TITAN: A Trajectory-Informed Technique for Adaptive Parameter Freezing in Large-Scale VQE
Authors: Yifeng Peng
Xinyi Li
Samuel Yen-Chi Chen
Kaining Zhang
Zhiding Liang
Ying Wang
Yuxuan Du
Venue: NeurIPS 2025 poster ()
Keywords: Barren Plateau
Variational Quantum Algorithms
Circuit Optimization
Measurement Overhead
VQE
Parameter Freezing
Primary area: other

Abstract: Variational quantum Eigensolver (VQE) is a leading candidate for harnessing quantum computers to advance quantum chemistry and materials simulations, yet its training efficiency deteriorates rapidly for large Hamiltonians. Two issues underlie this bottleneck: (i) the no-cloning theorem imposes a linear growth in circuit evaluations with the number of parameters per gradient step; and (ii) deeper circuits encounter barren plateaus (BPs), leading to exponentially increasing measurement overheads. To address these challenges, here we propose a deep learning framework, dubbed Titan, which identifies and freezes inactive parameters of a given ansätze at initialization for a specific class of Hamiltonians, reducing the optimization overhead without sacrificing accuracy. The motivation of Titan starts with our empirical findings that a subset of parameters consistently has negligible influence on training dynamics. Its design combines a theoretically grounded data construction strategy, ensuring each training example is informative and BP-resilient, with an adaptive neural architecture that generalizes across ansätze of varying sizes. Across benchmark transverse-field Ising models, Heisenberg models, and multiple molecule systems up to $30$ qubits, Titan  achieves up to $3\times$ faster convergence and $40$–$60\%$ fewer circuit evaluations than state-of-the-art baselines, while matching or surpassing their estimation accuracy. By proactively trimming parameter space, Titan lowers hardware demands and offers a scalable path toward utilizing VQE to advance practical quantum chemistry and materials science.

TITAN: A Trajectory-nformed Technique for Adaptive Parameter Freezing in Large-Scale VQE
Yifeng Pengl., Xinyi Li', Samuel Yen-Chi Chen', Kaining Zhang', Zhiding Liang', Ying Wang Yuxuan Du
Stevens Institute Technology
Wells Fargo Nanyang Technological University
Rensselaer Polytechnic Institute
ypeng21@stevens. edu, x1i215@stevens. edu, ycchen1989@ieee.org
kaining zhangōntu www.ggitttuuuu.u.e....m sg, zlianghahaha@gmail com, ywang60stevens. edu,
yuzuan. yuruan.duuntu. uuudtuu.ed..ag. edu.ag
Abstract
Variational quantum Eigensolver (VQE) a leading candidate for harnessing quan- tum computers to advance quantum chemistry and materials simulations, yet its training efficiency deteriorates rapidly for large Hamiltonians. Two issues underlie this bottleneck: (i) the no-cloning theorem imposes linear growth in circuit eval- uations with the number of parameters per gradient step; and (ii) deeper circuits encounter barren plateaus (BPs), leading to exponentially increasing measurement overheads. To address these challenges, here we propose deep learning frame- work, dubbed TITAN, which identifies and freezes inactive parameters of given ansätze at initialization for specific class of Hamiltonians, reducing the optimiza- tion overhead without sacrificing accuracy. The motivation of TITAN starts with our empirical findings that a subset of parameters consistently has negligible influence on training dynamics. Its design combines a theoretically grounded data construc- tion strategy, ensuring each training example is informative and B--resilient, with an adaptive neural architecture that generalizes across ansätze of varying sizes. Across benchmark transverse-ficke fiell Ising models, Heisenberg models, and multiple molecule systems up to 30 qubits, TITAN achieves up to 3x faster convergence and 40-60% fewer circuit evaluations than fate-off-theart baselines, while matching surpassing their estimation accuracy. By proactively trimming parameter space, TITAN lowers hardware demands and offers scalable path toward utilizing VQE to advance practical quantum chemistry and materials science.
Introduction
2 is widely believed to provide computational advantages in electronic- structure calculations (ESCs) G. pivotal task in catalyst discovery 41 drug design [5), and materials innovation 6 9.. The flagship protocol that leverages the power of quantum computers to advance ESCs is the variational quantum Eigensolver (VQE) [10 which estimates molecular ground-state energies by iteratively adjusting a parameterized ansätze lower the expectation value of fermionic Hamiltonian mapped onto qubits. Recent experiments have verified the potential of VQE in solving ESCs for small-scale systems [3) Despite the progress, scaling VQE from pedagogical circuits to chemically relevant systems confronts two intertwined training bottlenecks, i.e., (i) burren plateaus
Quantum computing
* * f Corresponding Author. The views expressed in this article those of the authors and do not represent the views of Wells Fargo. This article is for informational purposes only. Nothing contained in this article should be construed as investment advice. Wells Fargo makes no express or implied warranties and expressly disclaims all legal, tax, and accounting implications related to this article. Appendix is available at https://github. com wwt.m//ifeeggm///A/AAApppppaepep
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

(BPs) 77 18 and (ii) expensive measurement overhead. BPs describe the exponential suppression
of gradient magnitudes in the cost-function landscape as circuit depth or system size grows, thereby impeding effective optimization 21 Over the past years, huge efforts have been devoted to
overcoming BPs, where representative works include developing initialization heuristics [8] 22 23
layerwise training 24 25 architecture design [26]. and adaptive freezing (27,22]
In contrast to the extensive efforts focused on mitigating BPs, the equally fundamental challenge of measurement overhead has received comparatively less attention [30 31. This substantial overhead
arises from the fundamental constraint that quantum states cannot be cloned and only unitary evolution is permitted, necessitating external reconstruction of analytic derivatives [32]. In particular, the widely used parameter-she
rule requires two circuit evaluations per tunable gate angle, effectively doubling the number of shots per parameter [6). As result, the total measurement cost scales linearly with the number of parameters and increases with both circuit depth and qubit count. For instance, the benzene (C,Hs), a simple molecular Hamiltonian, requires 10--10a circuit evaluations (about hours for lon-trap quantum computers), even after grouping optimizations [37]. Thus, even in the absence of BPs, the scalability of VQE is fundamentally constrained by measurement overhead. Addressing this often-overlooked bottleneck is essential for translating the theoretical potential of VQE into practical quantum-computing workflows.
Initial attempts have been devoted reducing the measurement overhead in large-scale VQE. Prior strategies fall into several broad categories. Commuting-set partitioning and classical- shudow/drrnndomizztionn schemes compress the number of shots required estimate expectation values 388 39] Complementary Pauli-term grouping heuristics that exploit qubit-wise commutativity further reduce the sample count in VQEs [40. Quantum architecture search 45) QuACK 46.. and meta-learning for parameter initialization [7 are proposed to enhance the VQE perfor- mance and reduce the overhead. While these complementary approaches alleviate part of the overall computational burden, prior work typically treats them in isolation without unified perspective. critical open question remains: can jointly addressing multiple aspects the VQE optimization pipeline lead to further reductions in measurement overhead and overall computational cost?
We address this question from new perspec- tive: connecting optimization dynamics with the parameter initialization. This insight stems from
Inteasit
our empirical discovery of "frozen- parameter" phenomenon across diverse Hamiltonians, as
02(0)
8,
R.00
R
shown in Figure [ This constitutes our first contribution, which may inspire further inves- tigations into parameter dynamics within VQE. Building on this observation, we devise deep learning (DL) model, TITAN, aiming to harness the power of neural networks to capture and pre- dict the frozen parameters of VQE for a given family of Hamiltonians across varying scales. As in other deep learning applications, TITAN comprises three stages: dataset construction, model implementation and training, and model inference.
0,00)
R,00
R.(0)
R_00)
8,00)
Incensity
Figure 1: An illustration of the paaizen-arrmmtter" phc- nomenon of VQE. Intensity refers to the number of inactive sessions. More details are offered in Sec. [4]
Despite its promise, implementing TITAN poses notable challenges, especially for dataset curation and model architecture design. During the data construction phase, we proposed the Adaptive Purameter Freezing and Activation (APFA) to collect trajectories of inactive parameters from VQE instantiated on diverse Hamiltonians within the target family. A key requirement is that each trajectory must avoid BPs and convey informative training signals. To meet this criterion, we develop theoretical framework based Gaussian initialization (see Theorem D). exhibiting that our generatienntinnn protocol circumvents BP artifacts. This theoretical result forms a core technical contribution of our work and may be of independent interest. For the model implementation, introduce the voordinate-awar TitaN's adaptability, enabling it to accommodate variations in qubit count, circuit depth, and Hamiltonian structure without retraining. Our final contribution is a systematic evaluation of TITAN'S performance through extensive experiments on standard lattice and molecular systems with up to 30 qubits. The achieved results exhibit that TITAN attains 2 99% of the ground-state energy reached by the baseline model by freezing at least 40% of parameters.
fully-convolutional self-attention (CFCSA) technique to enhance

---

2 Background and Related Works
Quantum Computing Quantum computing leverages intrinsically quantum-mechamee phenom- ena- superppoppiiiionn entanglement, and interference [48 49]. In this framework, the fundamen- tal unit of information is the qubit, a two-level quantum system that can occupy any complex linear combination of computational basis states, i.e., (
[0] + 3 (11), with a. B E C and An n-qubit state represented by a vector in a 2" ddmenniional Hilbert space, ( = -x= Ca (x) -x=0 /cel² = 1. Here |æ)) denotes the computational basis for n qubits, where .........0 Zn) corresponds to binary string (Z1.72,..... Inn of length n, and is defined by the
|a|2 + |
lensor-produch rule [z) = (m1) @ [22) @ .... & (zn).
Quantum gates amount to the unitary operations that transform quantum state into another. Solo- vay-Kitaev theorem for standard constructions 50), which guarantees that any unitary operation can be decomposed into single and two-qubit gates. Measurement in quantum mechanies is described by a positive operator-valued measure (POVM) {Mm) that satisfies the completeness relation Lm M, = I. In the special-but ubiquitous case of a projective measurement in the computa- tional basis, the POVM elements reduce the projectors M, (a)) (ar x {0, 1)",, so that measur-
ing an n--ubit state ) = Σ²" Cp (x) yields outcome æ with probability pz = |<x| [ = |cx1?
after which the post-measurement state collapses to (z)).
Variational Quantum Eigensolver (VQE) quantum chemistry 51 one is often interested in finding the ground-state energy of a molecular system described by the Hamiltonian: H Σ, h, H,, where each H, is a tensor product of Pauli operators = (b)), X = (9)), Y= ( 0 ) Z =
( ). and the real coefficients h; encode system-specific molecular parameters 555 166 The VQE aims to approximate the ground state (W/x)) of H by employing parameterized quantum circuit (ansätze) (=)) = U()) (vo). where (ebo is an easily preparable reference state (e.g.. the Hartree- Fock state in chemistry), and U()) is unitary operator constructed from quantum gates whose adjustable parameters O- (0(1), ((2) )(P) are optimized minimize the energy expectation value. The generic form of an N N-qubit ansätze is
P 00)) II (W:V((()))
where V.(0(() denotes the i-th tunable gate and W; refers to the i-th fixed unitary operation. Existing ansätze used in VQEs can be classified into two classes: structured ansätze and unstructured ansätze. For structured ansätze, they are commonly designed with a layer-wise structure, comprising repeated blocks of parameterized gates and entangling gates. A typical example in this class is the hardware- efficient ansatz (HEA) (57) e.g., U())= = II (Wr _ RX(0(())) with RX = exp(-iiX/2))
We = - N-1 CNOT, and CNOT 2,i+1 being applying CNOT [0) (0) & I + 11 (1 @ X gate to the i-th and i + 1-th qubits. Other notable structured ansätze include SU2 58 59 and SEL (60] ansätze. In what follows, we denote an N-qubit VQE with structured ansätze as (fI, N, D)). where D denotes the number of variational parameters per qubit wire in each block and p T.x N. For unstructured ansätze, they generally encode prior knowledge of the explored Hamiltonian in (0). where the corresponding gate layout is denoted by S. Typical instances in this regime include unitary coupled cluster (UCC) [T and unitary coupled cluster with singles and doubles (UCCSD). We denote an N N-qubit VQE with a unstructured ansatz as (H, A(p, S)).
i=1
The objective of VQE is to approximate ground-state energy Eg of H by optimizing 0, i.e.,
(()) = (v(0)
Emin min E(0)= x Eo.
(2)
The optimization is often completed by the gradient-descent optimizer, where the update rule yields 8g < 8, - n 00/000 and 17 is the learning rate 19 20
Related works Improving the optimization efficiency of large-scale VQE remains central challenge
in quantum computing (62 63. Recent progress falls into three major directions: (i) Measurement
grouping. Learning-based grouping [64] and classical-shadow protocols (65 66 compress the
number of shots required to estimate large Hamiltonians, but may incur substantial classical post-
processing overhead for high-qubit systems. (ii) ansätze design. Hardware-efficient circuits [57]
3

---

suffer from BPs; remedies include quantum architecture search 43 67
pruning 27 68 and
ansätze [69. These improve expressibility and trainability at the expense of costly search procedures and domain expertise. (iii) Advanced optimizers. Quantum-
optimizers such as Quack [46] and meta-learning frameworks [70] 71 47 non-convex landscapes, but introduce additional hyperparameters and nested optimization loops. Collectively, these three lines of research are orthogonal to our contribution. TITAN addresses a fourth skiss-prramtter-ppaa reduction via predictive freezing-ther that can be layered on top of any preasurement-gruppee scheme, circuit-desinn strategy, or optimizer, thereby providing complementary boost to existing categories. More details of related works are in Appendix A.
3 "Frozen-parameter" Phenomenon and Implementation of TITAN
In this section, we first present APFA scheme, using exhibit the Frrozen-parament phenomenon in VQE. We then show the implementation details of the proposed TITAN.
3.1 Phenomenon of Frozen Parameters in VQE
VQE can exhibit srrameter-ffrezing effect in which portions of the variational parameters become static during optimization; however, the extent this phenomenon has never been quantified. Here introduce the APFA-to to knowledge, the first framework that records the full, time-resolved mask trajectory of every parameter, thereby providing concrete numerical evidence of how, when, and to what degree individual angles freeze over the course of training.
APFA The APFA mechanism dynamically identifies and freezes low-saliency parameters a VQE (at the (-th iteration) to record the freezing trajectory by executing the following four steps.
(Step i) For every coordinate keep an exponential moving average (EMA) 722 of the absolute gradient a(t) = a a + (1 a)
(2) is the instantaneous gradient g.. (4) |,0 0 < < 1, where Dr
0f/000) iteration t, and is smoothing factor controlling memory depth.
(Step ii) For a VQE with parameters O, iteration t, we define the stochastic gradient g: Vo f (0,) + É., where & is an isotropic noise term sampled from a multivariate normal distribution (O I to improve exploration. Let = ||g l (1la/((1gg)) ((ooll2 + =) be the pradient-deca ratio with a small constant E. Two scale factors, x17 (freeze) and X! (activate), are modulated by rt.
x
Asmin + (1 r2) (As.max >s,min), 0 <<> > < Affmaxx S
((t) Aa,min + (1 Tx) (Aa,max Aa.min); 1 << Na,min < Aa.max
where famin / max and Aa,min max are four hyperparameter. Then, let 9r = 1 EP 9t be the
mean EMA magnitude across p parameters. Following this and Eq. (3) the freeze and activate thresholds for APFA at the i-th iteration are (() = 2 Dt and Ta = Tf
(Step iii) Each parameter maintains two counters: of (t) i.e., the number of successive optimization iterations with ( < 7 (t) and ( i.e., the number of successive iterations gl Ta These counters together form binary mask m, E (0, 1} indicating its state with being active and 0 being
frozen, respectively, A parameter is frozen when cj Ny and reactivated when (4) > Na. where Ny and N., are hyperparameter referring to the patience lengths.
(Step iv) Denote the mask vector as m, and the learning rate at the 1-th iteration 7)- The first-order optimizer performs the Hadamard update 0,41 = 0, " (m, @ gg), where @ is the Hadamard product. As such, only active parameters are updated.
After exccuting the four APFA steps for T iterations, we stack the deration-wise binary masks {0, (T++)) where each row m, indicates, at iteration t, which
into Y {mi)"" of the P trainable parameters are frozen (0) or active (1). The cumulative sum of this sequence, C Ei= m; E R.,, serves as compact "intensity" measure of how often each parameter has been frozen during the entire training process (see Appendix B for further details).

---

Observation of frozen parameters Based on the proposed AFPA, we report the cumulative freezing counts of all parameters of HEA when applied to esti- mate the ground state energy of the isotropic Heisen- berg Hamiltonians. By varying the qubit size from O to 15 qubits, and ranging the layer number of HEA from O to 15, the achieved results are shown in Fig- ure 2 The color map ranges from white to dark blue (frozen in nearly every epoch). Empirical evi- dence shows that initialization decisively shapes the subsequent saliency landscape: parameters that start in poorly conditioned regions are rapidly driven to negligible gradients, whereas well-scaled directions remain trainable throughout the run. This tight cou- pling between initialization and training dynamics suggests that etrly-stage geometry can predetermine long-term redundancy patterns. Hence, the optimization trajectory naturally partitions the parameter space into persistently active and repeatedly redundant dimensions. This phenomenon delivers two key insights that connect the parameter initialization and the optimization dynamics. First, parameters identified as inactive can be frozen a priori, reducing the optimization overhead without sacrificing accuracy. Second, is possible leverage learning models to predict those inactive parameters.
Frozen occurrences
7800
3000
4000
Qubit index
2000
Frozee
2000
Layer index
Figure Statistics of the frozen parameters in- tensity in HEA optimized for isotropic Heisenberg Hamiltonian.
3.2 TITAN: Dataset Construction and Modeling
Motivated by these insights, we propose TITAN. It demonstrates that (i) frozen parameters can be predicted priori and (ii) this knowledge markedly reduces VQE measurement overhead without affecting the accuracy. At a high level, TITAN establishes deep predictive model to estimate freeze-intensity tensor Y returned by APFA, given the specified Hamiltonian and ansatz. After training. TITAN can predict frozen parameters given any new ansatz and Hamiltonian to reduce VQE measurement overhead. The implementation of TITAN comprises three stages: dataset construction, model training, and inference. We next describe each stage in detail.
3.2.1 Dataset Construction
The training dataset employed in TITAN takes the form as D= {X,,, C.) where M are data points and each pair consists of (i) an input tensor X, capturing the relevant features of circuit configuration, and (ii) the corresponding freeze-intensity tensor C, which indicates the activity of each parameter recorded by APFA.
critical issue in constructing D is ensuring that each data pair is informative. More precisely, when VQE instance suffers from BPs, the parameter-freezing trajectory returned by APFA fails to yield useful information. To address this, we refine the original Gaussian initialization scheme [18] to ensure the trainability of large-scale VQE for a broad class of ansätze. This result is formalized in the following theorem, with the corresponding proof provided in Appendix C.
Theorem (Enhanced Gaussian Initialization). Let H be the explored N N-qubit local Hamiltonian. Following notations in Eq. ( when the employed HEA (()) yields (Wze-1. V2 (0(2t-1)) (M,CZ,, (0(60(2))1))) (Wae, (oe(0())))
(I. (Y VE 1,..., L with M, I for t> 1 and M, be the tensor product of fixed single-qubit unitary [ independently sampled from (R2(= Ry ( (+ Rx Ry ))
Define the expectation ((0) (0lU (U)(0))) Then Ee, Ux....ux ((ggr..) 5) 2 O(1L)) when each element in 0 (0(1) O(2)) is sampled independently from (10 22) with 72 (1)LL)
Theorem I provides a non-vanishing lower bound on the square of partial derivative that scales worst as (1/L)) for the depth-dependent choice 72 c/ L with c 0. Therefore, the overall gradient norm remains 2(1) rather than decaying as 2 2-2(2) Hence, the parameter space does not collapse into an exponentially flat plateau at initialization. Compared to the original Gaussian Initialization [8] that considers Pauli matrix observables, Theorem hass milder conditions on the observable format, which suits the VQE for quantum many-body Hamiltonians.
5

---

Frozen
Parancter
Selection
Dataset Construction
Frozen Paramtery Irajectoy
Training Dataset: Label
Loss Function
TITAn
TITAn
Training Dutuset: Label
Large Size
TITAn
TITAN
MISA
Rechee-18
Handee
Antio Pannes
Franee Paren
Hlane & Dseel
Mil--bu Asurt
(b) ResNet-18 Based Model Training
(e) Applications VQEs
Figure 3: Overview of TITAN framework. TITAN predicts the frozen parameters of different ansätze scales. (a) Dataset construction, (b) Model Training, (c) Extension the multiscale ansätze.
3.2.2 Modeling Implementation and Optimization of TITAN
A deep neural network A (-) is employed to map X to an estimate C; = h (X.). We employ ResNet-18 18 backbone 73 augmented with multi-head self-attention (MHSA) [7] layer acting
channel-wise to predict the intensity tensor C. Let hw denote this mapping from X C. To learn
w, one minimizes loss function over the dataset D: min Nv EE=IC C || where || ||
denotes the Frobenius norm. Modern optimizers such as Adam are typically used to minimize the o loss function, and the training details are elucidated in the Appendix D.
CFCSA encoding for arbitrary Hamiltonians and ansätze layouts. The CFCSA (Coordinate + Descriptor Fully Convolutional & Self-Attention) encoding scheme can handle arbitrarily structured A(L, D) or unstructured VQE ansätze circuits (H, A(p, S)) introduced in Section
D-1 - N N where E (O....,A 1) tracks the layer index, d E {O...., D tracks the gate index within each layer, and g (0, ..... N form the first three channels, allowing the network to recognize positional and structural roles of each parameter (e.g., early vs. late layer, different rotation axes, distinct qubits).
For Coordinate Channels (3D), each parameter is associated with a triple -
tracks which qubit is targeted. These three normalized coordinates
For Descriptor Channels (K-D), set of external scalars = {81,.... [KK'' 8K) context (e.g. coupling constants, symmetry tags, UCC-specific hyperparameter), Each descriptor Sy is normalized to [0, 1] and then broadcast over the entire p grid, creating additional channels. Hence, changes in the Hamiltonian (S) or ansätze metadata are passed directly into the network as an extra (K) -dimensional descriptor input.
provides problem-specific
After stacking all channels, obtain input tensor: X ((++))XXX(D)), This tensor is processed by: Convolutional Layers (stride = 1) that preserve spatial dimensions, and then MHSA the flattened (L x DN) tokens. The key insight is that both the convolutional filters (stride 1) and the global MHSA mechanism are dimension-agnostic,, allowing the same backbone weights w to process any (I, N, D) combination. Thus, architectural modifications needed handle different depths, qubit counts, or gate sets.
3.2.3 Inference Phase and Complementary to Other Methods
TITAN can generalize across unseen Hamiltonians: changing S modifies descriptor channels, injecting new context without altering the network structure. Arbitrary Circuit Architectures simply changes the input shape, which the fully convolutional MHSA backbone can still process. Hence, CFCSA of TITAN enables single predictor operate vastly different VQE setups, offering

---

Qubits: Layers:
Qubits: 5| Layers: 10
Qubits:
Qubits: 15 Layers: 15
Ope Inder
Lóger Iodns
Figure 4: Frozen parameters intensity under APFA. Each panel shows the intensity (darker refers more often frozen) when applying VQE with HEA to the isotropic Hamiltonian.
Table 1: AE under Gaussian initialization. Frozen threshold (T 80) is marked with #; 90 is not. A < 0 is highlighted with green frozen parameters 2 5 is colored with cyan
N:5
N:
N:7
N:8
N:
N: 10
N:
N: 12
N: 13
N: 14
15
Proposed TITAN: Final Energy Comparison (AE
Ersantiam
Enandinn
1:5
0.104
+0.156
+0.324
0.068
-0.092"
00.500
+0.582
0.366
40,34
-0.350
-0.315
L:6
-0.081"
-0.069
-0.138
-0.033
-0.014"
+0.354
-0.181
+0.217
-0.021
-0.157
-0.087
1:7
0.222
0.025
-0.035
-0.138
0.054
+0.282
-0.002"
0.008
-0.111
-0.140
+0.538
L:8
0.015
0.022
+0.176
+0.147
+0.146
0.127
0.029
0.024
10.184
0.077
+0.426
L:9
-0004
0.024
-0.114"
-0.300"
00.11
-0.120"
-0.122
-0.114
-0.777
-0.088
-0.063
-0.081"
-0.074
Number of Frozen Parameters
L:5
3,50
3/60
3/70
2,80
2/90
1/100
29/118 110"
2/120
2/130
2/140
8/150
1:66
6/60"
1/72
1/108"
4/120
5/133
3/144
5/156
3/168
7/180
L:7
9/70'
15/84"
2/98
2/112
3/126
3/140
14/1541"
6/168*
6/182
12/196
4/210
1/80
8/96"
9/192
1/240
L:9
4/90
10/108"
9/126"
7/144"
2/162
11/180"
1/198
26/216"
14/234"
2/252
15/270"
7/100"
1/120
1/140
2/160
2/180
11/200
11/220"
1/240
2/260
5/280
1/300
scalability and transferability across the full design space of (ft, A(L, D)) or (H, A(p, pairs. In Section and Appendix E, show TITAN complementary other methods.
4 Experiments
Having established the validity of our training corpus, we now detail both the dataset-generation experiments and the subsequent validation protocol for TITAN. The empirical study is organized into two complementary suites: Heisenberg spin models and Quantum-chemisty benchmarks.
Hamiltonian with Varying Qubits and Layers We first consider a Heisenberg Hamiltonian H
(a + bYY4+ c eZ,4.41), where N 1 15) is the number of qubits. The coefficients are set as c) 5, 5]..
Quantum-chemistty benchmarks After Jordan-Wigner or Bravyi-Kitaev transformation, the molccular Hamiltonian Á becomes a weighted sum of Pauli strings. So (=))) V()) (ØHF), the energy function Eq. (2) becomes E(0) = (dur) (1) f (()) (ØHF).
Metric Throughout this work, we evaluate the performance of both Heisenberg Hamiltonian and quantum chemistry by the variational energy itself as Eq. (2):
lower E(O)
better approximation of the exact ground state.
4
Settings All experiments are implemented by Tencent Quantum Tensor Circuits 775 and Pennylane 766 with NVIDIA GeForce RIX 3060. More setting details are introduced in the Appendix F.
"Frozen-parane "arozen-prramtte"" Phenomenon We first exhibit that APFA enables a stable and interpretable sparsity pattern that persists across circuit scales as shown in Figure 4 when applying VQE with HEA to isotropic Hamiltonian with L E {5...., freezes almost all angles in the first two layers. Increasing the depth to 5 x 10 shifts the active region to the circuit tail, concentrating optimization near the output. With 15 qubits, there are also significantly more activation parameters the output side.
......,100 10) and N E (5..... 15). For 5 circuits, APFA
HEA (Isotropic Hamiltonian) We employ HEA with layers and qubits, T. E (5...., . 10) and N ( 15). In the upper block of the Table I more than 90% of the cells shaded
7

---

Lewts
Anute OLdda: Lagane
Arazz: Grbie:10 Ligen:
Amari: Leyon:
Anazt: Gublic Lagars
Final Eneegy
Final Energy
Enargy
Finll Emergy
Fiaal Energy
Initiaization Methods
Inttalization Methods
Inibalization Methods
Insbalization Methods
Intiiaizztion Mothods
Anstte S12 Qubb;: Logars
Arazz: SUC Guste: 10 Ligens:10
Arnzt: suz Cabta: Leyen:3
Araste SU2 Qutt:: Logets
Energy
Final Energy
Final Energy
Final Enwrgy
Finsl Energy
Intialization Mathods
Inttialization Mathods
Initialization Mathods
Initialization Mathads
Infiilization Methods
Baseline
Random
-TITAN
Figure 5: Comparison of the final energies obtained using different initialization methods (Gaussian 18]. Zero, Uniform, and TITAN) and optimization strategies (Baseline (Vanilla), Random Freeze, TITAN) with gradient descent optimizer. The results shown for the isotropic Hamiltonian with SEL (upper row) and SU2 (lower row), under the threshold 5 = 80. Lower final energy values indicate better optimization performance. Frozen parameters refer the Appendix F.
c 55
=
0
c=
bawiiee- titn
TITAN
54333812349
15331012145
64331012148
34921812543
54321012345
54321012545
34521012345
ca.5
<=3
Random
$4338812345
54331012345
14308812155
54331012343
34331012345
(c--5)
(c-3))
(c=-1)
Trusen Parnee o0)
Counts
54321012545
54331012245
Figure 6: Energy difference heat-maps for the HEA isotropic Heisenberg Hamiltonian with Gaussian initialization. Each panel displays (E b, c) Exaseiie Emn coefficient grid with horizontal axis a E [-5, and vertical axis E [-5, 5]: the seven columns sweep the ZZ coupling coefficient -5 3, 0, 1,3,5 5 from left to right. Top row: TITAN. Middle row: Random Freezing. Bottom row: Froze parameters counts (r 80). Positive (green) values indicate that the baseline converges to higher final energy than the compared strategy, whereas negative (magenta) values signify energy losses relative to the baseline. The color scale is clipped symmetrically about zero.
green (AF S 0), indicating that TITAN's data-driven mask either matches or surpasses the baseline across almost every (layers, qubits) setting where Baseline is without any parameters frozen. And almost 50% of the cells are shaded cyan Therefore, TITAN can also adjust the number of frozen parameters by adjusting the threshold 7 to achieve the energy need that is relatively lower similar the baseline (AE < 0). Additional threshold results appear in the Appendix F.
Test in SU2 and SEL (Isotropic Hamiltonian) On the other hand, to prove the generalization of TITAN, we also tested on other ansätze, i.e., SU2 and SEL, as shown in the Figure 5 We show how TITAN complements traditional initialization methods, such as Zero initialization, uniform initialization, and traditional Gaussian initialization [8]. Random Freeze never outperforms Baseline 2.2 and often incurs energy penalties of 0.3-1.2 Ha. This underscores that the benefit comes from pruning targeted rather than a simple reduction. The success of TITAN on unseen ansätze and Hamiltonians suggests that the learned mask captures Hamiltonian-nnvrrian directions in the parameter landscape.

---

Qubits: Frozen: 6/10
Oubits:: 10 Frozen: 8,20
Qubitt:: 15 Frozen: 10/20
Qubits: 20 Frezen: 15:40
Qubits: 30 Frozen: 23,60
IITAM Raceliv
TITA Massiie
mMM
Traa
Baselise
Beseira
Racelin
60
60
66
56
40
60
80
190
20
5|
Oubits:
8/20 TTMM
Qubits:
Quibits: 20
15/40
Qubits:
TITAN
201
TIWA
46
TIAA Randon
TITAN
Bexetee
Booelng
-15
01 U8e
40 de
withd
300
23 90 180
43 6 0) Use horstion with
20 48
Heeradin with
Figure 7: Convergence behavior of TITAN (solid red) versus the Baseline (blue dotted) and a Random-freet control (black dashed) TFIM with coupling J 3 and transverse field h = 2 with Gaussian initialization [18]. The top row plots the f2-norm, while the bottom row tracks f (final energy). Shaded envelopes denote to across 5 independent runs with threshold (T 80).
Table 2:
Energy differences between Molecules with three initialization strategies: Gaussian [8] ()), zero (V), and uniform (4) under TITAN freezing and Random freezing. Less energy is colored with blue and worse is red with (T 80).
H (4)
HF (4)
LiH (10)
BeH, (12)
H2O (10)
N, (12)
CO (12)
AE
ETTIA
Ensseline
0.000
0.000
0.018
0.008
0.000
-0.007
0.015
V 0.000
0.000
-0.017
-0.008
0.000
0.007
0.015
0.497
0.148
0.145
0.236
0.078
2.512
1.729
AE
ERaniom
Eudiie
2: 248
0.000
0.003
0.013
0.004
0.011
0.017
2.248
0.001
0.003
0.013
0.004
0.010
0.017
1.991
0.573
-0.016
0.112
0.621
1.524
1.748
Number of Frozen Parameters
1/3 1/3
10/24
21/92
4/54
47/117
77/117
HEA (Anisotropic Hamiltonian) terms of Anisotropic Hamiltonian, Figure 6 reports the fi- nal-energy gap A (o, b, Eraseline Ein for a 8-qubit, 5-layer HEA. Across the entire (a, b, c) hypersurface TITAN yields predominantly green cells, with improvements reaching AE Rs 0.1 Gains especially pronounced for |c| 2 3 (i.e. strongly ZZ dd-ominnte regimes) and for balanced couplings |a| Re [6], highlighting TITAN's robustness under both isotropic and anisotropic interac- tions. However, when considering random parameters freezing, it exhibits near-zero mean: green and magenta cells are interspersed without discernible structure, and the absolute deviations seldom exceed A < 0.03. This corroborates that the performance lift stems from TITAN's informed freeziig strategy rather than stochastic variance.
For 0 (pure XY model) TITAN's advantage narrows yet remains positive, whereas for Zz- dominated columns (c +55 improvements are both larger in magnitude and spatially wider. This pattern suggests that circuits with stronger longitudinal entanglement profit more from TI- TAN's carly dimensionality reduction. The heat maps are approximately symmetric with respect to (a, b) + (-a,- -b), in line with the underlying Hamiltonian symmetry; TITAN preserves this property, indicating that the predictor does not inject bias towards a particular sign of the couplings.
TFIM (Anisotropic Hamiltonian) Compared with both reference baselines, TITAN exhibits markedly superior optimization dynamics across all examined qubit counts. In the top-row panels of Figure 7 the norm of the gradient for TITAN (solid red) diminishes precipitously within the first ~ 10 training iterations, whereas the Baseline (blue dotted) and Rundom-freeze control (black dashed) maintains substantially higher gradient amplitudes for 20-40 iterations. This steeper decay implies that TITAN more rapidly enters a region of the parameter manifold where the objective landscape locally smoother, thereby reducing the variance of subsequent updates. The bottom-row panels reveal

---

correspondingly accelerated decrease in the objective value f (final variational energy): TITAN attains energies lower than the Baseline after fewer than 15 iterations for N I 15 qubits and sustains an advantage that widens with system size, reaching gap of ~ 1.5-2.0 energy units at N = 20. Moreover, the shaded envelopes are markedly narrower for TITAN, indicating reduced run-to-run
variability and greater stability.
Quantum Chemistry Molecules We also present the molecules experiment in the Table For most molecules (e.g., H2, HE, and LiH), the Gaussian initialization under TITAN yields negligible AE, suggesting that these methods closely match or slightly improve upon the baseline. In contrast, Uniform initialization can lead to larger deviations, indicating higher energy. The last row reveals that 30% 66% of parameters can be frozen a priori (e.g. 77/117 (65.8%) for CO) without degrading energy. validating TITAN's performance in VQE measurement overhead reduction.
5
Conclusion
This work introduced TITAN, the first end-to-end framework that learns to predict frozen- parzmmtrr intensity for VQE. By harvesting large-scale frozen parameter trajectories by APFA, we created labeled dataset that exposes persistent redundancies VQE ansätze. Combined with the enhanced Gaussian initialization, which provably circumvents BPs via depth-dependent variance and random local Clifford twirling during dataset construction. This study remains limited to classical simulators and benchmark circuits do not exceed ~ 100 qubits, however, in the future, plan to scale to deeper circuits in real-world quantum sensors, ultimately enabling sesource-efficiee VQE deployments for chemically and physically relevant systems.
10

---

References
[1] Michael A Nielsen and Isaac I. Chuang. Quantum computation and quantum information. Cambridge university press, 2010.
[2] Richard P Feynman. Simulating physics with computers. In Feynman and computation, pages 133-153, cRe Press, 2018.
[3] James R Chelikowsky, N Troullier, and Yousef Saad. finiie-dffeeenc--sssuupootttiddd method: Electronic structure calculations without a basis. Physical review letters, 72(8):1240, 1994.
[4] Seenivasan Hariharan, Sachin Kinge, and Lucas Visscher. Modeling heterogeneous cataly- sis using quantum computers: An academic and industry perspective. Journal of chemical information and modeling, 2024.
[5] Pei-Hua Wang, Jen-Hao Chen, Yu- Yuan Yang, Chien Lee, and Yufeng Jane Tseng. Recent advances in quantum computing for drug discovery and development. IEEE Nanotechnology Magazine, 17(2)266-30, 2023.
[6] Yudong Cao, Jonathan Romero, Jonathan P Olson, Matthias Degroote, Peter D Johnson, Mária Kieferová, Ian D Kivlichan, Tim Menke, Borja Peropadre, Nicolas PD Sawaya, et al. Quantum chemistry in the age of quantum computing. Chemical reviews, 119(19):10356-10115 2019.
[7] Barry Bradlyn, Luis Elcoro, Jennifer Cano, Maia G Vergniory, Zhijun Wang, Claudia Felser, Mois Aroyo, and B Andrei Bernevig. Topological quantum chemistry. Nature, 547(7663):298- 305, 2017.
[8] Raghunathan Ramakrishnan, Pavlo O Dral, Matthias Rupp, and O Anatole Von Lilienfeld. Quantum chemistry structures and properties of 134 kilo molecules. Scientific data, 1(1):1-7, 2014.
[9] Qi Gao, Michihiko Sugawara, Paul D Nation, Takao Kobayashi, Yu-ya Ohnishi, Hiroyuki Tezuka, and Naoki Yamamoto. A quantum--lasssiee method applied material design: Pho- tochromic materials optimization for photopharmacology applications. Intelligent Computing, 3:0108, 2024.
[10] Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, et al. Variational quantum algorithms. Nature Reviews Physics, 3(9):625-644, 2021.
[11] Scriva, N Astrakhantsev, Pilati, and G Mazzola. Challenges of variational quantum optimization with measurement shot noise (2023). arXiv preprint wwxiv:2308.00044
[12] Yu Zhang, Lukasz Cincio, Christian FA Negre, Piotr Czamik, Patrick J Coles, Petr M Anisimov, Susan M Mniszewski, Sergei Tretiak, and Pavel A Dub. Variational quantum eigensolver with reduced circuit complexity. npj Quantum Information, 8(1):96, 2022.
[13] Benchen Huang, Marco Govoni, and Giulia Galli. Simulating the electronic structure of spin defects on quantum computers. PRX Quantum, 3(1::010339, 2022.
[14] Robert M Parrish, Edward G Hohenstein, Peter McMahon, and Todd J Martínez. Quantum computation of electronic transitions using a variational quantum eigensolver. Physical review letters, 122023):33040, 2019.
[15] Byungjoo Kim, Kang-Min Hu. Myung- Hyun Sohn, Yosep Kim, Yong-Su Kim, Seung-Wo Woo Lee, and Hyang-Tas Tag Lim. Qudit-based variational quantum eigensolver using photonic orbital angular momentum states. Science Advances, r0443)ead03342, 2024.
[16] Hocheol Lim, Doo Hyung Kang, Jeonghoon Kim, Aidan Jarloa- Shane McFarthing, Rowan Pellow-Jarman, Hyeon-Nae Jeon, Byungdu Oh, June-Koo Kevin Rhee, and Kyoung Tai No. Fragment molecular orbital-based variational quantum eigensolver for quantum chemistry in the age of quantum computing. Scientific reports, 14(1):2422, 2024.
11

---

[17] Martin Larocca, Supanut Thanasilp, Samson Wang, Kunal Sharma, Jacob Biamonte, Patrick J Coles, Lukasz Cincio, Jarrod R McClean, Zoë Holmes, and M Cerezo. review of barren plateaus in variational quantum computing. arXiv preprint arXiv:2405.0078/1 2024.
[18] Kaining Zhang, Liu Liu, Min-Hsiu Hsieh, and Dacheng Tao. Escaping from the barren plateau via gaussian initializations in deep variational quantum circuits. Advances in Neural Information Processing Systems, 35:18612- 1611218667, 2022.
[19] Jarrod McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes. Nature communications, (1)):881, 2018.
[20] Marco Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J Coles. Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nature communications, 12(1))1771, 2021.
[21] Andrew Arrasmith, Zoë Holmes, Marco Cerezo, and Patrick Coles. Equivalence of quantum barren plateaus to cost concentration and narrow gorges. Quantum Science and Technology, 7(4):045015, 2022.
[22] Yifeng Peng, Xinyi Li, Zhemin Zhang, Samuel Yen-Chi Chen, Zhiding Liang, and Ying Wang. Breaking through barren plateaus: Reinforcement learning initializations for deep variational quantum circuits. arXiv preprint arXiv:2508. 18514, 2025.
[23] Yifeng Peng, Xinyi Li, Zhemin Zhang, Samuel Yen-Chi Chen, Zhiding Liang, and Ying Wang. Can classical initialization help variational quantum eircuits escape the barren plateau? arXiv preprint arXiv:2508. 18497, 2025.
[24] Kerstin Beer, Dmytro Bondarenko, Terry Farrelly, Tobias J Osborne, Robert Salzmann, Daniel Scheiermann, and Ramona Wolf. Training deep quantum neural networks. Nature communica- lions, 11(1):808, 2020.
[25] Ernesto Campos, Daniil Rabinovich, Vishwanathan Akshay, and Biamonte. Training saturation in layerwise quantum approximate optimization. Physical Review A, 104(3):L030401, 2021.
[26] N Cody Jones, Rodney Van Meter, Austin G Fowler, Peter McMahon, Jungsang Kim, Thaddeus D Ladd, and Yoshihisa Yamamoto. Layered architecture for quantum computing. Physical Review 2(3):031007, 2012.
[27] Edward Grant, Leonard Wossnig, Mateusz Ostaszewski, and Marcello Benedetti. An initializa- tion strategy for addressing barren plateaus in parametrized quantum circuits. Quantum, 3:214, 2019.
[28] Arthur Pesah, Marco Cerezo, Samson Wang, Tyler Volkoff, Andrew T Somborger, and Patrick J Coles. Absence of barren plateaus in quantum convolutional neural networks. Physical Review X. 11(4):04101, 2021.
[29] Jeihee Cho, Junyong Lee, Daniel Justice, and Shiho Kim. Enhancing circuit trainability with selective gate activation strategy, 2025.
[30] William J Huggins, Jarrod R McClean, Nicholas C Rubin, Zhang Jiang, Nathan Wiebe, K Bir- gitta Whalcy, and Ryan Babbush. Efficient and noise resilient measurements for quantum chemistry near-term quantum computers. npj Quantum Information, 7(1)233, 2021.
[31] Giuseppe Scriva, Nikita Astrakhantsev, Sebastiano Pilati, and Guglielmo Mazzola. Chal- lenges of variational quantum optimization with measurement shot noise. Physical Review A, 109(3):032408, 2024.
[32] William K Wootters and Wojciech H Zurek. A single quantum cannot be cloned. Nature, 299(588)):82 803, 1982.
[33] Kosuke Mitarai, Makoto Negoro, Masahiro Kitagawa, and Keisuke Fujii. Quantum circuit learning, Physical Review A, 98(3):032309, 2018.
12

---

[34] Maria Schuld, Ville Bergholm, Christian Gogolin, Josh Izaac, and Nathan Killoran. Evaluating analytic gradients on quantum hardware. Physical Review A, 99(3):032331, 2019.
[35] Gavin E Crooks. Gradients of parameterized quantum gates using the parameter-shift rule and gate decomposition. arXiv preprint arXiv: arXiv:/905.133: 13311, 2019.
[36] David Wierichs, Josh Izaac, Cody Wang, and Cedric Yen-Yo Yu Lin. General parameter-shift rules for quantum gradients. Quantuum 6:677, 2022.
[37] Pranav Gokhale, Olivia Angiuli, Yongshan Ding, Kaiwen Gui, Teague Tomesh, Martin Suchara, Margaret Martonosi, and Frederic T Chong. O(n3) measurement cost for variational quantum cigensolver molecular hamiltonians. IEEE Transactions Quantum Engineering, 1:1-24, 2020.
[38] Vladyslav Verteletskyi, Tzu-Ching Yen, and Artur F Izmaylov. Measurement optimization in the variational quantum eigensolver using a minimum clique cover. The Journal of chemical physics, 152(12). 2020.
[39] Kouhei Nakaji, Suguru Endo, Yuichiro Matsuzaki, and Hideaki Hakoshima. Measurement optimization of variational quantum simulation by classical shadow and derandomization. Quantum, 7:995, 2023.
[40] Andrew Jena, Scott Genin, and Michele Mosca. Pauli partitioning with respect gate sets. arXiv preprint 1907.09977 2019.
[41] Pranav Gokhale, Olivia Angiuli, Yongshan Ding, Kaiwen Gui, Teague Tomesh, Martin Suchara, Margaret Martonosi, and Frederic T Chong, Minimizing state preparations in variational quantum eigensolver by partitioning into commuting families. arXiv preprint 1XXv:1907.11625, 2019.
[42] Xianchao Zhu and Xiaokai Hou. Quantum architecture search via truly proximal policy optimization. Scientific Reports, 13(1):5157, 2023.
[43] Yuxuan Du, Tao Huang, Shan You, Min-Hsiu Hsich, and Dacheng Tao. Quantum circuit architecture search for variational quantum algorithms, npj Quantum Information, 8(1):62, 2022.
[44] Wenjie Wu, Ge Yan, Xudong Lu, Kaisen Pan, and Junchi Yan. Quantumdarts: differentiable quantum architecture search for variational quantum algorithms. In International conference on machine learning, pages 17745-37764, PMLR, 2023.
[45] Zhimin He, Maijie Deng, Shenggen Zheng, Lvzhou Li, and Haozhen Situ. Training-fre freee quantum architecture search. In I Proceedings of the AAAI conference artificial intelligence, volume 38, pages 12430-12438, 2024.
[46] Di Luo, Jiayu Shen, Rumen Dangovski, and Marin Soljacic. Quack: accelerating gradient- hased quantum optimization with koopman operator learning. Advances Neural Information Processing Systems, 36:25662-25692. 2023.
[47] Junyong Lee, JeiHee Cho, and Shiho Kim. Q-maml: Quantum model-agnostic meta-lcarning for variational quantum algorithms. arXiv preprint arXiv:2501.05906 2025.
[48] Emanuel Knill. Quantum computing. Nature, 463(7280):4414443, 2010.
[49] Thaddeus Ladd, Fedor Jelezko, Raymond Laflamme, Yasunobu Nakamura, Christopher Monroe, and Jeremy Lloyd O' Brien.. Quantum computers. nature, 464(7285)445-33, 2010.
[50] Christopher M Dawson and Michael Nielsen. The solovay-kitaev algorithm. arXiv preprint quant-ph/0505030, 2005.
[51] Giulia Galli and Michele Parrinello. Large scale electronic structure calculations. Physical review letters, 69(24)35547, 1992.
13

---

[52] Huaijin Wu, Xinyu Ye, and Junchi Yan. Qvae-mole: The quantum vae with spherical latent variable learning for 3-d molecule generation. Advances in Neural Information Processing Systems, 37:22745-22771 2024.
[53] Alberto Peruzzo, Jarrod McClean. Peter Shadbolt, Man-Hong Yung, Xiao-Qr Qi Zhou, Peter J Love, Alán Aspuru-Guzik, and Jeremy I. O'brien. A variational eigenvalue solver on photonic quantum processor. Nature communications, 5(1))4213, 2014.
[54] Yordan S Yordanov, Vasileios Armaos, Crispin HW Barnes, and David RM Arvidsson- Shuukk. Qubit-excitation-base adaptive variational quantum eigensolver. Communications Physics, 4(1):228, 2021.
[55] Sam McArdle, Suguru Endo, Alán Aspuru- Supuru-Guzik, Simon C Benjamin, and Xiao Yuan. Quantum computational chemistry. Reviews of Modern Physics, 92(1):015003, 2020.
[56] Haiyang Yu, Meng Liu, Youzhi Luo. Alex Strasser, Xiaofeng Qian, Xiaoning Qian, and Shuiwang Ji. Qh9: A quantum hamiltonian prediction benchmark for qm9 molecules. Advances Neural Information Processing Systems, 36:40487-40503 2023.
[57] Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M Chow, and Jay M Gambetta. Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets. nature, 549(7671):242-225 246, 2017.
[58] Sukin Sim, Peter D Johnson, and Alán Aspuru-Guzik. Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classica algorithms. Advanced Quantum Technologies, 2(12):190007) 2019.
[59] Ali Javadi-Abhari, Matthew Treinish, Kevin Krsulich, Christopher J Wood, Jake Lishman, Julien Gacon, Simon Martiel, Paul D Nation, Lev S Bishop, Andrew W Cross, et Quantum computing with qiskit. arXiv preprint arXiv:2405.088/00 2024.
[60] Maria Schuld, Alex Bocharov, Krysta M Svore, and Nathan Wiebe. Circuit-centric quantum classifiers. Physical Review A, 101(3):032308, 2020.
[61] Jonathan Romero, Ryan Babbush, Jarrod McClean, Cornelius Hempel, Peter Love, and Alán Aspuru-Guzik. Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz. Quantum Science and Technology, 4(1 4(1):014008, 2018.
[62] Samuel Yen-Chi Chen, Chao-Han Huck Yang, Jun Qi, Pin- Yu Chen, Xiaoli Ma, and Hsi- Sheng Goan. Variational quantum circuits for deep reinforcement learning. IEEE access, 8: 8:141007-141024, 2020.
[63] Samuel Stein, Nathan Wiebe, Yufei Ding, Peng Bo, Karol Kowalski, Nathan Baker, James Ang, and Ang Li. Eqe: ensembled quantum computing for variational quantum algorithms. In Proceedings of the 49th annual international symposium on computer architecture, pages 59-71, 2022.
[64] Guillermo Carefa-Pérezz Matteo AC Rossi, Boris Sokolov, Francesco Tacchino, Panagiotis KI Barkoutsos, Guglielmo Mazzola, Ivano Tavernelli, and Sabrina Maniscalco. Learning to measure: Adaptive informationally complete generalized measurements for quantum algorithms. Prx quantum, 244342, 2021.
[65] Hsin- Yuan Huang, Richard Kueng, and John Preskill. Predicting many properties of a quantum system from very few measurements. Nature Physics, 10500710000-155, 2020.
[66] Ho Lun Tang, VO Shkolnikov, George S Barron, Harper Grimsley, Nicholas Mayhall, Edwin Barnes, and Sophia E Economou. qubit-adapt-vge- An adaptive algorithm for constructing hardware-efficient ansätze on a quantum processor. PRX Quantum, 2(2):020310, 2021.
[67] Weiwei Zhu, Jiangtao Pi, and Qiuyuan Peng. A brief survey of quantum architecture search. In Proceedings of the 6th international conference on algorithms, computing and systems, pages 1-5, 2022.
14

---

[68] Hanrui Wang, Zirui Li, Jiaqi Gu, Yongshan Ding, David Z Pan, and Song Han. Qoc: quantum on- chip training with parameter shift and gradient pruning. In Proceedings of the 59th ACM/IEEE design automation conference, pages 655-660, 2022.
[69] Francisco JR Ruiz, Tuomas Laakkonen, Johannes Bausch, Matej Balog, Mohammadamin Barekatain, Francisco JH Heras, Alexander Novikov, Nathan Fitzpatrick, Bernardino Romera- Paredes, John de Wetering, et al. Quantum circuit optimization with alphatensor. Nature Machine Intelligence, pages 1-12, 2025.
[70] Timothy Hospedales, Antreas Antoniou, Paul Micaelli, and Amos Storkey. Meta-learning in neural networks: survey. IEEE transactions on paltern analysis and machine intelligence, 4409)5514-55199 2021.
[71] Rui Huang, Xiaoqing Tan, and Qingshan Xu. Learning to learn variational quantum algorithm. IEEE Transactions on Neural Networks and Learning Systems, 34(11):8430 8440, 2022.
[72] Antti Tarvainen and Harri Valpola. Mean teachers are better role models: Weight-averaged con- sistency targets improve semi-supervised deep learning results. Advances in neural information processing systems, 30, 2017.
[73] Kaiming Hc, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770-778, 2016.
[74] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention all you need. Advances in neural information processing systems, 30, 2017.
[75] Shi-Xin Zhang, Jonathan Allcock, Zhou-Quan Wan, Shuo Liu, Jiace Sun, Hao Yu, Xing-Han Yang, Jiezhong Qiu, Zhaofeng Ye, Yu- Yu-Qin Chen, et al. Tensorcircuit: a quantum software framework for the nisq era. Quanium, 7:912, 2023.
[76] Ville Bergholm, Josh Izaac, Maria Schuld, Christian Gogolin, Shahnawaz Ahmed, Vishnu Ajith, M Sohaib Alam, Guillermo Alonso-Linaje, AkashNarayanan, Ali Asadi, et al. Pen- nylane: Automatic differentiation of hybrid guantum-classice computations. arXiv preprint arXiv:1811.044 1,0466,, 2018
15

---

NeurIPS Paper Checklist
1. Claims
Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Question:
Answer: [Yes]
Justification:
In introduction and abstract we summarized our contributions and scope.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results
be expected to generalize other settings.
is fine to include aspirational goals as motivation as long is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification:
In the Appendix F, we discussed our limitations.
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those not discussed in the paper.
The authors are encouraged to create separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification. asymptotic approximations only holding locally). The authors should reflect how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low images are taken in low lighting. Or speech-to-text system might be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach address problems of privacy and faimess.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory assumptions and proofs
Question:
For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
16

---

Justification:
We have full proof of Theorem I in the Appendix C.
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch provide intuition.
Inversely, any informal proof provided in the core of the paper should complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that affects the main claims and/or conclusions of the paper (regardless whether the code and data are provided or not)?
Answer: [Yes]
Justification:
We have claimed the experimental settings in page 7 and the Appendix
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g.. in the case of large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) the contribution is primarily a new algorithm, the paper should make clear how to reproduce that algorithm.
(b) the contribution is primarily a new model architecture, the paper should describe the architccture clearly and fully.
(c) If the contribution is new model (e.g., a large language model), then there should either way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome to describe the particular way they provide for reprodueibility. the case of closed-source models, may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers have some path to reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
17

---

Answer: [Yes]
Justification:
We have included our codes Appendix and Anonymous Links.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurIPS code and data submission guidelines (https://nips... cc/
for more details.
While we encourage the release of code and data, we understand that this might not be possible, so "No" an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https
/ /pipp/.cc/pullc//uuuii wwwticccguie//eeeeeeeeeii CoodeSubiiii ssi ssionnolli onnolicy)))) for more details.
The authors should provide instructions on data access and preparation, including how access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information possible in supplemental material (appended to the paper) recommended, but including URLs to data and code permitted.
6. Experimental setting/details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification:
We have claimed the experimental settings in page 7 and the Appendix
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core of the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, or as supplemental material.
7. Experiment statistical significance
Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Answer: [Yes]
Justification: We have claimed the information in our paper.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by error bars, confi- dence intervals, statistical significance tests, at least for the experiments that support main claims of the paper.
The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call a library function, bootstrap, etc.)
The assumptions made should be given (e.g.. Normally distributed errors).
should be clear whether the error bar is the standard deviation the standard error of the mean.
18

---

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of enrors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out range (e.g. negative error rates).
If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Question:
Answer: [Yes]
Justification:
We have concluded the resources in Page 7.
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g.. preliminary or failed experiments that didn't make into the paper).
9. Code of ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https: //t:::///euis..eeeeee
Answer: [Yes]
Justification: Our paper conform, in every respect, with the NeurIPS Code of Ethics.
Guidelines:
The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special cireumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g., if there is a special consid- eration due to laws or regulations their jurisdiction).
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [NA]
Justification: There is societal impact of the work performed.
Guidelines:
The answer NA means that there is no societal impact of the work performed.
If the authors answer NA or No. they should explain why their work has no societal impact or why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that improvement in the quality of generative models could be used to
19

---

generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
there are negative socictal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification: This paper poses such risks.
Guidelines:
The answer NA means that the paper poses no such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards challenging, and many papers do require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [NA]
Justification:
The paper does not use existing assets.
Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package dataset.
The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (c.g., website), the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, wwppppsiipteeeee whw.ree.wthhhe....c.. com/ datasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that are re-packaged, both the original license and the license of the derived asset (if has changed) should be provided.
If this information is not available online, the authors are encouraged to reach out to the asset's is creators.
13. New assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
20

---

Answer: [NA]
Justification:
The paper does not release new assets.
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset used.
At submission time, remember to anonymize your assets (if applicable). You can either create anonymized URL or include an anonymized zip file.
14. Crowdsourcing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification:
The paper does not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Including this information the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included the main paper.
According to the NeurIPS Code of Ethics, workers involved in data collection, curation, other labor should be paid at least the minimum wage in the country of the data collector.
Institutional review board (IRB) approvals or equivalent for research with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification: The paper does not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Depending on the country in which research is conducted, IRB approval (or equivalent)
may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
16. Declaration of LLM usage
Question: Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, originality of the research, declaration is not required.
Answer: [NA]
21

---

The core method development in this research does not involve LLMs as any important, original, or non-standard components.
Justification:
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non- standadd components.
Please refer to our LLM policy hattps://neurips... 220p://eec///o.feeeeee/2/22///// for what should or should not be described.
22


Reviews:
[{"review_id": "vYp8gCg4Gs", "reviewer": "Reviewer_dadQ", "summary": "The paper first observes a phenomenon that in variational quantum circuits many parameters do not have drastic dynamics and hence can be frozen once optimized. Then the authors propose a learning-based framework to predict which parameters can be frozen to what values as initialization of variational quantum circuits. This prediction saves many resources in the training of variational quantum circuits and leads to faster convergence, which is demonstrated in experiments.", "questions": "For a frozen parameter, the reason that it does not move significantly is that the gradients tend to be zero for them. As observed in the paper, this phenomenon happens more in the shallower layers. I wonder if there is any connection between this phenomenon and the gradient vanishing problem in classical NN training.", "limitations": "There are limited discussions on the limitation of the proposed method in the paper.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Pros\n1. The observation of the phenomenon of frozen parameters is interesting and insightful.\n2. With the observation, it is also interesting to propose a learning-based method to predict frozen parameters from the description of the variational task.\n3. The experiments conducted are comprehensive and illustrative. \n\nCons\n1. The data presented in Table 1 shows only a small portion of the parameters are frozen, possibly because of the high threshold. I would like to see how these thresholds affect the performance of the proposed method.\n2. The color in Figure 6 is fainted for Random and TITAN, especially when printed. I suggest the authors make the transition smoother. \n3. The details on how TITAN complements Zero initialization and others are missing. I cannot fully understand what Figure 5 means, since I do not know the experiment setting.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate Reviewer dadQ for the thoughtful and constructive feedback. We have carefully addressed the points raised under **Weaknesses** [**W1-W3**], and the specific **Questions** [**Q1**]. We hope the following responses help clarify our contributions and assist in the reviewer's reassessment of our submission.\n\n**W1: How does the threshold $\\tau$ affect the performance of TITAN?**\n\n**Reply**: Thanks for the comments. We fully agree with the reviewer's perspective that exploring the varied threshold settings can offer deeper insights into TITAN’s effectiveness. For this reason, in the *original manuscript*, we reported *how the varied thresholds (i.e., $\\tau \\in$ {50, 60, 70, 80, 90}) affect the performance of TITAN*, as elaborated in Tables 3-5 (pages 19-23, Appendix). The key observation is that as the threshold $\\tau$ decreases, a larger proportion of parameters are frozen; however, this comes at the cost of reduced energy estimation accuracy. \n\nTo better address the reviewer’s concern, we conducted additional simulations to support Table 1 in our manuscript. These experiments follow the same settings as in Table 1, with the only difference being the variation of the threshold values. \n\nThe achieved results are shown in Table R1. In particular,  as $\\tau$ is reduced from 90 to 50, the number of frozen parameters rises from at most $5$–$11$ per layer to more than half of the total, and higher ground state energy solution. These observations confirm that the proposed method is sensitive to $\\tau$ in a predictable, monotonic fashion: lowering the threshold freezes more parameters and yields increasingly positive (i.e., worse) $\\Delta E$, while excessively low thresholds offer limited additional improvement in single testing. \n\n**Table R1** (Supplementary Table 1) The performance of TITAN with the varied threshold $\\tau$. The explored Hamiltonians are a family of 1‑D nearest‑neighbor Heisenberg models. The performance is evaluated by the energy difference, i.e., $\\Delta E= E_{TITAN} - E_{Gaussian}$, where each $E$ denotes the estimated energy returned by TITAN-frozen Gaussian initialization and with standard Gaussian initialization, respectively.\n\n| Qubits | Metric (L: 8)  | $\\tau$=90 | $\\tau$=80 | $\\tau$=70 | $\\tau$=60 | $\\tau$=50 |\n|:-------:|:----:|:---:|:---:|:---:|:---:|:---:|\n| **10** | $\\Delta E$ (Frozen Params) |-0.13 $\\pm$ 0.02 (5/160) | -0.08 $\\pm$ 0.02 (11/160) | -0.03 $\\pm$ 0.03 (30/160) | +0.49 $\\pm$ 0.15 (60/160) | +0.69 $\\pm$ 0.35 (81/160) | \n| **11** | $\\Delta E$ (Frozen Params) | -0.03 $\\pm$ 0.03 (4/176) | +0.04 $\\pm$ 0.01 (2/176) | +0.45 $\\pm$ 0.05 (57/176) | +0.69 $\\pm$ 0.27 (95/176) | +2.14 $\\pm$ 0.23 (122/176) | \n| **12** | $\\Delta E$ (Frozen Params) | -0.02 $\\pm$ 0.01 (9/192) | -0.03 $\\pm$ 0.03 (21/192) | -0.01 $\\pm$ 0.01 (42/192) | +0.04 $\\pm$ 0.14 (65/192) | +1.49 $\\pm$ 0.57 (101/192) | \n| **13** | $\\Delta E$ (Frozen Params) | -0.04 $\\pm$ 0.14 (2/208) | -0.07 $\\pm$ 0.00 (8/208) | +0.16 $\\pm$ 0.03 (19/208) | +0.31 $\\pm$ 0.14 (43/208) | +0.76 $\\pm$ 0.41 (78/208) |\n| **14** | $\\Delta E$ (Frozen Params) | -0.08 $\\pm$ 0.03 (5/224) | -0.08 $\\pm$ 0.01 (6/224) | +0.15 $\\pm$ 0.05 (20/224) | +0.43 $\\pm$ 0.21 (45/224) | +0.69 $\\pm$ 0.33 (96/224) |\n| **15** | $\\Delta E$ (Frozen Params) | + 0.02 $\\pm$ 0.24 (1/240) | -0.03 $\\pm$ 0.01 (8/240) | +0.58 $\\pm$ 0.18 (27/240) | +0.86 $\\pm$ 0.19 (72/240) | +1.50 $\\pm$ 0.21 (128/240) |\n\nIn the updated version, we have incorporated the above clarification into Section 4 (“Sensitivity to $\\tau$”) and reproduced Table 1 to make the relationship between threshold and performance explicit. \n\n\n**W2: The color in Figure 6 is faint for Random and TITAN.**\n\n**Reply**:  We have followed the reviewer's advice to redraw Figure 6 using a higher‑contrast and colour‑blind‑safe palette,  and increase both line weight and marker size to ensure clear differentiation in the revision. \n\n**W3: How does TITAN complement the zero initialization, and what does Figure 5 mean?**\n\n**Reply**: To fully address the reviewer's concerns, in this reply, we first show how Titan complements with other initialization methods, followed by elucidating the settings adopted in Figure 5 (page 8). \n\n$\\bullet$ *Complementary with other initialization methods.* We would like to clarify that TITAN complements Zero and other initialization strategies in a manner analogous to how Gaussian and Uniform initialization methods operate, as demonstrated in Figure 5 (Page 8). Specifically, TITAN predicts in advance which parameters of a given ansatz are likely to be frozen, before any initialization. After this step, any standard parameter initialization strategy (e.g., Zero, Gaussian, Uniform) can be applied to the remaining trainable parameters.  We also showed how TITAN complements the Layer-wise Greedy Gradient Descent (LGGD) [1] method in Appendix Page 9. All explanations are appended to the revised manuscript.\n\n[1] Grimsley, Harper R., et al. \"An adaptive variational algorithm for exact molecular simulations on a quantum computer.\" Nature Communications 10.1 (2019): 3007.\n\n$\\bullet$ *Detailed settings in Figure 5.*  We would like to first remind the reviewer that the purpose of Figure 5 is to evaluate the **domain generalization ability of TITAN**. Specifically, TITAN is trained on a family of isotropic Hamiltonians using the hardware-efficient ansatz (HEA), and then tested on two **structurally distinct ansätze**, SU2 and SEL, by predicting their frozen parameters. This setup allows us to assess whether *a single TITAN model, trained on one ansatz, can generalize to others*, thereby demonstrating its broader applicability across different quantum circuit architectures.\n\nThe experiment setup adopted in Figure 5 is as follows. For each ansatz (SU2 and SEL), we evaluate three commonly used initialization strategies, i.e., Gaussian, zero, and uniform, and examine how TITAN refines their corresponding freezing parameters. A purely random freezing mask is also included as a baseline for comparison. Five experimental settings are considered:  $(N,L)=(5,5),(5,15),(10,10),(15,5),(15,15)$, where $N$ and $L$ refer to the qubit count and the layer number, respectively.\n\nTo make the outcome transparent, we have extracted the raw data from Figure 5 and **present it as the Table R2**. An observation is that in almost every setting, TITAN yields lower ground energies than the random freezing, confirming its effectiveness. Moreover, the low estimation error of TITAN on both SEL and SU2 validates its domain generalization capability, which is crucial for enabling robust and transferable performance across diverse ansatz architectures. \n\n**Table R2**: Evaluating TITAN’s domain generalization across distinct ansätze (restatement of Figure 5). The explored Hamiltonians refer to a family of 1‑D nearest‑neighbor Heisenberg. TITAN is trained with the HEA ansatz and tested on the SEL and SU2 ansätze. Gaussian initialization is adopted. The definition of $\\Delta E$ follows the same meaning in Table R1. \n\n| Testing Ansatz | N=5 L=5| N=5 L=15 |N=10 L=10| N=15 L=5 | N=15 L=15 |\n|---------------------|--------|------------|---------|----------|----------|\n| SEL (TITAN Freezing)| +0.441 $\\pm$ 0.054 | -0.081 $\\pm$ 0.002 | -0.159 $\\pm$ 0.374 | -0.671 $\\pm$ 1.173 | -0.521 $\\pm$  0.229|\n| SEL (Random Freezing) | +0.689 $\\pm$ 0.269 | +2.280 $\\pm$ 1.261 | +0.650 $\\pm$ 0.178 | -0.055 $\\pm$ 0.715 | +0.479 $\\pm$  0.715|\n| SU2 (TITAN Freezing) | +0.123 $\\pm$ 0.368 | -0.096 $\\pm$ 0.301 | +0.119 $\\pm$ 0.215 | -0.311 $\\pm$ 1.714 | -0.301 $\\pm$ 0.035|\n| SU2 (Random Freezing) | +1.626 $\\pm$ 0.701 | -0.025 $\\pm$ 0.290 | +0.262 $\\pm$ 0.200 | +1.658 $\\pm$ 13.714 | +0.389 $\\pm$ 0.013| \n\n**Q1: Are shallow layers more prone to parameter freezing? Is this related to the gradient vanishing problem of classic NN?**\n\n**Reply**: For clarity, we address the reviewer's two concerns in turn in this response. \n\n$\\bullet$ *Phenomenon of frozen parameters*. We would like to emphasize that the distribution of frozen parameters is **problem-dependent**. While Figure 4 shows that frozen parameters tend to concentrate in the shallow layers when applying HEA to isotropic Hamiltonians, **this behavior is not universal**.  As shown in Figures 6, 7, 9, 12, 16, and 17 of the Appendix, the distribution of frozen parameters **varies significantly** across different ansatzes and problem Hamiltonians.\n\nDue to the problem-dependent nature of parameter freezing, we propose using a deep learning model to learn and capture this behavior. As demonstrated in Section 4 and the corresponding Appendix figures, TITAN generalizes well across diverse settings,  significantly reducing measurement costs while preserving the final ground-state energy.\n\n$\\bullet$ *Relation with vanishing gradients in classical neural networks*. TITAN is conceptually related to certain phenomena in classical NN, but *not to vanishing gradients*. In classical NN, vanishing gradients refer to the issue where shallow layers receive very small gradients, making them hard to train. In contrast, the frozen parameters identified by TITAN, as evidenced in Figures 6, 7, 9, and 17 of the Appendix, do not consistently appear in shallow layers and are not caused by gradient flow issues.\n\nA more relevant analogy is **neural network pruning**. In classical NN, some neurons may become inactive depending on the task or dataset. This is similar to our observation that certain VQE parameters remain inactive (i.e., frozen) depending on the Hamiltonian and ansatz. This connection may inspire further research into sparsity and parameter selection across classical and quantum learning models.\n \n[1] Hochreiter, Sepp. \"The vanishing gradient problem during learning recurrent neural nets and problem solutions.\" \n\n[2] Pascanu R, Mikolov T, Bengio Y. On the difficulty of training recurrent neural networks International conference on machine learning. Pmlr, 2013: 1310-1318."}, {"review_id": "VVeFHhj03o", "reviewer": "Reviewer_8G53", "summary": "To tackle challenges faced by Variational quantum Eigensolver (VQE), this paper introduces a deep learning framework called TITAN. The proposed method detects and freezes inactive parameters of a given ansatz at initialization for a targeted class of Hamiltonians.", "questions": "See Strengths And Weaknesses", "limitations": "Yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "S1. The proposed method TITAN significantly speeds up the convergence, uses fewer circuit evaluations, while maintaining the accuracy. The methodology and experimental results demonstrate a non-trivial contribution.\n\nS2. This paper is well-written and logically structured, making it accessible and easy to read even for readers who are not experts in the field.\n\nS3.  The appendix and codebase are clear and well-structured, providing rich and important information.\n\nW1. Can this method efficiently achieve accurate results for larger molecules?\n\nW2. The frozen parameter is innovative, but the model architecture shows limited novelty.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate Reviewer 8G53 for the thoughtful and constructive feedback. We have carefully addressed the points raised under **Weaknesses** [**W1-W3**], and the specific **Questions** [**Q1**]. We hope the following responses help clarify our contributions and assist in the reviewer's reassessment of our submission.\n\n**W1: Can this method efficiently achieve accurate results for larger molecules?**\n\n**Reply**: We agree with the reviewer's viewpoint to understand the scalability of TITAN. To address this issue, we have conducted extensive experiments to study the **scalability of Titan** in the original submission. In particular, the results in Figure 7 (Page 9) showed that TITAN could be transferred to models other than those trained up to **30 qubits** and still maintain the same performance level as the baseline. \n\n\n**Table R3**: Energy differences with the ground truth across different molecules over three initialization strategies and TITAN Freezing.\n|             | H₂ (4)   | HF (10)   | LiH (10) | BeH₂ (12) | H₂O (10)  | N₂ (12)   | CO (12)   |\n|-------------|:---------:|:---------:|:---------:|:----------:|:----------:|:----------:|:----------:|\n| Gaussian [1]| -0.896 $\\pm$ 0.106| -97.601 $\\pm$ 0.114 |  -7.882 $\\pm$ 0.085 |  -14.876 $\\pm$ 0.506 | -72.876 $\\pm$ 2.065| -103.411 $\\pm$ 2.639 | -107.669 $\\pm$ 3.095|\n| Zero  | -0.896 $\\pm$ 0.106 | -97.601 $\\pm$ 0.114 |  -7.882 $\\pm$ 0.085 |  -14.876 $\\pm$ 0.506 | -72.876 $\\pm$ 2.065| -103.411 $\\pm$ 2.639 | -107.669 $\\pm$ 3.095|\n| Uniform| -0.639 $\\pm$ 0.088| -97.389 $\\pm$ 0.201 |  -7.401 $\\pm$ 0.107 |  -13.743  $\\pm$ 0.932 | -72.281 $\\pm$ 2.107 | -102.616 $\\pm$ 4.898 | -107.217 $\\pm$ 3.209|\n| Gaussian [1] + TITAN | -0.896 $\\pm$ 0.106 |  -97.601 $\\pm$ 0.114 | -7.864 $\\pm$ 0.085| -14.868 $\\pm$ 0.522 | -72.876  $\\pm$ 2.065 | -103.418 $\\pm$ 2.891| -107.684  $\\pm$ 3.001 |\n| Zero + TITAN | -0.896 $\\pm$ 0.106 | -97.601 $\\pm$ 0.114 | -7.899 $\\pm$ 0.077 | -14.884 $\\pm$ 0.730 | -72.876 $\\pm$ 2.065 | -103.404 $\\pm$ 2.908| -107.654 $\\pm$ 3.031|\n| Uniform + TITAN | -0.142 $\\pm$ 0.577 | -97.537 $\\pm$ 0.197 | -7.256 $\\pm$ 0.131 | -13.979 $\\pm$ 0.984 | -72.359 $\\pm$ 2.299 | -100.104 $\\pm$ 5.424 | -108.946 $\\pm$ 4.025|\n| Exact| -1.175 [1] |-100.340 [2]| -8.023 [2]|-15.918 [3] | -76.428 [4] | -109.542 [5] | -113.326 [5] |\n| TITAN Frozen Parameters |  1/3  |  1/3  |   10/24  |   21/92 |   4/54 |   47/117|   77/117  |\n\nTo further address this issue, we explore the capability of Titan in seven quantum chemistry molecules under frozen threshold $\\tau = 80$ in **Table R3**.  For CO, which involves 12 qubits, TITAN succeeds in freezing more than $60\\%$ of those parameters while maintaining or improving the energy estimate. \n\n*The challenge towards larger molecules is the lack of **efficient simulators** for large-qubit circuits with optimization*. A detailed explanation in Reply to **Q1** for Reviewer cy7n. In words, with the development of tensor network [7] and CUDA-Q [8], computing resource limitations will be better resolved, and the applicability of TITAN can be further broadened.\n\nWe appended the discussion to the revised manuscript.\n\n[1] Sims J S, Hagstrom S A. High precision variational calculations for the Born-Oppenheimer energies of the ground state of the hydrogen molecule[J]. The Journal of chemical physics, 2006, 124(9).\n\n[2] Lathiotakis N N et al. A functional of the one-body-reduced density matrix derived from the homogeneous electron gas: Performance for finite systems[J]. The Journal of chemical physics, 2009, 130(6).\n\n[3] Goldberg M C, Riter Jr J R. Confirmation of the predicted Hartree-Fock limit in BeH2[J]. The Journal of Physical Chemistry, 1967, 71(9): 3111-3112.\n\n[4] Gurtubay I G, Needs R J. Dissociation energy of the water dimer from quantum Monte Carlo calculations[J]. The Journal of chemical physics, 2007, 127(12).\n\n[5] Chachiyo T, Chachiyo H. Understanding electron correlation energy through density functional theory[J]. Computational and Theoretical Chemistry, 2020, 1172: 112669.\n\n[6] Zhang K et al. Escaping from the barren plateau via gaussian initializations in deep variational quantum circuits[J]. Advances in Neural Information Processing Systems, 2022, 35: 18612-18627.\n\n[7] Kulshrestha et al. Qadaprune: Adaptive parameter pruning for training variational quantum circuits.\n\n[8] Schieffer et al. Harnessing CUDA-Q’s MPS for Tensor Network Simulations of Large-Scale Quantum Circuits. \n\n**W2: The frozen parameter is innovative, but the model architecture shows limited novelty.**\n\n**Reply**: We appreciate the reviewer for the positive affirmation about TITAN. In fact, we have systematically explored the effect of different architectures. The main conclusion is that: **Any reasonably expressive backbone** can be plugged into the pipeline and will learn the same gradient‑suppression pattern, confirming that the framework is **genuinely model‑free**. In order to help readers better understand this work, we weakened the discussion about neural architecture.  \n\n**Table R4**: $\\Delta E$ (Frozen Parameters) for a 1‑D nearest‑neighbor Heisenberg chain using the HEA ansatz, comparing different TITAN backbone architectures under frozen‑parameter evaluation.\n| Qubits | Architectures (L: 8)  | Frozen threshold $\\tau$ = 90 | $\\tau$ = 80 | $\\tau$ = 70 | $\\tau$ = 60 |  $\\tau$ = 50 |\n|:-------:|:-------------------|:---:|:---:|:---:|:---:|:---:|\n| **10** | ResNet-18 |-0.13 $\\pm$ 0.02 (5/160) | -0.08 $\\pm$ 0.02 (11/160) | -0.03 $\\pm$ 0.03 (30/160) | +0.49 $\\pm$ 0.15 (60/160) | +0.69 $\\pm$ 0.35 (81/160) | \n|        | Transformer  [1] | -0.15 $\\pm$ 0.00 (5/160) | -0.19 $\\pm$ 0.01 (12/160) | -0.07 $\\pm$ 0.02 (28/160) | +0.35 $\\pm$ 0.19 (57/160) | +0.77 $\\pm$ 0.28 (86/160) |\n|        |EfficientNetV2‑S [2]| -0.03  $\\pm$ 0.00 (4/160) | -0.04 $\\pm$ 0.02 (11/160) | -0.12 $\\pm$ 0.(23/160) | +0.22 $\\pm$ 0.07 (36/160)| + 0.43 $\\pm$ 0.24 (78/160) |\n|        | U-Net [3]  | -0.24  $\\pm$ 0.03 (7/160) | -0.03 $\\pm$ 0.03 (8/160) | +0.18 $\\pm$ 0.11 (27/160) | +0.55 $\\pm$ 0.21 (44/160) | +0.62 $\\pm$ 0.65 (85/160)|\n| **11** | ResNet-18   | -0.03 $\\pm$ 0.03 (4/176) | +0.04 $\\pm$ 0.01 (2/176) | +0.45 $\\pm$ 0.05 (57/176) | +0.69 $\\pm$ 0.27 (95/176) | +2.14 $\\pm$ 0.23 (122/176) |\n|        | Transformer | -0.16 $\\pm$ 0.03 (6/176) | -0.06 $\\pm$ 0.03 (6/176) | -0.07 $\\pm$ 0.41 (48/176) | +0.07 $\\pm$ 0.51 (69/176) | +1.56 $\\pm$ 0.57 (113/176) |\n|        | EfficientNetV2‑S | -0.05 $\\pm$ 0.00 (6/176)| -0.07 $\\pm$ 0.03 (6/176)| +0.33 $\\pm$ 0.08 (51/176) | +0.55 $\\pm$ 0.21 (69/176) | +1.71 $\\pm$ 0.44 (109/176) |\n|        | U-Net | -0.19 $\\pm$ 0.08 (4/176) | +0.05 $\\pm$ 0.00 (8/176) | +0.32 $\\pm$ 0.11 (48/176) | +0.37 $\\pm$ 0.49 (71/176) | +1.55 $\\pm$ 0.61 (116/176) |\n| **12** | ResNet-18   |-0.02 $\\pm$ 0.01 (9/192) | +0.06 $\\pm$ 0.01 (21/192) | -0.01 $\\pm$ 0.01 (42/192) | +0.04 $\\pm$ 0.14 (65/192) | +1.49 $\\pm$ 0.57 (101/192) | \n|        | Transformer| +0.01 $\\pm$ 0.01 (9/192) | +0.03 $\\pm$ 0.03 (21/192)| +0.12 $\\pm$ 0.08 (41/192) | +0.35 $\\pm$ 0.19 (63/192) | +1.71 $\\pm$ 0.34 (108/192) |\n|        | EfficientNetV2‑S   | -0.06 $\\pm$ 0.04 (8/192) | -0.01 $\\pm$ 0.07 (19/192) | +0.15 $\\pm$ 0.13 (42/192) | +0.28 $\\pm$ 0.11 (68/192) | +1.34 $\\pm$ 0.35 (96/192) |\n|        | U-Net   | -0.11 $\\pm$ 0.09 (9/192) | -0.05 $\\pm$ 0.00 (21/192) | +0.00 $\\pm$ 0.01 (43/192) | +0.35 $\\pm$ 0.15 (66/192) | +0.97 $\\pm$ 0.71 (108/192) |\n| **15** | ResNet-18   | +0.43 $\\pm$ 0.11 (1/240) | -0.03 $\\pm$ 0.01 (8/240) | +0.58 $\\pm$ 0.18 (27/240) | +0.86 $\\pm$ 0.19 (72/240) | +1.50 $\\pm$ 0.21 (128/240) |\n|        | Transformer   | +0.16 $\\pm$ 0.19 (1/240) | -0.13 $\\pm$ 0.06 (8/240) | -0.41 $\\pm$ 0.27 (18/240)  | +0.39 $\\pm$ 0.55 (61/240) | +1.45 $\\pm$ 0.19 (125/240) |\n|        | EfficientNetV2‑S   | -0.03 $\\pm$ 0.00 (1/240) | +0.06 $\\pm$ 0.01 (7/240) | +0.33 $\\pm$ 0.29 (25/240) | +0.65 $\\pm$ 0.33 (69/240) | + 1.79 $\\pm$ 0.19 (135/240) |\n|        | U-Net   | +0.23 $\\pm$ 0.01 (1/240) | +0.08 $\\pm$ 0.01 (8/240) | +0.42 $\\pm$ 0.23 (29/240) | +0.61 $\\pm$ 0.37 (69/240) | + 1.52 $\\pm$ 0.17 (128/240) |\n\nTo better address the reviewer's concerns, we retrained all substitutes—Transformer [1], EfficientNetV2‑S [2], and U‑Net [3] using the Adam optimizer with a fixed learning rate of $1\\times10^{-3}$ for $1,000$ epochs, keeping batch size, loss function, and data augmentations identical to the default convolutional backbone (ResNet-18 in Table R4) in **Table R4**, where ResNet-18 based TITAN (model size 46 MB), Transformer based (345 MB), EfficientNetV2‑S based (90 MB), and U‑Net based (124 MB).\n\nTable R4 shows that every backbone yields energy differences $\\Delta E$ that are **comparable** to the baseline TITAN model (ResNet-18), while freezing a similar fraction of parameters. The same trend holds across all qubit counts: swapping the backbone changes the numeric value by at most $2\\times10^{-1}$, yet the qualitative behavior—larger $\\tau$, more frozen parameters, and lower measurement cost with minimal loss in accuracy remains intact.\n\nWe have added the new ablation study and accompanying discussion (Section 4) to highlight this architectural agnosticism and to clarify that the contribution of the paper is orthogonal to the choice of neural backbone in the revised manuscript.\n\n[1] Vaswani A, Shazeer N, Parmar N, et al. Attention is all you need[J]. 2017, 30.\n\n[2] Tan M, Le Q. Efficientnetv2: Smaller models and faster training[C]. PMLR, 2021: 10096-10106.\n\n[3] Ronneberger O, Fischer P, Brox T. U-net: Convolutional networks for biomedical image segmentation[C], 2015: 234-241."}, {"review_id": "5Rg1zk5f0k", "reviewer": "Reviewer_27vu", "summary": "This paper focuses on the expense related to sampling overhead in VQE approaches on quantum computers, especially as it pertains in scaling with respect to number of parameters as general backpropagation is known not to be possible due to consequences of the no-cloning theorem.  Hence, reducing the number of actively trained parameters has the potential to dramatically reduce overall training costs for a commonly used method, which is exploited here.  Previous strategies have focused on simple parameter elimination or layerwise freezing.  This work introduces a new method based on machine learning from data on other optimization runs in order to predict optimized, adaptive freezing schedules to reduce costs while maintaining speed and accuracy as much as possible.  Across a range of systems up to 30 qubits, it achieves faster convergence and 40-60% fewer circuit evaluations to reach or surpass current accuracy.", "questions": "1. Can you try to better quantify in a single plot the typical expected improvement (or penalty) for time and energy when using Titan?  Right now it seems like there is substantial chance the energy gets worse in 50% of the cases for a modest decrease in cost, but I can't quite tell because it's in so many different plots for so many different models.  A summary statstic showing a high likelihood of maintaining quality, while clearly stating averaged reduced resources would be helpful for improving the significance and quality scores.\n\n2. Can you say more specifically what models and trajectories the Titan model itself was trained on, how much data was used, and what the training time was like?  Were the models exactly those tested in the paper, but with different parameters? Were they totally unrelated? If they were totally unrelated, or at much smaller sizes, this would strengthen the case this approach would generalize.\n\n3. The plotted freezing intensity suggests a somewhat deterministic pattern being discovered.  How does Titan compare to a freezing strategy that is not uniformly random, but rather. freezes early layers with much higher probability than later layers? Does Titan substantially beat such a baseline? See Figure 4 for my intuition.", "limitations": "The authors could perhaps benefit from\n1. Further discussing the costs of training their approach to get a holistic view of cost\n2. Mention the additional problems in variational circuits related to local minima in the cost function", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n\nThe paper is clear in the setup of the problem and motivation.  The conceptual novelty of using an ML approach to help decide the freezing strategy and optimize for cost is interesting, and different than previous metalearning approaches for guessing the parameters outright.\n\nWeaknesses\n\nThe major weakness of the paper is that for the added implementation complexity, the improvements are not as striking as one would hope.  In many of the presented results, Titan does a little bit worse or about the same as much simpler approaches, and only decreases the cost by a small constant factor.  It certainly appears to help in some cases, but it looks like it's not very obvious at the outset when it will, so for near experiments, one is likely to have to run the baseline anyways.  The results are presented in a way that make it a bit hard to see the potential advantages, and there is not much discussion of the size of training set or time required to reach the improvements reported.", "was_revised": true, "final_justification": "Initial concerns about if there was any generalization have been addressed, and it is more clear that the method offers broad improvements.  There is some room for additional quantification of end to end advancement, but broadly I would now lean towards accepting.", "rebuttal": "We sincerely thank Reviewer 27vu for handling our submission and providing constructive feedback. We have carefully addressed the concerns outlined under **Weaknesses** [**W1**], specific **Questions** [**Q1-Q3**], and **Limitations** [**L1-L2**]. We hope the responses below clarify our contributions and support a more informed reassessment of our submission.\n\n**W1: (1) The added implementation complexity of TITAN does not seem to yield significant performance gains over simpler baselines. (2) TITAN’s benefits appear inconsistent across settings, making it unclear when it will be effective. (3) The results lack clarity in highlighting TITAN’s advantages, and there is insufficient discussion of training set size and runtime overhead.**\n\n**Reply**: To ensure clarity, we address the reviewers’ three concerns in sequence below.\n\n(1) *TITAN does not seem to yield significant performance gains over simpler baselines*. We acknowledge that our current presentation may not have sufficiently highlighted the **statistical advantages of TITAN** over baseline methods, which might have contributed to the reviewer's impression that *the improvements are marginal*. To address this, we have reorganized the simulation results in the main text and now present the supporting statistics in Table R5. \n\n**Table R5**: A summary of the statistical performance of TITAN. **DG** and **ID** refer to *Domain Generalization* and *In-Distribution*, respectively. Isotropic means equal couplings; anisotropic means direction-dependent couplings Hamiltonian. Energy difference $\\Delta E$ is defined in Table R1. \n| Train | Test | Avg $\\Delta E = E_{TITAN}- E_{Gaussian}$ | Avg Saved Paras | Task type| \n|---------|-----|-------|----|:----:|\n| HEA (Isotropic Hamiltonian) | HEA (Isotropic)   | $ ‑0.0678 \\pm 0.3851$  |  $ 4.07 \\pm  4.29 $ (\\%) | ID |\n| HEA (Isotropic) |  SU2 (Isotropic)  |  $‑0.1982 \\pm 0.3880 $ | $ 2.96 \\pm  6.33$ (\\%)| DG |\n|  HEA (Isotropic) |  SEL (Isotropic) | $‑0.0933 \\pm 0.1911 $ | $ 3.89 \\pm  4.12$ (\\%) |  DG |\n|  HEA (Anisotropic) | HEA (Anisotropic)  |  $‑0.0078 \\pm 0.1851 $ | $ 4.05 \\pm  4.31$ (\\%)| DG |\n\n**Sub-Table R5**:  VQE benchmarks for molecules (i.e., H2 , HF, LiH , BeH2, H2O, N2, CO).\n\n| Train | Test  | Avg $\\Delta E = E_{TITAN}- E_{Gaussian}$ | Avg Saved Paras | Task type| \n|-------|------|--------|-------|:----:|\n| Molecules | Molecules | $ + 0.00057  \\pm  0.000094 $  |   $ 34.94  \\pm  3.226 $ (\\%)| ID |\n\nWe would like to clarify that the tasks in Table R5 fall into two broad categories: **in-distribution** and **domain generalization** tasks. The in-distribution tasks involve predicting frozen parameters for unseen Hamiltonians within the same family, using a fixed ansatz. In contrast, the domain generalization tasks evaluate TITAN’s ability to transfer across ansätze: the model is trained on a family of Hamiltonians using the HEA, and then applied to structurally distinct ansätze (i.e., SU2 and SEL) without retraining, by directly predicting their frozen parameters. While domain generalization is inherently more challenging, it demonstrates TITAN’s broad applicability better, enabling a single model to generalize across a wide range of VQE tasks.\n \nThe achieved results in Table R5 indicate that TITAN achieves *substantial improvements over baseline methods on in-distribution tasks*. Specifically,  more than 4% parameters are saved while maintaining more accurate ground energy, and saved parameters are up to an average of 36% for molecule tasks in Sub-Table 5. Even for domain generalization tasks, it provides notable improvements over baseline models that lower the ground energy with an average of 3.5% parameters saved.\n \n(2) *It is unclear when TITAN will be effective*.  The results in Table R5 indicate that an optimized TITAN model can benefit a wide range of VQEs across diverse ansätze and Hamiltonians. This broad applicability allows TITAN to reduce measurement overhead, with its advantages compounding as *the number of inference-time Hamiltonians increases*. \n\nA concrete example is advancing **quantum phase classification** [1]. For an $N$-qubit TFIM, we need to sweep $n_J$ couplings and $n_h$ field parameters to determine the critical point. As a result, the total number of TFIMs to be explored is  $n_{grid} = n_J n_h$. For each TFIM, the number of Pauli terms is  $K = 2N-1$. Assume the employed ansatz contains $P$ trainable parameters and the parameter-shift rule is used, which requires two circuit evaluations per parameter for each Pauli term. For each evaluation, $10^3$ shots are required. Taken together, the measurement cost for each TFIM is $ M_0 = 2KP_\\theta = 2\\times 10^3 (2N-1) P$ $. For $n_{grid}$ TFIMs, the vanilla VQE takes $ M_{tot} =  n_{grid} M_0 = 2\\times 10^3 (2N-1) n_J n_h P$ for a single iteration. We show the quantitative analysis in Table R6 below, which exhibits *the benefit of TITAN compounds rapidly as problem instances grow larger*.\n\n**Table R6**:  Baseline measurement cost for HEA ansatz in 1D-TFIM phase classification when layer number is $l = 6$ with 5% parameters freezing for TITAN. Here we only consider a single iteration.\n| Qubits N | $n_J$  | $n_h$ | $P $ | $ M_{tot} $ | $P^{TITAN} = 0.95 P$ | Reduced $ M_\\mathrm{tot} $ |\n|:------:|:------:|:-----:|:------:|:-----:|:-----:|:-----:|\n| 10 | 50 | 50 | 120 | $1.14×10^{10}$  | 114 | $5.7×10^8$ | \n| 100 | 50 | 50 | 1200 | $1.19×10^{12}$  | 1140 | $5.95×10^{10}$  |\n| 1000 | 50 | 50  | 12000 | $1.20×10^{14}$ | 11400 |  $6×10^{12} $  |\n\n[1] Herrmann et al. Nat Commun 13, 4144 (2022). \n\n(3) *What are the advantages of TITAN? How about its training set size and runtime overhead?*\n\nThe required training dataset size and runtime overhead of TITAN are summarized in Table R7. Following explanations in **W1 (2)**, the computation cost in the training is ignorable, particularly when amortized over a growing number of inference-time Hamiltonians. Moreover, TITAN’s advantages compound as the number of such inference tasks increases. \n\n**Table R7**: Training details of TITAN.\ntorch: 2.5.1+cu121| NVIDIA GeForce RTX 3060 12 GB\n| Dataset | Details| Samples per Case |Dataset Size | Training Time per Epoch |\n|---------------|--------|:-------:|-----|--------|\n| Isotropic Hamiltonian |  N(5-15), L(5-15) | 20 | 79.2 MB | 24.056 $\\pm$ 2.671 s |\n| Anisotropic Hamiltonian |  N(8), L(5-7), (a,b,c) in [-5,5]| 20 | 527 MB | 647.564 $\\pm$ 11.389 s|\n| Molecule |  H2 , HF, LiH , BeH2,  H2O, N2, CO |20 | 2.38 MB | 1.345 $\\pm$ 0.284 s |\n\n**Q1: Provide a single plot summarizing the typical expected improvement.**\n\n**Reply**: Due to figure upload restrictions, we present the results in Table R5. As noted in our response to **W1 (1)**, TITAN achieves an average parameter freeze of 3–5% on most **in-distribution tasks** while maintaining the accuracy of the minimum energy solution. Notably, for larger molecules or large-scale VQE instances, the freeze ratio can reach up to 60%. For **domain generalization** tasks, TITAN can still provide meaningful improvements. \n\n**Q2: (1) Specify training details used for TITAN. (2) Were the training models the same as those tested? If they were unrelated or smaller in scale, it would further support the generalizability of the approach.**\n\n**Reply**:  Let us address the reviewer's two concerns in sequence. \n\n**Concern (1)**. As suggested, we have included additional implementation details of TITAN and have provided the code and dataset in the supplementary materials. For illustration, Table R7 outlines the data provenance, network hyperparameters, and training time used.\n\n**Concern (2)**. The reviewer's understanding is correct. As shown in **Tables R2 and R5**, **TITAN exhibits strong domain generalization across ansätze**. Specifically, the model is trained on a family of Hamiltonians using HEA, and is then successfully applied (without retraining) to predict frozen parameters for structurally distinct ansätze such as SU2 and SEL, achieving competitive performance.\n \n**Q3: How does TITAN compare to a heuristic that freezes early layers with higher probability?** \n\n**Reply**:  We would like to clarify that the distribution of frozen parameters is **problem-dependent**. While Figure 4 illustrates a tendency for frozen parameters to concentrate in the shallow layers when using HEA on isotropic Hamiltonians, this pattern is **not universal**. As shown in Figures 6, 7, 16, and 17 of the Appendix, the distribution varies substantially across different ansätze and problem Hamiltonians.\n\nTo better address the reviewer's concern, we compare TITAN against the Early-Layer Freezing (ELF) heuristic, as suggested. The achieved results, as summarized in Table R8, indicate that the TITAN freezing strategy shows faster and more thorough gradient decay at all qubit scales (N=5–30).  While ELF beats plain Random, it still trails TITAN, and the gap widens as circuit size increases, around 4 times higher than TITAN in gradient norm and 3.1% higher final energy.\n\n**Table R8**: Comparison of gradient norm and final energy between TITAN and ELF for 30-qubit TFIMs. \n | Iteration |  0  | 40 |  80 | Final Energy |\n|-----------|----|---|----|----|\n| Baseline | 10.024 $\\pm$ 1.519| 3.924 $\\pm$ 1.104 |   0.182 $\\pm$ 0.095| -89.364 $\\pm$ 1.500  |\n| Random |9.107 $\\pm$ 1.262| 9.718 $\\pm$ 1.054 |   3.055 $\\pm$ 1.390|  -51.169 $\\pm$ 2.267  |\n| TITAN | 8.766 $\\pm$ 1.259 | 6.624 $\\pm$ 1.073| 0.383 $\\pm$ 0.200|  -86.156 $\\pm$ 2.022  |\n| ELF | 11.677 $\\pm$ 1.853 | 5.465 $\\pm$ 2.791 | 1.310 $\\pm$ 1.283 |   -83.494 $\\pm$ 3.758 |\n \n**L1 & L2: Discuss training cost and problems related to optimizing VQAs.**\n\n**Reply**: We have added a discussion on the training cost of TITAN; please refer to our response to **Q1** for details. In addition, we have included more references in the updated manuscript related to VQA optimization, covering topics such as barren plateaus, measurement reduction, and the local minima landscape."}, {"review_id": "39UCyAQMTQ", "reviewer": "Reviewer_cy7n", "summary": "TITAN is the work proposed in this paper to reduce the number of measurements required by VQEs by freezing some of the parameters statically and a priori so that the parameter shift doesn't have to be applied for their gradient calculation, saving circuit runs, time, and resources. The work also supports the prior work in reducing barren plateaus by showing that parameter freezing can also help reduce them in the optimization landscape. TITAN trains a RestNet model with input features as circuit parameters and their freezing intensities and outputs which parameters should be frozen before the VQE is executed.", "questions": "How were 100-qubit circuits evaluated using simulators when it is impossible to simulate them on even the largest supercomputers? Were they purely Clifford?\n\nWould a different type of TITAN model need to be trained for different types of ansatz? An alternative to HEA, for example?\n\nWhat is the overall overhead of training and all the measurements that need to be done for training? Is the saving in inference worth it if each trained model only serves a limited class of VQE instances?\n\nDid you observe any interesting trends in which parameters got frozen (e.g., beginning layers, end layers, etc.)?\n\nWould TITAN need a large training dataset? Does a new model have to be trained for different ansatz sizes, number of qubits, number of layers, etc.? Do different VQE domains require different Titan models?", "limitations": "The paper does not have direct negative societal impacts. Some limitations of the work are discussed, such as the scaling issues; others are highlighted above in the form of questions and weaknesses.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n\nParameter freezing of VQE is a bit of an underexplored area, and the thought of freezing the parameters a priori is especially interesting.\n\nThe empirical analysis indicates that there is a large scope for parameter freezing without losing performance in terms of achieving the ground state energy of the VQE Hamiltonian.\n\nWeaknesses:\n\nWe don’t know if the baseline technique reaches the minimum eigenvalue (the ground state energy in VQE) since only the difference between the baseline and TITAN is shown. Both techniques could be performing poorly and have a low differential. Please provide the results as raw numbers or provide the differential as a percentage of the raw minimum eigenvalue.\n\nNo error bars are provided for any values, even though they are all statistical quantities. For instance, the intensity of frozen parameters is a fractional quantity that varies from ansatz to ansatz, even if the two ansatzes have the same number of qubits (e.g., in Figure 2). So, some sort of variation metric or visualization should be provided, as these are not deterministic numbers. Also, Figure 2 just conveys that the number of frozen parameters increases linearly with the number of qubits and the number of layers. I'm not sure if this is a novel insight.\n\nShouldn’t the Hamiltonian itself be an input to the TITAN model training (currently, the input is only circuit features and freezing intensity of the parameters), because not just the circuit features but also the optimization loss affect the frozen parameters? So, the Hamiltonian itself should be included as an input? This might also help generalize the model. A similar case can be made for initial parameter values. They also affect the freezing intensity and should be included in the training.\n\nThe frozen parameters vary dynamically over time. I'm not sure if it is the best idea to freeze them statically a priori. \n\nIn most cases, the number of parameters frozen seems to be very low (Table 1), having a negligible impact on measured parameters.", "was_revised": true, "final_justification": "The rebuttal has provided additional results (the exact minimum energies, etc.), which have clarified many of my questions. Therefore, I have updated my score.", "rebuttal": "We sincerely thank Reviewer cy7n for the constructive feedback. We have carefully addressed the concerns raised under **Weaknesses** [**W1–W5**] and the specific **Questions** [**Q1-Q5**]. We hope the following responses effectively address the reviewer’s concerns and aid in the reassessment of our submission.\n\n**W1: Report the raw energy values or the difference as a percentage of the ground-state energy.**\n\n**Reply**: We have followed the reviewer's advice to exhibit how far TITAN and the baseline models deviate from the ground-state energy. For convenience, Table R9 summarizes the corresponding key findings. Please refer to Table R3 (reply to **W1** for Reviewer 8G53) for details. \n\n**Table R9**: Energy differences between Molecules with three initialization strategies and TITAN Freezing.\n|           | H₂ (4)   | HF (10)   | LiH (10) | BeH₂ (12) | H₂O (10)  | N₂ (12)   | CO (12)   |\n|---------|--------:|--------:|--------:|---------:|-------:|-------:|-------:|\n| Gaussian [1]| -0.896 $\\pm$ 0.106| -97.601 $\\pm$ 0.114 |  -7.882 $\\pm$ 0.085 |  -14.876 $\\pm$ 0.506 | -72.876 $\\pm$ 2.065| -103.411 $\\pm$ 2.639 | -107.669 $\\pm$ 3.095|\n| Gaussian [1] + TITAN | -0.896 $\\pm$ 0.106 |  -97.601 $\\pm$ 0.114 | -7.864 $\\pm$ 0.085| -14.868 $\\pm$ 0.522 | -72.876  $\\pm$ 2.065 | -103.418 $\\pm$ 2.891| -107.684  $\\pm$ 3.001 |\n| Exact| -1.175  |-100.340 | -8.023 |-15.918 | -76.428  | -109.542  | -113.326  |\n| TITAN Frozen Params |  1/3  |  1/3  |   10/24  |   21/92 |   4/54 |   47/117|   77/117  |\n\nThe achieved results in Table R9 provide the following insights: \n1) For most cases, the baseline model [1] and TITAN are very close to the exact results; \n2) For the molecule CO, TITAN achieves an additional energy reduction of approximately 0.015 while halving the number of parameters. \n3) The slight discrepancy between TITAN and the exact results stems from **the choice of ansatz**. For example, in the 30-qubit TFIM experiment, TITAN can freeze 30% of the parameters while still achieving convergence comparable to the exact result, as shown in Table R10. \n\n**Table R10**: Statistical performance of TITAN on diverse VQE benchmarks (Simplified version of Table R5) \n| Train | Test | Avg $\\Delta E = E_{TITAN} - E_{Gaussian}$ ( $\\Delta E / E_{Gaussian}$) | Avg Saved Paras | \n|------|------|-------|------|\n| HEA (Isotropic Hamiltonian) | HEA (Isotropic Hamiltonian)   | $ ‑0.0678 \\pm 0.3851 $ (-0.48 \\% $\\pm$ 0.92\\%)  |  $ 4.07 \\pm  4.29 $ (\\%) |  \n|  HEA (Anisotropic Hamiltonian) | HEA (Anisotropic Hamiltonian)  |  $‑0.0078 \\pm 0.1851 $  ( -0.80 \\%  $\\pm$ 1.28\\%) |   $ 4.05 \\pm  4.31$ (\\%)| \n\n[1] Zhang et al. Escaping from the barren plateau via Gaussian initializations in deep variational quantum circuits. NeurIPS (2022).\n\n**W2: No error bars are provided in Figure 2.**\n\n**Reply**: We would like to remind the reviewer that Figure 2 is intended **solely to illustrate the 'parameter-freezing' phenomenon that motivates our study**. For ease of visualization in the 3D plot, uncertainty bands were omitted. To support the illustration with statistical rigor, we have included detailed results in the Appendix, based on ten independent runs with different random seeds.\n\nIn addition, we would like to emphasize that all performance figures already include statistical variation in the submission, i.e., Figs. 5-7, as well as newly added results (Table R1-R11).\n\nRegarding the concern \"*the number of frozen parameters increases ...,  if this is a novel insight*\", we believe this is indeed a **novel** observation. To the best of our knowledge, the freezing phenomenon has not been explicitly characterized in prior work. Notably, *this novelty was also acknowledged by Reviewer 8G53*. Moreover, this observation serves as the foundation for TITAN, whose effectiveness has been systematically validated through extensive simulations.\n\n**W3: Shouldn’t the Hamiltonian itself be an input to the TITAN model training?**\n\n**Reply**: We would like to emphasize that whether the Hamiltonian itself should be encoded as input to TITAN depends on **the nature of the Hamiltonians under consideration**. In our submission, we focus on Heisenberg and TFIM models, which share a common Hamiltonian structure and therefore offer limited additional information to encode. As such, we do not include the Hamiltonian structure as input.\n\nHowever, we fully agree that for more complex lattice systems, e.g., triangular, hexagonal, or long-range interacting graphs, the Hamiltonian encodes rich structural information that may strongly influence parameter freezing behavior [1–2]. We have added a discussion of this direction to the updated version as a promising avenue for future work.\n\n[1] Yang et al. Scalable variational Monte Carlo with graph neural ansatz.\n\n[2] Roth et al. Group convolutional neural networks improve quantum state accuracy.\n\n**W4: Is it appropriate to freeze them statically at initialization?**\n\n**Reply**: We would like to emphasize that the choice between static and dynamic parameter freezing is **not unique**, as each strategy has its own merits and trade-offs. In particular, a dynamic freezing strategy may incur significant computational overhead, as updating the freeze mask at each iteration requires re-measuring gradients and re-synchronizing the optimizer. Accordingly, this cost scales linearly with both the number of parameters and the shot budget.\n\nWe consider TITAN's static approach a practical first step, supported by strong empirical results (e.g., Table R5). That said, dynamic or hybrid strategies remain promising directions for future work, now noted in the revised manuscript.\n \n**W5: In most cases, the number of parameters frozen seems to be very low.**\n\n**Reply**: We acknowledge that our current presentation may not have sufficiently highlighted the statistical advantages of TITAN over baseline methods, which might have contributed to the reviewer's impression that the improvements are marginal. To address the reviewer's concern, in the following, we briefly introduce the main reasons.  \n\n1) Regarding the marginal improvement in Table 1. The number of frozen parameters is **highly dependent on $\\tau$ and Hamiltonian**. With the results in Table R9, it is notable that for CO and N2, more than 50\\% parameters are frozen.\n\n2) As elucidated in Table R5, TITAN often attains an evident improvement for **In-Distribution tasks**. Even for **domain-generalization** tasks, TITAN can provide notable enhancement, enabling a broader applicability. \n\n3) As shown in Table R6, the benefit of TITAN increases rapidly as problem instances grow larger.\n\n**Q1: How were 100-qubit circuits simulated—were they Clifford-only?** \n\n**Reply**: We would like to point out that efficient simulation of large-scale quantum circuits, those exceeding 100 qubits, is an active area of research. Recent advances in *tensor-network simulators* and *Pauli propagator techniques* [Sci. Adv.10,eadk4321(2024)] have enabled the simulation of specific classes of non-Clifford circuits. For instance, [1] demonstrates the simulation of quantum convolutional neural networks with up to **1,024 qubits**.\n\n[1] Bermejo et al. \"Quantum convolutional neural networks are (effectively) classically simulable.\" \n\nThe main challenge in evaluating TITAN beyond 30 qubits lies in the **demands of VQE optimization**. While large-qubit circuits can be simulated, existing simulators are not yet efficient enough to support full optimization workflows, i.e., collecting the rich gradient trajectories required for training. Any progress in this direction would broaden TITAN’s applicability.\n \n**Q2: Would a different type of TITAN model need to be trained for different types of ansatz? An alternative to HEA?** \n\n**Reply**: TITAN demonstrates strong domain generalization capability. As shown in Table R5 (response to **W1** for Reviewer 27vu) and Table R11, in **domain generalization tasks**, TITAN is trained on a family of Hamiltonians using HEA and then applied (without retraining) to structurally distinct ansätze such as SU2 and SEL by directly predicting their frozen parameters without increasing final energy. For an anisotropic Hamiltonian, which is more complex due to the coupling strengths differing between directions, TITAN consistently attains better energy estimations even when parameters are frozen, underscoring its domain generalization ability.\n\n**Table R11** (Simplified Table R2): Performance of $\\Delta E$ on domain generalization tasks. \n| Testing Ansatz | N5 L5| N15 L5| N15 L15 |\n|:-----:|--------|--------|---------|\n| SEL | + 0.4415 $\\pm$ 0.0541 |  -0.6715 $\\pm$ 1.1730 | -0.5213 $\\pm$  0.2285|\n| SU2  | + 0.1231 $\\pm$ 0.3681 | -0.3110 $\\pm$ 1.7138 | -0.3013 $\\pm$ 0.0351 |\n\n**Q3: What is the overall overhead of training and all the measurements that need to be done for training?**\n\n**Reply**: We would like to accord Table R7 (**Q2** for Reviewer 27vu) to address the reviewer's concern. Once TITAN is optimized, no further training is required. As the number of test samples increases and the scale becomes larger, TITAN's benefits become more significant. \n\n**Q4: Any interesting trends?**\n\n**Reply**: We observe that the freezing phenomenon exhibits problem-dependent patterns. In Figure 4 (main text) and Figures 6, 7, 9, 12, 16, and 17 (Appendix), the distribution of frozen parameters varies across different ansätze and Hamiltonians. To capture these variations, we propose TITAN as a data-driven solution that adaptively selects freeze candidates, enabling it to generalize effectively across HEA, SU2, and SEL circuits (see the reply of **Q3** for Reviewer 27vu ). \n\n**Q5: Dataset & cross‑ansatz generalization**\n\n**Reply**:  Let us address the reviewer's concerns in sequence. \n1) TITAN does not require a large dataset as referred to in the response to **W1** for Reviewer 27vu. \n2) We further evaluated the HEA‑trained model outside the training distribution, as referred to in the response to **W2** for Reviewer dadQ."}]

Author rebuttal:

Meta-review: The paper reveals an insightful phenomenon: in variational quantum circuits, many parameters exhibit limited dynamics and can be frozen once optimized. Building on this, it proposes a learning-based framework to predict which parameters can be frozen and to what values, serving as initialization for variational quantum circuits. 
The proposed method, TITAN, significantly speeds up convergence and reduces circuit evaluations while maintaining accuracy. The methodology and experimental results demonstrate a meaningful contribution. The rebuttal further clarifies improvements in performance and generalization capability, effectively addressing reviewers' concerns.
Decision: Accept (poster)