Quantum-PEFT: Ultra parameter-efficient fine-tuning
Authors: Toshiaki Koike-Akino
Francesco Tonin
Yongtao Wu
Frank Zhengqing Wu
Leyla Naz Candogan
Volkan Cevher
Venue: ICLR 2025 Poster (2025)
Keywords: parameter-efficient fine-tuning
lora
quantum machine learning
orthogonality constraints
Primary area: foundation or frontier models, including LLMs

Abstract: This paper introduces Quantum-PEFT that leverages quantum computations for parameter-efficient fine-tuning (PEFT). Unlike other additive PEFT methods, such as low-rank adaptation (LoRA), Quantum-PEFT exploits an underlying full-rank yet surprisingly parameter efficient _quantum unitary parameterization_. With the use of Pauli parameterization, the number of trainable parameters grows only logarithmically with the ambient dimension, as opposed to linearly as in LoRA-based PEFT methods. Quantum-PEFT achieves vanishingly smaller number of trainable parameters than the lowest-rank LoRA as dimensions grow, enhancing parameter efficiency while maintaining a competitive performance. We apply Quantum-PEFT to several transfer learning benchmarks in language and vision, demonstrating significant advantages in parameter efficiency.

Published as a conference paper at ICLR 2025
QUANTUM-PEFT: ULTRA PARAMETER- FINE- TUNING
Toshiaki Koike-Akino" Francesco Tonin Yongtao Wu²
Frank Zhengqing Wu Leyla Naz Candogan² Volkan Cevher
Mitsubishi Electric Research Laboratories (MERL), 201 Broadway, Cambridge, MA, USA LIONS, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland
ABSTRACT
This paper introduces Quantum-PEFT that leverages quantum computations for parameter-efficient fine-tuning (PEFT). Unlike other additive PEFT methods, such low-rank adaptation (LoRA), Quantum-PEFF exploits underlying full-rank yet surprisingly parameter-efficien quantum unitary parameterization, With the of Pauli parameterization, the number of trainable parameters grows only logarithmically with the ambient dimension, as opposed to linearly as in LoRÁ- based PEFT methods. Quantum-PEFT achieves vanishingly smaller number of trainable parameters than the lowest-rank LoRA as dimensions grow, enhancing parameter efficiency while maintaining competitive performance. We apply Quantum-PEFI to several transfer learning benchmarks in language and vision, demonstrating significant advantages parameter efficiency.
INTRODUCTION
Fine-tuning large pre-trained models a cost-effective method to adapt a general-purpose model to additional domains and tasks in computer vision and natural language processing (Devlin et al., 2018; Liu et al., 2019; He et al., 2021b; Radford et al., 2019; Brown et al., 2020; Dubey et al., 2024). Yet, even the practice of fine-tuning for each application can be costly models scale to billions trillions of parameters. The substantial memory requirements, such as GPT-3'ss 350GB footprint (Brown et al., 2020), can pose significant resource challenges, restricting practical deployment.
Parameter-efficient fine-tuning (PEFT) addresses the resource challenges of task specialization for massive pre-trained networks without the need fine-tune parameters in full model weights dimensions (Aghajanyan et al., 2021; Hu et al., 2021; Edalati et al., 2023). For instance, low-rank adaptation (LoRA) (Hu et al., 2021) uses low-rank decompositions to modify weights, whereby reducing the number of trainable parameters. Despite its efficiency, there are limitations to the number of parameters, which include a compression ratio constrained by rank-1 decompositions and linear scaling of trainable parameters with weight matrix dimensions.
We introduce Quantum- PEFT, novel framework that achieves extremely parameter-efficient fine- tuning beyond LoRA-variants, e.g., (Zhang et al., 2023; Qiu et al., 2023; Liu et al., 2023b; Yeh et al., 2024), by leveraging quantum unitary parameterizations (Biamonte al., 2017: Schuld et al., 2015). Quantum circuits are way to represent unitary matrices product of smaller unitary matrices (i.e., quantum gates) with a total number of parameters growing logarithmically with the dimension, offering an extremely efficient framework for effective PEFT parameterizations. The core idea is to reparameterize the layers pre-trained networks as generalized quantum circuits capturing complex transformations, which then only require logarithmic number of trainable parameters. The ultra parameter efficiency is enabled by parameterizing the low-rank subspaces via Kronecker products of generalized Pauli rotations. The key contributions of our work include:
We introduce new quantum-inspinn modules based on generalized Pauli parametrization and quantum tensor network.
"Equal contribution. Correspondence to: Toshiaki Koike-Akino, Francesco Tonin <koike @ merl.com, francesco toniceooooonneeee epfl.ch...

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orthagonal
U
F
node
A
$
diagonal
W
W
W
node
B
orthogonal
VT
Vi
Q2
node
(a) LoRA
(b) AdaLoRA
(c) Quantum-PEn PEFT
Figure 1: Overview of Quantum-PEFT compared to LoRA and AdaLoRA for PEFT. W is the frozen pretrained weight, green boxes represent trainable parameters. LoRA updates W by training the low-rank matrices A, B. AdaLoRA introduces the SVD trainable form U, A, V with regularizer VTV -| In Quantum-PHFT, the matrices U, V are not trainable parameters, but rather computed through quantum mappings of ofders-of-magnitueee smaller intrinsic parameters. Contrary to AdaLoRA, U, V are left-orthogonal by construction in Quantum-PEFT.
- 7|
We propose a novel framework, named Quantum-PEFT, that leverages quantum uni- tary parameterizations for extremely parameter-efficient fine-tuning, achieving orders-of- magnitudes higher compression rates over state-of-the-ar PEFT methods.
Quantum-PEFF with Pauli parameterization enables logarithmic scaling of trainable param- eters with respect
ambient dimension of the model, realizing even smaller number of parameters than the lowest-rank LoRA.
Through extensive experiments on language and vision tasks, we show Quantum- PFFTTs significant advantage in parameter efficiency, achieving to 25-fold reduction in trainable parameters compared to LoRA, yet maintaining competitive performance.
2 RELATED WORK
Parameter-efficie fine-tuning (PEFT) Parameter-efficient fine-tuning (PEFT) methods allow significantly lower model training cost for different downstream tasks. A plethora methods have been proposed for PEFT (Houlsby al., 2019; Aghajanyan et al., 2021; Hu al., 2021; Edalati et al., 2023; Lester et al., 2021; ; Li and Liang. 2021; He et al., 2021a; Karimi Mahabadi et al., 2021; Chen al., 2022; Jie and Deng. 2023: Hao al., 2022: Houlsby al., 2019; Pfeiffer al., 2021), among which reparameterization. based techniques (Aghajanyan et al., 2021: Hu et al., 2021; Edalati al., 2023) bear the most relevance to our study, where the model architecture not changed but reparametrized with lower number of trainable parameters. LoRA (Hu et al., 2021) updates the pretrained weight matrix through the addition of a product two low-rank matrices with widespread adoption (Zi et al., 2023; Chavan et al., 2023; Hayou et al., 2024; Zhu et al., 2024). Many variants were introduced, c.g., tensor factorization (Edalati et al., 2023; Bershatsky et al., 2024; Chen et al., 2024a), nonlinear mappings (Liu al., 2023a), Hadamard (Ych et al., 2024) and Kronecker product (Edalati et al., 2023), layer sampling (Pan et al., 2024), embedding finetuning (Wu et al., 2024), and high-rank updates (Jiang et al., 2024b; Chen et al., 2024b). Additional discussions in Appendix A.6.
Unitary-constrainee PEFT AdaLoRA (Zhang et al., 2023) introduces dynamic rank adjustment during fine-tuning, with additional regularizer for orthogonality. Unlike AdaLoRA with inexact orthogonality and extra regularization, we directly parameterize full-rank unitary matrices via efficient quantum circuit embeddings. Orthogonal fine-tuning (OFT) (Qiu et al., 2023; Liu et al., 2023b) employs a unitary matrix to transform the pretrained weights. OFT typically requires more trainable parameters and rely expensive Cayley transform, highlighting the need for more efficient methods.
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x
Layers
X layers
RY
RY
8UMM
SUI4
SUy99
0111
BUN
D111
RY
RY
RY
5UNN
(r1)
RY
RY
50j11
Sujw
RY
SUNN1
SU(2)'
SU(4]
(a) Simplified two-design ansatz
(b) Generalized Network
Figure 2: QML components. (a) Simplified two-design ansatz as (2). It alternates RY and CZ. i.e., product of small unitary matrices. (b) Generalized quantum-inspired network for our unitary node. It generalizes the two-design ansatz to arbitrary dimensions by employing SU(N") blocks.
Unitary-constrained machine learning Unitary constraints can make training more stable and improve generalization, e.g. (Arjovsky al., 2016; Jing et al., 2017: Chang and Wang, 2021). Different parametrizations are used, e.g., orthogonal weights through the Cayley transform (Helfrich et al., 2018) and Householder reflection (Mhammedi et al., 2017; Huang al.. 2022). Stiefel manifold optimization in ML is widely studied (Wisdom et al., 2016; Bansal et al., 2018: Li et al., 2019). Riemannian optimization has been considered for low-rank neural networks (Schotthöfer et al., 2022; Zangrando et al., 2024; Schotthöfer and Laiu, 2024). where they directly optimize the low-rank network factors with manifold optimization. Our approach differs as we do not optimize over the orthogonal factors directly, but rather parameterize them through trainable intrinsic weights.
Quantum machine learning Relevant quantum machine learning (QML) concepts include express- ibility and entangling (Sim et al., 2019; Pérez-Salinas et al., 2020). Variational principles for quantum neural networks are in (Farhi and Neven, 2018), with extensions for quanvolutional networks (Hen- derson et al., 2020), quantum autoencoders (Romero et al., 2017), support vector machines (Suykens, 2013; Rebentrost et al., 2014), quantum graph neural networks (Zheng et al., 2021), and quantum generative adversarial networks (Lloyd and Weedbrook, 2018; Dallaire- and Killoran, 2018). Quantum circuits are analytically differentiable enabling gradient optimization (Schuld et al., 2019).
PRELIMINARIES
Notations: Let SU(N), sun. SO(N), O(N), and Vx(N) denote the special unitary Lie group of size N, its Lie algebra, special orthogonal group, orthogonal group, and real-valued Stiefel manifold having orthonormal K frames Vx(NT {X N X X IK respectively. We denote I, R. @. I. and J as identity matrix of proper size, real numbers field, Kronecker product, transpose, and imaginary number, respectively. Let L. q. N', K' be the number of alternating entanglement layers, the number of qubits, orthogonal node size, and intrinsic rank, respectively. We denote the derived unitary matrices Q. e.g. Qp denotes the unitary Buuli-arammterizztee matrix. 0 represents angles in Pauli parametrization, which are trainable parameters. RY and CZ represent the quantum RY rotation gate and controlled-Z gate, respectively. Detailed list of symbols is in Appendix A.7.
In QML, neural network modules are realized by embedding classical data and weight values as quantum variational parameters such as Pauli rotation angles to control measurement outcomes. provides universal approximation property (Pérez-Salinal et al., 2020) and exponentially rich expressibity (Sim et al., 2019). Any quantum circuit can be decomposed (Kitaev, 1997) into series of single-qubit rotations and two-qubit entanglements.
Pauli matrices Pauli operators play an important role generate any unitary rotations up global phase. The group SU(N) Lie group of unitary N x N matrices having determinant I-can be generated by the Lie algebra sun, i.e., the set of N N Aeem-ttemi matrices. For single-qubit rotations over SU(2), the Lie algebra is span of [yx,),, with Pauli matrices:
x [9 !J 7 Z ! -1 The exponential mapping of its linear combinations generates
(2).). For example, quantum RY rotation gate is given
RY(() exp(- exp (
0 0/2
0/2
cos(0/2 -in(0/
(1)
0
sin sin(0/2)
cos(0/2)
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which alone spans the special orthogonal group SO(2) and forms ((2) along with reflection Z.
Quantum circuits and matrix operations A quantum circuit consists of a sequence of quantum gates applied to quantum state. Mathematically, a quantum circuit can be viewed unitary matrix transforming one state (i.e., tensor) to another tensor. The quantum circuit is product of smaller unitary matrices (individual gates). We primarily employ two gates: single-qubit rotations RY gates, represented by 2 x 2 unitary matrices, and entangling 2-qubit CZ gates (4 x 4 diagonal unitary matrices). quantum circuit built from a series of gates and can represent any unitary matrix.
Two-design ansatz In QML, two-design ansatz (Cerezo et al., 2021) use small number of parameters in order of Ologg (NV] to represent unitary matrices SU(N) whose statistical properties identical to ensemble random unitaries. This property suggests that gradient optimization can uniformly adjust few-pane paraameeeeerr Pauli rotation angles along the unitary group SU(N). Comparing to the full degree of freedoms of dim(SU((N) 2 for any -kew-Hermitian matrices, the QML has great potential to realize parameter-efficien representation in its logarithmic order. In the following, we introduce generalized framework to extend the QMI features for extremely parameter-efficient neural network modules, which constitute the building blocks of
4 QUANTUM-PEFT
We propose to parameterize the matrix added to the pretrained weights W by leveraging an ultra compact representation based on quantum unitary parameterizations. Similarly to AdaLoRA (Zhang et al., 2023), we reparametrize the weight updates as a product of unitary matrices U E Vx(NN) (N) C R N V (M) C IR M and a diagonal matrix A R K x Specifically, the weight update AW for weight matrix W E NNN is given by: AW = U AVV. In Quantum-PEFT, the low-rank U, V are not optimization variables, but rather are expressed via efficient unitary parameterizations, i.e., Kronecker products of generalized Pauli rotations. In this way, the number of trainable parameters depends on the chosen underlying unitary parametrization. Our PEFT pipeline is shown in Figure 1.
4.1 QUANTUM-PEFT! PAULI, ORTHOGONAL AND DIAGONAL NODES
In this section, we introduce generalized quantum-insttnnt inspired modules as the core building blocks of Quantum- PEFT, i.e., product of RY and CZ. gates for trainable orthogonal nodes (akin to generalized RY modules), and trainable diagonal nodes (akin to generalized CZ modules). We additionally show how to solve the power-of-two scaling limitation of QML and address efficient computation.
Pauli parameterization The simplified two-design ansatz (Cerezo et al., 2021) visualized in Figure 2a in Appendix uses an alternating circuit composed of RY and controlled-Z (C7) entangling gates: CZ = diag 1, 1, 11], which is an element of reflection groups O(1)² = {+1]_. This ansatz is suited for neural networks as they are real-valued quantum operations over SO(N). i.c. not complex-valuee operations over SU(N) arising when using RZ or RX rotations. In Quantum-PERt PEFT, we propose to use the Pauli parameterization based on this construction, as given below:
L
Qp = II (1T (CZ E
RY
RY(0s,2)) 3 n)
(RY
(2)
where L is the number of alternating entanglement layers, and logg is the number of qubits: in (2) g odd s.t. (2 integer and the g even case can be trated similarly. In (2), the trainable parameters are the rotation angles 0z.1 associated with the employed RY gates (1). The 0z,, for each RY gates trainable, where the RY gate (1) has single parameter. This Pauli parameterization therefore has (2T. + 1) log, (N)- - 2I. parameters, increasing only logarithmically with the matrix size N. The entangling capacity controlled by and is further discussed in Appendix A.4. While the tensor rank is 2, thanks to the alternating CZ entanglement, the effective rank of the matrix QP is of full contrary to low-rank AdaLoRA (Zhang et al., 2023). Not only parameter efficient, but Pauli parametrization is also computationally efficient it takes O [Ng) operations compared to quadratic complexity for unitary matrix rotations. Motivated by this -oantuum-inspieed network, we further generalize the parameterization from SU(2) SU(N') with an arbitrary size of N' > 2 as
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Exponential, Taylor
I
N
(a) Trainable mapping Stiefel manifold
N
B
B
N'
Map
Q
Truncate
K'
Parameter
Skew-Symmetry
Unitary
Stiefel
(11,
1)
(b) Trainable diagonal nodes
Map
Diagonal Parameter
Identiry.
Rademacher
Figure 3: Quantum-PErl PEFT modules with corresponding tensor diagrams. (a) Trainable mapping onto the Stiefel manifold Vx (N'). Intrinsic rank K" top K' columns are trainable parameters in B. (b) Generalized CZ modules for diagonal nodes either O(1)*' R.
shown in Figure 2b a building block to represent a large unitary matrix SU(N) with a smaller number of factors SU(N") in a logarithmic scale of Oloog (N)]. To this end, we show how to (i) generalize RY gates by generating unitary matrices from eymmerree matrices, and (ii)
factors
use recursive cosine-sine decomposition to allow non-power-of-two N.
Mapping to Stiefel manifold Arbitrary unitary rotations of size N' can be realized by mapping skew-Hermitian matrices for SU(N") or skew-symmetric matrices for SO(N). We consider mapping methods for orthogonal nodes below. Let R.V' xN' be strictly lower-triangular matrix with non- zero trainable parameters from BR N with S N''N' - 12/ Given a skew-symmetric matrix A = B - B E N xN" we can generate a corresponding unitary (orthogonal) matrix, e.g..
with exponential mapping or Taylor series
Q = exp(A),
O
Σ
AP pl
where the mapping Qf is approximated version of Q' up to polynomial order P to avoid matrix exponentiation. Diverse unitary parameterizations are possible, e.g., Cayley transform. Householder reflection, Givens rotation; focus on QT as it shows good trade-off between accuracy, speed, and parameter efficiency. We refer to Appendix A. for detailed comparisons and discussions.
Figure 3(a) illustrates the pipeline generate matrices the Stiefel manifold Vx (N') from train- able parameters through the exponential/Taylor mapping. After mapping skew-symmetric matrix, truncating the square unitary matrix as Q.K.: generates righttorthogonal matrix. As all the mappings described above are differentiable, the Lie algebra can be trained via gradient methods, and we PyTorch's 's autograd compute the backward pass gradient. QML literature has employed the gradient method successfully (You et al., 2023; Bermejo et al., 2024) and we empirically observed no issues in loss convergence. While most mapping methods are studied in other literature (Qiu et al., 2023; Liu et al., 2023b; Chang and Wang. 2021; Wisdom et al., 2016; Bansal et al., 2018; Li et al., 2019), in a PEFT context we can further reduce the number of parameters by masking out the Lie parameters. For example, only the top K' columns of Bx are trainable, while the other parameters are frozen or null-out. We call K' an intrinsic rank to cover subset of Vx(N')
A naive implementation of the above mapping pipeline can use redundant memory before truncation. We resolve the memory redundancy by tensor contraction ordering (Pfeifer et al., 2014), except for Q e.g., multiplying unitary matrix with a feature vector æ E R.V' is recursively contracted as Qt Σ, (B BxZ, which does not require the full matrix Q' but only a series of low-rank multiplications with B. Computational complexity of the mapping and numerical accuracy considerations discussed in the next subsection.
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Table 1:
Memory requirements store trained LoRA and Qp
weights for DeBERTaV3 base, Llama 3.1 405B, and GPT-4.
N
K
N
LoRA (TTD)
LoRA
Quantum-PEE
K
Rank
Parameters Required Bytes
Parameters Required Bytes
36.9K
0.14MB
3.69K
0.01MB
BASE
(b) AdaLoRA (CP)
16
589.89
2.25MB
3,98K
0.02MB
N
256
9437.2K
36,00MB
9.7K
0.04MB
GPT-4 LLAMA
8.26M
31.5IMB
60.7K
0.23MB
(c) Quantum-PEFI (TD)
16
132. IM
0.50GB
64.5K
0,25MB
256
2188.2M
8.35GB
127.3K
0.49MB
Figure 4: Tensor diagram of LoRA variants.
1
36.7M
140MB
269.7K
1.03MB
16
586.6M
2.24GB
286.4K
1.09MB
256
9385.6M
35.80GB
565.
2.15MB
Overcoming the power-of-N' limitation The pipeline in Figure 3(a) can be seen as generalized RY modules for SU')",, then assembled to construct larger unitary node SU(NV as visualized in Figure 2b, meant to be used within Qp s.t. Quantum-PEFT can handle arbitrary dimensions N' 2. However, N should be power of N Using quantum Shannon decomposition (QSD) (Shende et al., 2005) i.e, recursive cosine-sine decomposition (CSD), any unitary matrix SU(N) can be constructed by SU(N) and (U2N2) for lower dimensions such that Ni N2 and Ni + N2 N for N 1:
C
-
U1 0
0
V1
0 V2
U
0
0
U2
C
S
where U E SU(N), U1 V/2 SU(N)), U2, Vi SU(N2), diagonal cosine and sine matrices such that C2 + 52 TE R Hence, power-of-N" rotations such as Kronecker products of Pauli rotations be still used for arbitrary size of matrices. It hence can solve the power-of-N' limitation.
Example 4.1. the simple case N' 2, N2 and N₂ are adjustable parameters s.t. N N2 N2 with non-power-of-two N. For example, when N = 12, we can use Ni 8 = 23 and N₂ = 4 2 where we can use four power-of-ree rotations. QSD allows recursive decomposition. For example, when N 28, decomposition twice to have N = 2 + + 22, where the first (N,, N2) = (24, 2,,2 + +22 and the second CSD for the N₂ part will be further decomposed (N1, N2) (23,2% 22).
unitary matrices of U1, U2. Vi, 1/2 as well as cos-sine RY
apply cos-sin
Diagonal node Generalizing CZ modules provides a few options: trainable diagonal matrix any real number IR. discrete number, and binary k= Trainable discrete diagonal ma- trix can be realized c.g. by Gumbel softmax or ReinMax trick (Liu et al., 2024). We refer to trainable binary diagonal matrix as Rademacher mapping, which can create perfect unitarity and reflection group in 0(1)K, Specifically. Rademacher mapping with ReinMax trick is given as QR = diaggReinMax, ([A, -]) [+1,- -1]] with a temperature 7 and diagonal parameter
R Fig. 3(b) illustrates diagonal nodes and its tensor diagram. When identity map is used, it can be used as singular values of any matrices under its singular-value decomposition (SVD). Therefore, the use of both trainable unitary matrices and diagonal matrices is sufficient for general representation of any matrix through its SVD, solving the only- mnlyuunitarity limitation of typical QML.
4.2 QUANTUM-PEFT: ANALYSIS
Quantum -EFT leverages a parameter-efficicn network by exploiting the new modules: trainable small orthogonal matrices parameterized by the Lie algebra to generate Stiefel manifold Vx via our generalized RY modules; trainable diagonal nodes either on R or 0(1) our generalized CZ modules; and the Pauli parameterization. We now analyze the orders-of- mfgggntde improvements in parameter efficiency w.r.t. existing LoRA variants and give more discussions.
Parameter efficiency LoRA uses two K K-rank matrices, having 2NK parameters. This is known as 2-mode tensor train decomposition (TTD). Any TTD can be transformed into its canonical form, hav- ing multiple unitary nodes and one diagonal node. AdaLoRAA uses approximated SVD, where unitarity

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is not perfectly imposed, leading K(K + 1) redundant parameters and extra regularization terms. From the tensor network perspective, AdaLoRA falls under Canonical Polyadic (CP) decomposition which does not strictly assume orthogonality. Using the Lie algebra, Quantum-PEFT can readily real- ize the non-redundant parameterization for trainable SVD) (i.e., 2-mode Tucker decomposition: TD). In this form, the number of trainable parameters depends the chosen underlying parametrization for the orthogonal matrices. Specifically, the Taylor parametrization Q4 for the maximum decomposition K' K'
size N² N, Qp achieves an extremely compact representation with only ((2 + (og)(N) 2L) K) parameters, scaling logarithmically with the matrix dimension N. The underlying parameterizations induced by our generalized RY modules spanning orthogonal group can effectively capture a full-rank weight update. This contrasts with A (Zhang et al., 2023), which uses approximate orthog- onality imposed by regularization terms in optimization, failing to reduce the number of trainable parameters being limited by the low-rank decomposition. Consequently, Quantum-PEFF enables magnitude parameter reduction compared to conventional LoRA-based approaches. The theoretical memory requirements of PEFT applied to query/value weights in comparison to LoRA shown in Table and Figure 4 shows tensor diagrams under tensor network interpretation. More discussions of other tensor networks are found in Appendix A.3.
K yields 2 K K2 trainable parameters, and the Pauli parametrization
Computational complexity Regarding computational complexity of computing the mapping, Pauli parameterization Qp has runtime O[ + log2 log2(N) (N)) with the efficient Kronecker Shuffle algo- rithm (Plateau, 1985), while the number of trainable parameters scales logarithmically with N. This shows that, up to small constant factors (e.g.. L can be set to for PEFT with sensitivity analysis in Ap- pendix A.4), the computational complexity remains highly competitive with LoRA's O(2N K) even with the mapping. Contrary to LoRA, the logarithmic scaling of parameters allowed by our construc- tion translates to a substantial reduction in memory footprint, which becomes increasingly important when dealing with very large models. The Taylor mapping Q' has complexity O(2(P + I). yielding a comparable time complexity order with LoRA O(2N at N N", K = K". We remark be used independently. namely QT, generate orthogonal
that the Taylor parameterization matrices from underlying small trainable weights as in Figure 3(a), which is computationally cheap and results in O(2N K²) parameters. This setup is preferred for reduced training time when more memory bandwidth is available, while Qp provides the highest parameter savings.
Quantization To further save memory, we can use standard integer quantization for trainable parameter: 0: 0g round( p)/ +,, jr, where scale value 3 = (0mmx - (2 1) and zero value μ = Omin for bitit quantization. The maximum 0max and minimum values 0min are obtained in chunk of group size g. When the quantization is applied on the Lie parameters, we employ the straight-through trick for quantization-awe detachh)) means gradient passing. Once trained, the required memory will be 32/g bits per Lie parameter when 3 and μ use floating-pott point (FP) 16 bits precision.
training (QAT), i.e., 8 == 0g + 8 8 P.dotach(), (), where
5 EXPERIMENTS
We evaluate our Crantum-PEFT for DeBERTaV3 (He et al., 2023), GPT-2 (Radford et al., 2019), ViT (Dosovitskiy et al., 2020), and Mistral-7B (Zhang et al., 2023) diverse fine-tuning. We fine- tune (1) DeBERTaV3 and Mistral-7B on the General Language Understanding Evaluation (GLUE) benchmark (Wang et al., 2019); (2) GPT-2 Medium on E2E Challenge following the original LoRA paper (Hu et al., 2021); and (3) VIT on CIFARIO 10 (Krizhevsky et al., 2009). Our experiments are not to claim that Quantum-PEFT always improves accuracy w.r.t. LoRA, but show that Quantum-PEFT can maintain competitive level of accuracy with orders-of-mant magnitude fewer parameters.
5.1 GLUE BENCHMARK
The experiment conducted on the GLUE benchmark, which consists of NLP understanding tasks. Our experiment follows the set-up in (Zhang al., 2023). The fine-tuning is applied on DeBERTaV3- base (He et al., 2023). We compare Quantum-PEFT with the following baselines: Full parameters fine-tuning (FT), LoRA (Hu et al., 2021), BitFit (Zaken et al., 2022), adapter tuning with Houlsby adapter (HAdapter) (Houlsby et al., 2019), adapter tuning with Pfeiffer adapter (PAdapter) (Pfeiffer al., 2021), A (Zhang et al., 2023), LoKr (Yeh al., 2024), LoHa (Hao et al., 2022), MORA
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Table 2: Results with V3BERTaV3 base GLUE benchmark. We present the Matthew's correlation for CoLA, the average correlation for STS-B, and the accuracy for other tasks. In each column, the best-performing PEFT approach is highlighted bold and the second best underlined.
# Trainable Parameters
Method
SST-2
CoLA
RTE
MRPC
STS-B
Avg.
Memory
FT
184M
95.63
69.19
83.75
89.46
91.60
85.93
14200x
BitFit
0.1M
94.84
66.96
78.70
87.75
91.35
83.92
7.69
HAdapter
0.61M
95.30
67.87
85.56
89.22
91.30
85.85
46.92
PAdapter
0.60M
95.53
69.48
84.12
89.22
91.52
85.97
46.15x
HAdapter
0.31M
95.41
67.65
83.39
89.25
91.31
85.40
23.85
PAdapter
0.30M
94.72
69.06
84,48
89.71
91.38
85.87
23.08
LoRA
0.33M
94.95
68.71
85.56
89.71
91.68
86.12
2
AdaLoRA
0.32M
95.80
70.04
87.36
90.44
91.63
87.05
x
LoHla
0.33M
95.50
66.52
80.43
89.95
89.46
84.37
25.38
LoKr
0,073M
95.07
69.46
85.20
89.71
90.76
86.04
5.62
MORA
0.49M
95.79
67.13
85.19
89.08
90.13
85,46
37.87x
QuanTA
0.093M
95.30
67.75
84.48
89.22
91.01
85.55
7.15x
Quantum-PEFT
0.013M
95.85
67.85
86.57
90.78
91.06
86.42
1x
Table 3: Results for different adaptation methods on the E2E benchmark and GPT2 Medium model.
Quantum- PEFT achieves similar performance LoRA with 4 times less trainable parameters and better performance than LoKr with same parameters.
# Trainable Parameters
Method
BLEU
NIST
METEOR
ROUGE-L
CIDEr
354.92M
68.2
8.62
46.2
71.0
2.47
LoRA
0.39M
66.88
8.55
45.48
68.40
2.31
AdaLoRA
0.38M
64.64
8.38
43.49
65.90
2.18
LoHa
0.39M
65.03
8.45
43.76
66.54
2.22
LoKr
0.098M
63.90
8.27
42.35
65.22
2.04
Quantum-PESt
0.098M
67.46
8.58
45.02
67.36
2.31
(Jiang et al., 2024b), and Quan'TA (Chen et al., 2024b). We fine-tune the query/key/value projection matrices, the output projection in the attention block, and the weight matrices two-layer MLPs. For all of the baselines, we follow the hyperparameters in (Zhang et al., 2023). For Quantum-PEIE PEFF, we use Qp with L = all tasks. We select the best learning rate by parameters sweep. We conduct five runs with different random seeds and report the mean. We use the same number of training epochs as AdaLoRA. Due to limited computing resources, we focus on tasks with training instances less than 100k, including SST-2, CoLA, RTE, MRPC, and STS-B. Detailed setups are given in Appendix B.
The results are summarized Table 2. We can see that both SST-2 and MRPC tasks, Quantum- PEFT can outperform AR.. On other tasks, Quantum-PPFR PEFT can still achieve comparable performance with other baselines. Notably, Quantum-PEFT only requires 0.013 million parameters, which are 25 times fewer than LoRA.
5.2 E2E BENCIMARK
We fine-tune GPT-2 (Radford et al., 2019) Medium on the common E2E natural language generation
benchmark (Novikova et al., 2017), following the setups of (Hu al., 2021). GPT2-Medium has 354M parameters with 24 transformer layers. The E2E benchmark consists of 42,200 samples for training, 4,600 for validation, and 4,600 for testing. We compare LoRA (Hu et al., 2021),
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Table 4: Efficiency comparison on GPT2-Medium.
Resource
LoRA
AdaLoRA
LoHa
LoKr
Quantum- PEFT
Training Time (ms/batch)
1719.06
1795.91 4.03x
1874.04 4.03x
1790.24
1723.39
Memory Ratio
4.03x
1x
1x
Table 5: Results with Mistral-7B on GLUE Benchmark.
The reported metrics are as in Table 2.
Trainable Parameters
Method
SST-2
CoLA
RTE
MRPC
STS-B
Avg.
LoRA AdaLoRA
3.54M
96.21
68.83
87.46
87.74
90.70
86.
3.54M
96.90
70.38
87.53
89.52
90.96
87.06
Quantum- PEFT
0.758M
96.67
70.61
88.10
89.94
91.63
87.39
AdaLoRA (Zhang et al., 2023), LoKr (Yeh et al., 2024), LoHa (Hyeon-Woo et al., 2022), and full FT with Quantum-PPl PFFFunt the simple independent Taylor parameterization QT. P for efficient computations at larger model size. Full FT results are sourced from prior works (Zi et al., 2023). For fair comparison, we use the same training settings and hardware, i.e., 4 NVIDIA A100 GPUs, for all methods. We train the baselines using the code provided by the respective authors using the peft library from Hugging Face. We apply PEFT to the query and value projection layers in each attention block and use the same number of training epochs, batch size, and LoRA scaling, except different learning rate. Table with all hyperparameters is provided in Appendix B.
Table 3 shows the results for E2E Challenge dataset. Quantum-PEFT's performance is on par or better than LoRA A with approximately times less trainable parameters, and significantly beats LoKr with the same number parameters. For the BLEU metric, our method obtains 0.58 gain compared with LoRA, with comparable results on the other metrics. We report results from the final epoch, whereas Hu al. (2021) presented the best performance observed during training, and used GPUs rather than 1 due to time constraints, which may contribute to the observed variances w.r.t. the reported performance in (Hu et al., 2021). These results demonstrate that Quantum-PPr PEFT can achieve comparable level of accuracy to the baselines while using significantly fewer parameters. In Table 4, we evaluate the training time and memory benefits of our method over LoRA while fine-tuning GPT2-Medium. We find that using Quantum-PEFT results in similar training time to LoRA. W.r.. memory requirements, we observe a 4x reduction in storage with Quantum
Briittu--EFFToooooo PEFT on GPT2-Medium.
5.3 LARGE-SCALE PINE-TUNING
To assess the effectiveness of Quantum-PEFT at larger model scales, we fine-tune the Mistral-7B model (Jiang et al., 2024a). Mistral-7B is a recent language model exhibiting strong performance at its size, outperforming the larger Llama 2-B3B on many benchmarks (Jiang et al., 2024a; Zhang and Pilanci, 2024). We experiment our method with this new language model on the GLUE benchmark for natural language understanding problems.
use the LoRA-related hyperparameters as with the DeBERTaV3 experiments. We use the optimization setups from (Zhang and Pilanci, 2024), where for all methods we use 4-bit quantization, employ Adam optimizer over 5 epochs, and also fine-tune significantly outperforms
the gate projection matrices. Table 5 presents the results. Quantum- the strongest methods on GLUE from Scction 5. on almost all datasets, while using 4.67x fewer trainable parameters than LoRA. This further highlights Duantum-PEFT's parameter efficiency and even superior-than LLoA performance when scaling to billion-scale large language models.
.4 IMAGE CLASSIFICATION BENCHMARK
We evaluate transfer learning task of the VIT model ggooglee/iit-baae-ppathh 6-2224 pre-trained on ImageNet-21k (Deng et al., 2009) towards CIFAR10 dataset (Krizhevsky et al., 2009). Detailed settings are found Appendix B. The base model is frozen after being quantized with 3 bits, and adapters for query and value projections are updated. For PUET, we use Qp parameterization for = 1. Table shows the comparison of full FT, LoRA, and Quantum- applied, the classification accuracy of the original ViT is poor, and
PEFT. When fine-tuning thus fine-tuning is important. Compared to the full FT which requires 95.81M parameters, PEFT can significantly reduce the required number of trainable parameters, especially with our Quantum-

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Published as a conference paper at ICLR 2025
Table 6: Results for ViT transfer learning from ImageNet-21k to CIFARIO. Base ViT 3-bit quantized.
Method
Original
FT
LoRAx=1
LoRAx=4 LoRAK=2
Quantum- PFFT
# Parameters
85,81M 98.05% %
0.037M 98.14%
0.074M 98.30%
0.147M 98.39%
0.007M 98.46% %
Accuracy
76.21%
Table 7: Quantization impact on Lie parameters with Taylor parameterization for ViT transfer learning from ImageNet-2Ik to CIFARIO. I0. Base ViT not quantized.
Quantization
FP32
INT8
INT4
INT3
INT2
INTI
# Bits per parameter
32
8.25
4.25
3.25
2.25
1.25
Accuracy (Uniform Loading)
98.81%
98.79%
98.78%
98.75%
98.67%
97.96%
Accuracy (Adaptive Bit Loading)
98.81
98.78%
98.87 %
98.80
98.77 %
98.64 %
PEFT. For example, Quantum-PEFT has 21-hold fewer parameters than LoRA with rank 4. More importantly, Quantum-PEFT shows superior performance despite the fact of the fewest parameters.
Quantization Table 7 shows the QAT performance with different number of bits per the Lie parameter for Taylor parameterization (QT, K = K' = 4 and P 18). Here, the base ViT model is not quantized, while only adapters are quantized. We use = 128 and FP16 for scale and zero values and μ.. We observe that reducing the precision for the Lie parameterization can gradually degrade. Nevertheless, thanks to QAT, no significant loss can be seen even with 1-bit integer quantization from FP32: i.e., 0.65% degradation. We also evaluate the performance of mixed- peeeedd-reiiion Taylor parameterization. One can see that adaptive bit loading can significantly improve the performance at few-bit quantization regimes. For instance, adaptive 1-bit quantization of Lie parameters has just 0.17% loss from FP32 and 0. 2.28% improvement from uniform 1-bit quantization. This may come from the effective pruning gain. More details of quantization are given in Appendix A.3 and A.S.
Sensitivity analysis of intrinsic rank Our introduced intrinsic rank K' reduce the trainable parameters than the specified rank K, by masking the top K' columns of Lie parameters. Table 8, show the impact of varying K' in the same settings Table on the ViT transfer learning task. Decreasing K' gradually reduces the required number of parameters. While the subspace rank is K 8, the number of parameters can be effectively K' K. The performance degradation from K' = 8 to = 1 is only 0.49%, and more importantly the accuracy is still better than LoRA in Table For example, LoRA with K 1 has an accuracy of 98. 14%. while Quantum- PEFT QT parameterization with K = 8 and K" = 1 has 08.38%, at the comparable number of parameters. It shows the great potential of masking out the Lie parameters while keeping higher subspace rank.
Table 8: Impact of intrinsic rank K' for ViT transfer learning from ImageNet-21k. to CIFARIO.
Intrinsic rank K'
2
4
Parameters
0.037M 98.38%
0.074M 98.52%
0.11
0.147M 98.74%
0.184M 98.63%
0.22 98.79%
0.257M 98.81 %
0.294M 98.87%
Accuracy
98. 76%
CONCLUSIONS
In this work, we introduced Quantum-PEET a novel framework leveraging quantum machine learning principles to achieve extremely parameter-efficient fine-tuning of large pre-trained models. Through reparameterization generalized quantum circuits, Quantum-PEFT represents weight updates using highly compact unitary matrix embeddings. Quantum-PEFT can achieve even lower parameter number than the lowest-rank LoRA; unlike prior low-rank adaptation methods bottlenecked by linear parameter growth, the employed Pauli parametrization seales logarithmically with the model size. unitary node
By use of QSD, language and vision benchmarks validate Quantum-PERt PEFT's excellent capabilities, achieving orders- of of-maagissdes higher compression rates than LoRA while maintaining competitive performance.
non- ooor-porr-of-two dimensions. Our experiments across
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ACKNOWLEDDEMENTS
LIONS-EPFL. was supported by Hasler Foundation Program: Hasler Responsible Al (project number 21043). LIONS-EPFL was sponsored by the Army Research Office under Grant Number W91INF- 24-1-0048. LIONS-EPFL was supported by the Swiss National Science Foundation (SNSF) under grant number 200021 100021_205011. 205011.
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A FURTHER DETAILS AND DISCUSSIONS ON QUANTUM-PEFT
To further elaborate on Quantum-PEFT, we provide the tensor network diagrams in Figure 5 exempli- fying its mechanism w.r.t. other LoRA-based methods.
(TRA(TTD)
N/N'
N/N
N
2NK
KronA WN/N++N
K
LoKr
N
2N'K-NF/N"
LoDA
N/N
N/N"
N'
2NK+
K
AdaLoRA
2NK*
SuperLoRA (TD) Nº
N
K
K
N
Quantum-PEFT (SVD)
2NK-K2
Naro
NPIo
K
LoTR (TTR(
K
NaID
Nar
NRIO
NeIK((2+(D-2))
Quantum-PEFII (PoKO-KO-DKIK+10
Neo
Quantum-PEFT (TTD)
Ne
Nal
NaOo
Quantum-PEFT (TRD)
DNPKK-K(0K+122
Quantum-PEE (STD) (2L+1)log(N)
Na
NºD
NPID
Figure 5: Tensor diagrams of Quantum PEFT and LoRA variants in tensor network perspectives for a matrix size N and rank K. The number of parameters are also present. Circle denotes dense multi-linear tensor node. Slashed open circles denote diagonal node. Half-closed circles denote unitary node. Delay symbols denote nonlinear nodes.
A.1 COMPARISONS OF DIVERSE UNITARY MAPPINGS
Unitary mappings Various methods to generate unitary matrices from skew-symmetric matrices possible. In Section 4.1, we focused on exponential and Taylor mappings. Given skew-symmetric matrix A = B T E IR N'N' N' we can generate a corresponding unitary (orthogonal) matrix, e.g., with exponential mapping, Cayley transform, Householder reflection, Givens rotation and those
variants, respectively. as follows:
QE exp(A), Qc (C= A)-1,
QH
II k=1
(5)
N QG = 11 II GBuk(Bnk), k=1n=k+1
K
P Σ p=0
F
Qr
AP,
QN= = (T+ A)
AP
(6)
p=0
where 9][-] is normalization operator for canonical coset decomposition (CCD) (Cabrera al., 2010). and (0) denotes the Givens matrix which is identity except that the n and (n + 1) 1)-th diagonal block is replaced with RY rotation. The mappings of QT and QN are respectively approximated versions of QE and Qc to avoid matrix exponentiation and inversion via Taylor series and Neumann series approximations up to polynomial order P. Note that Qp, QE and QG are identical to RY at N' 2.
Comparison of unitary mappings Fig. 6 shows the comparison of different unitary mapping methods over different matrix size for a rank K 4. We examined the unitarity test and speed bench on RTX6000 GPU for forward and backward processing. The unitarity error measures averaged to norm of [[QQ - Il]oc over a batch size of 32 and 10 random seeds. The exponential mapping uses torch. linalg matrix exp. and matrix inversion for Cayley transform torch. 1 linalg. solve. We assume = 18 polynomial order for Taylor and Neumann series. was found that Neumann series and exponential mapping become inaccurate the matrix size is
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Exponential
Exponential
Cayley
Cayley
Householder
Householder
Givens
10
Tayfor
Taylor
Neumann
Processing (Uo)
Naumsnn
Paull
109
Rademachar
10
10'7
10
10
10
10
100
1000
10
100
1000
Matrix Size
Matris Size
Figure 6: Unitarity error analysis and speed bench including forward and backward passes for different unitary mapping methods function of matrix size of N for rank of K = 4 NVIDIA RTX6000 GPU 24GB.
increased. While Pauli parameterization has relatively higher error than the rest of methods, can be much faster large matrix size. Householder reflections and Givens rotations had slower behaviors due to sequential nature. Although Rademacher diagonal matrix of [=1} 1 has a low complexity and perfect unitarity (here, we used ReinMax trick), it alone does not cover the Stiefel manifold Vx (N). Overall, Taylor series method showed good trade-off between accuracy and speed. Note that most large foundation models use thousands for matrix size of N per weight. Therefore, the accuracy and speed large matrix size regimes are important. With these trade-offs in mind, the experiments we evaluate the Taylor QT and Pauli Qp parametrizations. where Pauli gives logarithmic number of trainable parameters in the ambient dimension and Taylor shows satisfactory speed for larger models.
A.2 QUANTUM-INSPIRED PEFT MODULES
Generalized measurements As well generalized-Ri -Y gates and CZ gates, introduce gen- eralized measurement module. Although quantum operation is linear, quantum measurement can be nonlinear in general. Hence, motivated from the quantum measurement solve the linearity constraint, we can impose nonlinearity using activation functions. Using log-softmax after squaring corresponds measuring quantum state probability. For our case, such nonlinear activations can be imposed at any mid-circuit operations, In Fig. 5, we introduce a new tensor diagram with delay symbols representing the nonlinear node. Nonlinear mapping can also trainable when using another multi-layer perceptron (MLP) as used in LoDA. Letting f-)) be such nonlinear function,
tensor contraction can done via nonlinear Einstein sum: fout (Σ ( ( for parent tensor nodes {Ql(), ), where rout and frn denote outer nonlinearity and inner nonlinearity, respectively. Note that the nonlinear nodes can only pass the data after tensor contraction from ancestor nodes.
A.3 TENSOR NETWORK IMPLICATION
Fig. 5 shows tensor diagrams for various LoRA variants. Our Quantum-PERt PEFT framework unify them with reduced number of parameters by exploiting trainable orthogonal nodes, trainable diagonal nodes, and trainable nonlinear nodes. As mentioned, LoRA uses 2-mode tensor train decomposition (TTD) which is also known as matrix product state (MPS) tensor network. However, LoRA parameter redundant as any tensor network via TTD can be reformulated into its canonical form having multiple unitary nodes and only one diagonal node, resulting into fewer parameters. LoDA introduced the nonlinear node tensor network. AdaLoRA based on CP decomposition, which has parameter redundant. LoTR extends LoRA towards higher-mode TTD. SuperLoRA uses another tensor network based on higher-oder Tucker decomposition (TD), while nonlinear mapping is optionally introduced. In fact, TTD and TD can be normalized except one node (i.e., the canonical form), and hence our Quantum-PEFT based on the Lie algebra can eliminate the redundant parameters to improve the efficiency for LoRA, LoTR and SuperLoRA. Similarly, framework provides parameter-efficien unitary nodes in most other tensor networks including tensor ring decomposition (TRD), hierarchical Tucker decompostion (HTD) a.k.a. tree tensor network
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SU(M)
VN(
V
N
Truncate
N
K
K
N
N
N K
SU(Nia)
Parameters: NK-K(K-1)/2
Parameters: (3+2))
Quantum-PEFT STD Renormalization for N'=N1/a
Figure 7: STD renormalization step example when N' N//3 The total number of parameters is reduced from 729 180 for unitary node when K 1, N 3 N' 32, 1.
Table 9: Accuracy on the ViT CIFARIO transfer learning task with varying entanglement layers L.
2
3
16
32
Accuracy 96.09%
96.54%
96.71%
96.71%
96.71 1%
96.71% 1%
96.71
(TTN), multi-scale entanglement renormalization ansatz (MERA), and projected entangled pair states (PEPS). As descussed, Pauli parameterization based on STD ansatz can further reduce the number of parameters for those tensor networks into logarithmic scale. Note that STD parameterization can be regarded as a renormalization step of each orthogonal node tensor networks. Fig. 7 shows an example of the STD renormalization step when N 3-folded into N' 1/3 The total number of parameters to represent the unitary node for Vx (N) can be reduced in logarithmic order of logx=((N (N). When K = 1, .N- = 3 N' = 32, = 1, it becomes 180 from 729. Reducing the size of N' can further improve the parameter efficiency.
Table 10 shows an example result for ViT CIFARIO transfer learning task, using Taylor parameteriza- tion (with K K" 4 and P 18) for different tensor networks, including CP, TRD, HTD (TTN), TD, and TTD (MPS). We find that all tensor networks offer competitive performance to LoRA.
A.4 FURTHER ANALYSIS OF ENTANGLEMENT LAYERS
The relationship between circuit depth and entanglement capacity has been explored in quantum information theory, providing insights into the expressive power of quantum circuits. For instance, Sim et al. (2019) analyzed the entangling capabilities of various quantum circuits, establishing connection between the number of layers L and entanglement. It was shown that generally deeper cireuits exhibit increased entanglement capacity, which contribute richer representations in the context of quantum machine learning. In our proposed Quantum-PPF PEFT framework, the number of alternating entanglement layers L in the Pauli parametrization Qp (Equation (2)) governs the circuit depth. Deeper circuits, while potentially more expressive, they also introduce additional trainable parameters and computational overhead. Our empirical findings suggest that circuits with L 1 provides good balance between performance and efficiency for PEFT tasks. Increasing L can lead to moderate performance improvements, but the gains tend diminish with larger values of L,, indicating saturation effect. To further investigate the impact of L, we conducted sensitivity analysis on the ViT CIFAR 10 transfer learning task described Section 5.4 We evaluated the performance of Quantum- PEFT across various values of L, while keeping other hyperparameters fixed and with the base ViT model quantized to 2 bits. The results, summarized in Table 9, demonstrate the saturation behavior, as further gain is attained beyond L = 3. Overall, the optimal value of L is task- dependent, depending on the complexity of the target task. where the trade-off between performance gains and increased computational complexity needs to be carefully considered.
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Table 10: Different tensor network results with Taylor parameterization for VIT transfer learning from ImageNet-21k to CIFARIO. Base VIT is not quantized.
Method
CP
TRD
HTD (TTN)
TD
TTD (MPS)
Parameters
0.074M 98.53%
0.147M 98.14%
0.026M 98.11%
0.074M 98.05%
0.11IM
Accuracy
98.81%
Bit Loading
Allocation Bits 4
QU
O
1, o bit
K
K
3,2
16bits
Range
Neik
N233
Na13
N21
Quantum-PPEi PEFT (TRD)
K
Q
QIII
A
Q
Group
(a) TRD Tensor Network
(b) Mixed-Precision Bit Loading
(c) Adaptive Allocation/Pruning
Figure 8: Mixed-precision Quantum-PEFF in 3-dimensional TRD tensor network. Each tensor node and tensor parameter can have non-uniform bit assignments. Adaptive bit loading depends group range A. Assignment bit corresponds to adaptive structural pruning.
A.5 BROADER IMPACTS AND FUTURE WORK
It is interesting investigate how further reduce the memory for trainable parameters by employing quantization or pruning.
Mixed -eexed-preisioon tensor network One could consider a mixed-precision tensor network, where each tensor node and its parameter group have different precisions. Fig. 8(a) shows example of Quantum-PEFT in 3-dimensional TRD tensor network. The TRD formulated by unitary nodes {Q and diagonal node Specifically the (i, k))th element is given by nonlinear Einstein Q (Q
sum: W fout ( L. fin (Qiii. mm...n An,n Q )). As shown in Fig. 8(b), each node has trainable parameters 0, and we can adaptively assign more bits or fewer bits depending on the group range A, 0g,max Og.min for the i-th group. For example, the bit loading may use the following strategy: q: round(q (A A)) with an average range Á E[4Ø] where q bits are assigned for the i-th group with exponent &. >= O. When x 0, it reduces to uniform bit loading: i.e., % q for all group i. More sophisticated but time-consuming strategy error of the weight matrix min [W, WI, which requires combinatorial optimization.
consider the quantization
When the bit allocation zero (i.e., A; is close to zero) as shown Fig. 8(c), it corresponds to structural pruning except that the masked group can still hold non-zero values Further fine-grained pruning is also possible by nulling out if the value magnitude is smaller than a threshold. Therefore, it can accomplish adaptive rank mechanism similar to AdaLoRA.
Pretrained model compression In fact, Quantum-PEFFT framework can also be applicable to compress the pretrained model before adaptation. Tensor rank decomposition, quantization and pruning can be applied to pretrained model before transfer learning tasks, similar to Q-LoRA, R- LoDA, and S-LoDA. For ViT transfer learning task, we evaluated 3-bit quantization of pre-trained models.
A.6 FURTHER COMPARISONS WITH RELATED WORK
We provide the following remarks elaborating on the difference between the proposed Quantum-PPE PEFT method with recent related works PEFT. Several recent works explore alternative approaches to parameter-efffcien fine-tuning. Quantum-PEES is a new technique for low-rank based PEFT. Some recent works explore alternative approaches to A-Ra-bde fine-tuning. For example, Pan
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et al. (2024) fine-tune some important sampled layers while freezing the rest for some certain iterations, an orthogonal approach to LoRA. Their results show that Pan et al. (2024) share similar memory requirements LoRA; in contrast, Quantum-PEFT achieves significantly higher parameter efficiency than LoRA. Other works adapt the intermediate embeddings learned by the models, which differs from LoRA-based methods that adapt the weights directly. In this sense, Wu et al. (2024) intervene on a low-rank subspace of the intermediate model embeddings rather than model weights. One disadvantage of this method that it creates multiple additional hyperparameters, i.c. prefix and suffix positions to intervene on and which layers to intervene growth of hyperparameters. Therefore, their claimed higher efficiency comes the cost of higher computational burden of hyperparameter tuning for each task. On the other hand, Quantum-PEFT mainly considers two hyperparameters, i.e., the intrinsic rank K" and the number of entanglement layers L. Quantum-PEFT considers highly-efficient unitary quantum paramatrization, which is effectively full-rank. Recent work (Jiang et al., 2024b) has investigated high-rank fine-tuning, employs a learnable square matrix M E RR x for full-rank fine-tuning and compatibility mappings to ensure that the dimensions match with the weight W E RNN They set K = LV(N + M) to achieve the highest rank with square matrix the same total number of trainable parameters of LoRA with rank K. (Jiang et al., 2024b) is therefore not able to have lower-than-LoRA scaling, which is instead achieved by Quantum-PEFT. (Chen et al., 2024b) is a tensor-network- inspired parameterization without consideration of unitary gain, where their parameterization leads to parameter redundancy. Our Pauli parameterization under Lie algebra can strictly maintain unitary constraint without parameter redundancy. In fact, (Chen al., 2024b) is based on tensor folding, requiring in principle that the matrix size is factorizable as d = d1 x d2 x ... x dN. Therefore, it is not readily compatible for arbitrary size. For example, if the matrix size to fold into multiple axis. We clearly provided the way to solve this issue by quantum Shannon decomposition, which enables to decompose into sum of powers-of-two: i.c., N₁ 256 and N₂ 1. Furthermore, Quantum- -EFT provides more general applicability and insight to any arbitrary tensor network to reduce the parameter number as shown Figures 5 and 7, and provides more flexibility with adjustable entangling layer size L, which is not explored in (Chen et al., 2024b).
creating a combinatorial
257, it is difficult
A.7 LIST OF SYMBOLS
For ease of reference, we provide a table of notations used in this work in Table 11.
B DETAILED EXPERIMENTAL SETUPS
B. GLUE BENCHMARK
Below, we provide a summary of the tasks in the GLUE benchmark that are used in this work.
SST-2: stands for The Stanford Sentiment Treebank, a dataset on sentiment analysis tasks with two labels. The size of the training set 67k, and the size of the test set is 1.8k.
CoLA, represents The Corpus of Linguistic Acceptability, a dataset on sentence classification with two labels. It consists of 8.5k training data and Ik test data.
RTB: stands for The Recognizing Textual Entailment, including 2.5k training data points and 3k test data points.
MRPC: represents The Microsoft Research Paraphrase Corpus, a dataset on pairwise text classification with 3.7k training points and 1.7k test points.
STS-B: represents The Semantic Textual Similarity Benchmark, a task on measuring text similarity with 7k training points and 1.4k test points.
We select the same number of epochs for Quantum- PEFT as in AdaLoRA. We perform a hyper- parameters sweep for the learning rate over (0.01. 0.03. 0.06, 0.001, 0003, 0.006). We select the best learning rate and the best checkpoints over each epoch. We present the hyperparameters for Quantum-PEit -FFT in Table 12.
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Table 11: List of symbols.
Notation
Description
(U(N)
Special unitary group of size N Lie algebra of ST)
su(N)
SO(N)
Special orthogonal group of size N
O(N)
Orthogonal group of size N
Vk(N)
Stiefel manifold of K orthonormal frames in R-
IN
Identity matrix of size N
R
Field of real numbers
60
Kronecker product Transpose
D
Imaginary number Submatrix of A with rows
Ammn
A.
k-th column of matrix A
diagi-
Creates diagonal matrix
L
Number of alternating entanglement layers Number of qubits
N
Orthogonal node size Intrinsic rank
K'
P
Taylor expansion order Pauli paaulpaarameterizteeddd unitary matrix Unitary matrix from exponential mapping Unitary matrix from Taylor series expansion Quantum RY rotation gate with angle 0 Quantum controlled-Z. gate gate Pre-trained weight matrix Weight update matrix
QP
Q
O
(Y
CZ
W
AW
U
Left singular vector matrix Right singular vector matrix Diagonal matrix of singular values Lower triangular matrix and its parameter matrix Cosine and sine diagonal matrices from CSD Number of bits for quantization Quantization group size Quantization scale and zero-point Trainable parameter
V
A
B, Bx
C,S
B, 14
Og
Quantized trainable parameter
Table 12: Hyperparameter configurations for Quantum-PEFT on the GLUE benchmark.
Hyperparameter
SST-2
CoLAA
RTE
MRPC
STS-B
# GPUs
1
Optimizer
AdamW Linear
AdamW
AdamW
AdamW
Adam W
Learning Rate Schedule
Linear
Linear
Linear
Linear
Weight Decay
0.01
0.01
0.01
0.01
0.01
Batch Size
256
128
128
128
128
Epochs
24
25
50
30
Warmup ratio
0.1
Max sequence length
128
64
320
320
128
Rank K
3
3
3
a
32
32
32
32
32
Learning Rate
0.006
0.01
0.06
0.01
0.03
Unitary Parametrization
Qp (L = 1)
Qp (L
1)
Qp (L
1)
Qp (L = 1)
Qp (L = 1)
B.2 E2E BENCHMARK
Table 14 lists hyperparameters for the experiment transfer learning task of E2E benchmark.
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Table 14: E2E benchmark for GPT2 Medium.
Table 13: CIFAR-10 transfer learning for ViT.
Hyperparameter
LoRA
Quantum- PEFT
Hyperparameter
LoRA
Quantum-PER PEFT
GPUs
4
# GPUs
Optimizer
AdamW Linear
Adam Linear
Optimizer
AdamW Constant
AdamW Constant
Learning Rate
Learning Rate
Schedule
Schedule
Weight Decay
0.01
0.01
Weight Decay
0.01
Batch Size
8
8
Batch Size
32
Epochs
5
5
Epochs
100
100
Warmup Steps
500
500
Patience
Label Smooth
0.1
0.1
Rank K
1,2,4
1, 4
Rank K
4
(K' 1)
Learning Rate
0.001
0,003
32
32
Unitary
Qp (L 1),
Learning Rate
0.0002
0.002
Parametrization
QT (P 18)
Unitary
QT
Parametrization
B.3 VIT CIFARIO TASK
Table 13 lists hyperparameters for the experiment on transfer learning task of VIT. The base ViT model (google/ wit-base satt-bas--ptcch1 00066-2220 224)* pretrained on ImageNet-21k has 12 layers of multi-head attention modules, each of which has heads, 768 features, and a token length of 769. CIFARIO is an image classification dataset having 10 classes of 32 x 32 colored images with 50k training samples and 10k test samples. We use up-sampling to 224 x 224 resolutions with random resized cropping and horizontal flip. The original classifier head has 1000 class output, and selected 10 outputs based on the prediction score of CIFAR10 training data in prior to PEFT process. All weights and biases of the base ViT model including the classifier head are frozen after being quantized with 3-bit integers via rounding as described in Appendix A.5. Therefore, the base model is compressed from floating-point 32 bits to integer bits (with auxiliary scale and zero values 3 and μ for g 128 group), i.e., from 330MiB to 34MiB storage. It was confirmed that less than 3-bit quantization for the base ViT model compression had poor performance: 56.0% accuracy with bit and 97.4% with bits. The required run-time on GPU A40 40GB was about 3.37 second per iteration, and 5284.16 16 second per epoch.
hnttp:://ugggingaaac..o e/vi ttevvvitbbaaee base bat-b--ptcchhll 1attch1-2222 224
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Reviews:
[{"review_id": "ooKU1tod8L", "reviewer": "Reviewer_nb9o", "summary": "The paper introduces Quantum-PEFT, a quantum-inspired parameter-efficient fine-tuning (PEFT) framework for large language and vision models. Unlike traditional methods such as LoRA, which require low-rank adaptations of model weights, Quantum-PEFT uses quantum unitary parameterizations, resulting in a logarithmic scaling of parameters. Key contributions include the development of quantum-inspired modules using Pauli parametrization, significantly reducing the number of trainable parameters, while retaining competitive performance in various transfer learning benchmarks.", "questions": "See above. It would be great if authors could add some self-contained introduction to the necessary quantum concepts used in the paper.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Quantum-PEFT’s application of quantum unitary parameterizations to PEFT is novel, differentiating it from conventional LoRA models.\n2. It drastically reduces trainable parameters without performance loss and it is very important for the field, especially for resource-constrained training.", "weaknesses": "1. I'm not familiar with quantum ML, so it's a bit hard for me to understand the core concept. I think it's also hard to digest for someone who is not familiar with quantum ML. But this is fine because I don't think the intended audience of this paper includes someone not familiar with quantum ML.", "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": "oY89328lyq", "reviewer": "Reviewer_XPHL", "summary": "This paper introduces Quantum-PEFT, a novel parameter-efficient fine-tuning method that uses quantum-inspired unitary parameterizations. The key innovation is using Pauli parameterization to achieve logarithmic scaling of trainable parameters with matrix dimensions, compared to linear scaling in traditional LoRA methods. The method is evaluated on language and vision tasks, demonstrating comparable performance to LoRA while using significantly fewer parameters.", "questions": "1. How does Eq. 4 work?\n2. What is the optimal number of alternating entanglement layers L for different model scales?\n3. For the quantum Shannon decomposition approach to handle non-power-of-two dimensions, how does the decomposition choice (N₁, N₂) affect model performance?\n4. Does the proposed method have a large computational overhead from the entanglement layers?\n5. How does the proposed method compare with other methods mentioned above? My understanding of the main difference between this method and [4] is the use of unitary and diagonal matrices. Is this correct?\n6. The proposed method focuses on reduction in parameters. However, [2] and [4] claim both reduction in parameter and performance improvement. Does the proposed method use even fewer parameters?\n6. How is the proposed method related to quantum machine learning?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper introduces a parameterization based on quantum circuits that require only (2L+1)log₂(N)-2L parameters, a significant improvement over LoRA's 2NK parameters.\n2. The reduction in parameters is experimentally verified.\n3. The authors proposed generalized RY and CZ quantum gates for arbitrary dimensions beyond power-of-2.\n4. The proposed method seems robust under quantization.", "weaknesses": "1. Analysis of how L layers affect entanglement capacity is limited. There is no evaluation of entanglement entropy between layers.\n\n2. The paper only focuses on benchmarking against LoRA/adapter variants on small models (such as GPT2). Benchmarks on larger (and newer) models such as LLaMA seem to be quite standard and more related to practical use cases. \n\n1. Some of the recently proposed PEFT methods are not adequately addressed/discussed (at least mentioned somewhere in the paper), e.g.\n\n[1] https://arxiv.org/abs/2403.17919 (NeurIPS)\n\n[2] https://arxiv.org/abs/2404.03592 (NeurIPS)\n\n[3] https://arxiv.org/abs/2405.12130 \n\n[4] https://arxiv.org/abs/2406.00132 (NeurIPS)\n\nBoth [3] and [4] seem to be able to model high-rank matrices, and [4] appears to be also based on quantum circuit.", "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": "lUZh1OFVRX", "reviewer": "Reviewer_og4k", "summary": "This paper proposes Quantum-PEFT, a novel parameter-efficient fine-tuning method inspired by quantum computing concepts. The core contribution lies in achieving logarithmic parameter scaling through Pauli parameterization while maintaining orthogonality via Stiefel manifold mapping.", "questions": "* In Figure 5, which provides Intuitive illustrations of idea of Q-PEFT, is relegated to the appendix. Given its importance for understanding the method's foundations, especially for researchers from non-quantum backgrounds, innovations within this figure should be moved to the main text.\n\n* The mathematical notation requires better organization and clearer presentation. A dedicated notation section would significantly improve readability and accessibility.\n\n* What are the practical limitations and deployment considerations for real-world applications?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Novel theoretical framework combining quantum-inspired parameterization with PEFT\n- Significant reduction in parameter count compared to existing methods\n- Comprehensive experiments across multiple tasks and architectures", "weaknesses": "1. The paper lacks clear delineation of learnable parameters in the mathematical formulations. While extensive comparisons with LoRA are provided, the fundamental step of identifying and justifying which parameters are learnable is overlooked. This impedes understanding of the method's core mechanism.\n\n2. While the parameter efficiency is well-demonstrated, the computational complexity analysis is insufficient:\n- The introduction of Stiefel manifold mapping introduces additional computational operations beyond standard matrix arithmetic\n- Limited discussion of practical computational bottlenecks\n- Experimental results show no significant advantage in fine-tuning time efficiency\n- Absence of detailed analysis on computational overhead", "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": "FJfJ3XvYgo", "reviewer": "Reviewer_1BuM", "summary": "This paper proposes a new formulation for low-rank based PEFT, based on quantum mechanical notation. \nThe authors demonstrate that their ansatz using Pauli Parametrization of low-rank adapters provides a superior paramters-accuracy ratio in numerious contemprary benchmarks", "questions": "- How does your factorized update scheme relates to robust Riemannian optimization schemes on Stiefel Manifolds, e.g. as described in \n[1] for Tucker Tensors and in [2,3] for Matrix Factorizations of the Type USV (as in Adalora)?\n\n- Is there an option to make the method rank-adaptive, as e.g. in [1,2,3] or AdaLora?\n\n[1] Emanuele Zangrando, Steffen Schotthöfer, Gianluca Ceruti, Jonas Kusch, and Francesco Tudisco.\nRank-adaptive spectral pruning of convolutional layers during training. In Advances in Neural\nInformation Processing Systems, 2024.\n\n[2] Steffen Schotthöfer, Emanuele Zangrando, Jonas Kusch, Gianluca Ceruti, and Francesco\nTudisco. Low-rank lottery tickets: finding efficient low-rank neural networks via matrix differential equations. In Advances in Neural Information Processing Systems,\n2022. URL https://proceedings.neurips.cc/paper_files/paper/2022/\nfile/7e98b00eeafcdaeb0c5661fb9355be3a-Paper-Conference.pdf.\n\n[3] Steffen Schotthöfer and M. Paul Laiu. Federated dynamical low-rank training with global loss\nconvergence guarantees, 2024. URL https://arxiv.org/abs/2406.17887.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "I find the authors approach of unification of various low-rank adapter notations based upon quantum-mechanical notations interesting. \nThe paper has strong numerical evaluations.", "weaknesses": "While I enjoyed reading most of the paper, a few things are missing in the explanation of the method, causing some neccessary revisions. \n\n- The paper describe the compute ansatz in eq. (2), although the notations is quite heavy for readers that are not familiar with the field. The ansatz governs the compute instructions of the forward pass of the neural network. However, the backward-pass, especially computing the gradient update of the parameters is not covered. Especially with the dense notation, an algorithmic description is nessecary to explain how the gradient update is performed and a discussion of its computational feasibility should be added. \n\n- Updating a tensor-factorization as Tensor-Trains or Tucker Decomposition naively may induces unexpected perils for the low-rank optimization. In particular (in combination with the lacking update scheme), it needs to be discussed if a gradient descend on the Factors of the proposed parametrization indeed decreases the overall loss function of the method.", "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)