21951
0.21
0.20622
L.M40PE-T
0.05088
0.03493
0.03006
0.05252
0.03586
0,03082
0.05217
0.03476
0.03012
0.05250
0.03641
0.03084
DK
0.16018
0.25256
0.17826
0.13727
0.28337
0. 0.23080
0.11138
0.22104
0.32314
0. 00072
0.19026
0.30268
RBPX
0.06071
0,05645
0.05867
0.06005
0.05197
0,05662
0,05878
0.01933
0,05125
0,03586
0.01701
0,01833
0.07138
0.07351
0.07447
0.06529
0.06501
0.06614
0.03630
0.06252
0.05392
0.05122
0,01739
0.01836
0.04624
0.032199
0.02812
0.04663
0. 0.03930
0.02850
0.04073
0.03256
0.02800
0.04583
0.03283
0.02932
Ridge
0.04473
0.03173
0.02807
0,04561
0.03226
0.02839
0.045508
0.182277
0.02883
0.04570
0.03225
0.02911
Table 8: The RMSE result on entanglement entropy prediction of (SHB) with varied N and nan- M is fixed to 64. The best results are highlighted in boldface while the second-best results are distinguished in underlined. The results are averaged over 5 independent runs with different random seeds.
N
48
N = 63
.V 100
N 127
Methods
Psa
20
Pla. 60
150 100
150 20
P&. 60
1%6
100
20
1ua 60
100
20
60
12.0
100
CB
0,59882
0.61001
0.61184
0.61650
MLE 1 Layer
0.713112
0.51571
0.47069
0.66398
0. 0.60608
0.54498
0.76347
0.70521
0.06799
0,80014
0.76523
0.73077
CNN-1 layer
0.24949
0,22327
0.22088
0.25458
0.21880
0.20351
0.24914
0.25079
0.20831
0.27188
0,27863
0.26329
LLM-QPE- T
0,21397
0,18371
0.18092
0,23130
0.20676
0.19969
0.22633
0.20559
0.20089
0.26101
0.21311
0,23770
DK
0.26032
0.33764
0.38542
0.28784
0.35502
0.40298
0.27239
0.34513
0.30357
0.29700
0.35879
0.40720
RBPX
0,36122
0,27159
0.27155
0.35311
0. 0.26111
0.25779
0.311449
0. 0,23926
0,21543
0.31927
0.25187
0,25981
NTK 0,28779
0,26236
0.27971
0.30155
0. 0,200300
0. 0.25857
0.30701
0. 0.26255
0.27414
0.33246
0.27910
0,20076
Lasso
0,21108
0.18358
0.178811
0.23791
0.20990
0.200308
0.22278
0.20289
0.19857
0.25034
0,233338
0,24901
Ridge
0.21067
0.18263
0.17684
0.29754
0.20823
0.20168
0.222446
0.20327
0,19869
0,25034
0,23244
0.23210
Table 9: The RMSE result on entanglement entropy prediction of (&TPIS)) with varied N and nuat M is fixed to 64. The best results are highlighted in boldface while the second-best results are distinguished in underlined. The results are averaged over 5 independent runs with different random seeds.
N
NN
63
N
100
N 127
Methods
20
60
Dut 100
20
Nutt 60
Pltt: 100
Tich
20
Tiet
60
Matts 100
20
Tren 60
1ett 100
CS
0.36902
0.35881
0.36627
0,38715
MLF-1 Loyer
0,14876
0.17630
0.18955
0,15318
0.18284
0.18883
0.15078
0.18142
0.18649
0.14145
0.18481
0,18929
CNN-1 Layer
0.19267
0,24437
0.18010
0.17748
0. 0.22321
0.22642
0.25218
0.19731
0.15335
0.14577
0.12622
0.21737
LLM-QPD T
0.19218
0.17967
0.17475
0.17554
0. 15933
0.15170
0.18015
0.17421
0.168-11
0.18827
0.17332
0.16969
DR
0,19181
0.23056
0,25753
0,19911
0. 0.23567
0. 0,25800
0.19231
0. 2.22661
0,25297
0.18365
0,23063
0
XRFX
0.10420
0.14273
0.16264
0.10064
0.14708
0.16492
0.101 63
0.14128
0,16184
0.09946
0.14579
0.1 0.16301
NTK
0.20000
0.18048
0.18137
0.20849
0.18775
0.18466
0.20252
0.17762
0.18075
0.19118
0.18432
181
Lasso 0.17399
0.16189
0.16383
0.17336
0.17096
0.16667
0.16970
0.16189
0.16366
0,16131
0.16856
0,16330
Ridge 0.17369
0.14441
0. 0.10335
0.17533
0.17081
0. 0.16003
0.10083
0.16204
0.16379
0.16150
0.169
16447
and (UTFIM). The experiments setups (including N, nat and M) are the same with those in Sec. 4.1. Different from the correlation prediction task, ML models are not always superior to DL models, as LLM4 achieves the best performance on 63-qubit (UHB) system. While Ridge is the second-bess best, the two models have an average relative difference of 1.47%.
C.4. Are measurement outcomes redundant?
In this subsection, additional experiments and results are provided, to confirm whether measurement outcomes are redundant input representations for GSPE tasks. We conduct two independent experiments for supervised learning paradigms and based models, the same as those introduced in Sec. 4.1. The only difference that what we focus on is moved from
22

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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
predicting correlation Ci predicting entanglement entropy S2.
Fig. 10 demonstrate the achieved RMSE of e(S2) by supervised learning models. As with the results in the main text, no input representations of learning models cannot improve model performance generally. A slight difference is that when applied to predict entropy S of ((YFAM) and (tHB), ML models seem to be less sensitive to whether measurement information is included, compared to CNN.
matter predicting correlation or entropy. employing
0.3
Laco
Nigg Bidge-A
RSPS
CNN CAN-A
0.9
Lecoo
Midas
HoPS RSPSA
CAN CSNA
Lacoro A.
RSPSA
Locoo
Rodge-A
20
0.2
Lacoo
Rosge
CNN CANA
02
Lacoo
Rigge RiGae-A
RSF% RSFSA
CAN CNN-
LaceovA
Lacoo-A
Triiiing slow
Traniig dizen
(a) M is fixed to 64.
(b) M is fixed to 512.
Figure 10: Role of measurements as input representation towards the entropy prediction task. Panel (a) (or Panel (b)) exhibits the results of 127-qubit (CHB) and (РTFι<) when QSI, models are applied to predict the entanglement entropy with M = 64 (or M 512).
The notation "a- A" (or "a")) refers that the learning model uses does not use) v auxiliary information.
Fig. 11 shows the performance of the SSL-based model on predicting entropy when using real and randomized measurement outcomes as token embeddings. The trend of prediction performance similar to that of predicting correlation, which presented in Fig. 2 (the right panel). That is, in GSPE tasks, the employment measurement outcomes as input features does not improve the learning performance.
Ground truth
Random
Entropy
0.26
14 64
RMSE
0.24
M
128
0.22
M
0.20
20
80 Training size n
40
60
100
Figure 11: Randomization test of LIM4QPE-T on predicting S of 127-qubit (WaI)). The performance of LLM4QPE-T varies with training size and snapshots M. The labels "Ground truth" and "Random" refer to the use of real or random measurement outcomes as LIM4QPE's input features.
C.5. A case study: when neural networks take ad advantages?
This subsection provides two cases on the potential scenarios where neural networks may excel, including dealing with out-of-distribution (OOD) Hamiltonian parameters as well as OOD Hamiltonian models.
For the first case, we consider Lasso and MLP as ML and DL models, respectively. Here the problem setting is similar to the one used in C.2, where the task is predicting correlation. The feature dimension of Lasso is x 25 The employed MLP only contains one hidden layer, where the corresponding dimension is 1024 and its regularization weight À is fixed to 40. The snapshot M for all cases fixed to 128. After training, we apply the trained model predict the correlation of
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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
2-dimensional x 5-scale (/uB) with Hamiltonian parameters sampling from interval [0, uniformly, which is originally out of the distribution of the training set.
The achieved results are summarized in Tab. 10, indicating that MLP could achieve or even exceed the performance of Lasso on this OOD task.
Table 10: A case study Lasso and MLP being evaluated on the OOD 2-dimensionall 5 5-scale [éHn)- M is fixed to 128.
T
20
Y =
40
=
60
80
Yt
100
Lasso
0.14809
0.15412
0.13780
0.13759
0.13396
MLP-1024 -width
0.14130
0.13770
0.13697
0.13657
13.4488
For the second case, consider GB, XGB, MLP and CNN ML and DL models, respectively. The experiment setting is the same as the exploration of scaling behavior in Sec. 4.2. The feature dimension of all models is 4. MLP and CNN is the origin settings as Tab. 5 and Tab. 6. Tree models are tuned with hyperparameters via Optuna (Akiba et al., 2019). The training size n E {20, 60, 100} and the snapshots M E {64, 128, 256]. The learning models are trained on 25-qubit (sRya)), and evaluated on the higher 31-qubit system. The raw training sets are not upsampled to achieve label balance.
Results in Tab. present that neural networks excel most cases. extreme cases when M = 256 and n = 20, the training set has only one or two samples that belong to Z2 or Z3 ordered phase, neural networks can easily overfit these few samples and generalize the evaluation set.
Table 11: The classification accuracy (%) of quantum phases when considering system transferring and long-tailed label distribution, varied M and The best results are highlighted in boldface while the second-best results distinguished in underlined.
M 64
M 128
M 256
Methods
Y = 20
n = 60
n = 100
n = 20
= 60
n = 100
n = 20
n = 60
n = 100
MLP
92.76
92.82
93.72
93.85
92.
92.63
93.85
92.18
92.63
CNN
93.65
94.36
95.32
94.81
93.65
94.42
94.81
93.65
94.42
GB
92.37
93.91
95.83
92.62
93.53
95.96
87.37
92.95
95.19
XGB
93.01
90.13
94.36
89.68
90.19
93.65
82.76
90.19
93.40
C.6. Do results hold if neural networks go deeper?
This subsection provides the experiment results on GSPE as neural networks become deeper. The neural architectures include MIP and CNN as introduced in Sec. B.3. We evaluate six learning models introduced in the main text to estimate the correlation C' of (ынн)) and (итFим) with the system size N ranging from {48, 63, 100, 127}, and varying n from (20, 60, 100). The depth DL models is increased up to 100 layers. All other experimental settings are aligned with those of benchmark experiments in Sec. C.3.
Results are summarized in Tab. 12 and Tab. 13. As model depth increases, performance on predicting Ö of (UHB) and (чннм) initially improves but then degrades, yet is still inferior to classical ML models.
C.7. Do results hold if the number of measurement shots goes infinity?
In this subsection, we consider the GSPE task on the 8-qubit (НнB) under non-real-world setting where the number of measurement shots is assumed to be infinite (M oo), resulting in noise-free supervised labels. Simultaneously. the amount of training data samples is large, with E {10", 10 10a 10º3. We conduct experiments using four learning models of increasing model sizes, including Ri Ridge, MI.P-4 layers, CNN-4 layers, and I.I.MQPE-F.
Results are shown in Tab. 14 and Tab. 15. As training data amounts exponentially increases from 102 to 10 the performance of Ridge on predicting and Sg 8-qubit (/HB) is superior to that of other advanced DL models. Compared to the best model CNN-4 layers among the DL models the table, the model size of Ri dig is 100x smaller, but the average
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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
Table 12: The RMSE result on predicting C' of (dmB) with varied system size N and finetuning training size Nattt M is fixed to 64. MLP (CNN)- -x layers represents neural network MLP (CNN) that composed of layers with residual connection. The best results are highlighted in boldface while the second-best results are distinguished in underlined.
NV 48
NV
63
100
127
Methods
Righ
Run
60
Pan 100
Raltt
20
60
Alan
100
20
Pun 60
Aln 100
Pen 20
Pen 60
Rett 100
CS
0:21113
0:21257
0,21399
0.21/117
M1P-2 layers
0.08282
0.07752
0.06616
0.12055
0.08776
0.07086
0.10848
0,08158
0.07 10G
0.10091
0.10088
0.082 16
X1P-3 layers
0.06211
0.01853
1.01191
0.07256
0.05506
0.01167
0.07740
0.06196
0.070888
0.08535
0.08280
0.08691
M1P-4 Tayers
0.05428
0.03825
0.03524
0.06463
0.04435
0.03833
0.07532
0.05952
0.060110
0.07971
0.00173
0.08608
XLP-5
layers
0,07228
0.04721
0:03764
0.07308
0.03857
0.05091
0.08646
0.07146
0.07174
0
0.08650
0.08
CNN-2 layers
0.07160
0.04721
0.03795
0:07176
0.04066
0.03842
0.06549
0.04566
0.03464
0.06468
0
0.07404
CHN-3
layers
0.08089
0.03422
0:03435
0.09003
0.03401
0.03159
0.07603
0.03245
0.03295
0.084220
0.03179
0
CHN-1
layecs
0.06181
0.01699
1:03156
0.06621
0.03608
0,03100
1.06136
125
0.02808
0.07111
0.03196
0.05221
CNN-10
layars
0.06388
0.05577
0.038.5
03633
0.08897
0.09836
0.05456
0.03361
0.03555
0.05273
0.08775
0.03523
CNN-20
layers
0.15740
11911
1807
0. 1313363
0. 0.10532
0.07100
0.11759
0.09081
0.07022
0.10187
0.08780
0.07183
CNN--5
layers
0,16399
0.12271
0.07735
0.16071
0.1
0.09655
0.14741
0.11789
0.09367
0.13:20
0.12921
0.10086
CNN-10C layers
0.207977
0.20659
0:203394
0. 18882
0.1 19880
0.173333
0.14762
0.144002
0.13628
0.13465
0.13356
0.13150
LIM4OPE-T
0.06189
0.03368
0:03197
0.06111 11
0.03364
0.02863
0.05050
0.03227
0.02726
0.05079
0.03184
0.026334
RBPK
0.064552
0.04176
0.0-1101
0.04726
0.038229
0,03922
0.0-4096
0,03259
0.03282
0.03850
0.03115
0.03086
Lasso
0.04221
0.02636
1.02
0.048556
0.02791
0.02526
0.01219
0.02602
0.026-16
0.01137
0.03292
0.02083
Ridg
0.04247
0.02884
0.02475
0.04216
0.02816
0.02402
0.04191
0
0.02251
0,04110
0.02620
0.02161
Table 13: The RMSE result on predicting C of (éTFIM) with varied system size N and finetuning training size ru. M is fixed 64. MLP (CNN)- -x layers represents neural network MLP (CNN) that composed of layers with residual connection. The best results are highlighted in boldface while the second-best results are distinguished in underlined.
Methods
N
48
N
63
- 100
-
127
20
1%0
60
P50
100
1%.6
20
7%.
60
| PsB
100
7%.0
20
PAa 60
PPn
100
PLA
20
Pan -0
PPnh 100
CS
0.20924
0.20990
0,21092
0,21150
XLP-2 Layers
0.07899
0.06371
0.05524
0.07986
0.05279
0.04283
0.08293
0.05303
0.04630
0.07908
0.05006
0.04333
XLP-3
layers
0.06080
0.05664
0.08074
0.06514
0.08828
0.08914
0.06301
0.06358
0.07317
0.06324
0.06510
0.07327
X1P-1
layers
0.05912
0.05791
0.05980
0.05899
0.05705
0.06163
0.05678
0.05628
0.06977
0.05535
0.06196
0.07197
51P-5
layers
0.07422
0.06545
0.05739
0.07311
0.06921
0.08922
0.06648
0.06556
0.070:14
0.06911
0.07222
0.06867
ONN-7
Tayers
0.12845
15030
0.08935
0. 02227
0.16086
0. 0.10315
0. 0010884
0.08879
0.05177
0.10495
0.08535
0.04647
CNN-3
Layers
0.13545
0.17135
0.12004
0.12545
0.17126
0.11778
0.11433
0.11267
0.05027
0,13312
0,03562
0.06347
CNN-4
Loyars
1363
0.17178
0.12015
02668
0.17103
0.19609
0. 0.12221
0.11046
0.06586
0.13757
0.10498
0.05556
CXN-10
layera
0. 10861
0.14012
039999
0. 10899
0.14113
0.136-0
0.08386
0.10294
0.0G330
0.07107
0.00196
0.01910
CXN-20
layers
0.06796
0.07030
0.08552
0.09565
0.03168
0.03917
0.17631
0.10762
0.01129
0.05152
0.03588
0.01086
CNS-50
layera
0.05084
0.03783
0.204409
0.29550
0.27408
0,23003
0.27766
0.037066
0.04305
0.28359
0,26455
0,22790
CHN-100 layers
0.31883
0.31729
043
0.31156
0.31115
0.30988
0.30174
0.30196
0.30013
0.29768
0,29570
0,29139
LIMEGFE-T
0,05088
0.03499
0.03008
0.05252
0.03568
0.031882
0.05217
0.03476
0.03012
0.05259
0.03641
0,83084
Lasso
0.01624
0.03219
0.02812
0.016633
0,03980
0.02839
0.04073
0.03256
0.02899
0.01583
0.03263
0.02932
Ridge
0.04473
0.03113
0.02807
0.04561
0.03226
0.02839
0.04598
0.03277
0.02883
0.04570
0.03285
0.02911
Table 14: The RMSE results on predicting C of (еHн)) with varied training size n. System size
8. The number testing sets
fixed to x 10 Labels are noise-free (M- + oc). The best results are highlighted in boldface
M + 00
# Params
n = 102
n == 103
n == 10
n == 105
Ridge
0.01M
0.00780
0.00528
0.00367
0.00660
ML.P-4 layers
0.09M
0.04219
0.04172
0.03961
0.03956
CNN-4 layers
1.14M
0.01987
0.02078
0.02056
0.02054
LLM4QPE-F
9.89M
0.03966
0.04304
0.04916
0.04659
performance is 3x better. In such a non-real-world setting, the empirical performance of QSL. models is not ruled by the scaling behavior we have observed in Sec. C.2. Despite the exponentially increased training data, the RMSE results fail to show a declining tendency.
On the other hand, we fix the training data amount n = 10 and vary the system size N E {8, 10, 12, 16, 25. As
25

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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
Table 15: The RMSE results on predicting S of (tHB) with varied training size n. System size N = 8. The number of testing sets is
fixed to 2 x 10 Labels are noise-free (M + oc). The best results are highlighted in boldface
M + 00
# Params
=
10
= 10a
10*
= 10
Ridge
0.01M
0.01563
0.00947
0.00753
0.00851
MLP-4 layers
0.09M
0.10817
0.09142
0.05398
0.05302
CNN-4 layers
1.14M
0.04334
0.02410
0.03520
0.02073
LIM4QPE-F
9.89M
0.10648
0.11171
0.10895
0.10826
Table 16: The RMSE results predicting C of (HB) with varied N. The training and testing set both have 10 samples, with noise-free labels (M x.). The best results highlighted in boldface
M- +
N
N 10
N 12
N 16
N
25
N 31
R Ridge
0.00367
0.00444
0.00566
0.00636
0.00599
0.00579
MLP-4 layers
0.03961
0.03677
0.03460
0.03129
0.02769
0.02625
CNN-4 layers
0.02056
0.03710
0.03432
0.03050
0.02582
0.02381
LIM4QPE-F
0.04666
0.04385
0.03969
0.03728
0.03083
0.02951
demonstrated in Tab. 16, the performance of Ridge predicting Ö of (ынв) of different system sizes is much superior to that of other advanced DL models in all cases. The average RMSE of Ridge is 1/5 times than that of CNN-4 layers, the latter of which is best among the DL. models the table.
C.8. Additional experiment results on measurement outcomes as embeddings
In this subsection, we further explore how measurement outcomes as embeddings play a role in transformer- QSL models like LIM4QPE. We consider two cases: (i) an idealized setting where the number of measurement shots approaches infinity, resulting in noise-free supervised labels; this case, we use randomly generated measurement outcomes as embeddings to preserve the integrity of the learning process; and (ii) a realistic setting, where we vary the number of actual embedded measurement outcomes while maintaining consistent label fidelity. The number of embedded measurement outcomes Memh varies from 64,5)..
For the first case, the embeddings follow the same randomization way introduced in the main text, with system size N E {8, 10, 12, 16, 25, 31}. Tab. 17 shows that model performance generally improves when fewer random outcomes are emhedded.
Table 17: The RMSE results of LLM49PE-F predicting Ö of N-qubit (&HB), with embedding Menb random measurement outcomes. The training and testing both have 10" samples, with noise-free lahels (M 20). Memb is the actual number of emhedded measurement outcomes.
M + 00
N
N 10
N
= 16
N 25
31
Memb 1
0.04666
0.04385
0.04126
0.03728
0.03083
0.03125
Meub 8
0.04746
0.04926
0.03969
0.03984
0.03408
0.02951
Momb 64
0.04795
0.04791
0.04785
0.04043
0.03637
0.03524
Memb = 512
0.04913
0.04521
0.04506
0.03905
0.03406
0.03268
For the second case, we adopt the same experimental setting as that in Sec. C.2. M is fixed to 512. As exhibited in Tab. 18, the performance of DL models remains relatively stable, suggesting that the LLM-like embedding approach empirically renders the measurement outcomes largely redundant as features. This observation highlights a fundamental limitation of the explored embedding methods in correlation prediction tasks.
26

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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
Table 18: The RMSE results of LIM4QPE-F on predicting C' of N-qubit [vHB), with embedding Memb real measurement outcomes. testing size is set to 200, M is fixed to 512. Memb I M is the actual number of embedded measurement outcomes. Natt the training size over the finetuning phase.
63
100
127
Pat 20
Fast = 60
Pan = 100
nsn = 20
Myh = 60
f.hh =
100
Matt = 20
P&t = 60
P1.ft
100
Memh+
0.02555
0.02104
0.02019
0.02307
0.01872
0.01760
0.02239
0.01739
0.01635
Momn+ 8
0.02556
0.02106
0.02019
0.02309
0.01873
0.01760
0.02242
0.01739
0.01635
Mcmb = 64
0.02556
0.02104
0.02019
0.02309
0.01872
0.01759
0. 0.00239
0.01739
0.01636
Memb = 512
0.02560
0.02104
0.02019
0.02309
0.01872
0.01759
0.02240
0.01740
0.01635
Table 19: The RMSE results of predicting C' of N N-qubit [&нB), using MLP (with two embedding strategies), Lasso and Ridge as learning models. Measurement outcomes are embedded as input features MLP in two ways: raw tensor directly characterizing, or averaging (Avg.) over M measurement outcomes for each qubit (M N + x N). E {63, 100, 127]. Training size
n E {20,80) 100}. Measurement shots M E {64, 128, 256, 512}. The best results are highlighted in boldface
N 63
N
100
N 127
7. = 20
n = 60
1 = 100
n2 20
60
n = 100
n1 20
60
n 100
Raw
0.08964
0.05522
0.04872
0.08666
0.04949
0.04055
0.08878
0.05068
0.04076
Avg.
0.05572
0.03522
0.02984
0.05525
0.08972
0.02801
0.05505
0.03951
0.03242
M- 64
Lassc 0.04856
0.02791
0.02326
0.04219
0.02602
0.02646
0.04137
0.03292
0.02083
Ridge 0.04216
0.02816
0.02402
0.04191
0.02711
0.02251
0.04110
0.02620
0.02161
Raw 0.10921
0.05905
0.04835
0.10966
0.06137
0.04485
0.10408
0.06359
0.04554
Avg. 0.04403
0.03034
0.02552
0.04699
0.03561
0.02603
0.04435
0.03421
0.03007
M 128
Lassc 0.03168
0.02171
0.01905
0.03127
0.02045
0.01735
0.03041
0.01980
0.01647
Ridge
0.03169
0.02178
0.01921
0.03069
0.02067
0.01786
0.03053
0.02087
0.01726
Raw 0.14088
0108316
0.06015
0.12558
0.086488
0.05983
0.11720
0.08232
0.06089
Avg. 0.03581
0.02673
0.02272
0.04022
0.02966
0.02168
0.03883
0.03188 88
0.02893
M = 256
Lassc 0.02556
0.01749
0.12125
0.024166
0.01741
0.01467
0.02283
0.01542
0.01324
Ridge 0.02556
0.01751
0.01572
0.02408
0.01697
0.01494
0.02286
0.01576
0.01377
Raw 0.15943
0.11187
0.08246
0.13586
0. 0.10826
0.08329
0.12608
0.10324
0.08330
Avg. 0.03020
0.02475
0.02211
0.03713
0.02864
0.02272
0.03644
0.02962
0.02618
= 512
Lassc 0.02037
0.01586
0.11038
0.01892
0.01403
0.01263
0.01702
0.01257
0.01117
Ridge
0.02036
0.01583
0.01436
0.01891
0.01404
0.01271
0.01798
0.01285
0.01186
C.9. Do results depend on how measurement outcomes are embedded?
We further evaluate two additional embedding strategies commonly used in DL models for QSL tasks, beyond the three methods discussed in the main text. Specifically, these additional strategies include (i) using raw normalized measurement outcomes input (Raw) and (ii) using averaged normalized outcomes input (Avq. .). We evaluate DL models using these two embedding strategies for predicting Č' of (UнB).
The results, summarized in Tab. 19, show that while the averaged embedding remains inferior to classical ML models such as Lasso and Ri Rige it dramatically outperforms the other model using Raw embedding strategy in the predicting correlation task. On the issue how to better design the feature map (or proper embedding) of measurement information for neural networks, we leave this as future work.
C.10. A case of visualization
This subsection visualizes the prediction results of Ò for a 48-qubit (ıHB) system, using a single Hamiltonian sample and Ri Ridge regression as the learning model. The training set consists of n. 100 samples, each measured with M = 128 shots.
27

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Rethink the Role of Deep Learning towards Large-scale Quantum Systems
Exact (testing label)
Approximate (classical shadow)
Prediction value
RMSE=0.00000
RMSE=0.15411
RMSE=0.02183
Figure 12: A case of comparison between Ridge and baseline (CS) on predicting Ci with 48-qubit (WHB). We take one Hamiltonian sample as an example, the left is the exact ground truth, the middle is the estimation by CS, and the right is the estimation by Ridge, where RMSE is 0. 0.15411 and 0.02183, respectively.
Figure 12 compares the performance Ridge regression and the baseline estimator, classical shadows (CS), in predicting the correlation matrix Ci for the selected sample. The matrix C is of size 48 x 48, with each element Ci E (0, 1]. Compared to the ground truth (i.e., the testing label), Ri Ridge exhibits stronger generalization: its RMSE is significantly lower than that of CS, which primarily reflects the noise level present in the training labels.
28


Reviews:
[{"review_id": "sjzSSoaCtY", "reviewer": "Reviewer_HcSt", "summary": "The authors study supervised machine learning in the framework of quantum tasks in particular the Ground State Properties identification and Quantum Phase Classification. The authors evaluate a set of shallow and deep classifiers in both tasks and evaluate the computational cost as well as accuracy. The authors study three types Hamiltonian specification of problems, the Heisenberg models, Ising transverse models and the neutral Rydberg atoms models. The tasks under consideration are the estimation of the ground state and the quantum phase classification. The authors provide a set of experiments showing that in the average the DL models are not much better than than the used shallow classifiers.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank Reviewer HcSt for insightful comments. We have addressed all your concerns in the detailed responses below. For clarity, questions in `Claims and Evidence`, `Theoretical Claims`, `Strengths and Weaknesses`, and `Questions for Authors` are referred to as `C&E`, `TC`, `S&W`, and `QA`, respectively. All newly added simulations are attached to [*LINK*].\n\n[*LINK*] https://anonymous.4open.science/r/ml4quantum-C80F/Rebuttal_icml_25.pdf\n>**Q1 [`C&E`, `TC`] The evaluation criteria are problematic.**\n\nWe respectfully disagree with the reviewer’s concern regarding the validity of our evaluation criteria, particularly for dataset size $n$. **While many learning tasks assume access to large datasets, this assumption is impractical in quantum system learning (QSL) due to the high cost of quantum data generation**. As detailed in Appendix A.4, a rough estimation reveals that collecting $10^6$ examples would take about 35 years on current ion-trap quantum chips.\n\nDespite this, prior works often overlook quantum resource constraints and benchmark DL models using unrealistically large $n$, raising concerns about their realistic applicability. To address this gap, we **intentionally constrain** $n$ to enable a fair and realistic comparison between current ML and DL models in QSL, as stated in Line 44, Page 1.\n>**Q2 [`C&E`, `TC`] Are DL models unfairly compared by limited data?**\n\n As noted in **Q1**, we **have to use restrictive datasets in QSL** because of the practical constraints of current quantum chips, i.e., one must choose between a high-quality dataset with small $n$ or a low-quality dataset with large $n$. To address the latter, we conducted simulations with $n=10^5$. As shown in Table 3 & 4  [*LINK*], together with those in the main text, show that ML models **consistently outperform existing DL models** across commonly studied QSL datasets.\n> **Q3 [`C&E`] Would specialized DL models perform better here?** \n\nYes, it is possible. In Appendix C.5, we have numerically shown that DL can outperform ML models for long-tailed datasets. Notably, there is **ongoing research** on exploring scenarios where DL models can surpass ML models in QSL tasks.\n\nHowever, the **primary goal** of our study is to revisit **existing DL and ML models on standard QSL tasks** under realistic conditions **instead of exploring new scenarios where DL models advance**. The achieved findings in this submission not only inform DL model design but also motivate further exploration into when and how DL can outperform ML models in practical QSL scenarios as the reviewer is concerned.\n> **Q4 [`C&E`] How do CNNs perform as their depth increases?**\n\nWe have conducted additional experiments to address this concern. We apply the four models introduced in the main text to estimate the correlation $\\bar{C}$ of $|\\psi_{\\rm{HB}} \\rangle$ and $|\\psi_{\\rm{TFIM}}\\rangle$ with the qubit count $N$ ranging from $48$ to $127$, and varying $n$ from $20$ to $100$. The depth of DL models is increased up to $100$ layers. Other settings are the same as those in the submission. Results are summarized in Tables 1 & 2 [*LINK*]. As the depth of DL models increases, **their performance underperforms shallower ones, widening the gap with ML models**.\n> **Q5 [`S&W`, `QA`] Use generators to scale data and assess performance trends.**\n\nAs shown in Table 3 & 4 & 5 [*LINK*], we consider infinite measurement shots and include additional simulations with $n=10^5$ for 8-qubit $|\\psi_{\\rm{HB}}\\rangle$, and $n=10^4$ for larger-scale systems. Even under these settings, the achieved results continue to demonstrate the advantage of ML models.\n\nLet us kindly remind the reviewer that while classical simulations can generate large noise-free datasets, they contradict QSL’s ultimate goal, which is using the proposed models to facilitate **real QSL tasks**. Due to the high cost of querying quantum systems explained in **Q1**, the accessible QSL dataset is very limited in practice. The use of classical simulators here is a convention, allowing researchers to bypass the need for scarce quantum resources and explore the fundamental capabilities of different models.\n> **Q6 [`QA`] Would DL models outperform ML with more data, or is the task fundamentally better suited to ML?**\n\nIf 'f/ classifier' refers to ML models, the results in **Q2** and **Q5** confirm that **current DL models do not outperform ML models with more data**. \n\nMoreover, the reasons why current DL models underperform ML models remain largely unexplored. The results presented in this submission **further highlight the urgency of addressing this gap**. Besides the potential impact of global features as noted by the reviewer, there are two other possible factors: \n- Most DL models overlook the challenges posed by limited data availability, whereas many ML models explicitly account for this with theoretical guarantees; \n- DL models are often adapted from unrelated learning tasks and typically employ poorly tailored feature maps.", "claims_and_evidence": "The authors claim that the DL models are either equivalent or worse when trained and evaluated along with a set of shallow classifier on the two main quantum learning tasks. However I feel that in particular the evaluation criteria being problematic. The reasons that DL work sis because the amount of data became large enough to allow the averaging of features and updates and adjustment of millions of parameters. In this particular work I feel that DL are unjustly compared in conditions that are not well suited to the basic principles of training DL, that is the amount of data is not really a parameter available to manipulation. In addition the there are specific paradigms such as few shot learning, single shot learning, reduced precision networks) that would probably behave better in the conditions described in this paper. On the other hand the models evaluated are using single layer convolutional layer and as such are not exploring the full potential of the convolutional features.", "methods_and_evaluation_criteria": "Yes the general idea to evaluate the machine learning methods is consistent with the authors claim.", "theoretical_claims": "There are not any theoretical claims, but my problem remains with the fact that models that are compared might not have been compared on a proper basis.", "experimental_designs_or_analyses": "Experiments are designed well for this type of work", "relation_to_broader_scientific_literature": "The work is well situated in the current works on the learning of quantum properties from classical observations.", "essential_references_not_discussed": "Most of relevant literature has been cited.", "other_strengths_and_weaknesses": "I feel that this work is not deep enough for the experiments. The authors had a generator and thus should use the full potential to generate large amount of data and relate the amount of data to the performance of the models on a larger scale", "other_comments_or_suggestions": "I would suggest to augment all experiments beyond reasonable real-world data availability and consider the problem space as a method for estimating the limits and the requirements for these methods to work to an expected error", "questions_for_authors": "So if one had more data would the f\\shallow classifiers be under-perform or is the nature of this data so global that local filters fail to extract the necessary information for learning?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "R1N892he6J", "reviewer": "Reviewer_Wxrf", "summary": "This paper considers the state properties of quantum systems problems. In order to deal with the issue of unfair comparison, this paper benchmarks DL models against traditional ML approaches across the Hamiltonian.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate Reviewer Wxrf's thoughtful review. For clarity, questions in `Strengths And Weaknesses` are abbreviated as `S&W`. All newly added simulations are attached to [*LINK*].\n\n[*LINK*] https://anonymous.4open.science/r/ml4quantum-C80F/Rebuttal_icml_25.pdf\n\n> **Q1 [`S&W`] The motivation is not clear in deriving the observation that the outcomes are largely redundant as input representations.**\n\nThanks for the comment. Let us first kindly remind the reviewer that as indicated by Reviewer m6FX, the key motivation of our work is *\"...benchmarks deep learning models against traditional machine learning approaches while maintaining equivalent quantum resource usage...\"*. To this end, we conducted systematic experiments to explore the capabilities and limitations of current learning models on standard QSL tasks.\n\nAs the reviewer rightly noted, a key perspective in understanding the power of DL models lies in unveiling how data features influence their performance. In particular, assessing the necessity of using measurement outcomes as input features is important for the following three reasons: \n- (1) Recall that quantum resources are expensive as shown in Appendix A.4. If DL models with and without measurement outcomes as input features achieve similar performance, measurement-free DL models are preferred, as they **avoid quantum hardware interaction during inference** and substantially reduce quantum resource consumption.  \n \n- (2) Feature selection impacts model size and computational efficiency. Removing unnecessary features of the model helps reduce the model size.  For instance, in the case of $8$-qubit $|\\psi_{\\rm{HB}}\\rangle$ with $512$ shots, the used MLP size is reduced over $2$ times when removing raw measurement outcomes as input features, leading to a substantial reduction in memory and inference costs on the classical side.\n\n- (3) The proposed randomization test helps identify effective embedding strategies for measurement outcomes. As noted in **Q1** by Reviewer 3fBm, our results suggest that a well-designed embedding can improve the performance of DL models. Following this approach, any newly proposed embedding strategy can be evaluated using the randomization test to assess its contribution to learning performance.\n \n> **Q2 [`S&W`] It is unclear whether this phenomenon is based on the quantum system or deep learning.**\n\nBefore presenting the details to address the reviewer’s concern, we would like to respectfully highlight that **our work is the first to systematically evaluate the performance of current ML and DL models on QSL tasks under realistic quantum resource constraints**. The obtained empirical findings offer concrete evidence to guide the community in identifying **new scenarios where DL models may excel**, while also shedding light on **why existing DL models often underperform compared to ML models** in standard QSL settings. Below, we outline two possible explanations for the latter case.     \n\n- (1) Under realistic and constrained quantum resource budgets, collecting large-scale and high-quality datasets for QSL is often prohibitively expensive. This introduces an inherent trade-off between the number of training examples $n$ and the quality of labels associated with each sample. When $n$ increases, a fixed total query budget necessitates allocating fewer measurement shots per example, resulting in increased label noise. In this context, expanding the input feature space by including raw measurement outcomes does not necessarily improve performance, since they are noisy and high-dimensional, which may introduce **redundant or irrelevant information**.\n\n- (2) As noted in our response to **Q6** of Reviewer HcSt, **current DL-based QSL models lack well-designed feature mappings**, limiting their ability to extract useful information from the measurement outcomes to enhance the learning performance. As a result, it is **an active research field** about developing novel embedding maps with respect to the measurement outcomes. \nWe have conducted additional experiments on embedding measurement outcomes for MLP shown in Table 8 [*LINK*]. The achieved results reveal that a good pre-processing of measurement outcomes brings out performance improvement. This motivates the exploration of **more effective embedding techniques** and **novel neural architectures** to enhance DL models' performance in QSL tasks.\n\n> **Q3 [`S&W`] In the deep learning experiments, a deeper network is not considered in the setting, which does not clearly validate the findings.**\n\n We have followed the reviewer's advice to extensively evaluate much deeper neural networks, such as MLP and CNN, incorporating well-known techniques such as dropout, residual connections, and appropriate $l_2$ regularization to ensure effective training. Results are summarized in Tables 1 and 2 (see [*LINK*]). The achieved results indicate that compared to DL models with increased depth, ML methods consistently retain their advantages.", "claims_and_evidence": "yes", "methods_and_evaluation_criteria": "yes", "theoretical_claims": "yes", "experimental_designs_or_analyses": "yes", "relation_to_broader_scientific_literature": "benchmark DL models", "essential_references_not_discussed": "yes", "other_strengths_and_weaknesses": "This paper considers the problem of how to learn information from deep learning under quantum resources. In the process of revisiting, the total number of queries is fixed, and the randomization test is designed. The motivation is not clear in deriving the observation that the outcomes are largely redundant as input representations. Moreover, it is unclear whether this phenomenon is based on the quantum system or deep learning. Furthermore, in the deep learning experiments, a deeper network is not considered in the setting, which does not clearly validate the findings.", "other_comments_or_suggestions": "no", "questions_for_authors": "4", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YUg7ifu4DI", "reviewer": "Reviewer_m6FX", "summary": "The paper examines the necessity and effectiveness of deep learning in quantum system learning (QSL), particularly in estimating ground state properties (GSPE) and quantum phase classification (QPC). \nThe paper systematically benchmarks deep learning models against traditional machine learning approaches while maintaining equivalent quantum resource usage.\nThe paper shows that machine learning and deep learning models both improve with more training data and measurement snapshots.\nMoreover, machine learning models often achieve performance comparable to or exceeding that of deep learning approaches in QSL tasks.\nLastly, a proposed randomization test shows that measurement outcomes have minimal impact on deep learning models’ performance in GSPE but are important for QPC.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the Reviewer m6FX's positive affirmation of our work. Below, we provide detailed responses to the remaining concerns. For clarity, questions in `Strengths and Weaknesses` and `Questions for Authors` are referred to as `S&W` and `QA`, respectively. \n\n> **Q1 [`S&W`, `QA`] The paper should present the evaluation protocol limitations of the prior works.**\n\nThanks for the comment. We have followed the reviewer's suggestion to append further explanations about the limitations of evaluation protocols adopted from prior literature in related works and Appendix A.3. For self-consistency, let us first briefly recap how our work differentiates from previous studies. In Line 47 of our submission, we explicitly state that our study *\"... heuristic advantages of DL models often ignore their dependence on quantum resources, resulting in an unfair comparison with ML approaches\"*. Besides, as indicated in Line 787, prior works have primarily focused on proposing new neural network architectures for QSL tasks, which laid the groundwork for our benchmarking efforts.  \n\nIn what follows, we provide a summary of our main revisions to further highlight the limitations of the evaluation protocols used in prior work.\n\nPrior studies on DL models in QSL typically rely on classical simulators to generate ideal labels [Wang et al., arXiv:2211.16943; Zhang & Di Ventra, PRB 107, 075147 (2023); Tang et al., ICLR 2024.], without accounting for the limitations imposed by a finite number of measurement shots. This overlooks a critical aspect of practical quantum settings, where obtaining accurate labels requires a substantial number of costly and limited quantum measurements, e.g., collecting $10^6$ examples with 1000 shots per example would take about 35 years on current ion-trap quantum chips. In contrast, ML-based QSL studies more often incorporate quantum resource constraints by using labels derived from classical shadows or from a limited number of actual quantum measurements, thereby better reflecting realistic deployment scenarios [Huang et al., Science 2022.; Cho et al., Nat. Commun. 2024.; Lewis et al., Nat. Commun. 2024.]. **This inconsistency in evaluation practices hampers fair comparisons between DL and ML models and risks overstating the practical effectiveness of DL approaches**. As stated in Line 47 of our manuscript, such inconsistencies lead to the problematic claim that DL models offer heuristic advantages over ML models.\n\nIn our study, we explicitly address these limitations by **unifying quantum resource constraints across all models and evaluating their performance under this consistent cost framework**. This evaluation protocol allows meaningful comparison and motivates the development of DL models that are viable under practical quantum constraints.\n\n> **Q2 [`QA`] Provide a theoretical explanation or proof of this phenomenon.**\n\nThanks for the insightful comment. Let us first address that our work represents the first systematic empirical study evaluating the performance of existing ML and DL models in quantum system learning (QSL) under realistic quantum resource constraints. Much like the influential study *\"Rethinking Generalization in Deep Learning\"* [Zhang et al., ICLR 2017.], our aim is not to theoretically dismiss the potential of DL models in QSL, but rather to **initiate and guide rigorous theoretical investigations into their capabilities and limitations in this setting**. The empirical findings reported in our work reveal the current challenges faced by standard DL architectures and offer concrete evidence that motivates the development of QSL-oriented DL models. These results underscore the need to better understand when and how deep learning can be effectively applied in QSL. \n\nWhile our study is primarily empirical, the observed phenomenon can be partially understood through known theoretical principles in statistical learning theory.\n- (1) Under fixed quantum resource budgets, there exists an inherent trade-off between the number of training examples $n$ and the number of measurement shots $M$ per example. Increasing $n$ reduces $M$, leading to higher label noise. DL models, which typically require large amounts of high-quality data to generalize well, are particularly sensitive to this noise. In contrast, ML models with hand-crafted feature maps are often more robust in such regimes and come with provable theoretical guarantees.\n- (2) The use of raw measurement outcomes as input features substantially increases input dimensionality while often introducing redundancy. This combination—high-dimensional and noisy input with limited supervision—poses a considerable challenge for DL models. Without task-specific architectural design, deeper networks may overfit or fail to extract meaningful structure from the data.", "claims_and_evidence": "The paper provides extensive experimental results to enforce the claims:\n\n- The paper ensures fair comparisons between the deep learning and machine learning models by maintaining equivalent quantum resource usage while testing across three distinct Hamiltonian families.\n\n- The paper presents extensive empirical results on GSPE and QPC, showing clear trends in model performance when the data size and measurement snapshots are increased. Moreover, the experimental results show that machine learning models can outperform deep learning models under resource constraints.\n\n- The paper conducts a randomization test to analyze the impact of measurement outcomes as input features, providing strong statistical evidence that deep learning models do not effectively leverage this information in GSPE tasks. Meanwhile, for QPC tasks, measurement outcomes significantly affect model performance.", "methods_and_evaluation_criteria": "Since the paper presents benchmarking settings to evaluate the role of machine learning and deep learning models, the provided benchmarks, as discussed in \"Claims And Evidence\", make sense for the problem.", "theoretical_claims": "The paper does not provide any theoretical proof of the claims.\nThis is somewhat acceptable since the paper focuses on extensive empirical proofs.", "experimental_designs_or_analyses": "The experimental designs and analyses look reasonable.", "relation_to_broader_scientific_literature": "Many studies have applied DL to quantum tasks, assuming it provides superior performance [1,2,3,4].\nThe paper's empirical results question the claimed advantages of deep learning models in quantum tasks.\n\nThe findings in the paper show that machine learning methods may be more suitable for real-world QSL applications due to the limited availability of quantum resources.\n\n\n[1] Wang, Haoxiang, et al. \"Predicting properties of quantum systems with conditional generative models.\" arXiv preprint arXiv:2211.16943 (2022).\n\n[2] Tran, Viet T., et al. \"Using shadows to learn ground state properties of quantum hamiltonians.\" Machine Learning and Physical Sciences Workshop at the 36th Conference on Neural Information Processing Systems (NeurIPS). 2022.\n\n[3] Zhang, Yuan-Hang, and Massimiliano Di Ventra. \"Transformer quantum state: A multipurpose model for quantum many-body problems.\" Physical Review B 107.7 (2023): 075147.\n\n[4] Tang, Yehui, et al. \"Towards LLM4QPE: Unsupervised pretraining of quantum property estimation and a benchmark.\" The Twelfth International Conference on Learning Representations. 2024.", "essential_references_not_discussed": "There are no additional related works that are essential to understanding the key contributions of the paper.", "other_strengths_and_weaknesses": "Besides the experimental results enforcing the claim that the machine learning models outperform the deep learning models, the paper should present the evaluation protocol limitations of the prior works.\nThis could be crucial since the paper's results go against the prior results.\nThe reader might want to see the reasons, and the difference could be the evaluation protocol.", "other_comments_or_suggestions": "No other comments or suggestions.", "questions_for_authors": "I am kind of positive about the paper. Two things that I am concerned about are:\n\n- the theoretical explanation or proof of this phenomenon;\n\n- and the limitations of the prior evaluation protocol.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "MSBKZ53ha6", "reviewer": "Reviewer_3fBm", "summary": "The authors conduct a thorough investigations of Deep Learning (DL) vs Machine Learning (ML) methods and all of the design choices surrounding them for the tasks of quantum system learning (QSL).\n\nThey consider the tasks of quantum phase classification (QPC) and ground state property estimation (GSPE) and study three Hamiltonians.\n\nThe authors point out that DL methods require more quantum resources.\n\nThe authors conduct thorough experiments to test scaling laws of DL and ML methods, the effect of measurement as data features in in GSPE and QPC tasks, and offer some observations to the QSL community.\n\nML: linear regressors, kernel methods and tree methods.\nDL methods: multilayer perceptrons, convolutional neural networks, and self-supervised learning.\n\nReal outcomes as embeddings show little improvement to DL model via a randomization test.\n\n## update after rebuttal\n\nI have monitored the rebuttal and will maintain my score. Thank you to the authors for their responses.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank Reviewer 3fBm for the positive recognition of our work. Below, we address the remaining concerns. For clarity, questions in `Comments Or Suggestions` and `Questions For Authors` are referred to as `COS` and `QA`. All newly added simulations are attached to [*LINK*].\n\n[*LINK*] https://anonymous.4open.science/r/ml4quantum-C80F/Rebuttal_icml_25.pdf\n\n> **Q1 [`COS`] Do results depend on how measurement outcomes are embedded?**\n\nAs noted by the reviewer, an important future direction in quantum system learning (QSL) is the exploration of task-specific embedding strategies, which we have discussed in Line 437 of the main text. That is, **our empirical findings are not intended to dismiss the potential of using measurement outcomes as input features in QSL but rather to motivate the development of novel embedding strategies that can enhance learning performance**.\n\nTo further address the reviewer’s suggestions, we have added two additional simulation tasks, as detailed below.\n- (1) For the three embedding strategies explored in the main text, we vary the number of measurement shots $M$ from $1$ to $512$ and evaluate the performance of DL models when applied to predict the correlations of $|\\psi_{\\rm{HB}} \\rangle$ with $N\\in \\{63, 100, 127\\}$. The results are summarized in Table 7 [*LINK*], where the performance of DL models does not vary too much. These results suggest fundamental limitations of the explored embedding methods in correlation prediction tasks.\n- (2) We further evaluate two additional embedding strategies commonly used in DL models for QSL tasks, beyond the three methods discussed in the main text. Specifically, these additional strategies include (i) using raw normalized measurement outcomes as input and (ii) using averaged normalized outcomes as input. We evaluate DL models using these two embedding strategies for correlation prediction of $|\\psi_{\\rm{HB}} \\rangle$. The results are summarized in Table 8 [*LINK*].  Among all embedding methods, we find that the averaged embedding, while still underperforming ML models, achieves better learning performance than the others in predicting the correlations of $|\\psi_{\\rm{HB}} \\rangle$. \n\n> **Q2 [`COS`] Is there a more intrinsic measure to the simulations, akin to FLOPs in standard analyses in deep learning scaling?**\n\nWe would like to address this question from two aspects. \n- (1) Quantum resources versus classical resources. In QSL, quantum resources (i.e., the total number of queries to the quantum system) are **significantly more scarce and expensive than classical resources** like memory, compute time, or FLOPs. This scarcity motivates our decision to unify quantum resource usage across all evaluated models, enabling fair and realistic comparisons.\nAt the same time, as the reviewer rightly noted, classical resources are also important. Metrics like FLOPs remain valuable for assessing the computational complexity of different learning models. **Since quantum and classical resources are complementary**, future benchmarking efforts could benefit from jointly considering both to more fully characterize learning efficiency in hybrid quantum-classical systems.\n- (2) The way of acquiring information from quantum systems. Recall that most current QSL models rely on **incoherent and local measurements**, where the number of measurement shots is a natural and practical metric. However, if a QSL model is instead based on coherent measurements or intermediate quantum memory, a more comprehensive cost model becomes necessary. In such cases, one must also account for the differing costs associated with acquiring information from the quantum system.\n\n> **Q3 [`QA`] Do your findings hold without quantum resource constraints?**\n\nYes, our findings remain valid even when the quantum resource constraint is lifted. As noted in our response to **Q5** from Reviewer HcSt, we conducted additional simulations using $n=10^5$ training examples for 8-qubit systems, with **an infinite number of measurements per sample** (see Tables 3 and 4 [*LINK*]). Even under these settings, the achieved results continue to exhibit the advantage of ML models.\n\n> **Q4 [`QA`] In Fig. 1, why is there a difference between the left half and the right half?**\n\nWe would like to clarify that the apparent discrepancy between the left and right halves of Fig. 1 primarily stems from differences in hyperparameter settings across tasks. In particular, the learning models were independently tuned for correlation and entropy prediction, and thus their performance is not directly comparable across the two objectives. As a result, interpreting the reversal in scaling trends as a meaningful comparison between the tasks can be misleading.\n\nTo further support this explanation, we include additional simulations. As summarized in Table 9 [*LINK*], by properly adjusting the input feature dimension of Ridge regression, the performance gap between TFIM and HB systems in the correlation task can be eliminated.", "claims_and_evidence": "The authors have supported the claims of their work, showing that ML methods are on par with DL methods and rethinking the role of measurement outcomes as embeddings, scaling laws, etc.", "methods_and_evaluation_criteria": "The methods and evaluations in the paper are appropriate and have captured the existing literature well.", "theoretical_claims": "This is an empirical study, so N/A", "experimental_designs_or_analyses": "The design and analyses are sound and well-documented.", "relation_to_broader_scientific_literature": "The paper is an important comparisons of established methods in the field and the key contributions are observed trends that will inform future research in the field.", "essential_references_not_discussed": "Not that I am aware of.", "other_strengths_and_weaknesses": "Very thorough experiments:\n* scaling laws\n* comparisons of ML and DL methods under the same quantum measurement bugdets\n* effect of measurements on predictions\n* scaling the size of the model as vs the strength of the regularization.", "other_comments_or_suggestions": "The role of the measurement outcomes as embeddings should be discussed, e.g. maybe some exploration of the way these embeddings go into the model is important, and your studies may change when controlling for that.\n\nIdeally the quantum resources shouldn't be only measured as number of measurements and examples. Is there a more intrinsic measure to the simulations, akin to FLOPs in standard analyses in deep learning scaling?", "questions_for_authors": "How does your analysis change if you lift the restriction on the quantum resources? Do your findings still hold?\n\nIn Figure 1 why interestingly why is there a difference between the left half and the figure and the right half? It seems that the scaling comparative trends reverse as we go from left to right. Why the different prediction objectives (correlation vs. entropy) provide different ordering of the scaling curves? Could you elaborate on this?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper examines the necessity and effectiveness of deep learning in quantum system learning (QSL), particularly in estimating ground state properties and quantum phase classification. Through systematic benchmarks, they demonstrate that the classical machine learning approaches often outperform deep learning approaches in QSL tasks under the realistic scenario of a fixed (and limited) quantum resource budget. Their observation suggests a two-sided problem: Deep learning models may require more data than what quantum systems can realistically provide, and at the same time, existing deep learning models and embeddings may not be suitable for QSL tasks. This paper makes a valuable contribution by highlighting the risk of the field advancing in a potentially misguided direction. It serves as a timely reminder to approach such topics with more careful consideration, and it could also motivate further research aimed at understanding the limitations of current approaches and developing more effective deep learning models and embeddings that could eventually make these methods practical for QSL.
Decision: Accept (poster)