AQER: A Scalable and Efficient Data Loader for Digital Quantum Computers
Authors: Kaining Zhang
Xinbiao Wang
Yuxuan Du
Min-Hsiu Hsieh
Dacheng Tao
Venue: ICLR 2026 Poster (2026)
Keywords: quantum data loading
Primary area: other topics in machine learning (i.e., none of the above)
TL;DR: We introduce AQER, a scalable approximate quantum loader to efficiently encode classical and quantum data.

Abstract: Digital quantum computing promises to offer computational capabilities beyond the reach of classical systems, yet its capabilities are often challenged by scarce quantum resources. A critical bottleneck in this context is how to load classical or quantum data into quantum circuits efficiently. Approximate quantum loaders (AQLs) provide a viable solution to this problem by balancing fidelity and circuit complexity. However, most existing AQL methods are either heuristic or provide guarantees only for specific input types, and a general theoretical framework is still lacking. To address this gap, here we reformulate most AQL methods into a unified framework and establish information-theoretic bounds on their approximation error. Our analysis reveals that the achievable infidelity between the prepared state and target state scales linearly with the total entanglement entropy across subsystems when the loading circuit is applied to the target state. In light of this, we develop AQER, a scalable AQL method that constructs the loading circuit by systematically reducing entanglement in target states. We conduct systematic experiments to evaluate the effectiveness of AQER, using synthetic datasets, classical image and language datasets, and a quantum many-body state datasets with up to 50 qubits. The results show that AQER consistently outperforms existing methods in both accuracy and gate efficiency. Our work paves the way for scalable quantum data processing and real-world quantum computing applications.

Published as a conference paper at ICLR 2026
AQER: A SCALABLE AND EFFICIENT DATA LOADER FOR DIGITAL QUANTUM COMPUTERS
Kaining Zhang', Xinbiao Wang', Yuxuan Du 2: Min-Hsiu Hsieh DDacheng Tao
"Generative Al Lab, College of Computing and Data Science, Nanyang Technological University, Singapore 639798, Singapore School of Physical and Mathematical Sciences. Nanyang Technological University, Singapore 639798, Singapore
'Hon Hai (Foxconn) Research Institute, Taipei, Taiwan yuxuan, duêntu. u. edu. sg, min-hs uu..iu.. ha i eh foxconn. com, wwc.tng..tt@tttu...... eaddnntu.edu...
ABSTRACT
Digital quantum computing promises to offer computational capabilities beyond the reach of classical systems, yet its capabilities are often challenged by scarce quantum resources. A critical bottleneck in this context is how to load classical or quantum data into quantum circuits efficiently. Approximate quantum loaders (AQLs) provide a viable solution to this problem by balancing fidelity and circuit complexity. However, most existing AQL methods are either heuristic or provide guarantees only for specific input types, and a general theoretical framework is still lacking. To address this gap, here we reformulate most AQL methods into a unified framework and establish information-theoreee Our analysis reveals that the achievable infidelity between the prepared state and target state scales linearly with the total entanglement entropy across subsystems when the loading circuit is applied to the target state. In light of this, we develop AQER, scalable AQL method that constructs the loading circuit by systematically reducing entanglement in target states. We conduct systematic experiments to evaluate the effectiveness of AQER, using synthetic datasets, classical image and language datasets, and a quantum many-body state datasets with up to 50 qubits. The results show that AQER consistently outperforms existing methods in both accuracy and gate efficiency. Our work paves the way for scalable quantum data processing and real-world quantum computing applications.
bounds their approximation error.
INTRODUCTION
Digital quantum computers (Feynman
1982) promise to deliver computational capabilities that
surpass those of classical systems in diverse fields (Shor 1994 Harrow et al. 2009 Cao et al. 2019
Liu et al. 2021 | Huang et al.) [022 Liu et al. 2024). After four decades of exploration (Martinis
al. 1985 1 et al. [2000] Makhlin al. 2001
C0orCllare Clarke and Wilhelm 2008 DiCarlo et al.) 2009
Isenhower et al. 2010; Arute et al. 2019 Wu et al.
2021 Graham et al. 2022), substantial progress has been achieved, with superconducting (Acharya al.. 2025) and neutral atom platforms (Xu
ct al. 2025) demonstrating quantum ad advantages on synthetic benchmarks. Nevertheless, the available quantum resources, such as the number of high-quality qubits and coherence time, would remain severely constrained in the foreseeable future (Jiang et al. 2025). This limitation underscores the need consistently improve the efficiency of the three foundational modules of quantum computing: quantum state preparation (Girolami) 2019 Gonzalez-Conde et al. 2024). quantum processing (Bharti et al., 2022; Cardama et al. 2025). and readout (Zhang et al. 2021 Schreiber et al. 2025). Maximizing resource utilization across these stages is essential to advancing the practical utility of quantum computers. In this context, efficient quantum state preparation, which amounts to constructing quantum gate sequences that encode classical quantum inputs into quantum states, emerges as a critical prerequisite (Ranga et al. (2024). This step remains notoriously challenging, as theoretical results indicate that in the worst case, preparing arbitrary quantum state
"Corresponding authors.

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Published as a conference paper at ICLR 2026
within provable error tolerance may require exponential number of quantum gates or ancillary qubits (Zhang et al. 2022c Gui et al. 2024).
Recently, new concept known as the approximate quantum loader (AQL) (laconis and Johri 2023) has emerged, offering a promising direction for efficient quantum state preparation. Unlike earlier approaches that aimed to achieve provable accuracy guarantees, embraces trade-off between preparation fidelity and circuit complexity. The central insight driving this paradigm is that many quantum algorithms can tolerate some imprecision in the input state, which allows substantial re- ductions in gate count and resource overhead. For instance, in quantum machine learning, small perturbations in input features often have negligible impact on classification accuracy (Nguyen et al.) 2020) and do not compromise the demonstration of quantum advantage, particularly in terms of sample complexity (Huang et al. 2023). Motivated by this observation, considerable effort has been devoted to designing efficient AQL schemes for various data types. These methods broadly fall into two categories: tensor network bbaee approaches (Jobst et al.] [2024] laconis al.) 2024). which employ matrix product state (MPS) and other TN representations, and circuit-based approaches (Mit- suda et al. 2024, Shirakawa et al. 2024). which directly optimize quantum gate sequences (see Fig. [ for general framework and Sec. 3.1 for details). Despite this progress, most existing techniques are either heuristic or provide theoretical guarantees only under restrictive scenarios. Con- sequently, a systematic understanding of the fundamental limits on approximation error achievable by AQL. remains elusive. Addressing this question would provide critical insights for the principled development of AQL algorithms and for the resource- efffcietefficientt use of digital quantum computers.
Here we narrow the above knowledge gap by first reformulating a wide range of AQL meth- ods into a unified framework. An intuition is illustrated in Fig. where the AQL construction amounts to identifying a sequence of (tunable) quantum gates that minimizes the distance be- tween the state evolved by this sequence and the target quantum state. Building this frame- work, we derive information-theorete lower and upper bounds for the approximation er- ror of AQL (Theorem [3 which are indepen- dent of specifie AQL strategies. Specifically, we demonstrate that the approximation error de- creases linearly with the newly proposed entan- glement measure, which amounts to the sum- mation of the single-qubit entanglement entropy of the target quantum states after the inverse of the AQL gate sequence. This result provides the key insight that AQL performance can be fundamentally characterized by the degree entanglement reduction achievable during the AQL, with larger entanglement reduction leading to more accurate loading.
Too
U(0,AA) 2
Approximate Quantum Loader
TN-based method U Racw)
Circuit-bused method
Variational: 0 - 0 40
U(
10000000 U. Non- variatiiooo::
The general framework of AQL. Typical AQLS separated into categories: TN-based meth- ods and circuit-bant based methods. both of which aim to construct quantum circuits from a given gate set U that approximately prepare the target state.
Figure 1:
Motivated by the theoretical importance of entanglement in determining the performance of AQL, we develop AQER, a scalable and efficient AQL method that constructs the gate sequence guided by the principle of maximal Entanglement Reduction. In particular, AQER leverages this principle to reduce the entanglement of target states by progressively adding parameterized single- and two-qubit gates, and employs an explicitly constructed single-qubit gate sequence to further reduce the approximation error. Compared to existing AQL methods, AQER offers two key advantages. First, AQER is flexible and universal, supporting efficient approximate loading of both classical data and unknown quantum states. Second, AQER is robust and easy to optimize. Guided by entanglement-reet reduuctoon optimization, it not only achieves low approximation error but also mitigates vanishing gradient problems during the parameter training (Larocca et al., 2025), thereby ensuring scalability to large-qubit systems. To validate the effectiveness of AQER, we conduct systematic experiments to benchmark its performance on different quantum state loading tasks, ranging from synthetic quantum states, real-world image and language datasets, to quantum many-body systems with up to 50 qubits. The achieved results reveal that AQER consistently outperforms existing methods. Our work paves the way for scalable quantum data processing and practical applications of quantum computing in real-world tasks.
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In summary, our contributions are threefold. (i) We propose unified framework for wide range of AQL methods and derive two information-theot theooretc bounds on the approximation error of AQL with respect to the entanglement measure. To the best of our knowledge, this is the first study to establish theoretical limits for AQL from an information-theorele perspective. (ii) Motivated by these theoretical results, we develop AQER, scalable and efficient AQL. method that features principled entanglement reduction optimization to efficiently utilize gate resources. (iii) We conduct extensive numerical simulations across diverse datasets with up 50 qubits, and compare AQER against reference AQL methods. The results validate the effectiveness of AQER, demonstrating superior performance with lower approximation error and reduced gate count. The corresponding code is available at GitHub for reproducibility and benchmarking purposes.
2 PRELIMINARY
Here, we introduce the basics of quantum computing, quantum entanglement, and variational quantum algorithms, followed by reviewing typical AQL methods. Refer to Appendix
A for more details.
Basies of quantum computing. The pure state of a qubit can be written as (bb) al0) + b/1),, where
a, b E C satisfy |a|2 + 16|2 1, and [0] (1,0)², [1] (0, 1)T The -uuii space is formed by the lensor product of N single-qubit spaces. For a pure state (eb). the corresponding density matrix is defined as p [,b] where (1bl = (l1))). Mixed states are represented by density matrices of
the form p = Σx Cj (/s) (1xx). where C& 2 0 and Σ. 4 = 1. For an N-qubit state [19). the reduced density matrix of subsystem A C [N] is obtained by the partial trace Pa = T aaooiaalllll ally>>>>]] A quantum gate is represented by a unitary matrix acting on the state, and can be depicted in the circuit model as - in quantum circuit notation. Typical quantum operations include fixed gates such as CZ diag(1, 1, -1), and tunable single-qubit rotation gates given by R. (0) e
e-100 /2 with a E (X,Y } and X = (1) Y = ( ) Z = (d -1 being Pauli-X, -Y, -Z operators. The rotation gate could be generalized the two-qubit case, for example Rzz with ZZ Z & Z. The quantum measurement refers to the procedure of extracting classical information from a quantum state. It is mathematically specified by a Hermitian matrix H called the observable. Measuring the state (ي)) with the observable H yields random variable whose expectation value is (1b) H (b) or Tr p). respectively. To quantify the similarity between two quantum states, we use fidelity defined as F (whn m/2) 12 or F = Tr(pip2]. with the corresponding infidelity given by 1 - -F (Nielsen and
Chuang 2010). Both of them can be evaluated by taking the density matrix of either [R/1) or (/22) as
the observable.
Quantum entanglement. Quantum entanglement is a unique property of quantum systems that distinguishes them from classical systems and serves as pivotal quantum resource for achieving quantum computational advantages. Specifically, an N --qubit quantum state (&)) of a composite system tensor product
is called entangled across two subsystems A, B c [N] if it cannot be written
of states of the subsystems, i.e., (ıb) * (( @ (Db) for any states |wba), : (&b) in the respective subsystems. Various metrics have been developed quantify the degree of entanglement in quantum states. One commonly used measure is the Renyi-2 entropy, defined as ((())) = log, Tr(p2]]
where P.A the reduced density matrix of (ı)) on the subsystem A. When (A())) 0, the quantum state (15) is separable and can be written as a product state (v)) = -) Ø /VB).
Variational quantum algorithm (VQA). VQA (Cerezo al. 2021a) is hybrid quantum-classical paradigm in which a parameterized quantum circuit (PQC) is trained by a classical optimizer. It has become a core framework for quantum machine learning (Schuld and Killoran (2019] [uu et al., 2025) . with applications spanning discriminative learning, generative learning (Lloyd and Weedbrook 2018
Gao et al. 2018 Tian et al. 2023), and reinforcement learning (Chen et al., [020)] Cheng et al. 2023 2) Concretely, a PQC is written as ( = Vm (0m) (i(O)). ). and the parameters 0 are optimized to minimize a cost function defined with respect to an input state Pin and an observable i.e. (0) Tr O V (O))aa
The parameters are updated by a classical optimizer such as gradient descent or Adam, which requires accessing (exact or estimated) gradients. When the variational circuit is simulated on classical computer, gradients can be computed directly via automatic differentiation (Bergholm et al. 2018). When executed on a quantum device, gradients are typically estimated by the parameter-shift rule Wierichs et al. 2022):
0s = f00+) - {0_).
30,
where 0g differ from 0 by + /2 the j-th parameter.
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2.1
RELATED WORK
Existing approaches to AQL can be broadly categorized into TN-based and circuit-based methods.
TN-based methods. TN-based methods exploit tensor networks to efficiently represent and prepare low-entanglement quantum states. For example, matrix product states (MPS) with bond dimension k can be prepared exactly with (LV two-qubit gates (Schön et al., 2005). and approximate encoding is feasible for states with compact MPS representations (Ran 2020). In light of this,
TN-based strategies have been applied construct approximate quantum state encodings for classical data (Holmes and Matsuura, 2020, Iaconis and Johri, [202]] Jobst et al. 2024] laconis et al. (024). These approaches provide a principled framework with controlled approximation error for low- entanglement inputs, but their utility to quantum data or highly entangled classical data is limited.
CGreuit-basee methods. Cireuit-based methods directly optimize the quantum gates that generate the target state, often without relying on an explicit low-entanglement representation. These methods can be broadly categorized into variational and non-variational strategies. Variational approaches train parameterized circuits to minimize infidelity with respect to the target state (Nakaji et al. 2022 Rudolph et al. 2023b; Mitsuda et al., 2024), while non-variational methods iteratively optimize local two-qubit gates along the circuit, progressively improving accuracy without explicit parameter training (Rudolph et al. 2023a Shirakawa et al., (024). Circuit- based methods are flexible and broadly applicable, but lack rigorous theoretical guarantees and suffer from barren plateaus.
3 APPROXIMATE QUANTUM LOADER
In this section, we first reformulate the main AQL approaches within unified optimization framework and derive two coormation-theoretiio bounds of the approximation error of AQLS in Sec. 3.1 We then present the implementation details of AQER in Sec. 3.2
3.1 UNIFIED FRAMEWORK OF AQL AND THEORETICAL ANALYSIS
Existing AQL methods can be broadly categorized into TN-based methods and circuit- basee methods. For clarity, first elucidate unified framework for AQL and explain how different methods can be formulated within this framework. Building on this framework, we then provide an information- theoretic analysis of the approximation error achievable by AQL..
unified framework of AQL. We begin by recalling the fundamental definition of approximate quantum loader (AQL). An AQL amounts to preparing the target state with controllable accuracy by generating quantum gate sequences from a given gate set U, as shown in Fig. | Specifically, let (0; .A) L denote a circuit composed of m gates, where and A represent tunable parameters and the circuit architecture, respectively. With the aim of identifying the optimal @ and A, AQL can be reformulated as a unified framework for solving the following optimization problem
arg min 0,A
1 KKrargee (0: A) (t)))eeepee (prodduc))
(1)
where Utargett) is the target quantum state and pépouut)) is an casily preparable product state. Within this framework, various AQL methods differ in how they construct the circuit U/ (0),.), either by the way of updating 0, the design of A. or adjusting both. A brief overview of how TN- and circuit-based methods fall into Eq. (1) is provided below, with further details given in Appendix A.3
TN-based methods. For these methods (Ran) 2020), the circuit U(e, A) is constructed incrementally by sequentially appending local unitaries obtained from the TN representation. Each unitary is further decomposed into hardware-availe available gates using the KAK decomposition [Tucci] 2005). which yields the final circuit. In terms of Eq. (T). the optimization proceeds by extending the eircuit as (O: ( new AU Anew), ), while keeping the previous parameters and architecture fixed.
Circuit-based methods. Cireuit-based AQL methods can be divided into variational and non- variational approaches. In variational approaches (Nakaji al. 2022 Rudolph et al.) 2023b Mitsuda et al. 2024), the circuit ((; .A) is constructed by optimizing a variational quantum circuit with cost function (()) 1 - (Vtaaret AU) | productt) 12 where 0 are tunable and AA is fixed. By contrast, non- voriviiiaiinaal approaches (Rudolph et al. 2023a Shirakawa et al. (024) gradually update all two-qubit unitaries in prescribed circuit following zigzag schedule. This involves sequentially
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(a) General tramework of AQER
(b)
Entanglement Reduction
(a)
Step Entangicment Reducction
P2] Punget
New Gate Block
Y-fe)
(
(1)(
Entrage
(1)(
Eutropy
(
(a)
V(((2))
Va
4..
Vie
Ves(ar 2)
(a
(as((a)
Product Approximaation
(c) Product Approsimation
Parameter Refineeent
Uxqea (0)
(9r)
W(B.y)
Vi(a))
-R,(8,)-B,,0%>>
10)
[0)
(.)
Jundelity
R,(3;))-R,11>>
10)
10)
[0)
[0]
R2
(ey(ar)
Optimizer
Voo(a,)
10y
RR
10y
10)
(R,
IIl: Parameter Reficemctt
B.
10)
10)
Figure 2: The workflow of the AQER algorithm. (a) An overview of AQER, which consists of three steps. (b) Step I: entanglement reduction. This step iteratively appends two-qubit gate blocks to progressively reduce the entanglement of the input (Usarges) with the circuit (r) (c) Step II: product state approximattio,. This step approximates the low-entanglement state (er) by applying single-qubit rotations { {Rz (B,))) b and (7a) | to the initial state (0) 8X (d) Step III: parameter refinement. This step finetunes all circuit parameters in 0 (3, minimize the infidelity and obtain the final AQL UAQER((")
adjusting both the parameters and the architecture AA of the circuit U(; and thus fits within the general optimization framework in Eq. (T).
Information-theoretii analysis. benefit of the unified framework in Eq. ①I is that it enables an algorithm-independeer theoretical analysis of AQL. Here, we establish two information-theorrtiia bounds of the approximation error achievable by AQL using an entanglement measure, as stated in the following theorem with the proof deferred Appendix B.2
Theorem 3.1. Denote the entanglement measure for an N-qubit state (b) S(1))) EN E Then, for the state (Urarget) and a circuit U with S(U (Vcarge))) S, the infidelity between (Dtarger) and the state generated from U any product state (v/product) is lower bounded 1 V(targe UU (U(oouuc))))
== [15)==((1V21--/1 -1). Moreover, given access to p. we can construct a product state W (prroduct) such that the infidelity is upper bounded as (eargee (1/roducc)) S f2(5):= = (1 1n2 S + O($2) and f2(S) fi( + 2N
[S]). When +
- 2 OSS))
ln2
The above results indicate that the approximation error of AQL fundamentally governed by the entanglement measure S of the evolved states U' U/argee In particular, the infidelity between the target state (earget) and the prepared state U [yproduct) scales linearly with the entanglement measure value S. Hence, a smaller S guarantees lower infidelity and smaller approximation error. Since S depends on both (Daargot)) and the circuit U, reducing infidelity through parameter and architecture optimization in AQL is equivalent to minimizing the entanglement measure S.
Remark. Theorem 3.1 can generalized to noisy channel cases as provided in Appendix
3.2 AQER: AN EFFICIENT AND SCALABLE AQL
The theoretical role of the entanglement measure S in determining the approximation enror suggests that it can serve as a practical indicator the quality of an AQL. Driven by this insight, we propose AQER, an efficient and scalable AQL that solves the optimization problem in Eq. (I) by constructing the gate sequence guided by the principle maximal entanglement reduction. In the remainder of this subsection, we present the implementation of AQER.
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Overview of AQER: As illustrated in Fig. a AQER consists of three key components: (I)
suppressing the entanglement measure in the imput data by incrementally adding quantum gates; (II) applying single-qubit rotation gates for correcting leading errors; and (III) optimizing the circuit parameters for further refinement. Next, we explain these procedures separately.
Step I: Entanglement Reduction. The goal of this step is to construct a gate sequence that reduces the entanglement of the target input by monitoring the entanglement measure S. As illustrated in Fig. b)., this achieved by iteratively appending two-qubit gate blocks Vz, (cx;), where the tunable variables include the acting qubit pair Ie = (jr, kt) with 1 S Je j ki S N and the gate parameters 0g, Specifically, the blocks {Vz, (at) have the identical structure Ry Rz with single-qubit rotation gates applied to both qubits. Let (a-1(Q1:-1) = Vz,,, (a( 1) Vi, (a1) ) be the gate sequence generated after (1 1) iterations. At the 1-th iteration, the acting qubit pair I, and gate parameters ag of the block Vz, (au) are obtained by solving the following optimization problem
I, at = arg min S (Vz(a)((t-))), I,6
where S is the entanglement measure defined in Theorem 3.1 and (t-1) == (a- Branget)- This iterative process repeated for T times such that the entanglement of the state (OT) VT (rargge))) is sufficiently small, where
Q1.7
Step II: Product State Approximation. After Step I, AQER aims to further suppress the approximation error by applying single-qubit rotation gates, as illustrated in Fig. c)). The motivation for this step is that, as suggested in Theorem 3.1 the state (er) with low entanglement can be well approximated by a product state. To prepare such a product state from standard initial state (0) @) apply additional single-qubit rotations 7) = N (Rz (3:) Ry )) Note that the optimal parameters , = (B)..... , By) and 7 = (Ti.... 7YN) can be explicitly derived without numerical optimization, as indicated in the corollary below. Refer to Appendix B. for the explicit form and derivation.
Corollary 3.2 (informal). Given constant access the quantum state (T)) with reduced entanglement, the explicit form of each parameter in (B.7) can be derived without optimization.
Step III: Parameter Refinement. The final step of AQER further enhances the accuracy by fine-tuning the parameters of the gate sequences constructed in Steps I and II, as illustrated in Fig. dd) For clarity, denote the combined circuit as UAQER ( V/r(ar) (W)+ 7), where O = (a, 3, 7)
collects all tunable parameters. Parameter refinement then amounts to minimizing the infidelity between the target state (taaget)) and the state UAQER (0) NN i.e.
arg min - Wrarget U AQQER (0)0))
After the optimization of Eq. global phase that does not affect any measurement outcomes.
obtain the AQL: (Uload) e e U9UAQE (0))))) where is
Remark. (i) An important feature of AQER is the usage of the entanglement measure S proxy for the approximation error. For quantum data, evaluating and optimizing S is efficient since it involves only local measurements. For classical data, AQER can be simulated classically to construct UAQER- (ii) By first suppressing the entanglement measure S, AQER not only reduces the approximation error but also distinguishes itself from prior circuit-based methods by mitigating barren plateau issues, thereby enhancing trainability and scalability. See Appendices D and G for additional discussion and the complexxxy analysis of AQER. (ii) In general, AQER is heuristic algorithm. However, for certain structured vector or state families, AQER admits efficient performance guarantees. In particular, as shown in Appendix H AQER provably generates an optimal loading circuit for IQP states with polynomial resource cost.
4 EXPERIMENTS
We conduct extensive numerical simulations to evaluate the performance of AQER in loading both classical and quantum datasets. Further implementation details and additional results are provided in Appendices B and respectively.
4.1
DATASET CONSTRUCTION FOR AQL
We briefly introduce datasets used in this work with more details in Appendix E.1

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Table 1: Infidelity (4) of different AQL methods on MNIST, CIFAR-IO, SST-2, S-RQC, and GS- TFIM datasets. We compare AQER with G E {20, 40, 80) against reference methods, where the latter use equal or slightly larger G due to feasibility constraints detailed in Appendix E.2 Values are reported with the mean (and standard deviation) over M samples. The hest and second-hest results are highlighted in blue and orange, respectively.
MNIST
CIFAR-10
SST-2
S-RQC
GS-TFIM
G
36
54
90
G
30
60
90
C7
36
54
90
C
27
54
81
G
36
72
90
MPS
0,330 0.187
0.257
0.068
0,056
0,049
0.901
0,870
0.814
0.746
0.663
0,605
0.056
0,039
0.041
(0.101) (0.089) (0.076)
(0.038)
(0.031) (0.028)
(0.022)
(0.022) (0.022)
(0.100)
(0.118) (0.127)
(0,004)
(0.003)
(0.002)
G 20
40
80
G 22
G 20
80
01430
0.234
0.103
0.081
0,000
0.030
0.892
0.759 0.546
0.731 0,484 0.367
0,168 0.020 0.007
(0.089) (0.079) (0.042)
(0.039)
(0.026) (0.016)
(0.016)
(0.012) (0.007)
(0.097) (0.133) (0.161)
(0.090) (0.013) (0.002)
20
40
G 30
20
AQCE
0.296 0.145
0,051
0.068
0.018
0.024
0.891 0.761 0.518
0.534
0,363
0.267
0.108
0.068
0.056
(0.083) (0.053) (0.023)
(0.036)
(0.026) (0.014)
(0.017) (0.018) (0.013)
(0.149) (0.156) (0.110)
(0.026) (0.024) (0.015)
G 20
40
80
G 20
40
80
G
20
20
80
80
AQER
(Ouns)
0.195 0.090
0.034
0.043
B.029 0.088
0.819 0.652
0.285 0,128 0.067
0.009 0.003
(0.060) (0.004) (0.015)
(0.023) (0.016) (0.010)
(0.017) (0.013) (0.008)
(0.152) (0.106) (0.069)
(0.011) (0.005) (0.001)
Classical data. We use three standard classical datasets: MNIST (Lecun et al. 1998), CIFAR- 10 (Krizhevsky et al.] 2009), and SST-2 (Socher al. 2013). These datasets have been widely employed to benchmark AQL methods on diverse data types across vision and language tasks. Specifically, MNIST contains 28 x 28 grayscale handwritten digits: CIFAR-10 includes 32 x 32 RGB images of natural objects such as cats and cars; and SST-2 is a sentiment classification dataset, for which we use a pretrained Sentence-BEi -BRT model (Reimers and Gurevych] 2019) to obtain 1024- dieeniinelional sentence embeddings. Each dataset is preprocessed into M = 50 normalized
vectors, which are then encoded as target states using either amplitude encoding (v) = Σ} 2
or compact encoding (Blank et al. 2022) [v) = -)=1(0; + +0)+2* )bi) with N E 110, qubits.
Quantum data. We construct two types of quantum datasets: (i) synthetic states generated from random quantum circuits (RQCs); and (ii) ground states of the one-dimensional transverse-field Ising model (ID TFIM) (Pfeuty) 1970). denoted as S-RQC and GS-TFIM, respectively. These datasets represent typical examples of states arising from quantum circuit evolutions and many-body quantum systems. S-RQC contains M 50 states generated by applying different ROCs to the state (0) &N Each RQC is sampled from the set RandomShuffle (LC'ppe.x> 1 k=1 U { (pp. 14W 11= which includes W CZ gates and 3W single-qubit rotations on randomly chosen qubits {pk, (pk.gx). } Here, we set N= = 10 and NRQC = 40. GS-TFIM contains ground states of the ID TFIM Hamiltonian H - ΣNN11 J7,7 ΣNV gX,- We consider coefficients g = and E (0.8, 0.9, 1, 1.2)
to construct datasets of size M = 5 for each N E (10, 20, 30, 40,50%
4.2 EXPERIMENTAL SETTINGS
Reference AQL methods. We select three typical reference AQL methods to provide a comprehensive comparison with AQER. The three reference methods are: (i) TN method based on ID MPS (laconis and Johri 2023). which represents approaches with guarantees on low-entanglement data; (ii)
hardware-efficient circuit (HEC)- method (Nakaji et al. 2022). where the circuit is commonly used in VQAs; (iii) automatic quantum circuit encoding (AQCE) (Shirakawa et al. 2024), which illustrates recent advances in vorivaaiational AQL.
Evaluation metries. To quantify the accuracy of an AQI., we use infidelity 1- KMoodd ( 12 as the measure of approximation error. The infidelity ranges from 0 to 1, with smaller values indicating lower approximation error and higher accuracy. To evaluate the efficiency of an AQL, we consider the quantum resources to prepare the state from the encoding circuit. In particular, we record the number of employed two-qubit gates (e.g., CZ and Rzz) in the encoding circuit. This quantity, denoted by G. serves as the measure of quantum resource consumption, since it dominates the circuit runtime (Ma and Li 2024). A smaller G corresponds to a more efficient AQL.
Hyperparameter settings of AQER. The number of iterations in Step of AQER is set to T e (5, 10, 20, 40, 60,80, 100} by default. For GS-TFIM with large qubit numbers (N > 20), we use
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MNIST
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Figure 3: Performance of AQER across MNIST, CIFAR-I0, SST-2, S-RQC, and GS-TFIM datasets, distin- guished by different colors and markers. (a) Infidelity versus the entanglement measure value S after Step II of AQER, averaged over M samples. Color hars from light dark indicate increasing T. Dashed lines indicate the linearized upper (U.B.) and lower (L.B.) bounds in Theorem 3.1] which neglect higher-order terms. (b) Infidelity versus different T values after Step III of AQER across all datasets. (c) Infidelity versus different measurement shots for the GS-TFIM dataset, with different T E {10, 20, 40, 100).
T E {20, 40, 60, S0, 100, 120, 160, 200), since larger systems generally require more gates to capture quantum correlations. In AQER, the iteration count 7' controls the two-qubit gate count G, with one iteration introducing one two-qubit gate. In the f-th iteration of Step I, the parameters a; Eq. (2) first initialized zero and then optimized using the Nelder- tolerance of 10-2 The qubit index set Z, is optimized by selecting the qubit pair that minimizes S through adjusting or,- Step III performs optimization using the Adam optimizer with a learning rate of 10 for T3- = 2000 iterations. For quantum datasets, quantities such as S and gradients are estimated from 10 simulated measurement shots by default.
method with convergence
4.3 EXPERIMENTAL RESULTS
We evaluate AQER on both classical and quantum datasets verify its accuracy and efficiency, as well as its trainability and scalability on large systems, and to compare its performance against existing AQI. methods. Additional numerical results are provided in Appendix]] F.
AQER outperforms all reference AQL methods for both classical and quantum data. We first compare the performance of various AQL methods by measuring their approximation errors (infidelity) for different two-qubit gate counts G across multiple datasets. Table lists these results for MNIST, CIFAR- 10, SST-2, S-RQC, and GS-TFIM (with with G E {20, 40, 80). For the referenced AQL methods, G is set to the same slightly larger values, as determined by the fcasibility constraints explained in Appendix E.2 It can be observed that AQER consistently surpasses existing AQL methods by achieving the lowest infidelity with the same or even smaller G. The most pronounced improvement is observed on S-RQC, where AQER reduces the infidelity by more than 60% relative the second-bes method (AQCE) for G E (40, 80). and further achieves lower infidelity while using 50% fewer two-qubit gates than other methods. These results validate the advantage of AQER, which achieves lower infidelity with equal or even fewer two-qubit gates than existing AQL methods.
10). Results for AQER are shown
AQER consistently decreases the infidelity by reducing the entanglement measure S. We next examine the effectiveness of the entanglement entropy reduction of Step in decreasing the approximation error of AQER. To do this, we record the value of the entanglement measure S in Step I with different settings of the iteration times T E (5. 10, 20, 40, 60, 80, 100). and also record the corresponding approximation error after Step II. These results are shown in Fig. 3(a). For all datasets, increasing T' generally shifts points toward lower S and infidelity, which stay within the theoretical upper and lower bounds given by Theorem 3.1 demonstrating that AQER progressively expands the circuit to reduce entanglement and improve AQL accuracy. For example, increasing T from 5 to 100 reduces S by roughly fourfold for the MNIST dataset, with the corresponding infidelity decreasing by similar factor, which validates the effect of entanglement reduction on lowering infidelity.
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(b)
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Figure 4: Performance of AQER the GS-TFIM dataset. (a) Infidelity during Step III optimization across different values with = 50 qubits. (b) Infidelity for different qubit numbers N and Step iteration times T. (c) Expectation values of the averaged magnetization (X) measured on AQER-loaded GS-TFIM states for different g/ values with N 10. Each curve corresponds to a different T E [10, 20, 403.
Effect of two-qubit gate count and shot number on AQER performance. We further investigate how the number of two-qubit gates and measurement shots influence AQER. In particular, we conduct
experiments with varying T E {5, 10, 20, 40, 60, 80, 100} and shots {10 10 10 10%). The infidelity versus varying T after Step III of AQER for different datasets is shown in Fig. [), while the effect of different measurement shots for the GS-TGI ES-TFIM dataset is illustrated in Fig. B(c). Larger T values lead to significant reduction in infidelity. For example, increasing T from 5 to 100 decreases the infidelity for the GS-TFIM dataset from above 2- to below 2 Similarly, increasing the number of shots generally reduces infidelity by suppressing statistical noise in circuit generation and optimization. This effect is more pronounced for larger T. with the reduction being less than 4 for T = 10 and more than 16 for T 100. These results quantitatively demonstrate that larger circuits combined with sufficient measurement shots effectively improve AQER performance.
The trainability of AQER. We then demonstrate the trainability of AQER large systems via experiments on the GS-TFIM dataset with N = 50 qubits. The parameter optimization in Step III of AQER for varying T' E {20, 40, 60, 100, 200} is shown in Fig. aa). The optimization curves do not exhibit barren plateaus, which would otherwise trap the process at high infidelity near 1. The initial infidelity is already far from 1, consistent with Theorem 3.1 For instance, with T' 200, the infidelity starts around 0.3 and decreases effectively to around 0.I. These results demonstrate that the entanglement-reductte mechanism AQER successfully mitigates barren plateau effects in Step III, ensuring trainability and serving as prerequisite for its scalability.
Scalability of AQER. To validate the scal- ability of AQER, we conduct experiments on the GS-TFIM dataset with varying qubit numbers N {20, 30, 40, 50} and T E (40, 80, 120, 160, 200). As illustrated in Fig. 4b) for all system sizes, infidelity consistently decreases as increases. particular, AQER maintains roughly con- stant infidelity across different when T scales linearly with N, specificallyy follow- ing T' = 4N - 40, highlighting favorable scalability with respect both qubit num- ber and two-qubit gate count.
(a)
(b)
Exact
T= 40
Error
2
80
Target
2
2
2
Figure 5: Downstream performance of AQER on classi-
cal data. (a) Reconstructed MNIST and CIFAR-10 images using AQER with T' E (20, 40, 80). (b) Classification ror on the SST-2 dataset using AQER-loaded states with T E (10, 20, 40, 60, 80, 100), compared to exact loading (black dashed line).
Downstream performance of AQER. Fi- nally, we evaluate the performance of AQER in downstream tasks on both quan- tum and classical datasets. We begin with the detection of quantum phase transitions in the TFIM, which is measured by the averaged magne- tization (X) := 1 > (X,) of the ground state as the order parameter. We record the values of (X) for the loaded states with varying T values function of g/ as shown in Fig. 4c).
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These results demonstrate that AQER-loaded states capture the transition from the ferromagnetic phase with g/ < to the paramagnetic phase with g/d J > 1, with a rapid change near the critical point at g/ J = 1. Increasing T generally results in a more accurate approximation of exact values, indicating that larger circuits can encode more correlations relevant for the order parameter. Notably, states with moderate T 10 capture the overall trend, suggesting that AQER-loadee states effectively capture quantum phase transitions with limited quantum resources. Next, for classical data, we reconstruct MNIST and CIFAR-I0 images with AQER using varying T values. As illustrated in Fig.5la) (), reconstructions approach the targets as T increases. We also evaluate binary classification on SST-2 using a quantum kernel method with details in Appendix [E.3 As shown in Fig. (5)). the error decreases with larger 7 and approaches near the exact-loading error of 2 at T = 100. These results demonstrate improved downstream performance with larger
CONCLUSION
In this work, we introduced a unified framework for AQLs and derived information- -hheoreec bounds showing that the infidelity is fundamentally controlled by the entanglement of quantum states. Based on this insight, we proposed AQER, a scalable and efficient method that systematically reduces entanglement to achieve low infidelity with efficient gate usage. Extensive benchmarks on classical and quantum datasets show that AQER consistently outperforms existing methods both accuracy and circuit efficiency. These results provide both theoretical guarantees and practical approach for efficient quantum data loading, enabling broader applications in data-dependent quantum algorithms.
ACKNOWLEDGMENTS
This project is supported by the National Research Foundation, Singapore, under its NRF Professor- ship Award No. NRF-P2024-001.
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A PRELIMINARY
A. NOTATIONS
Here, we unify the notations used throughout this manuscript. We denote by [N] the set {1..... .N} The symbol a, denotes the j-th component of vector a. The tensor product operation is denoted as "0". The conjugate transpose of matrix A is denoted as At, The trace of matrix A is denoted as Tr[A]. The notation |x|] denotes the largest integer that is smaller than or equal to æ. We employ O to describe complexity notions. The phase of a complex value r is denoted by arg(z).
For vectors, the notation || ll2 represents the f2 norm. We use Schatten norms for operators: the trace [|,a,,l2. which equals the norm || == Tr and the operator norm || lo == =ll== ||
largest singular value of X. In particular, any density operator p satisfies || = 1 and ||ell S 1.
For linear map (quantum channel) T acting on operators, we denote its diamond norm by || ||0// == supa21 su [[ where idg denotes the identity channel on a d-dimensional ||XI|,
ancilla system. We use / (or La) to denote the identity matrix and id to denote the identity channel on the underlying system. A quantum channel completely positive and pnce-presnrving (CPTP).
A.2 RELATED WORK
The problem of quantum data loading has been extensively studied across multiple lines of research. For example, early investigations established that exact amplitude loading of an arbitrary N N-qubit pure state requires exponentially many quantum resources. Specifically, it was shown that without assuming prior structure, ((2%) single- and two-qubit gates are necessary for generic state prepara- tion (Grover 2000; Kaye and Mosca, 2001; Long and Sun 2001 Bergholm et al., 2005 Plesch and Brukner 2011) b. This scaling matches the intrinsic complexity of generic states: an N N-qubit pure state corresponds to a normalized vector in C² with 2 degrees of freedom. Even with auxiliary qubits introduced to reduce circuit depth (Sun et al. 2023), the total gate count remains exponential. For structured instances, more efficient constructions are possible: for instance, an S-sparse vector can be exactly loaded using (SSN) CNOT gates (Gleinig and Hoefler 2021). However, most states derived from classical datasets, such as images (Lecun et al. [998] Krizhevsky et al. 2009), are neither sparse nor unstructured, motivating approximate upproaches.
Tensor-Network (TN)-Based Approximate Loading, Tensor-network methods leverage low- entanglement structure to efficiently represent quantum states and construct corresponding loading circuits. The most widely applied TN family for quantum data loading is the matrix product state (MPS), which provides a sequential representation of an N-qubit state using a bond dimension Given an MPS, one can construct a circuit of O(N)3) two-qubit gates that exactly prepares the state (Schön et al., 2005), while truncated singular value decomposition (SVD) allows controlled approximation for more general target states (Schollwöck [011: Ran 2020). Recent works have applied based methods to both synthetic distributions and real-world classical datasets. For example,
(Icoonis Jaconis and Johri (2026) presented a method to convert MPS representations of images into quantum circuits with logarithmic scaling in the number of pixels, experimentally demonstrating amplitude encoding of complex images on trapped-ion device. Jobst al. (2024) exploited the decaying Fourier spectrum of classical images to construct approximate MPS encodings, which reduce quantum circuit complexity and can be further refined using simple sequential circuits inspired by MPS structure. Other works have focused on efficiently preparing real-valued smooth probability distributions with linear-depth circuits derived from MPS (Holmes and Matsuura 2020 Iaconis et al.) 2024). These procedures combine classical preprocessing (e.g.. Fourier feature compression) with MPS-based circuit synthesis to achieve high-fidelity state loading suitable for near-term quantum hardware. While TN-based methods offer principled and resource-efficient constructions for low- entanglement states, their applicability is limited when the target state exhibits high entanglement or strong long-range correlations, as the required bond dimension, which corresponds to the circuit depth and two-qubit gate count, can grow rapidly.
Circuit- based Approximate Loading, Circuit-basse methods directly construct quantum circuits to approximate a target state. These methods can be broadly categorized into variational and non- variational approaches.
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Variational approaches. Variational methods employ parameterized quantum circuits (PQCs), such as hardware-efficiens or sequential ansatzes, and optimize their parameters to minimize infidelity with respect to the target state (Nakaji et al. 2022; Mitsuda al. 2024 Rudolph et al. 2023b Wang et al., 2024a). For real-valued data vectors, classical routines and measurements in the computational basis and, when necessary, in the aaatamaarddtranfommmn basis are used to capture both amplitude and sign information (Nakaji et al. (2022). For complex-valuee data vectors, fidelity can be used directly as the cost function, with classical shadow techniques employed efficiently estimate the fidelity and its gradient during optimization (Mitsuda et al. 2024). Moreover, hybrid strategies are proposed to utilize TN-based initialization: the target state is first approximated using a tensor-network representation (e.g., MPS), mapped to a parameterized circuit, and then variationally fine-tuned (Rudolph et al. 2023b). This approach retains the structural advantages of TNs while extending applicability beyond strictly low-entanglemen states.
Non-variational approackes. Non-variational methods iteratively refine local gates to approximate the target state. Shirakawa et al. (Shirakawa et al. 2024) introduced the AQCE algorithm, which sequentially updates two-qubit unitaries to improve fidelity. Rudolph et al. (Rudolph et al. (023a) enhanced this framework by initializing AQCE with MPS approximation rather than trivial identity, demonstrating faster convergence and higher accuracy. Hybrid workflows that start from an MPS-based circuit and perform local iterative refinement combine the controlled approximation of TNs with the adaptive flexibility of circuit-level updates, providing efficient and accurate loading for moderately entangled datasets.
Despite their flexibility and broad applicability, they can suffer from barren plateaus and optimization traps, and unfavorable scaling for large qubit numbers or highly entangled target states (You and Wul 202T; Wang et al. 2023) You et al. 2022 Liu et al. [2033] You et al., 2023). Moreover, circuit-based methods remain heuristic. Rigorous theoretical guarantees on fidelity and resource costs are generally
absent, limiting systematic design and predictability for practical quantum data loading.
A.3 OVERVIEW OF TYPICAL AQL METHODS
In this section, we summarize several representative methods for approximate quantum loading (AQL) and show how they can be unified within the general AQL framework introduced in Section 3.1 in the main text. Recall that in this framework, an AQL procedure constructs a quantum circuit U(0; AA that loads the target state (target)) from an initial product state [prroduct), and optimizes it by adjusting the parameters 0, or the architecture A, or both.
A.3.1 TENSOR-NETWORK (TN) BASED METHOOS
Here we introduce the AQL method based on the matrix product state (MPS) (Ran) 2020). which forms one-dimensional tensor network. The MPS method iteratively constructs an MPS approximation of the current state with bond dimension 2. From this approximation, sequence of N 1 two-qubit unitaries is obtained, which are inverted and applied to the current state to generate an updated state. Specifically, in the (i + 1)-th iteration, the MPS method considers the current state
I MPs) = G (N-1)) ... G (earget):
(4)
from which a new set of two-qubit unitaries
G(N-)66+1),-.. G(N-1)++1
is extracted and appended to the existing circuit. By repeating this procedure sufficiently many times, the state in Eq. approaches the product state 10)8.. Consequently, the collection {G,} defines the encoding circuit for AQL. We note that each two-qubit unitary G, is further decomposed into hardware-native gates via the KAK decomposition (Tucci) 2005), resulting in CNOT (or CZ) gates for real matrices and 3 gates for general complex matrices.
Within our framework, MPS-based AQL corresponds to sequentially updating both the circuit architecture and parameters:
U U(0 + U(O 0 new; U Anow),
while previously added gates remain fixed. Increasing the number iterations systematically reduces the infidelity with respect to the target state.
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A.3.2 CIRCUIT-BASEE METHODS
Variational: Here we provide an example of the variational methods, i.e., Hardware- (HE) circuits NNkajj et al. 2022). which are commonly used as AQL circuits. The HE circuit UHe (0) consists repeated layers parameterized single-qubit rotations and CNOT gates acting on adjacent qubits. Each layer applies rotation layer Ry followed by Rz, and then CNOT layer. The CNOT pattern alternates between even-odd and odd-even qubit pairs: in the i-th layer, CNOT gates are applied on pairs (2n, (2n 1)%N) for mod 2 = 0 and on pairs (2n + 1, (2n + 2)%N) for i mod 1, where S 2m v N or S 2n I
2 as appropriate. The parameters are typically updated using gradient-baie based optimizers to minimize the loss function
( = - (rrrgen Unn(() (0))))))))
Within our general framework, HE circuits represent a variational approach which only the parameters 0 are optimized, while the circuit architecture A is fixed.
Non-variational: The Automatic Quantum Circuit Encoding (AQCE) method (Shirakawa al. 2024) is variational approach that iteratively updates two-qubit unitaries in prescribed circuit forward-backward
forward- backward fashion. Suppose the current encoding unitary list [G...., Gm). The locally optimal update for the m-th two-qubit gate is obtained as follows. For given choice of the two-qubit subsystem 2m, define
F Tr INTie..
(targes) Pharges\\No/G] ......GG--.) Gmm-]
(8)
where (db)) the initial product state. Performing a singular value decomposition (SVD)
F =XY,
(9)
the updated two-qubit unitary
Gnew XY.
(10)
In practice, the algorithm considers all possible choices of the two-qubit subsystem Qm and selects the one that maximizes [TT[D]] This choice can be understood as follows: the fidelity between the encoding state corresponding to the updated unitary and the target state is
F = (WolG ... G Gmm11 (new(y Giu+1 ... ...G'r (ttagge))
Tr Cmewt Gt
Gh (rarros)) (Wul) Gt C.mm
(Gitt Oxrluearge) G ... Gt -m ]]}²
Tre.
Tre. [Gee Fm
Tr [YXXXDDY]] 2
[Tr [D]]".
Thus, maximizing [T] corresponds to maximizing the fidelity F.
After several bawward-bckwward sweeps, new gates are added the circuit following the same procedure by treating G 1. The final sequence of two-qubit unitaries [G, is then further
decomposed into wonddware-friendde gates via the KAK decomposition (Tucei 2005). resulting in
CNOT (or CZ) gates for real matrices and 3 gates for general complex matrices.
Within our framework, AQCE fits naturally as non-variational method that sequentially updates both the circuit parameters 0, obtained from the KAK decomposition of the updated two-qubit unitaries, and the circuit architecture A, determined by the qubit pairs on which the updated unitaries act.
B PROOF OF THEOREMS
B. TECHNICAL LEMMAS
Before the proof main theorems, we provide technical lemma, which gives the maximum fidelity approximate mixed single-qubit state by any single-qubit pure state.
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Lemma B.1. Denote by p a single-qubit mixed state. Then
max Tr(l (olp) 1 S(1-
(II)
2
where Sp) denotes the Renyi-2 entropy of the state p.
Proof. For convenience, denote by p E5=1 Myby the spectral decomposition of p. where 21. IR and + X2 Tr(p] due the density matrix property. Without loss of generality, assume X 2 22. Thus, the left side of Eq. (11) achieves the largest value when (Φ) = [1/1,, i.e.
max Tr[ (ole) = Tr(le)) = A
(12)
Additionally, the spectral decomposition could also be employed in the Renyi entropy
S(p) = loga Tr(p²]
2
loga Tr
A N (13
k=1
logz Tr
(
lloga (Af + 12)
(13)
By combining Eq. 13 with d1 + A2 = 1, we could obtain the formulation of X with respect to the Renyi entropy S(p) as follows,
A1 1 / S(m) -
(14)
2
Comparing Eqs. 12 and 114. we have
+ 221-8(m)
max (olp))
2
Thus, Lemma B.1 is proved.
Lemma B.2. Denote by Pij the (i.j)-th element in the density matrix form of p. Then, the maximum fidelity in Lemma 1 could be achieved by (0) = (y) where B arg(pw) and
arcsin Bn Vlpp
Proof. First, we expand the state (Ø) terms of B and y:
(6) = 2 -3/2100 =
e
cos 3
sin 2
cos
sin 7
0
1
!
=
Te 1 cos 2 et sin
(15)
2
Then, the fidelity between and (d) is
Tr(lo)(0lp) T+
2 +
cos 7
e
10
sin
Poo
Pa1
sin y
cos 7
P10
P11
1
1
(1_1 cos
+
Cos 7
Poo +
e
sin
711 + LB sin 7Po1
P1
2
2
+
Puo
P11
008
Re
[e
18
pio]
sin
(16)
2
2
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where Eq. (16) follows from density matrix properties Tr[p] 1 and p p². Subsequently, by
considering the formulation of and y with respect to p. we have
(r(lo) =
Pro
P1
Cos 7 + lPtol sin
2
Poo
P11
V
+ 1p1ol3.
(17)
Moreover, the Renyi entropy S(p) could be formulated in terms of p follows
S(p)
logg Tr(p²)
loga (póo +pi1 + 222p1ol?)
Therefore, we have
2 21-3(0) (pão + + - +21pral))
=2pho + 2pi: (Pon + Pp1)²² + 4|p1ol
(Po0 PP1) 4/p1ol
(18)
Thus, the Lemma B.2 is proved by combining Eqs. Lemma B.
and (88 with the maximum fidelity value in
Lemma B.3, Let N be an N N-qubit CPTP map and let == Utpli denote the unitary channel associated with an N -qubit unitary Ug for each E [L]. Given an initial state Pa, define the ideal and noisy output states by
p (Ur o (.)))))))))(
(19)
ò = (Nous oN o.... Nou; o (P))
(20)
where ğ is obtained from po by inserting + 1 layers of noise N alternating with the unitaries. Then,
| - pll S (L + 1) I| - id
(21)
where id denotes the identity channel and
I|o is the diamond norm.
Proof. For convenience, we denote the noiseless channel by e(L) = U₁ o oll
(22)
and the fully noisy channel by
E No o N ....o Noll N.
(23)
By definition,
e(L) (po).
(24)
Thus,
|| p|) - 1 2 (pv) 11, < || e Il.-
(25)
where the inequality follows from the definition of the diamond norm, which upper bounds the trace-norm difference of the outputs any input state.
In the following, we derive an upper bound on || - c6( 11. There are + noise positions in the noisy channel e(L), one before U1, between each consecutive pair U, and 1, and one after UL. We construct sequence of intermediate channels (((() ( such that
(0)
e(L)
(L.11) (LL) Ex
(26)
and for each k E {1,..., L}. the channel (kk) is obtained by applying the first k noise layers. Accordingly, we have the telescoping decomposition
c(L) E/1 = (((+1) (0)
(b) ())-
(27)
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Each ((()) - (((k-)) corresponds to replacing single identity channel by V,, while all other positions remain unchanged. Therefore, there exist CPTP maps A, and A& (given by compositions of unitary channels Ue and noise/identity channels) such that
(h)) (k-11) AR
(k) (N id) o AL (8)
(28)
Since both A/" and AR are CPTP maps, their diamond norms satisfy
(R)
| Il llo = | l1 = 1.
(29)
Thus, we have
| - 11 S || llo II - -ddllo ||A = I id].-
(30)
Using the triangle inequality and summing over all (L + 1) noise positions, obtain
L+1
| cf 11.
Σ (11)- 11 11.
S (L + 1) |NN - -idlo--
(31)
Combining Eqs. 25 and (31). we obtain
|| 1l1 S || -%/ 11. s (L+ 1) I -idlle idllo-
(32)
Thus, have proved Lemma
B.3
B.2 PROOF OF MAIN THEOREM
Here we present the full versions of the theorem in the main text along with their proofs.
Theorem B.4. Denote the entanglement measure for a N-qubit state (w) as S(w)) = EN ((g((b))) Then for the stute (Waagee)) and the circuit U with ((U"i)ranga)) = S, the in- fudelity between Bragges) and the state generated from U on any product state (Pproduct) is lower bounded as
1 1 (U [VPprouuc)) 2 f((S) = 22-
(33)
2
Moreover, given access U'iWuarge)) we can construct product state 1 product)) such that the infidelity is upper bounded as
1- (Veagge [U (1//
(product)
12 fz(S)
1 (1 V2 81 - -1+1s1
(34)
2
where [.] denotes the floor function. We remark that fi(S) O by calculating the Taylor expansion.
1n s + O(S*) and f2(S) = In2 5
2N
Proof. For convenience, we denote by p the density matrix formulation of the state U1 (earget)) We denote by Pa := Tr/] [p] the density matrix of the n-th single-qubit subsystem of and denote the Renyi entropy S. S(,.) for simplicity. Since Pa is single-qubit density matrix, S, E [o, 1]. We construct general set of single-qubit projectors (Iv,) [I with [ı) & Wa). Accordingly,
we define a set of random variables (yn) where y, if the measurement on the n-th qubit yields (ı5..) and yn 0 otherwise. Thus, can reformulate the infidelity between |Utarget) and Ule) as
(Vearged (V = Tr Ip (o ( (V.. (v...) = - Pr(y.. 1, Vn E [N]). (35)
First, we derive the lower bound on the infidelity in Eq. (35) as follows. We have
1- Pr = 1, Vn E [N]) 2 1 min Pr Pr(yn
[=[N]]
1 min Tr [p (1 (1 @ (1)) @ /6(N-))] (wt.////////000
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min
Tr [px (1))
ME[M]
+ /2 S.
min M
(36)
where the last inequality follows from Lemma
B.1
We further proceed to deal with the terms in decreases for x E [0, 1]. Thus,
Eq. 36 In particular, the function f(x:) = v 2 21
N
min 211-8. -1 -1- ME[N]
min f(8.) S ME[N]
Σ
Sn = f N = V2
(37)
N
n=1
Combining Eqs. 36 and (37
we obtain
2
| (target U 11))
(38)
2
For the case of + 0, Eq. 388 tends to 2N S (S5a) by calculating the Taylor expansion.
Next, we derive the upper bound of the infidelity. We consider specific choice of projectors {(om)) ) where each (6,)) is the pure state approximation of P(x} with the largest fidelity given
by Lemmas B.1 and B.2 Similarly, we define set of random variables {=m)n=1> where T., 1 if
measuring the n-th qubit of yields the state (n)) and 0 otherwise. Based on Lemma B.1 we
have
Pr(z.- 1) 1 v
Pr(x,, = 0) = 1 2
(39)
2
2
Thus, the infidelity between states (target) and U U( can be reformulated
Maagge U 2 = Tr 1
-r 1, Vn E [N]).
(40)
In the following, we proceed with the derivation from the probability in Eq. 40 to obtain the upper bound of the infidelity. We have
Pr(xn- = 1, Vn E [M]) = 1 - Prr(rr n E [N], s.t. x, = 0)
N
2 1
Pr(x, =0)
n=l
221
(41)
where Eq. 41 yields from Eq. (3). The function f(x)
2 - I is concave for [0, 1] since
has the second-order derivative f"(x) (x)
(In2) 2' -2)
< 0 for x E [0, Therefore, we have
4(2 1)
((a)+ {
(f(a f(0) + f(a b)
when a, b, bb 1).
(42)
when a, [0, and bbbe [1,2].
due to the property of concave functions. Eq. 42 can be employed to Eq. (4T) by selecting qubit pair (i.j) such that S, S, & {0, 1} and pushing the new value S', S² towards the boundary [o. 1]. By conducting the above procedure for at most N - times, we could bound the probability as follows
N
- [S]) + ((1 (5.)/ + -
Pr(a. 1, Vn E [N]) 2
(N - -[S]-
1) (())
2
2
2 -1 [S] [1]+1).
(43)
where Eq. 443 yields from calculating the value of function f. Thus, we obtain the upper bound for the infidelity:
Veergee U U1)) = 11Pr( = 1, Vn E [NT) S (1- 22- [S!)
(44)
For the case of + 0, Eq. we have proved Theorem B.4
tends to ln2 + (S2)) by calculating the Taylor expansion. Thus,
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C THEORETICAL RESULTS OF NOISY CASES
While AQL is designed for both noiseless and noisy regimes, its behavior in the noisy case is particularly critical for near-term applications. In practical NISQ settings, hardware noise perturbs the circuit evolution, and its impact on the error beyond the noiseless bounds needs be analyzed. Here, we extend the results in Theorem B.4 into the general CPTP channel case.
Theorem C.1. Denote the entanglement measure for an N -ubit (pure or mixed) state as S(p) = E (p)) Let ((E(ara))))) S. where & is CPTP map. Then, for any product state (product), the infidelity between E (Ptarges) and (pproduct) is lower bounded as
1 (Wrroducct | (Prarges) (/product) 2 f(S) 1 - V2 3s
(45)
2
Moreover, given access to E (Prarget), we can construct a product state W Wproducc such that the infidelity upper bounded as
1 22 8 8118] - -1 1+ 1S]) 11111)
1 1- (V E (Prargoa) product) S fe(() ==
(46)
2
where I.] denotes the floor function. We remark that fi(
1n S + OSS) and [2(5) = ln S 2N
2
( by calculating the Taylor expansion.
Next, we consider a layered noisy circuit model and derive noise-dependent bounds, as stated in Theorem C.2
Theorem C.2. Denote the entanglement measure for an N-qubit state p as S(p) >N (p)) Let (earge))) be an N N-qubit pure target state and set Pearget = |Uearget) (Vrarges Let M,N be two CPTP noise channels and (p(p) UtpUi for unitary Ug. Consider two L-layer noisy circuits ((L) == M ur, M o... M Un M and = o .N o.... N o UL N. Suppose (Pearge))))))) S. Then for any product state (s/product), the infidelity between the target Pearget and the state {(L) (1V (oeiduuct((///oouce (Vopruuee)) is lower bounded
1- Tr Traaggget E(4, ([pproduct) (Wrrondteee/WWWWeeeeelee 1) 2 fi(S) - I) idll. + IIW -idlill))). (47)
Moreover, given access to FFL)) infulelity is upper bounded as
Fli (Peargot), can construct a product state b (prrodue)) such that the
- TrlPaarggtt ((T) (W product) (V/ product D) S + + (IIM idll. + II - -idll). (48)
Functions fi and fa follow the definitions in Theorem
C.I
The above results show that the entanglement-govereee setting, with an additional noise term that scales linearly with the depth T. and the noise strength. More precisely, the noise strength is captured by the accumulated quantity (L+1) [I--iddl+ For concrete CPTP noise models, this correction is easy to interpret. For example, a depolarizing channel D, (p) = (1 plppp)pp//// / with error rate p satisfies [DD, id|l Op)), leading to correction of order (L + p).. Similar linear scalings hold for other common noise models such as dephasing and anplitude-damping channels. When both the entanglement measure and the accumulated noise remain moderate, AQL can achieve small approximation error on NISQ hardware.
terms in Theorem B.4 persist in the noisy
We remark that for noisy quantum channels, the entanglement measure is generally higher than that in the corresponding noiseless case. Here we provide an example of the depolarizing channel in Theorem C.3 Therefore, due to the exactly same bounds formulations in Theorems B.4 and C. the infidelity achieved by noisy circuits would be worse than that of noiseless circuits.
Theorem C.3. Denote the entanglement measure for an N -qqubit (pure or mixed) state S(p)
>N (p). Let Dpp (1 - -p)p pl /d. be the depolarizing channel with error rate p. Then
(1-
P) S(p) + Np < S(D,(p)) < <S(p) + N Wogg 1 ++(1- p)2
2
(49)
14
In
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C.I PROOF OF THEOREM C.I
Proof. For convenience, we denote by the N N-qubit state & (Ptargat) and by := TrinI/to /tn] [e] the density matrix of the 1-th single-qubit subsystem of p. We denote the Renyi entropy S, (p,)) for simplicity. Since P.. is a single-qubit density matrix, S, E [0, We construct general set of single-qubit projectors (1v,) with product) @ (n)- Accordingly, we define set of random variables { ) where yn. = if the measurement on the n-th qubit yields (vh,) and Yn = otherwise. Thus, we can reformulate the infidelity between p and [/rroduct) as
produet (P] Pp Pproduet)
Tr (o
= Pr(yn = Vn E [N]).
(50)
First, we derive the lower bound on the infidelity in Eq. (50) as follows. We have Pr - 1)
1 --Pr(y. 1, Vn [A])
min mE(N]
min Tr [p (1(10000) (10(n-1) (1)) @0 [o(N-n))]
ME(M]
min
Tr [pm (1)) (vhml)
RE[N]
+ 211-5
min ME[N]
(51)
2
where the last inequality follows from Lemma B. single-qubit density matrix and a pure state in terms its Rényi entropy.
which gives the maximal fidelity between
We further proceed to deal with the terms Eq. (51) In particular, the function f(x) V21-= decreases for z E [0, 1]. Thus,
min =c[N]]
821-8.-1=
- -1 min S(S,)
S
1 N
Σ
Sn
=
=
2 $ -1
(52)
eC[N]
N
n-1
Combining Eqs. 51 and 52
obtain
121- -1
(Uproduer pll p product)
f(()).
(53)
2
For the case of + 0, Eq.
53 tends to
2n2 S + O(5)) 2N
) by calculating the Taylor expansion.
Next, we derive the upper bound of the infidelity. We consider specific choice of projectors {(o.) -= { Lemmas [.] and B.2 That is, (6x)) is chosen maximize Tr(pn (T.)) (ø.1). Similarly, define set of random variables (- where Fn if measuring the n-th qubit of yields the state (o..)
where each (o,,) is the pure-state approximation of Ptnz. with the largest fidelity given by
and x.p otherwise. Based Lemma B.1 have
1 1+ 221- -1
Pr(xn P)( 1 - /
Pr(x.. =
(54)
Thus, the infidelity between p and the product state (ø) &N (o.) can be reformulated as
1 - (á)p))) = Tr [p (N ( (0.l))
1 - Pr(x,, = 1, Vn [N]).
(55)
In the following, we proceed with the derivation from the probability in Eq. (55) to obtain the upper bound of the infidelity. We have
Pr(xn = 1, Vn [N]) Pr s.t. v. = 0)
N 2 Σ
Pr(c. =0
==1
V21-5.
(56)
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where Eq. (56) follows from Eq. (54) and the union bound.
The function f(x) 221-= 1 is concave for x E [0, 1] since it has the second-order derivative
["(x) (In 21=2)211- 21-z (21-= 2) 00
(57)
4 a 1)8/2
for x E [0, 11. Therefore, we have
{ ʃ(0) + ((a+b) when a, b, b [0,1],
f(a) f(b)
(58)
b( f(1) when a, b [0. 1] and
[1,
due to the property of concave functions. Eq. (58) can employed to Eq. (56) by selecting qubit
pair (i, such that Sy {0, 1 and pushing the new values Sy, Sj towards the boundary of [0, 1] while keeping Sy + S² = + Sj. By conducting the above procedure for at most N - times, we can bound the probability
follows:
Pr(z,- = 1, Vn E [M]) 2 1 + [S][(1) + f(S (S]) + (N - [SJ -5)1-11-00 f(0))
1)
VV2 S+18| -1- -1- 1---11111) 11)+
(59)
where Eq. (59 yields from calculating the value of the function / at and 1. Thus, we obtain the upper bound for the infidelity:
(olpl)) = Pr(xn- = Vn E [N])
(1 1 11- 1 [s] =: f2(S).
(60)
For the case of S + 0, Eq. 60 we have proved Theorem C.1
tends to 2 + 0(S)) by calculating the Taylor expansion. Thus,
C.2 PROOF OF THEOREM
Proof. For convenience, we denote by = =( and p := == o., (Ptargtt) the state obtained from Prarget with and without noise channels, respectively. Next, we focus on the lower bound in Eq. (4), while the upper bound in Eq. (48) be derived similarly. By employing
Theorem we have
1- (Vproduct /PlPuuudtt? = Tr Tr product] 2 fi(S)
(61)
for all product state pe/puuue))) where product producc))) ) (2) product. Thus, the infidelity between the target state Puarget and the state recovered by ((4) is
1 Tr Ptarger
E g(L) producc)]
Tr product| + Tr Trp product] [ripor Telppoprodaie product| + Tr product) Tr Peaarete (L)) ( (produut))
2 fi(S) [rippo [poppuuuuuct produccll
produccll
Tr Prarger ((4) (o (ppodduc)) )|
(62)
The second term in Eq. 62 can be bounded as
[T((paroduuc] Tr[paprodues]] < 1 plh product Il S (L+ 1)I - id|lor
(63)
where the last inequality yields from Lemma B.3 and the operator norm | I 1.
The third term in Eq. 662 bounded in the similar way
Tr product)
- E (4) (a (Oproduct))
Tr .... U4, (Ptarges) ) product] Tr [Ptagge g(L) producc))
Tr Praageell
U4 (1 (pproduct)] [r(aaaget E (LL) (Oproduct))
< Mr. o .... Ui (aproduct) g(L) (Aproduct)) |/1 |/1 Peargett 1lo
< (L + I) -i id]
(64)
Combining Eqs. 63) and (64) with Eq. 62 we obtain
Tr Piarrgee g(L) (( (aprodues)) 2 (S) - (L + 1) (IIM id]l + || - -id|(a),
which is Eq.
47
Eq. (48) can be derived similarly. Thus, we have proved Theorem C.2
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C.3 PROOF OF THEOREM
Proof. For convenience, we denote by Pm and on the reduced density matrix of p and o = Dp(p) at the n-th qubit, respectively. Then, we have
n TM///m6 [Px(e]] = TTp]/(6) [11 - pppp + pT/22 =
(1 -1-20 ppppp + PT.
(65)
2
Thus, the Renyi entropy of the state d, be obtained as
S(o,)) = logg Tr(o"l
log, Tr ((1 - (1 + _ n인
+ p(1 - p)Tr(pn]++ + _² T
logz (1-p)) p)
[p Tr
logz { (1 Tr + 1 2 }
(66)
Next, we derive inequalities for S(o,,) based on Eq. 66 We have
S(ox)) - log, (1 p) Tr 1
+ p
P)
S -loga (1 Tr poa +
1
2) Tr[ph]
p)
1
S(m)) -l082
1
1
70²
2
=S(pp) log2
(67)
1 (1 p)2
where the inequality follows from Tr[ph] < 1.
On the other side,
S(an) = logg { (1 Tr + +D
2 }}} }
(1 p))² P(1 5)
logg Tr[p"] - logg
Tr(pi]
- P)
1
S((N)) logg
1 1
2 Th[pi]
2
2 s(p.)
In
1
2
P
1
2
p)
2
Tr|p2]
1
1
S(pw)
2
1n
Tr[
S(w)) + P
(2 25()))
In
S(w,) + In4
(2 (1 S(ww))
(1 1n
P S(pm.
p
(68)
'n
where the first inequality follows from logg (1 I²) < . the second inequality follows from
p E 1]. and the third incquality follows from 2
1 x for x E [0, 1]. By summing Eqs. (66
and 688 over [N], respectively, we obtain Eq.
(49
D MORE IMPLEMENTATION DETAILS OF THE AQER ALGORITHM
In this section, we provide detailed discussion of the technical aspects of AQER, including the computation and optimization of the entanglement measure S, the explicit construction of product- state approximations, and the trainability of variational circuit optimization in AQER. We also discuss why S serves as an efficient proxy for the global approximation error and how it facilitates scalable training on large quantum systems. Finally, we explain how employ AQER with classical data.
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D.1 STEP I: COMPUTATION AND OPTIMIZATION OF THE ENTANGLEMENT MEASURE
The computation and optimization of the entanglement measure S in Step of AQER requires only limited amount of quantum resources. Since S, which defined as the sum of single-qubit Renyi entropies, consists solely of local terms, it can be efficiently evaluated using local mcasurements. Specifically, the Renyi entanglement entropy of any subsystem of constant size can be estimated via quantum state tomography its reduced density matrix, which requires only (1/c²)) measurements to achieve an c-precision estimate. Besides, for a candidate 2-qubit gate block in each iteration of Step I, only the single-qubit Renyi entropies of the two involved qubits are affected. Therefore, the change in S due to adding a new gate block equals the change in the sum of these two single-qubit entropies. Thus, in each iteration of Step I, the gate block that induces the largest decrease in the relevant single-qubit entropies is selected as the new structure. The parameters within these blocks optimized using classical methods such as the Nelder-Mead algorithm. Through this procedure. entanglement is systematically reduced using only local information. This reduction allows efficient scaling to large quantum systems.
D.2 STEP II: PARAMETER COMPUTATION
The parameters for single-qubin rotations in Step II of AQER can be computed explicitly and efficiently via single-qubit state tomography. Based on Theorem 3.1 the state (Ur) after Step has reduced entanglement and can be well approximated by product state, which is prepared from |0) @ by using single-qubit rotations. In Step II, the parameters (B,,7)) for the single-qubit rotations (2 (n:) can be explicitly computed from the elements of the single-qubit reduced density matrices obtained via tomography. Specifically, according to Lemma B.2 each B; and Y/ is directly computed from the matrix elements Poo- P11, and Pro of the corresponding reduced density matrix. As mentioned in the discussion regarding Step I, the tomography for each qubit requires (11/c²) measurements to achieve an e-precision estimate, which is independent of the total system size N. Once obtained, these parameters provide an explicit product-state approximation of (or) without any iterative optimization. This yields high-fidelity initialization for Step III.
D.3 STEP III: THE TRAINABILITY OF PARAMETER OPTIMIZATION
By reducing entanglement in Step I and explicitly constructing single-qubit gates in Step II, AQER mitigates the effects of barren plateaus on subsequent parameter optimization in Step III. Generally, in the training of VQAs, barren plateau phenomena occur when the loss function is initialized near its average value with exponentially small gradients. This makes parameter optimization extremely challenging. For loss functions defined as the infidelity with respect to an N -uqubit target state, the average loss for randomly initialized parameterized circuits scales as 1 1-0(1/2N), In contrast, in Step III of AQER, the entanglement measure S is suppressed before full optimization. By Theorem 3. smaller S values correspond to states with smaller infidelity loss. Therefore, the initial point of the variational circuit effectively positioned away from the barren plateau region. This ensures that subsequent optimization starts from a well-conditioned region where gradients meaningful, mitigating the impact of barren plateaus and improving trainability. This mechanism differs from previous strategies for avoiding mitigating barren plateaus that primarily reshape the optimization landscape or amplify gradient signals (e.g., via architectural constraints (Pesah et al. [021] Zhang
al. 2022a). smart parameter initialization strategies (Grant et al. [019 Zhang et al. 2022b Wang
et al. 2024c; Peng et al. 2025), or local cost functions (Cerezo et al. (022bb) while not necessarily guaranteeing a low-loss starting state. Moreover, AQER mainly mitigates parameter spacee barren
platcaus, which does not resolve data-induced barren plateaus that can still cause severe trainability
and generalization issues (Wang et al. 2024b Zhang al. 2024).
D.4 AQER: ENTANGLEMENT MEASURE AS A PROXY FOR APPROXIMATION ERROR
Using the entanglement measure S as proxy for the global approximation error provides multiple advantages. Directly optimizing the infidelity loss requires global measurements, which are com- putationally expensive and sensitive statistical variance, especially in large systems. In contrast, can be efficiently evaluated and reduced using only local measurements. By first reducing S, AQER not only lowers the approximation error but also prepares the circuit in a favorable regime for
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further global optimization. This strategy enhances both the efficiency of evaluation and the overall trainability and scalability of AQER on large-qubit quantum systems.
D.5 AQER WITH CLASSICAL DATA
For classical data, AQER is implemented through classical simulation while maintaining the same three-step structure. In particular, the quantum state corresponding to classical data is represented in the form of classical vectors or tensor networks.
In Step I, the entanglement measure S is computed by classically evaluating reduced density matrices and their Renyi entropies. Gate optimization proceeds through classical simulation of candidate blocks, with parameters optimized using cruddeent-bsee methods with exact gradient information. In Step II, single-qubit rotation parameters (,3,, 7%) are extracted directly from computed reduced density matrices according to I Lemma B.2] Step III utilizes classical simulation for exact loss evaluation and gradient computation without statistical noise.
Classical Simulation:
E EXPERIMENTAL SETUP
In this section, we first describe the construction and preprocessing of both classical and quantum datasets. We then present the hyperparameter configurations of the reference quantum data loaders employed in experiments.
Numerical simulation settings. All experiments presented in this work were performed using classical simulators. For computational tasks involving fewer than 20 qubits, employed Penny- Lane (Bergholm et al., 2018), a Python-based based package. For larger quantum systems with 20 or more qubits, we utilized PastaQ (Torlai and Fishman 2020), Julia-based package that employs tensor network techniques to support efficient simulation of large-scale quantum systems.
E CONSTRUCTION OF CLASSICAL AND QUANTUM DATASETS
Classical data. For MNIST, each image is zero-padded 32 32, flattened into a (124-niimnsio vector v, and normalized to have unit f norm. The corresponding target state is defined by the amplitude encoding v)) -ド 4y () with N 10 qubits. For CIFAR-IO, each 32 x 32 RGB image is flattened by concatenating the three color channels into 3072-dimensional vector, zero-pade prro-pddddd to 4096 dimensions, and then normalized. The second half of the vector is further treated as the imaginary part and combined with the first half to construct the compact encoding (Blank al.) 2022): (n) Σ (u1 + + 2204))3)) with N = qubits, which serves as the target evaluate AQER under different encoding schemes. For SST-2, each sentence initially represented by a 1024- dimensional embedding vector obtained from pretrained Sentence- Centence-BERTT model. The embeddings are then normalized, and the corresponding target states are defined via amplitude encoding using N 10 qubits.
Quantum data. We consider two types of quantum datasets: synthetic states generated from random quantum circuits (RQCs) and ground states of the one-dimensional transverse- field Ising model (ID TFIM). These datasets serve as typical examples of physically relevant states arising from quantum circuit evolutions or quantum many-body systems. Specifically, the first dataset is consists of M 50 states generated from different RQCs on the state (0). Each RQC is sampled from the set Random,Shu Shuf. fle (LC2, IW ( (Ro.., p.. (0x)) =1)) where < Pv. + q < N are randomly sampled from [.V]. Each R. a single-qube rotation on the qubit pı with o Uniform{X, Y. Z} and 0₂ ~ Uniform (0, 2Tr.. Such circuits produce highly entangled states when the number of two- qubit gates W > O(N) that serve as representative synthetic quantum data. practice, we use N 10 and W 40. The second dataset consists of ground states of the 1D TFIM, which is defined by the Hamiltonian HTFIM - Σm=1 N- JZ,Z++1 >
D gX,,- We consider coefficients near the to construct datasets with the size
phase transition point, i.c. = 1 and J E (0.8, 0.9, 1, M 5 for N E {10, 20, 30, 40, 50).
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Table 2: Feasible numbers of CNOT/CZ gates used in different encoding methods. The term N the number of qubits, and the term &o e N can be any positive integer.
Method
two-qubit gate count G
AQCE
G 15k
AQCE (R)
G 10k
MPS (C)
G 3( 3(N 1) 1)k
MPS (RR)
G 2 1 1)k
HEC
G T N.
AQER (Ours)
G
E.2 HYPERPARAMETER SETTINGS OF REFERENCE QUANTUM DATA LOADERS
Here, we provide the hyperparameter settings of the reference quantum data loaders used in our experiments. The descriptions of these methods can found Section A.3
Automatic quantum circuit encoding (AQCE). The AQCE method performs the two-qubit unitary updation sequentially in forward-backward manner several times before adding new gates into circuit. In the experiment, we add 5 new unitaries for each adding-gate step, followed by 200 rounds of forward-backward gate updation.
Hardware-efficient circuits (HEC). We adopt layered hardware-efficient (HE) architecture, where each layer consists of an Ry rotation layer, an Rz rotation layer, and a CNOT layer act- ing on adjacent qubit pairs. Specifically, for the i-th layer, CNOT gates applied to qubit pairs (2n, (2n + 1)%N), 0 < 2n I N when 2%2 = and (2n + 1, (2n + 2)%N), 0 < 2n S N-2 when 1%2 1. Parameters in HEC are initialized randomly from the uniform distribution [0, 2m.. Training is performed using the Adam optimizer for 2000 iterations, which is consistent with the setting in AQER.
We remark that reference AQL methods introduce different constraints on the number of hardware- available two-qubit gates, such as CNOT and CZ. We summarize the feasible two-qubit gate count under these methods in Tab. Specifically, the decomposition of an arbitrary two-qubit unitary requires two and three CNOT7CZ gates in the real and general complex cases, respectively (Vatan and Williams, 2004
E.3 QUANTUM KERNEL METHOD WITH CROSS-VALE VALIDATION
We employ the quantum kernel method (QKM) (Rebentrost et al. 2014 Wang et al. 2021) for binary classification. QKM employs quantum feature map to embed classical input vectors | into quantum states (()(())) )}, which are used to construct a kernel matrix K₃ (()()())))) (xx >) that quantifies pairwise similarities between data points. Denote by E [, 1} the label of data (9) Then, the training of support vector machines (SVM) corresponds solving the dual optimization problem
max Ca.-
1 Σ - Laaayu
s.t. 0 5 a, S C. C.E.. a,y: = 0,
where C' is the penalty parameter that is set to by default. Once the parameter a of the QKM is trained, the predicted label for a new sample æ is computed using the decision function
N
① = sign
a.y
where ((x((), x)) (x) (1)()))))) sample æ.
(=))))
2 is the kernel between the training sample (0)) and the new
In the experiment, compare the approximate encoding from AQER with the exact amplitude encoding. We employ stratified k-fold cross- cross-validation (with = 5) on the dataset with size 200 to estimate classification performance. For each fold, the SVM is trained the training subset using
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MNIST
CIFAR-10
SST-2
S-ROC
GS-TFIM
(b)
T= - 40
2
T= .... 10:
2
2-2
2
23
24
2
26
27
10
10
10
10
T
Shots
Figure 6: Performance of AQER on entanglement suppression across MNIST, CIFAR-10 10, SST-2, S-RQC, and GS-TFIM datasets, distinguished by different colors and markers. (a) Entanglement measure value S versus different T (5, 10, 20, 40, 60, 80, 100) after Step AQER across all datasets. (b) Entanglement measure value S versus different measurement shots for the GS-TFIM dataset, with different {10, 20, 40, 100).
the precomputed kernel matrix, and predictions made the held-out validation subset using the corresponding submatrix of K. The reported error is the mean over all folds.
F ADDITIONAL EXPERIMENTAL RESULTS
In this section, we present additional numerical results about the performance the AQER on both classical and quantum datasets.
Step of AQER effectively reduces entanglement with increasing iteration times. We first examine how AQER progressively lowers the entanglement measure value S for MNIST, CIFAR-10, SST-2, S-RQC, and GS- TFIM datasets. Fig. 6 () shows after Step varying 7 {5, 10, 20, 40, 60, 80, 100}. For most datasets, increasing T consistently decreases S, indicating that AQER effectively reduces the entanglement of the target state by gradually expanding the circuit. The GS-TFIM dataset exhibits minor fluctuations due to statistical noise caused by a limited number measurements, but the overall decreasing trend remains clear. These results corroborate the effectiveness of AQER in progressively mitigating entanglement as T' grows.
AQER for all datasets with
Effect of shot number on entanglement reduction. We next study the effect of different mea- surement shots entanglement reduction. b(b) illustrates the entanglement measure S for the GS-TFIM dataset, with 7' {10, 20, 40, 100}. Increasing the number of shots generally reduces S, mirroring the trend observed in the main text for infidelity. This reduction is more pronounced for larger T. For instance, increasing the measurement shots from 102 to 10t reduces S by roughly factor of for T 10. whereas for T 100 the reduction reaches approximately 16-fold. These results indicate that sufficient measurement shots significantly enhance the capability of AQER to suppress entanglement.
Step III of AQER effectively reduces infidelity with increasing iteration steps. We then investigate the effect of different numbers of Step iterations T3 E (40, 80, 160, 320, 640, 1280, 2000) on AQER performance. Fig. shows the infidelity for MNIST, CIFAR-10. 10, SST-2, S-RQC, and GS-TFIM datasets, with each subtigure corresponding to T = 10, 20, 40, and 100. For all cases, increasing T3 steadily decreases the infidelity, indicating that longer Step III optimization effectively improves the approximation quality. This reduction is more pronounced for larger T, demonstrating that AQER benefits from combining larger initial circuit with more extensive optimization in Step III.
Binary classification with AQER. We further examine the classification performance of QKM using AQER-loaded states. Specifically, consider binary task MNIST digits 0 using
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MNIST
CIFAR-10
SST-2
S-RQC
GS-TFIM
2
2-3
Infidelity
2
10
T=
40
T= 100
2-9
2
2
2
21
2
2
2
2
2
2
2
21
2
2
2
2
T3
T3
T3
T3
Figure 7:
AQER infidelity with different optimization iterations in Step III, i.c.
T3
(40, 80, 160, 320, 640, 1280, 2000}. Each subfigure corresponds to T = spectively.
10, 20, 40, and 100, re-
2
Exact
2-5
2 2-6
Error
2-7
2 2-8
2 2
23
24
25
26
27
T
Figure 8: Binary classification
MNIST digits vs using AQER-loaded states with E (5, 10, 20, 40, 60, 80, 100}, compared to exact loading (black dashed line).
AQER-loaded states with different T values. The results are shown in Fig. where the error rates for T E (5, 10, 20, 40, 60, 80, 100} are compared against exact loading. We observe that the classification error decreases steadily as T increases, and from T > 20 onward the performance already matches the exact-loading error. This indicates that AQER can achieve near-exact downstream performance on classification tasks with relatively small circuit sizes.
Resource-accuracy trade-offss of AQER versus AQL baselines. We further compare the accu- racy-resource trade-off's of different AQL methods the S-RQC dataset. Fig. plots the infidelity achieved by AQER, HEC, AQCE, and MPS as a function of four resource metrics: circuit depth, number of trainable parameters, total number of measurement shots, and classical training time in seconds. Specifically, the circuit depth is defined as the number of sequential gate layers in the compiled quantum eircuit under full parallelization of commuting gates on different qubits. We count only layers that contain at least one two-qubit gate, since in practice two-qubit gates are typically an order of magnitude slower than single-qubit gates. The parameter count is the number of independent continuous parameters in the quantum circuit (e.g.. rotation angles). The measurement shots are the number of quantum measurements used in the entire AQL procedure. The training time measured laptop equipped with an Apple M2 chip and 8 GB of RAM. Overall, AQER achieves lower infidelity than the other methods under comparable or lower circuit depth and parameter counts, indicating more efficient of circuit expressivity. For measurement shots and training time, the
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AQER
HEC
AQCE
MPS
20
2-2
Infidellty
2
2
2
2
2
27
2
2
2
2
2
2
2-6
2
2
2
20
Circuit depth
Parameter count
Measurement shots
Training cost (secands)
Figure 9: Infidelity the function of metrics for different AQL methods the S-RQC dataset. We consider four metrics: the circuit depth, the parameter count, the number of measurement shots, and the training cost in seconds. Each data point is the average over 5 samples.
10
gradient norm
10
10-6
10-8
10-10
10-12
2-
- 40
T= 10
T 60
10-14
20 20
+ 10=
10
20
30
40
50
Figure 10: Scaling of AQER gradient norms with the number of qubits for the TFIM dataset. The plot shows the gradient norm at the initial stage of Step III training, as a function of system size N {10, 20, 30, 40, 50), for different circuit sizes T {10, 20, 40, 60, 100). The black dashed line depicts an exponentially decaying reference curve proportional to 2-
behavior is more nuanced. The MPS baseline, owing to the simplicity of its algorithm, requires fewer shots and shorter training time, but this comes the cost of substantially worse performance in terms of circuit depth, two-qubit gate count, and parameter count when realized as circuit, as well as significantly higher infidelity. When compared against circuit-based baselines (AQCE and HEC), AQER consistently achieves lower infidelity with comparable or lower mcasurement overhead and training cost. Therefore, AQER achieves favorable resourcy-accuracy trade-off among Atrcuit-based AQL methods.
Gradient scaling of AQER with large qubit number. We analyze the scaling of AQER gradient norms on the TFIM dataset with large qubit numbers to demonstrate that the Step III optimization in AQER is free from the barren plateau issue. For system sizes N E {10, 200,00.200 30, 40, 50) and circuit sizes T E {10, 20, 40, 60, 100), we compute the gradient norm with respect to all trainable parameters at the beginning of the optimization. The result shown in Fig. TO Across all system sizes and circuit sizes, the initial gradient norms remain the order of 10 and do not exhibit any exponential decay with N, in clear contrast to the exponentially vanishing reference curve N shown in the figure. These observations provide numerical evidence that the trainability of AQER remains stable as the system size increases from 10 to N = 50.
Robustness of AQER under noisy channels. We perform noisy simulations the N 10 TFIM task to assess the performance AQER in realistic NISQ regimes. In particular, after each single-qubit and two-qubit gate layer, apply depolarizing channel with emror rates pi
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2-1
2-
Infidelity
2-3
2-
P==10-1
10
2-5
x1=3x10
-2103100
10-1
12-1003
2
24
2
26
T
Figure 11: Infidelity of AQER under global depolarizing noise for the N = 10 TFIM task. The plot shows the final infidelity function of the two-qubit gate count T for three noise levels, where single-qubit and two-qubit gate error rates are (pI.P2) {(10 10 ?), (3 x 10 4,3 x
10 2), (10 10 })}
23 (a)
20 (b)
20
Infidelity
-
-
2 2
2-12
20
2
2
2
24
2
2
2
2
24
T
T
Figure 12: Performance of AQER the N 10 GHZ state. (a) Entanglement measure S function of the two-qubit gate count T. (b) Infidelity as a function T..
(10-3,3 10-1, 10-1,10-1) and y2 10p, respectively, which corresponds representative ranges reported for current quantum devices. For each noise setting and two-qubit gate count T E 7, 10, 12, 15, 20, 24, 28, 32, 36, 40, 50, 60, 80, 100). we run AQER and record the final infidelity. The result is shown in Fig. II Across all three noise levels, we observe a noise-dependent optimal circuit size T" T increases from very small values, the infidelity decreases, which indicates that entanglement-guide becomes dominant. Beyond T", further increasing T leads gradual increase in infidelity as the effect of noise outweighs the expressivity gains. Moreover, as the physical enror rates decrease from (Pt, P2) = (10-3 10-2) to (10-4, 10-3), the best achievable infidelity improves from above 2 to around 2 and the optimal T increases. Altogether, these results demonstrate that AQER remains effective and robust in NISQ-level noise regimes.
circuit growth improves approximation quality before noise accumulation
AQER for preparing the GHZ state. We evaluate AQER on the N 10 GHZ state as study its performance on highly entangled yet structurally simple states. The GHZ state is highly entangled in the sense that each single-qubit reduced density matrix is maximally mixed and the initial entanglement measure satisfies
N. Nonetheless, AQER is able to reduce this entanglement very efficiently. As shown in Fig. 12 the entanglement measure S decays rapidly with the two-qubit
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4x4
4x5
5x5
(a)
(b)
2
2
21
Infidelity
2
5 20
2-1
2
2
2
24
2
2
2
2 T
2
Figure 13: Performance of AQER on 2D random circuit states. Target states are generated by
depth-4 nearest-neighhor circuit on 4, x and 5 x lattices. (a) Entanglement measure versus the two-qubit gate count T E (10, 20, 40, 60, 80, 100). (b) Infidelity versus 7 for the same set of 2D random circuit states.
4x4
4x5 - 5x5
(a)
(b)2*
2
Infidelity
2-1
5 2
2-2
2
2 T
26
24
2 T
26
Figure 14: Performance of AQER on 2D XXZ ground states at the critical point. (a) Entanglement measure S as the function of the two-qubit gate count T {10, 20, 40, 60, 80, 100) for 4 x 4, 4 x 5. and lattices. (b) The infidelity as the function of T.
gate count T and drops below 2-3 already at T = 9. The corresponding infidelity is reduced to below 2 2 using only 9 two-qubit gates. This example highlights that quantum state can be highly entangled while still being structurally simple, and that in such cases AQER can efficiently find low-depth circuits that achieve very low infidelity despite the large initial value of S.
AQER for preparing 2D random circuit states. We consider preparing 2D random circuit states rectangular lattices to evaluate AQER on physically relevant targets. The target state is generated by depth-4 circuit composed of alternating single-qubit and CZ layers. In each single-qubit layer, every qubit is acted on by a randomly chosen rotation from [Rx, Ry, Rz with a uniformly sampled angle. In each CZ layer, CZ gates are applied to nearest neighbors along horizontal or vertical directions according to one of four distinct tilings of the 2D grid (two horizontal and two vertical patterns). These four CZ patterns are cycled over the four layers so that, across the entire circuit, all nearest-neighbor couplings are activated. We consider 4 x 4. 4 x 5, and 5 5 lattices and apply AQER with two-qubit gate counts infidelities are shown in Fig- 13 For all lattice sizes, the entanglement measure S decreases by at least a factor of 8 as T increases from to 100, indicating that the entanglement-rednetioon strategy remains effective for the 2D random circuit state. The corresponding infidelities also decrease as T increases. For example, on the 4 x lattice the infidelity drops from above 3 at = 10 around 2 T = 100, and the 5 x 5 lattice it decreases from above 2 to below 2 These results demonstrate that AQER can substantially and consistently decrease both the entanglement measure and the approximation error for shallow 2D random circuit states the circuit size T increases.
{10, 20, 40, 60, 80, 100). The resulting entanglement measures and
AQER for preparing 2D XXZ ground states at the critical point. We consider preparing the ground state of the spin- XXZ. model on lattices evaluate AQER 2D many-body quantum
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000
000
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92
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400
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Figure 15: Loading circuits generated from AQER with T' 20 for the MNIST dataset. Green and red blocks denote Ry and Rz gates, respectively, Blue circles connected by lines denote Rzz gates.
states. The Hamiltonian is given by
Hxxz
Jay of +o(0²) + Jg ofo; (.55)
(69)
where (i,j) runs over nearest-neighbor pairs the 2D grid and =... denote Pauli matrices. We focus on the critical point Jry = J, and consider 4 x 4, 4 x 5, and 5 x 5 lattices. For each lattice, we apply AQER with two-qubit gate counts T' {10, 20, 40, 60, 80, 100). The entanglement measures and infidelities are shown in Fig. [4] For all lattice sizes. the entanglement measure S decreases steadily T' increases, typically by factor about 2-3 when T grows from to 100 (e.g.. from above 2 to below 2 on the x 4 lattice and from above 2 to below 2 on the 5 x 5 lattice). The corresponding infidelities also decrease monotonically with by more that factor of 2. These results show that, AQER can reliably compress entanglement and reduce the approximation error for 2D quantum many-body systems, with both and the infidelity decrease systematically as the circuit size T increases.
Quantum circuit visualization. We visualize the quantum circuits generated by the AQER algorithm with T 10 for the MNIST, CIFAR- 10, SST-2. S-RQC, and GS-TFIM datasets in Figures 15 19 respectively. Each figure shows the circuit layouts five examples, including single-qubit rotation gates Ry, Rx and two-qubit Rzz gates.
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86
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Figure 16: Loading circuits generated from AQER with T 20 for the CIFAR-10 10 dataset. Green and red blocks denote Ry and Rz gates, respectively. Blue circles connected by lines denote Rzz gates.
G COMPUTATIONAL COMPLEXITY OF AQER
Here we provide the computational complexity of AQER. We separate the analysis into the classical data setting, where the target state is represented explicitly as state vector classical computer, and the quantum data setting, where one has access only to copies of a target quantum state prepared on quantum device. Throughout, we denote by N the number of qubits and by d 2 the dimension of classical data vector. The number of iterations in the Step of AQER is denoted by T. and the number of iterations in the Step III of AQER is denoted by Tz.
G.1 CLASSICAL DATA
We first analyze the cost of AQER when the target state is given explicit state vector Cd on a classical computer. In this regime, all quantities required by AQER (reduced density matrices, entropies, gradients) can be computed exactly from
Cost of applying gates. Each gate used in AQER either diagonal (Rz and Rzz) or 2-sparse (for Ry). Therefore, applying single- two-gubit gate U to the state vector Cdd can be implemented as matrix-vector multiplication with diagonal or 2-sparse matrix, and thus costs (dd) arithmetic operations. A forward pass of the circuit with ( then costs Odd)).
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900
92
93
44
230
en0
45
0
46
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400
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47
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000
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96
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600
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000
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omaona
0
000
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000
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600
400
97
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98
000
q9
40
400
400
oUB
400
e00
O00
oma
onm
ommonm
onn
000
99
omm
onm
onm
Figure 17: Loading circuits generated from AQER with 20 for the SST-2 dataset. Green and red blocks denote Ry and Rz gates, respectively. Blue circles connected by lines denote Rzz gates.
Cost of computing reduced density matrices and entropies. tional basis be written
Let the state vector in the computa-
Di... Ci....de'
(70)
Then, the reduced density matrix (RDM) of qubits (1,2) is given by
Σ Ui.chahan.n Te, Redds.nddd
j1,22,k,, E {0.1).
(71)
Computing this RDM requires summing over all 2 assignments of (ia...., (x), and hence costs 0(2*)= (d) operations. The same complexity applies to RDMs of any constant number of qubits and any choice qubit indices. Once the RDMs are available, the single-qubit entropies and the entanglement measure used in AQER can be obtained by diagonalizing constant-size matrices, which incurs only negligible additional cost. Thus, the cost of computing the RDMs and corresponding entropies for constant number of qubits is O(d).
Step I: greedy entanglement reduction. In Step I, AQER greedily builds circuit with 5T gates by iteratively adding single- and two-qubit gates that reduce the entanglement measure. At each iteration, the algorithm evaluates most ((N2) candidate qubit pairs, and evaluating one candidate
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400
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92
ODEOU
Q3
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Figure 18: Loading circuits generated from AQER with T 20 for the S-RQC dataset. Green and red blocks denote Ry and Rz gates, respectively. Blue circles connected by lines denote Rzz gates.
requires (dd time. Thus, the total cost of Step is
OdT
(72)
Step II: product state approximation. In Step II, AQER computes the single-qubit RDMs for all N qubits and uses these to initialize the parameters of single-qubit rotations. The cost of computing N single-qubit RDMs OdN)) and the subsequent calculation of the 2N gate parameters requires only O(N) operations. Therefore, the complexity of Step II is
(dd)..
(73)
Step III: gradient-basee paramter fine-tuning. Step III performs T3 iterations of gradient- based optimization over all trainable parameters in the AQER circuit. The total number of parameters is
P= 57 + 2N,,
(74)
where 5T comes from the single- and two-qubit rotations in the T iterations of Step I, and 2.N comes from additional single-qubit rotations determined in Step II.
To estimate the complexity of one iteration in Step III. consider the cost of computing the gradient of the loss function with respect to all P parameters. Using either the parameter-shhe shift rule or automatic differentiation the state-vector simulation, computing each partial derivative involves numbers
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0
92
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ODEOE
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Figure 19: Loading circuits generated from AQER with 20 for the GS-TFIM dataset (NV 10). Green and red blocks denote Ry and Rz gates, respectively. Blue circles connected by lines denote Rzz gates.
of forward evaluations of the whole circuit. Each such forward evaluation costs O(d(T +(((+++)) N)), so the
cost of computing one partial derivative is Odd(T + NN). Since there are P O + N) parameters, a full gradient evaluation costs Odd(T + N))). Performing T3 gradient steps Step III therefore leads to a total cost of
OddTs(++N)))
(75)
Overall complexity for classical data. Combining Eqs. 72 tional complexity of AQER in the classical-date setting is
and 75 the overall computa-
N2 + dN + (T+
(76)
which can be summarized as
OdT, T polylog(d)).
(77)
This scaling is linear in the Hilbert-space dimension d (as unavoidable for exact state-vech vector simula- tion), and polynomial in the Step iteration count T and the number of gradient steps Tg.
G.2 QUANTUM DATA
We now consider the quantum data setting, where the target state p is available on a quantum device, and explicit classical description of p is assumed. In this case, all RDMs, entropies and gradients
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required by AQER estimated from finite number of measurement shots. For convenience, denote by M the number of shots used to estimate each expectation value or each entry of an RDM.
Step I and Step II for quantum data. As in the classical case, Step I adds (TT) gates that reduce the entanglement measure. At each iteration, the number of candidate gates is at most N The cost per candidate (MT)) and therefore the total cost of Step in the quantum-dat, setting scales as
O(MT2N3).
(78)
In Step II, AQER estimates single-qubit RDMs. This requires (MMN) shots and O(N) classical operations, so the complexity of Step II is
((MN).
(79)
Step III: gradient-based parameter fine-tuning. For gradient-based optimization with quantum data, we can use the parameter-shift rule to calculate the gradient. For each parameter, estimating one partial derivative costs (M + N)) operations. As before, the total number of parameters is P 57 + 2N OT + N). Therefore, one full gradient evaluation in Step III requires
(TM(++ND
(80)
operations. Performing T3 gradient steps then yields the total complexity of Step III:
OMMT,(T N)?).
(81)
Overall complexity for quantum data. Combining Eqs. 78 (79) and 81
the overall computa-
tional complexity of AQER for quantum data is
(82)
H THE EFFICIENT LOADING OF IQP CIRCUIT STATES VIA AQER
In this section, we analyze the performance of AQER on classical vectors or quantum states arising from Instantaneous Quantum Polynomial (IQP) circuits. We first show Theorem H. that, when the IQP angles lie on known discrete grid and we have classical access to the full state vector, AQER can exactly reconstruct a loading circuit in at most E iterations of Step I. We then relax the discrete-grid assumption and prove in Theorem H.2 that for generic IQP states with i.i.d. continuous angles, AQER can, with high probability, find a loading circuit that reduces the entanglement measure S to at most using most [E] iterations of Step in AQER. Finally, Theorem H.3 establishes an analogous exact-loading guarantee for discrete IQP state family with fixed angle 7/8 and unknown interaction graph, where we only have quantum access the IQP state. Let 1) be the degree of the graph E. In this case, AQER still recovers an exact loading circuit with probability at least 1 - using most E iterations of Step and a total quantum access number that is polynomial in N, log(1/8), 2D, and [E
Theorem H.1 (Exact loading of discrete IQP state vectors). Let N 2 2 be the number of qubits, and let K N be a known grid size. We consider a fully-comnected IQP circuit state
(u(w)) HNN
II
2,2,2
HN 10*)
(83)
fiijeE
where E. is the unknown interaction graph of the IQP circuit, and each parameter is unknown and is restricted to the discrete grid
ky
Wi
ky E (-2K, -2K 1,..., 1,2KK
(84)
2K +1'
Assume have classical access to the state vector of (w(W)) Then, up an overall global phase, the AQER algorithm can reconstruct an exact loading circuit for the IQP state using T iterations of Step where T [E)
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Proof. For the case where the target state is known be an IQP state, restrict the optimization in Step Ipplot AQER as follows. For convenience, let Iet _ [N] denote the set of qubits involved in the new two-qubit gate block added at the t-th iteration of Step I, and define the cumulative set T. == U.= I... At iteration + consider a candidate two-qubit gate block acting on a pair {i, For each qubit q (i.j), if E Js, we restrict the corresponding single-qubit part of the block to be the identity via (2) = If q É J. we set the single-uubit part to be Rz (TT) (-y(-/22
-HH, which
implements Hadamard up to a global phase. For the entangling gate Rzz on {i, we restrict its rotation angle to the grid an / + 1) with a, E f-2K, + -....... 2K 1, J.2.). Therefore, the optimization of the entanglement measure is performed by grid search over the choice of the qubit pair (i. j} and the discrete values of Oij-
Since the entanglement measure S is defined the sum of single-qubit Renyi entropies, it is invariant under arbitrary single-qubit unitaries. Under the optimization rule for Step described above, optimizing S for the target IQP state is therefore equivalent to optimizing S for the residual IQP state [w(ww) defined as follows,
(d((w) = II
Fw,, 2,2122 HaN 10N)
II
e 2,, H+)
(85)
(UJER
(KJ)ER
where ()) H (0) (10) 11))/V2.
Next, we focus on the first iteration of Step I. Since [E| > 0, there exists a candidate gate block on qubits {p.g} E E, and the optimization problem is
arg min (c an
ia 2,,, (,,
arg Sin Oz
-, 22
tw., 2,2,12 II (5,))E
|+)8N
arg min S
(1SIIc w. 2,2,2,12 |+) (8)
arg min& S
e
Euncr uunce By (, 22,122 11)
(86)
where we define the effective angles
Wiys
{i.j} + (p.4),
(ap(
(87)
Wij + appg {i,}} {p.4).
The entanglement measure in Eq. (86 defined in terms of single-qubit Renyi-2 entropies S(p)
ΣN S,, (p). which depend only on the single-qubit reduced density matrices (RDMs): (p)
S(p,) loga Tr(p%), where Pa is the RDM of an --qubit state p on the n-th qubit. Therefore, it
suffices to compute the single-qubit RDMs of the state in Eq. (86). We remark that any single-qubit z,, Zn), where H. Tr(,,. yn Tr Y,), and x, Tr(pn 2..) are the expectation values of the corresponding Pauli operators on qubit n. Thus, computing the single-qubit RDMs reduces to evaluating H. Yn and 2, In partieular, for the state in Eq. (86). we have
RDM has the Bloch decomposition P.. ] + I,, X. yo
In In (p)p)) (+
e & Emare 8.la,) X,,E EmEIEe Bulo Aulo 1
= (+| ON E Emnrenn By (a, (7., X,e Emnrenn 8.
(88)
II (is))N(()
cos cos(Bi,(app))
8N 8N, X,,+)
(89)
II (13)EN(()
(0s(3,,,a.m))
(90)
In Eq. (88). we define (mm) S E as the set of edges incident on n, i.c., each {i,}} E (n)) satisfies
{i,)), and use the fact that for (i, (n))) the operator 7, commutes with X,, and hence
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cancels in the conjugation without affecting the expectation value. Eq. (89) then follows from the Heisenberg -icctureeuueeeee identity for a single incident edge, for example e¹ 2.... Xn, 18 2.,/2. X, cos 8yn Yn sin 8mm together with the fact that any term containing Zm has expectation on ++) + because (+] O. Eq. (90) then follows from (+|[ = (+|X++) 1. A similar
analysis applied to Y, and Z,. which yields yv = 2n = 0.
The entanglement measure is therefore function of appy and takes the form
N
+ 2n (App))
S(apy) = Σ
log2 (apg)
log2
(9))
n=1
where nn (app) is given by Eq. (90.. By construction, the effective angles B.y (ap(ppg) in Eq. (87 depend
on Opg only when {i,5} {p. 7)- Consequently, only z:p( and zy (Ppg)))) carry the dependence pp--
Ty xpp(pp)) cos (apr) Cp.
O2
II (p.m) (PPP)((aa))
COs 3pms
xg (ppp) = cos (5pg (ppp) Cq.
Cg :=== =
II
Cos Bpm:
(q.m) ((p.a))
where C. and C, are constants with respect to apy and Cp. Cg # 0 by the choice of the angle grid. Plugging these expressions into Eq. (9T) and grouping the constant terms, we can write
+ C cos" Bpg Ppg(Opg)
1 + C22 cos" Bpg (Gtpg)
S(am)) = const - loga
- logg
2
2
so that minimizing S()mp)))) over the grid am E fa / + 1)} equivalent to minimizing the function
1 + C2 cos² Bpg(Apg)
1 C² cos² (pp((pp)
g(apg)) == logz
- log2
(92)
2
is easy to verify that oy Upg i.e. (pg((gnn) = Wpg + 4m = is the unique optimal
value of the above objective, and it can be found by grid search over Osj = a:j /2K + 1) with ai {-2K,- -2 1,..,, 2K 1,2K].
Having identified, for a fixed edge (p. E F., the unique optimal choice 0p = wpg such that
(pg(app) Upg + app 0, we now iterate this construction. At the 1-th iteration of Step let ( denote the effective angles on the IQP edges and define the residual edge set
p
[(i.j) E E 8 #0).
(93)
By the discussion above, if F(t) is nonempty we pick any {p, o} E F(t)) and minimize S over the grid of angles apy. Uniqueness of the minimizer implies that the optimal choice ag satisfies
g(t+1) g(t) + (t) 0,
(94)
while all other angles remain unchanged, i.e., (((+1) = / for all {i,}} * {p.g). Therefore the residual edge set strictly shrinks at each iteration.
|
| 1.
(95)
Since |F(0| < [E), after T | < E iterations we reach a configuration with F(T) = Ø. i.e., ((57) 0 for all (i.j} E F. In this case all two-qubit entangling rotations are cancelled, and the resulting state has vanishing entanglement measure = By definition of as the sum of single-qubit Renyi entropies, this implies that the state obtained from (o(w)) after T iterations of Step I is a product state, which we denote by (ppeod)-
Finally, by Lemma B.2 we can construct a single-qubit circuit composed of R and Ry rotations that maps (pprod) exactly to the computational basis state (0^) up to a global phase. Let Uj be the product of all Step gate blocks obtained by the above procedure and Un be the single-qubit circuit from Lemma B.2 Then
UnU (e(w)) 1
(96)
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Published as a conference paper at ICLR 2026
up global phase. Taking adjoints and reversing the order of gates, obtain an explicit loading circuit
Ulad (Un)U)) U
(97)
such that Ujoad 10*) |m(w)) up to a global phase. This completes the proof of Theorem H.1
Theorem H.2 (Approximate loading for generic IQP state vectors). We follow the notations in Theorem H.7 and consider the IQP circuit state
[u(w)) HON
II (ije
-i.,2,2,12
HON 10")
(98)
where the angles { (L.))CE are i... random variables sampled from [- x) uniformly. Let D be the maximum degree of E. We assume the classical access to the state vector of [o(w))) Then for any 8, > O, AQER uses at most T S {E) iterations of Step by searching the grid with size K
DN
produce circuit Uj satisfying S(Ui)o(w))) with probability least 1 - 8.
Proof. First, we demonstrate that with high probability all parameters wiy satisfy lower bound on the cosine value. Since each Wiy sampled from -T,, π] uniformly, we have
6
8
5
Pr coswwyl
aresin
(99)
E
E
E
By a union bound over {i.}} E E, follows that with probability at least we have
1
[2
cos Wiyl
for all {i,}} E E.
(100)
In the following, we condition on this high-probabilityy event.
Next, we follow the same optimization procedure of Step as in the proof of Theorem H.T the two-qubit gate on {i,j} has angle Ciyy restricted to the grid
Specifically,
GK
am
E { 2K, 2K
2K - 1K-,,2K0 1,2K}
2K 2 +1
where K
DN
We also impose that at most one two-qubit gate block is added for each pair
2
{i.s} S [M]. Under this restriction, Step has at most one update per edge, so the total number of iterations satisfies T < [E]. For each {p. 9) E, the grid search in the optimization problem of Eq. (92) finds value Ciny whose distance to -Wyy is at most T 72(2 + 1). Thus, after T = [|] iterations, the remaining residual state be written
(o(B) Euare 8,2,2,12 +
(101)
where each effective angle satisfies 13y,1 S T/(2K +
The entanglement measure of the state (u(3)) can be obtained via calculation analogous to Eqs. (88 (1). In particular,
1 + x2?
S(I((B))) = loga
2
==1
N
(102)
N
cos' Bij
(103)
#=1 [L)))N(()
< N - 2D
π
(104)
2K +1
44

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Published as a conference paper at ICLR 2026
27
N - N
(105)
2K 1
N - N
1 D
(106)
2K
π
= DN
S
(107)
2K +
Here Eq.
102 follows from the inequality - logg z
I x for x. E [0, 1] by setting x = x
3
Eq.
103
follows from Eq. (90), which gives x" = =1 (L)) cos" Beg. Eq. (104) is derived using
13gg
(2K 1) and [n) S D, that each product is lower bounded by cos +1). uses cos" = sin
Eq. 105)
1 x² for |a|| S 4/2. Eq. (106) is obtained from the Bernoulli inequality (1 - x)" 2 1 Dr for E [0, 1]. Finally, Eq. 1077 follows from the choice
K
DN
which ensures DN 2K+1
S This completes the proof of Theorem H.2
Theorem H.3 (Exact loading of discrete IQP states), Let N be the number of qubits. We consider fullyyconnected IQP circuit state state
M ((E)) = HON
II
iz2,2,/8
HaN 10")
(108)
(L,)EE
where E. is the unknown interaction gruph of the IQP circuit, with the maximum degree D. Assume we have quantum access to the state (u(w)). Then, up to an overall global phase, the AQER algorithm can reconstruct exact loading circuit for the IQP state using T iterations of Step I, where T S E
with probability at least 8 8 by using at most O (1E) log N' calls to the given IQP state.
except that we restrict the grid search for each pair (p. g) to the angles 0ppg E (0, -x/44/4/ x/4). We remark that now only have quantum (rather than classical vector) access to the IQP state. Our goal is to guarantee that, in each iteration, the E/, where the failure probability
Proof. We follow the Step I optimization as in Theorem
grid search selects qubit pair in E with probability at least comes solely from the finite number of measurement shots. Consequently, after T = =EE iterations of Step I, all qubit pairs in E are cancelled with probability least 8.
Next, we focus on the first iteration. Similar to the proof of Theorem written
the residual state can be
(e((E)) e Ea 22, |+) &N
(109)
For any candidate pair {p, q} and any choice of apy, the single-qubit Bloch coefficients satisfy
(app)
II cos (os(Ba)(pp)))
yn (n O,
(app)) 0.
(K)))N()))
In Theorem H.1 we showed that Step optimizes the entanglement measure
xr,, ((pp)))
S(app)
loga
(110)
2
and that a decrease in S is achieved via an increase in |x (Apy)) Since [[(n) < D and (ay))) {0, +/44), we have
{(
E. (amg) E
12
:0 :osdsD 12-
0-1/ 1].
Thus, for each (p. 9} S [N], by using
2
D log
8
45

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Published as a conference paper at ICLR 2026
quantum shots, estimate the relevant zy (App)) and (:g(0 with additive smaller than
constant multiple of 2 and hence determine whether both [xpp and |x,p| increase when switching ap from to -7/4 with probability least - ( then identify at least one {p.g) E with probability least 8/ E| Repeating this procedure for all T = [E] iterations, a union bound implies that the overall success
{p. E E. By going over all
{p.g) C [N],
probability least 1 6. The total number of quantum shots is
(
o 2 log
N2 5
N E O (18)N22) log N' 6
This completes the proof of Theorem H.3
I THE USE OF LARGE LANGUAGE MODELS
In this work, large language models (LLM) were used only as general-purpose tool for minor language polishing and expression refinement. The LLM did not contribute to any scientific ideas, experiments, or results, and all content generated under its assistance was carefully reviewed and edited by the authors. The authors take full responsibility for all contents of the paper, including those influenced by LLM-assisted text.
46


Reviews:
[{"review_id": "Mieemyzn33", "reviewer": "Reviewer_ctwW", "summary": "This paper proposes a structured method for synthesizing circuits for quantum state preparation, with a primary focus on loading classical data onto a quantum computer as quantum states (e.g., encoding a classical vector into a quantum vector). The core insight behind the proposed method, AQER, is supported by a theoretical upper bound on the best-case infidelity (over initial product states), expressed in terms of the sum of single-qubit entanglement entropies of the target state evolved under a circuit.\n\nBuilding on this insight, the method proceeds in three stages:\n- Circuit search to reduce the entanglement of the target state.\n- Approximation of the resulting low-entanglement state by a product state.\n- Parameter refinement to optimize performance.\n\nThe authors perform simulation-based benchmarks on both classical and quantum data-loading tasks, demonstrating that their method consistently outperforms existing approaches.", "questions": "Can the theoretical characterization of infidelity be extended to general noise models, where the unitary evolution is replaced by a generic quantum channel (CPTP map)?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The approach is grounded in a clear and well-motivated theoretical insight, which lends credibility to the method and provides a room for future extensions. Given that quantum state preparation is, in general, an infeasible problem in terms of computational complexity, it is reasonable that the paper does not pursue purely theoretical guarantees.\n\n- Numerical benchmarks across multiple datasets demonstrate consistent performance improvements over prior methods.", "weaknesses": "- Since the target hardware setting is the NISQ regime, the lack of experiments on real quantum devices, or even simulations under realistic noise models (e.g., depolarizing noise), makes it difficult to assess how the proposed method would perform in practice. This limitation is particularly relevant given that several prior works in this area include evaluations on real hardware.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "QBYZ4AhaTX", "reviewer": "Reviewer_LZZQ", "summary": "This paper introduces AQER, a novel three-stage algorithm for approximate quantum loading (AQL). The core idea is to construct a quantum circuit by greedily minimizing an entanglement measure, the sum of single-qubit Renyi-2 entropies. The authors provide a theoretical bound linking this measure to the final approximation error and demonstrate through extensive simulations on classical and quantum datasets that AQER outperforms several existing methods in both accuracy and gate efficiency.", "questions": "See weakness", "limitations": "", "rating": 6, "confidence": 4, "soundness": 1, "presentation": 2, "contribution": 2, "strengths": "The paper's primary strength is Theorem 3.1, which establishes a formal information-theoretic bound between the approximation infidelity and the proposed measure. This provides a solid theoretical justification for the algorithm's design, moving beyond purely heuristic approaches.\n\nThe entanglement-reduction-guided strategy for building the circuit is a novel and intelligent heuristic. It offers a structured method for ansatz construction that aims to position the optimization in a favorable region, which is a key challenge in variational algorithms.\n\nThe authors have conducted extensive and compelling numerical experiments across a diverse set of benchmarks, including classical data and quantum many-body states up to 50 qubits. The consistent and significant performance gains shown in Table 1 strongly support the practical effectiveness and scalability of the proposed method.", "weaknesses": "1. The paper compellingly argues that AQER mitigates barren plateaus in the final optimization. However, a critical discussion or numerical experiment is missing on the optimization landscape. The algorithm's success hinges on this greedy step being efficient. The paper would be significantly strengthened by a discussion of why the local structure of this cost function makes the optimization in Step I tractable.\n\n2. The experimental comparison relies heavily on the two-qubit gate count, which is a good but incomplete metric. To make a fairer and more robust assessment of AQER's scalability, I suggest the authors consider the gate depth, the number of trainable parameters, measurement overhead, and training cost.\n\n3. The study is performed under ideal, noise-free conditions. Given that the practical utility of any near-term algorithm depends on its noise resilience, the paper would be substantially more impactful with a discussion on this aspect. For instance, how might hardware noise affect the measurement of entanglement and the subsequent greedy circuit construction? Even a qualitative discussion or a small-scale simulation under a simple noise model would add significant value.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RiqRvxfohC", "reviewer": "Reviewer_Ki6d", "summary": "The paper introduces a unified framework for analyzing the accuracy of approximate quantum loaders (AQLs), by bounding the achievable infidelity in terms of an entanglement measure S (the sum of single-qubit Rényi-2 entropies after applying the inverse loader to the target state). Building on this, the authors propose AQER, a data loader that constructs an encoding circuit by first reducing the target state’s entanglement, then applying single-qubit product-state corrections with closed-form parameters, and finally refining all parameters jointly. Experiments on classical datasets (MNIST, CIFAR-10, SST-2) and quantum datasets (random quantum circuits and Ising model ground states) reaching up to 50 qubits, show lower infidelity than MPS-, HEC-, and AQCE-based baselines for the same or fewer two-qubit gates.", "questions": "- Can the authors specify what happens, in terms of the algorithm’s computational complexity, if the target states have a high level of entanglement?\n\n\n- Can the authors provide theoretical guarantees that the optimization step in Eq. (2) is efficient?\n\n\n- How is the method applicable to high-dimensional classical data that require a large number of qubits?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The paper presents a clear, useful bound for all AQLs, formulated in terms of the entanglement measure S.\n\n\n- The AQER algorithm is intentionally designed to minimize the S, which the theory connects directly to infidelity.\n\n\n- Once target states are available, training AQER relies only on local measurements.\n\n\n- Experiments provide encouraging evidence that supports the paper’s central claims", "weaknesses": "Although the above strength, I do not think the current version of the paper matches the ICLR criteria for the following reason:\n\n- My main concern is the computational cost required to reach low infidelity in the general case. The theoretical bound in the paper is meaningful only in the limit S goes to 0. For target states with moderate or high entanglement, achieving a small S requires deeper circuits U, and it is unclear whether this can still be done with circuits of polynomial size.\n\n\n- Also, the upper bound is stated in terms of the existence of a product state with certain properties. This does not guarantee that a generic product-state ansatz will satisfy the same bound in practice.\n\n- Another concern I have is that computing the entanglement measure requires access to the target state. For quantum data, this may be possible if multiple copies of the state are available, though the number of required copies is not made explicit. For classical data, however, one must first classically construct the full target state, which is not tractable at large numbers of qubits. In fact, the classical-data experiments in the paper are limited to around 10 qubits.\n\n\n- Even for quantum states, as far as I understand, the simulations proposed in the paper rely on tensor-network methods, which inherently assume limited entanglement. This suggests that highly entangled quantum data are effectively out of scope for the demonstrated results.\n\n\n- Finally, it is not clear that the optimization described in Eq. (2) can be solved in polynomial time.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "GiSU7kdDr8", "reviewer": "Reviewer_eQiP", "summary": "AQER relates approximate-state-loading error to entanglement entropy S and derives corresponding information-theoretic bounds", "questions": "1.\tCould 2-D random circuits or critical 2-D spin systems be included to probe performance when entanglement is less easily reduced?\n2.\tCould the authors provide gradient-magnitude statistics across N=10–50 qubits?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1.\tProvides information-theoretic bounds that relate approximation error directly to a locally measurable entanglement proxy, guiding circuit construction for state preparation.\n2.\tConsistently outperforms MPS, HEC and AQCE at equal two-qubit gate budgets on both classical and quantum data sets.", "weaknesses": "1.\tTheorem 3.1 is general, but the evaluated states appear to be low- or modestly entangled; tougher, higher-entanglement regimes are not covered.\n2.\tNo scaling analysis of gradient variance with qubit number; trainability claim remains heuristic.\n3.\tThe paper claims code availability but does not provide the anonymous GitHub link in the main text, which could hinder reproducibility efforts.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)