QuanTA: Efficient High-Rank Fine-Tuning of LLMs with Quantum-Informed Tensor Adaptation
Authors: Zhuo Chen
Rumen Dangovski
Charlotte Loh
Owen M Dugan
Di Luo
Marin Soljacic
Venue: NeurIPS 2024 poster (2024)
Keywords: LLM
Language Model
PEFT
Finetuning
High Rank
Primary area: natural_language_processing
TL;DR: We propose Quantum-informed Tensor Adaptation (QuanTA), a novel, easy-to-implement, high-rank fine-tuning method with no inference overhead for large-scale pre-trained language models.

Abstract: We propose **Quan**tum-informed **T**ensor **A**daptation (**QuanTA**), a novel, easy-to-implement, fine-tuning method with no inference overhead for large-scale pre-trained language models. By leveraging quantum-inspired methods derived from quantum circuit structures, QuanTA enables efficient *high-rank* fine-tuning, surpassing the limitations of Low-Rank Adaptation (LoRA)---low-rank approximation may fail for complicated downstream tasks. Our approach is theoretically supported by the universality theorem and the rank representation theorem to achieve efficient high-rank adaptations. Experiments demonstrate that QuanTA significantly enhances commonsense reasoning, arithmetic reasoning, and scalability compared to traditional methods. Furthermore, QuanTA shows superior performance with fewer trainable parameters compared to other approaches and can be designed to integrate with existing fine-tuning algorithms for further improvement, providing a scalable and efficient solution for fine-tuning large language models and advancing state-of-the-art in natural language processing.

QuanTA: Efficient High-Rank Fine-Tuning of LLMs with Quantum-Informed Tensor Adaptation
Zhuo Chen
Rumen Dangovski
Charlotte Loh 13
Owen Dugan 12
Di 124 Marin Soljačié
124.
NSF Al Institute for Artificial Intelligence and Fundamental Interactions
DDpaatmment of Physics, Massachusetts Institute of Technology
3 *Department of EECS, Massachusetts Institute of Technology
'Department of Physics, Harvard University
{chenzhuo, rumendd, cloh, odugan, solj,,,ljiii edu
Abstract
We propose Quantum-informed Tensor Adaptation (QuanTA), novel, easy-to- implement, fine-tuning method with no inference overhead for large-scale pre- trained language models. By leveraging quantum-inspired methods derived from quantum circuit structures, QuanTA enables efficient high-rank fine-tuning, sur- passing the limitations of Low-Rank Adaptation (LORA)-lw--tt approximation may fail for complicated downstream tasks. Our approach is theoretically supported by the universality theorem and the rank representation theorem to achieve effi- cient high-rank adaptations. Experiments demonstrate that Quan'TA significantly enhances commonsense reasoning, arithmetic reasoning, and scalability compared to traditional methods. Furthermore, QuanTTA shows superior performance with fewer trainable parameters compared other approaches and can be designed integrate with existing fine-tuning algorithms for further improvement, providing a scalable and efficient solution for fine-tuning large language models and advancing state-of- -ft----r-te--tttt in natural language processing.
Introduction
Pre-trainied large language models (LLMs) have revolutionized natural language processing (NLP) by achieving state-of-the-ar performance across various tasks (12). Traditionally, these models adapted to specific downstream applications via full fine-tuning, where all model parameters are retrained. However, as model sizes increase, the computational cost and memory requirements for full fine-tuning become prohibitive, especially with models like GPT-3 [3] with 175 billion parameters, Mixtral 4 with 8 22 billion parameters, and more recently the LLaMA series (577). containing soon up to 400 billion parameters (8). These constraints have spurred the development of parfieter-efffcieent fine-tuning (PEFT) methods, which aim to adapt LLMs by updating only a small subset of parameters, thereby reducing resource demands
Among PEFT methods, Low-Rank Adaptation (LoRA) 10 has gained prominence due to its simplicity and effectiveness. LoRA fine-tunes LLMs by introducing low-rank matrices into the pre-trained model's weight updates, pragmatically reducing the number of trainable parameters while maintaining performance close to full fine-tuning in many tasks. However, LoRA's reliance on low-rank approximations can sometimes lead to a performance gap compared to full fine-tuning, particularly for complex tasks, as it may not capture all necessary task-specific adaptations (
38th Conference Neural Information Processing Systems (NeurIPS 2024).
"Corresponding author.

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LoRA
Base Model
Reshapa
QuanTA (Ours)
Performant nca
Full
Fine-
LoRA
Tuning
Ti
Bi
Pretrained
Weights
Trainable Parameters
Quantum Circuit Inspiration
Av.i
Wi
Jal
Copy
Roshapp
d2
Universe
Low Rank Limitation
High Rank Representetion
Figure 1: Conceptual comparison of QuanTA and LoRA methods. LoRA parameterizes the weight matrix update outer product of two low-rank matrices, limiting its capacity. QuanTA, inspired by quantum circuits, uses tensors that operate on specific axes of the (reshaped) input, enabling high-rank parameterization. Supported by the universality theorem and rank representation theorem, QuanTA can represent arbitrary matrices effectively, allowing it to achieve performance comparable or sometimes even better than full fine-tuning, with only a fraction of the parameters. Note: the performance graph is conceptual illustration.
Recently, there have been many attempts to generalize LoRA using tensor-based methods [12 13 However, these approaches primarily focus on reducing the number of trainable parameters within the low-rank framework yet they continue to face the same limitations of restricted representation. In Quantum mechanics, quantum circuit provides a natural realization of unitary matrix which is full rank, motivating us to develop new schemes for high-rank fine-tuning.
Inspired by these advancements, we propose Quantum-informed Tensor Adaptation (QuanTA) a novel, easy-to-implement, fine-tuning method with no inference overhead inspired by quantum circuits (Fig. D. QuanTA enables efficient high-rank adaptations by utilizing tensor operations analogous to those in quantum circuits, addressing the limitations inherent in low-rank methods like LoRA.
In summary, our contributions are as follows:
1. We introduce QuanTTA novel, easy-to-implement, PEFT method with no inference over- head inspired by quantum circuits, enabling efficient high-rank fine-tuning without additional inference latency and offering the potential for integration with other existing PEFT methods for further enhancement.
2. We present the universality theorem and the rank representation theorem, theoretically proving that Quan'TA can efficiently parameterize high-rank matrices, overcoming the limitations of low-rank methods.
3. We validate QAan'TAs performance through extensive experiments, demonstrating significant improvements various reasoning tasks and efficiency compared to traditional methods.
2 Related Works
Parameter-Effice Fine-Tuning (PEFT) methods aim to address the computational burdens associated with fine-tuning large-scale models by adjusting a relatively small fraction the total parameters to fit specific downstream task. Roughly speaking, there are three existing categories of PEFT methods:
1. Adapter-based methods. These methods introduce additional trainable modules into the structure of a pre-trained, otherwise frozen, model. These modules can be integrated in various ways: series adapters are interposed between existing layers like attention MLP components [ [4H16), while parallel adapters coexist alongside these components [17). In general, these methods tend to increase the inference load due the extra components that are not readily integrated into the original model weights.
htttp:://githubb.comqquuuuu wttp:/cccc//aaataa fine- wwwaieiwiinnnnnnnieeeeee tuning/uunntaa quanta

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2. Prompt/Prefix-baee methods. These methods employ additional prompts or soft tokens at the beginning of the input sequence, focusing fine-tuning efforts on these newly introduced vector embeddings while maintaining the original model weights static [18 19 However, this approach can suffer from suboptimal performance and increased inference times. In addition, the soft tokens take up space of real tokens and therefore reduce the effective context size available for the model.
3. Reparameteriiation--aan methods. These methods modify the existing weights with some parameter-efficien parameterization during the fine-tuning phase. Among these methods, Low-Rank Adaptation (LoRA) [T0] and its variants, such as DoRA 20 and VeRA [21]. are particularly noteworthy for their widespread adoption and robust performance across various tasks. In addition to LoRA, many other PEFT methods also belong to this category, including more sophisticated approaches such as Hadamard [22), Kronecker product [23] reparameterizations as well as many other methods (2427) Crucially, methods in this category do not impose additional inference burdens after fine-tuning as the modified weights can be merged into the pre-trained model weights prior to deployment.
Besides these three categories, there are additional PEFT methods such LoTA [2], where tensor decompositions are performed across multiple weights, LoRETTA [13,, which uses tensor train decomposition for each weight matrix and has both adapter-base based and reparamnterizzaioo--asss
variants, MPO-based fine-tuning (28), and very recently LISA (29), ReFT [30] and MoRA [1..
Physics-inspired machine learning In parallel, there have been various attempts to integrate physics- based priors into machine learning for many years. Symmetries and physics structure have been incorporated into the neural networks architecture and training in various applications to achieve notable performance (32.9]] Various classical and quantum physics processes have been utilized to design new neural networks 40 41 and generative models 42 48)
3 Motivation: Low Rank is not Always Sufficient
LoRA operates under the hypothesis that parameter updates during fine-tuning exhibit low "intrinsic rank." For a pretrained weight matrix Wo gdxk LoRA parameterizes the weight update as W Wo+ AW Wi + BA, where R and Rdxr are low- low-rank matrices. In this configuration, only and B are trainable, while Wo remains fixed. Consequently, the rank of the weight update AW is limited to r.
RTE Dataset
DROP Dataset
0.75
0.50
0.25
0.00
64
96
128
32
128
Figure 2: Subspace similarities between two LoRA experiments of different ranks (64 and 128) for two datasets. Each point (i,j) rep- resents the subspace similarity between the first right singular vectors of the 64 ex- periment, and the first j right singular vectors of the r 128 experiment. Only points for S j are plotted. DROP dataset has a signifi- cantly high "intrinsic rank" than RTE dataset.
Although the original LoRA paper shows empirical evidence support the low-rank hypothesis, recently it has been found that this hypothesis may still fail for more complex tasks, especially for those that sig- nificantly differ from the pre- trainingg dataset, leading to suboptimal performance [I] 31). To assess the general applicability of the low-rank hypothesis, we examine two datasets varying difficulties: the RTE
dataset (49), a classification task where the model is tasked to verify the correctness of statements, and the DROP dataset [50], generation task where the model performs discrete reasoning over paragraphs. We posit that the RTE dataset is simpler, thus more likely to conform to the low-rank hypothesis, whereas the DROP dataset presents greater challenge.
Model
Accurucy/Fi -Soore (1)
RTE
DROP
1LaMA2m Base
61.0
MaaM22g b LoRA,.54
86.0
LLaMA22a LoRA.ars 85.8
56.2
Table 1: Performance of base and LoRA fine- tuned LLaMA2-7B on RTE [49] and DROP [50] datasets. We use accuracy and Fi-score as the metrics for them respectively.
As shown in Table the LLaMA2-7B model (6 in general can achieve a better score on the RTE dataset than the DROP dataset. In addition, as we increase the rank from 64 to 128, LoRA's performance the RTE dataset remains the same, consistent with
3

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the low-rank hypothesis, while the performance on the DROP dataset improves, suggesting the DROP dataset may require a higher "intrinsic rank."
To further measure the "intrinsic rank" of weight updates for these datasets, we follow the methodol- ogy outlined in [10] and compare the subspace spanned by the right singular vectors of the resulting weight updates between the = 64 and r = 128 experiments. Figure 2 shows the subspace sim- ilarities between the query weight updates of the two ranks at layer 16 for both datasets. In the figure, each point (i,j) represents the subspace similarity between the first i singular vectors of the r 64 experiment and the first j singular vectors of the r 128 experiment. A subspace similarity close to I indicates significant overlap, suggesting that the subspace is crucial for fine-tuning, while similarity close O suggests orthogonality, implying that the vectors represent noise. For the RTE dataset, subspace similarity is large only for very small i values, and quickly decays to 0 for larger i, indicating that fine-tuning on the RTE dataset has a low "intrinsic rank." Conversely, for the DROP dataset, subspace similarity remains large across all 64 singular vectors, demonstrating high "intrinsic rank." " Additional details of subspace similarity and addition data are provide in Appendix [A]
These findings demonstrate the necessity of high-rank fine-tuning in complex tasks, challenging the effectiveness of LoRA. This naturally prompts the following question: How can we design efficient methods to facilitate high-rank updates during fine-tuning?
Preliminary: Quantum Circuit
The behavior of quantum mechanical systems, especially those involving particles with discrete degrees of freedom, is well described by matrix theory. Quantum circuits naturally realize unitary matrices whose sizes grow exponentially with the number of particles, providing potent framework for high-rank representation. Here, we review some fundamental concepts of quantum states and quantum circuits to motivate approach.
Quantum state and vector representation. An N-qubit quantum state - -, th (t) C2 is 2 "dddmesiiiinal complex-valuee vector in Hilbert space, with e/: the components and (a) the basis vectors (similar to e; in vector notation). Since quantum states typically consist of qubits with local dimensions of 2, it is instructive to view the quantum state as multi-dimennional tensor with different indices labeling different qubits: (&b) = .aa..... 1., f2,.... iN), where é1, i2,..., iN is the binary representation of i. This be equivalently viewed as reshaping the quantum state from vector C²*
to a tensor x2x2xxxX2
Quantum circuit and matrix representation. A quantum cir- cuit is unitary matrix U E U(2 C C² x2" that transforms one quantum state into another: (0) = 4 (æ)). These circuits are constructed from smaller unitary matrices known as quantum "gates," which operate on one or two qubits. A one-qubit gate unitary matrix (1)) E U(2*), while a two-qubit gate a unitary matrix (22) UU22 1 These gates are applied to specific qubits as follows:
u
Figure 3: Any unitary matrix can be decomposed into a quantum circuit using one- and two-qubit gates.
( (v) = Eul
Unisn de 1 22,,2,,... ................................ iN)
In
for one-qubit gate applied to qubit n, and
(
(+b)
Σ
(2)
| i2..... iN))
Jm
for a two-qubit gate applied to qubits m and n. (Note that m and n do not need to be consecutive qubits.)
quantum circuit comprises a series of these one- and two-qubit gates {U(a)] applied sequentially to the quantum state:
u (V) II (la) (+))
TTypcallly, quantum circuits and quantum gates are considered within the group SU( However, the groups SU(2*) and L2(2%) differ only by ((1) factor, which does not affect the results presented in this paper.

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Since quantum circuits are unitary, they inherently represent full-rank matrices in snite-dimensional systems.
Universality of quantum circuit. Similar to the universal approximation theorem for neural networks, it has been established that any quantum circuit on N qubits can be decomposed into quantum circuit using only one- and two-qubit gates [51H53], as shown in Figure 3 This is particularly relevant for reparameterization
fine-tuning methods, where we aim parameterize matrix matching the shape of the base model's weight matrix using small number of parameters.
5
Quantum-informed Tensor Adaptation
Since quantum circuits offer an elegant parameterization for large unitary matrices of shape 2 2N, by relaxing the unitarity constraint and allowing for arbitrary local dimensions, we can develop an effective tool for high-rank, parameter-efficeen fine-tuning, Inspired by this, we propose QuanTA: Quantum-informed Tensor Adaptation, which parameterizes the parameter updates in way analo- gous to a quantum circuit.
Construction. To illustrate the construction of QuanTTA we focus on the case of square weight matrices W E Rddd in the main paper and defer the general case Appendix B In addition, we assume the hidden dimension d can be decomposed as d = d1 x d2 x x d This condition is often satisfied for large language models. By reshaping x E _2 vector can be interpreted as a quantum state with N "qudits," with the nth axis corresponding to qudit with local dimension d,..
RR xg -xy the hidden
Similar to a quantum circuit, QuanTA consists of "gates" (or tensors) that apply to only specific axes. Since single-axis gates are subsets of two-axis gates, it suffices to consider parameterizations using only two-axis gates. Let (()) be tensor of shape T(a) Rnn xd (n)) (,(0) that operates the ( th and (h) axes with corresponding dimensions dmto) and drto). Analogous to applying a two-qubit gate to a quantum state, applying this tensor to the hidden vector is defined as
(T(() r)
where the a labels are dropped for simplicity, but it should be noted that different T'a)'s can be defined on different axes. Equivalently, this operation can be viewed matrix-vector multiplication with all but the th and (²) axes created as batch dimensions.
QuanTTA is then constructed by sequentially applying a collection of such tensors {T(a)] in the same manner as a quantum circuit:
Tx = T (a)x.
(5)
Although it is difficult to write the full Eq. (5) in index notation for an arbitrary set of tensors, we demonstrate in Appendix G that the einsum expression for this operation can be systematically generated.
As concrete example of translating Eq. 5 to index notations and einsum, consider the case of N 3; (T(a)) consists of three tensors, each applied to two axes (as depicted in Fig. I.. In this case, it is easy to express in index notation the application of the QuanTA operator to the hidden vector;
(Tx)
Σ
(()))
Σ
(2)
Σnhh ( Kathaa Ja.de
(6)
k2.82
J1.ka
f2.J0
as well as the calculation of the full QuanTA matrix;
Tas Testaakadidaad
Σ
(2) Enhe Aa Ractaddss
(3)
by k1
kg
Although Eq. 6 and 7 may look complex in their formulation, in practice they can be easily
implemented respectively using einsum
Each dn does need be prime and the decomposition does not need to be unique
5

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abb torch. einsum (". abc, efbc, gihde...gg...... ...ie.....hhi,, T_3, T_2, T_1)
and
torch. einsum einsum("efbe, ("efbe,dian) seeett,,dii.,ggddd-ggiaaaae ghde-2ghiab"", T_3, T_2, T_1)
Initialization method. At initialization, the adapted model should be the same the base model and all the weight updates should be O. However, enforcing Tx = 0 requires setting one or more T(a) = O, impeding gradient propagation through the tensors and negatively impacting training performance.
To address this issue, we use another set of tensors {{S(a) (with the corresponding QuanTA operator S) that are initialized to the same value as (T(a)} } but remain frozen throughout fine-tuning. We then define the adapted layer as
Wox Wax Tex: Sx.
(8)
where the subscript 0 to denote tranable paraemters. At initialization, the terms Tex and -S exactly cancel out, ensuring the adapted layer reduces to the base model.
It is important to note that this initialization method does not introduce additional costs. After initialization, the full matrix can be explicitly constructed, allowing us to redefine W = Wo + S and simplify the adapted layer to
y Wex W/2 + Toz.
(9)
Theoretical Results
Here, we list few important theorem and provide the proofs in Appendix
Theorem 6.1 (Universality of QuanTA), Let W be arbitrary matrix of shape 2 x 2 For any collection of local dimensions {dn} such that each da is power of and II, dn 2 it is always possible to decompose into a finite sequence of tensors (T(a),, where each tensor applies on two axes with local dimensions ditta and dytas.
We note that the fine-tuning method KronA considered special case of QuanTA.
23 can be incorporated into our framework and
Theorem 6.2 (Rank representation). Let R r(T) be the rank of the full QuanTA operator, R(a) = r(T(m)) be the rank of individual tensors, d be the total dimension of d(a) dmnt dats) be the total dimension of the individual tensor T(a), and NT be the total number of tensors. The following inequality always holds
(R(a) d(a) d(Nr R < min
dR a d(a)
Σ
(10)
In the special case when all the tensors are full rank (Rt = d(a) for all a), the full QuanTA operator also full rank (R d).
Theorem 6.3 (Composition openness). There exists set = {M&} matrices generated from fixed QuanTA structure and two matrices M, E S such that M, & S.
We note that the composition openness condition because, for any two low-rank matrices of the same rank, their composition remains of the same rank. While LoRA may mitigate this limitation by introducing nonlinearity, constrained by closure under composition. In contrast, Quan'TA satisfies the composition openness condition even in the absence of nonlinearity, which suggests that its expressivity can continue to grow as the depth of the neural network increases, even if the network is nearly linear.
satisfied by low-rank matrix decomposition
expressivity still

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averaged No inference overhead. As reparameteriiztiio-buss methods, QuanTA does not impose any inference latency, since the trained T operator can explicitly constructed as matrix and merged into the base model weight matrix.
Memory and computational complexity during fine-tuning. In the forward pass, only hidden vector of size d is kept in the memory as we sequentially apply the tensors to it. Each tensor operation can be viewed as a batched matrix-vector multiplication and has computational complexity of d dmdd for tensor applying on the mth and nth axes, so the total computational complexity for
QuanTA layer is d E. am(s) dn(o)) In addition, each tensor contains (dmd elements. Therefore, each Quan'TA layer contains Ea ((a) trainable parameters that need be stored in the optimizer. As an illustrative example, suppose dmm d'/N for all m and there is one tensor for every two axes, the computational complexity be simplified to N(N - 1)/2 dd*2//N, and the parameter count becomes N(N 1)/2 d'/N When N 2, QuanTA reduces full fine-tuning.
Experiments
To benchmark QuanTAA against other fine-tuning methods, we performed experiments on a wide range of datasets (see Appendix D for details). For all experiments, hyperparameters on the test set. Instead, we create a validation set from the train set and optimize the hyperparameterrs the validation set. All the results reported in this section multiple experiments with varying random seeds, and the term "parameters" and "# params" this section always refer to the trainable parameters. Details the experiments and hyperparameters shown in Appendix
avoid optimizing the
Model
PEFT Method
# Params (%)
F, Score (1)
FT
100%
59.4
Series
0.747%
58.8
60
Parallel
0,747%
59.0
LoRA
0.062%
$4.0
58
LLaMM22
LoRA,wa2
0.249%
54.8
LoRA,1125
0.996%
56.2
56
Quan CaunTAp A1 (Ours)
0.041%
59.5
QuanTA (Ours)
QuanTArsausa (Ours)
0.261%
59.6
54
LoRA
LoRA,-s
0.050%
61.0
Series
QuanTAnsssss (Ours)
0.029%
69.0
52
Parallel
LLaMA2ne
LoRAcus
0.024%
74.3
10-4
10-3
10-2
10-1
10
Quan 16 (Ours)
0.014%
79.4
Params (%)
Table 2: Benchmark of various fine-tuning methods the DROP dataset using LLaMA2 7-70 billion parame- ter models the base model. In each case, report the average of Fi score over 2-4 experiments with dif- ferent random seeds.
Benchmark of different fine- tuning methods on the DROP dataset as a function of training parameters using LLaMA2 7 billion parameter model as the base model.
Figure 4:
DROP Dataset. We begin benchmark with the DROP dataset [50]. chosen representative example that requires high-rank fine-tuning. In Table we compare our QuanTA method with LoRA of different ranks, as well as series and parallel adapters, by fine-tuning LLaMA2 6 with up to 70 billion parameters.
As shown in Table 2 LoRA consistently underperforms compared to other fine-tuning methods. While increasing the rank improves performance, LoRA still falls short, suggesting the necessity of high-rank fine-tuning for this task. In addition, QuanTA achieves performance on par with, or better than, full fine-tuning using only a small fraction of the parameters, demonstrating the effectiveness of Quan high-rank fine-tuning capability.
To investigate how these methods scale with the number of trainable parameters, we conducted experiments varying the number of trainable parameters LLaMA2-7B model. The results
7

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Accuracy (*)
Model
PEFT Method
Parums (%)
BoolQ
PIQA
SIQA
HellaS,
WinoG,
ARC-e
ARC-c
OBQA
Avg.
GPT-3129
60.5
81.0
78.9
70.2
68.8
51.4
57.6
PaLMsue
82.3
81.1
76.6
53.0
53.4
ChatGPT
73.1
85.4
68.5
78.5
66.1
89.8
79.9
74.8
77.0
FT
100%
71.3
82.1
78.6
90.2
79.0
82.9
67.2
76.8
78.5
Prefix
0.11%
76.8
73.9
42.1
72.1
72.9
54.0
60.6
Series
0.99%
63.0
79.2
76.3
67.9
75.7
74.5
57.1
72.4
70.8
Paralle
3.54%
67.9
76.4
78.8
69.8
78.9
73.7
57.3
75.2
72.3
LLaMAgg
LoRA*
0.83%
80.7
77.4
78.1
78.8
77.8
61.3
74.8
74.7
DoRA
0.43%
70.0
82.6
79.7
83.2
80.6
80.6
65.4
77.6
77.5
DoRA
0.84%
69.7
83.4
78.6
87.2
81.0
81.9
66.2
79.2
78.4
QuanTTA (Ours)
0.041%
Prefix
0.03%
65.3
75.4
72.1
55.2
68.6
79.5
62.9
68.0
68.4
Seriex*
0.80%
71.8
83.0
79.2
88.1
82.4
82.5
67.3
81.8
79.5
Puralle]
2.89%
84.8
79.8
92.1
84.7
84.2
71.2
82.4
LLaMApa
LoRA*
0.67%
72.1
83.5
80.5
90.5
83.7
82.8
68.3
82.4
80.5
DoRA
0.35%
72.5
85.3
79.9
90.1
82.9
82.7
69.7
83.6
80.8
DoRAT
0.68%
QuanTA (Ours)
0.029%
73.2
85.4
82.1
93.4
85.1
87.8
73.3
84.4
83.1
FT
100%
72.9
83.0
79.8
92.4
83.0
86.6
72.0
80.1
81.2
LoRA!
0.83%
LLaM
DoRA
0.43%
72.0
83.1
79.9
89.1
83.0
84.5
71.0
81.2
80.5
DoRA
0.84%
71.8
83.7
76.0
89.1
82.6
83.7
68.2
82.4
79.7
Quan'TA (Ours)
0.041%
LoRA
0.67%
73.3
85.6
80.8
91.6
85.5
84.2
73.7
83.3
82.3
LLaMA2 QuanTA (Ours)
0.029%
75.8
86.9
81,2
94.4
87.0
89.6
77.9
85,2
84.8
LLaMA3ga
LoRA
0.70%
85.2
79.9
91.7
84.3
84.2
71.2
79.0
DoRA
0.35%
74.5
88.8
80.3
95.5
84.7
90.1
79.1
87.2
85.0
DoRA
0.71%
74.6
89.3
79.9
95.5
85.6
90.5
80.4
85.8
85.2
Quan'TA (Ours)
0.035%
87.3
Table 3: Benchmark on various commonsense reasoning tasks. All results of models and PEFT
methods labeled with *** are from (54), and results with *t" are from [20]
shown in Fig. 4 Each point in the figure represents an average of four experiments with different random seeds, and the standard deviation across these experiments is shown as error bars
As illustrated in the figure, QuanTA achieves performance comparable using a small fraction of trainable parameters. Conversely, LoRA only achieves subpar performance with small number of trainable parameters, though its performance improves with an increase in parameters. Other PEFT methods, such as series and parallel adapters, achieve results close to full fine-tuning but significantly
better than full fine-tuning
parameters than Quan'TA.
We continue to evaluate our method on a collection of commonsense reasoning datasets. Following the methodology in [54], first fine-tune the model the COM- MONSENSE/70K dataset [54), a comprehensive collection of commonsense reasoning questions, and subsequently evaluate it eight different downstream tasks.
Commonsense Reasoning.
In Table we benchmark our QuanTA method against other fine-tuning techniques using and 13 13-billion-parameter I.LaMA and LLaMA2 models, as well as the 8-billion-pares parameter LLaMA3 model. Alongside prefix tuning, adapter methods, and LoRA, we also compare our approach to the recently proposed LoRA variant, the DoRA method (20] The results clearly indicate that our QuanTA method outperforms LoRA in all cases and surpasses the DoRA method in most benchmarks, using less than one-tenth of the parameters.
Arithmetic Reasoning. We further test our method on arithmetic reasoning tasks by fine-tuning the model on MATH10K dataset [54] and assessing its performance on four tasks. We note that while (54] includes additional downstream tasks the arithmetic reasoning benchmark, some test data was later found to have leaked into the training dataset. In this study, we only benchmark the four downstream
*e vary both the random seed for model initialization and the sampled train, dev, and test datasets, which could be the reason of large standard deviations.

---

Accuracy (1)
Model
PEFT Method
# Params (%)
AQuA
GSMSK
MAWPS
SVAMP
Avg. W/O AQuA
38,9
56.4
87.4
69.6
71.1
LLaMA21B
FT
100%
19.3
65.2
92.0
80.7
79.3
LoRA
0.83%
17.5
65.7
91.2
80.8
79,6
QuanTA (Ours)
0.19%
J6.7
67.0
94.3
80.3
80.5
LoRA
0.67%
16.7
72.3
90.8
84.3
82.5
LL
QuanTA (Ours)
0.13%
18.9
72.4
94.5
84,8
83,9
Table 4: Benchmark on various arithmetic reasoning tasks. GPT-3.5 (labeled with "**) results are taken from 54
tasks unaffected by this data leakage. Additionally, evaluation procedure differs slightly from that in [54] (see Appendix B for details).
Table 4 presents the evaluation results on the four downstream tasks. Notably, all questions in the AQuA dataset are multiple-choice with mostly five options, and all models except GPT-3.5 failed to achieve accuracy higher than 20%. Therefore, we conclude that all models perform equally poorly this task and exclude it from the average accuracy computation. This phenomenon is also consistent with previous findings [54] 20) The results show that Quan TA significantly outperforms LoRA and even surpasses full fine-tuning with small number of parameters. It is surprising that QuanTT exceeds full fine-tuning in these tasks, which may be due overfitting the challenges of optimizing hyperparameters for full fine-tuning.
In Appendix include benchmarks with additional fine-tuning methods and additional datasets.
Limitations. QuanTA currently requires applying the tensors sequentially to the hidden vectors, which may result in underutilizing the GPU when the tensors are too small. will be helpful to develop a more efficient implementation to fully utilize GPU resources. The hyperparameters in QuanTA, such the number of tensors applying on the same axes, have not been optimized. Choosing an optimal set of tensors could further enhance the performance of QuanTTA. In the current experiments, we only consider LLaMA model series and thorough study on different models will be beneficial if more computational resources are available.
Conclusion
In this paper, we introduced QuanTAA a novel, easy-to-implement, PEFT method with no inference overhead for large language models. QuanTA leverages quantum-inspiree techniques to achieve high- rank adaptations, addressing the limitations of existing low-rank methods. QuanTA introduces high- rank fine-tuning through the universality theorem and rank representation theorem. Our extensive experiments demonstrate the efficacy of QuanTA across various tasks, including commonsense reasoning, arithmetic reasoning. and scalability. QuanTA consistently outperforms traditional fine- tuning methods and other PEFT approaches, achieving superior performance with significantly smaller number of trainable parameters. This highlights the potential of quantum- -nnioummd--- techniques in enhancing the adaptability and efficiency of large language models.
QuanTA offers scalable and efficient solution for fine-tuning large language models, advancing the state-of-the-art in natural language processing. There are several promising directions for future research and development of Qua Expanding its application to wider range of tasks and specialized domains could demonstrate its versatility and robustness. Combining QuanTA with other PEFT methods incorporating it into ensemble models might further enhance performance, particularly for complex tasks. The parameter efficiency of QuanTA may also imply a lower chance of overfitting. Additionally, exploring advanced optimization techniques tailored specifically for Quan TA could improve convergence rates and overall efficiency. Further design based on principles from quantum computing, such as entanglement and superposition, may lead to even more efficient fine-tuning methods. Our work paves the way for further exploration of quantum-informed methods or even future quantum technologies for machine learning, making a valuable approach for both research and practical applications with broader impacts.

---

Broader Impacts
The development of QuanTA represents an important advancement in the fine-tuning of LLMs, with profound societal implications. By leveraging quantum-informe methods, QuanTA reduces computational and memory demands, making advanced NLP capabilities more accessible and cost- effective. This democratization of Al technology can facilitate its adoption in essoucce-cnstraingg environments, bridging technological disparities. Additionally. the integration of quantum techniques could spark interdisciplinary innovations, enhancing healthcare diagnostics, financial risk assessment, and personalized education. Furthermore, QuanTTA's efficiency aligns with global sustainability efforts by reducing the energy consumption associated with Al training, contributing to the reduction of Al's carbon footprint. Thus, Quan'TA not only advances NL.P but also promotes inclusive, sustainable, and impactful AI technologies across various sectors. However, the deployment of such powerful AI models raises concerns about data privacy, security, and the potential misuse of Al technologies. Addressing these ethical and societal challenges is crucial to ensure that the benefits of QuanTA are realized responsibly and equitably.
Acknowledgements
The authors acknowledge support from the National Science Foundation under Cooperative Agree- ment PHY-2019786 (The NSF Al Institute for Artificial Intelligence and Fundamental Interactions, http://iaifi. org/). This material is based upon work supported by the U.S. Department of Energy, Office of Science National Quantum Information Science Research Centers, Co-design Center for Quantum Advantage (C2QA) under contract number DE-SC0012704. The research was sponsored by the United States Air Force Research Laboratory and the Department of the Air Force Artificial Intelligence Accelerator and was accomplished under Cooperative Agreement Number 118750-19-2-1000 The computations in this paper were run on the FASRC cluster supported by the FAS Division of Science Research Computing Group at Harvard University.
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Appendix
A
Additional Details on Subspace Similarity
In the main paper, we use the subspace similarity to measure the "intrinsic rank" of fine-tuning specific dataset. In this section, we provide more details on
While it is tempting to compute the rank by performing singular value decomposition (SVD) on the weight matrices of fully fine-tuned models, such measurement generally overestimates the intrinsic rank due to random parameter updates during fine-tuning that are irrelevant to the performance on the downstream tasks. The authors of [TO] proposes better way to measure the intrinsic rank, which describe as follows.
First, LoRA fine-tuning for two different ranks 71 and 72 and obtain the LoRA weight updates AW = ((())((( (mr) and AW(())) B(r2), ((r2), Then, we perform singular value decompositions on the weights to obtain (W = (()) Next, let's denote the first right singular vectors of (w(r) (first i columns of ((ri) as (+ and the first j right singular vectors of AW(r=> (first j columns of V(+z) as denoted as V(rzs), The subspace similarity between these two subspace is defined as
|(nes) T v(ras) 2
((11,22,,,))
[0,
(A.1)
min(i,5)
This function equals if any subspace can be contained in the other, equals if the two subspaces are orthogonal, and in general measures the overlap between and between the two subspaces. FOr fine-tuning that has low "intrinsic rank"; only the subspace spanned by the first few singular vectors (that correspond to the largest few singular vectors) should be similar, with the rest nearly perpendicular originating from random noise during fine-tuning. Thus, the subspace similarity should be close to 1 only when either or is small, and quickly decays to 0 for large and j. On the other hand, when the "intrinsic rank" is high, all the singular vectors one subspace can be important and therefore would appear in the other. In this case, the subspace similarity can remain high for all values of and j.
In the main paper, we choose the RTE dataset 49 and the DROP dataset [50]. and reported the values corresponding the query weight matrix of the 16th layer. In this section, we include results corresponding to additional weight matrices. In Fig. and A.2 we show the subspace similarities for the value weight matrix at layer 16 and 23. We observe that the similar behaviors appear for these two weight matrices as in the main paper, where the RTE dataset exhibits low "intrinsic-rank", while the DROP dataset has a high "intrinsic-rank".
64 and 128, and measure the subspace similarity for both
RTE Dataset
DROP Dataset
1.00
0.50
0.25
64
0.00
32
64
96
128
0
32
64
96
128
Figure A.I: Subspace similarities between two LoRA experiments of different ranks (64 and 128) for two datasets for the value weight matrix at layer 16.
15

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RTE Dataset
DROP Dataset
1.00
0.75
32
0.50
0.25
32
64
96
128
0
32
64
96
128
Figure A.2: Subspace similarities between two LoRA experiments of different ranks (64 and 128) for two datasets for the value weight matrix at layer 23.
B Constructing General QuanTA Operators
In the main paper, we discussed QuanTA operation when the layer weight is square Wo e Rdxd and we assumed d to be composite number that can be decomposed into d = d1 x d2 x ... x dy. Here, we will consider more general cases.
Let's first consider general rectangular matrices Wo E Rd but still assume d and k to permit decompositions in the form d = dj d2 x dy and i = k; k2 ... KN, without loss of generality, let's assume d 2 k (transpose the matrix if d < k). In addition, assume and & has simple ratio and let di /k; d/k: Note that this requirement may seem strict, but in many cases, can be as simple as a multiple k. (For example, LLaMA2-70B model contains many 1024 x 8192 and 8192 1024 weight matrices, in which case d/k 8/1). Then, we can set dn k, for all 1. Among the collection of tensors, let's assume 7 Rdidd xkıd, (d,, k;,) is "rectangular" tensor that applies on the first and nihh axes. After applying this tensor, the hidden vector changes shape from R xd2 xxxxxxxxxx. xdy to R x xdxx making the hidden vector into the correct size. Then, one just needs to make sure that all the tensors subsequent to this tensor needs to have the correct shape when applying the first axis. pictorial representation is shown in Fig. B.3
Figure B.3: Illustration of how to parameterize rectangular matrix with QuanTA
In the special case where the "rectangular" tensor is the last tensor applied to the hidden vector, this operation can be equivalently written as applying "square" tensor of shape Rdid, x th, resulting in an output hidden vector of shape E TRd. Xd2 X-- z IRd and then slicing the first & elements from it. In the case when < k, this operation needs to be reversed, and it can be expressed as first padding the hidden vector to appropriate size, before applying the tensor circuit.
More generally, it is possible to choose any set of {dr j and {kn J (where there products may or may not equal to d and k), as well as any of tensors that transforms [Rd Xdg x.. xdy - R& x x.. Kky,
Then, one can always truncate or pad the input vector of size d to length II, dn. and the output vector from size II kn k. We note that while this method will always work, it recommended choose J and ( such that there products are close d and k. achieve the best performance and avoid unnecessary cost. We furhter note that having d * II d, and A + IIN Ain still allows the full QuanTA operator T to be merged into the original weight matrix, by padding and truncating the T into ædxk
16

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C Additional Theoretical Results and Proofs
In the main paper, we have listed a few important theorems. In this section, we provide the proof for these theorems. We will first need the following lemma, which is fundamental to modern quantum computation. The proof of this lemma can be found in any modern quantum computation textbook such as Ref. [53] or in Ref. 56].
Lemma C.1 (Universality of two-qubit gates), Any 2M x 2M unitary matrix can be written as quantum circuit using just two-qubit gates of size x 4.
This immediately gives us the following corollary. Corollary C.1. Any II, dn x II du unitary matrix with d, a power of 2 can be written as a quantum circuit using two-qudit gates of size d dn dn
Proof. It is possible to reshape the matrix into 2M 2M and use Lemma C.1 to obtain a two-qubit gates representation. Since two-qubit gates subset of two-qudit gates, this already concludes the proof. However, one can group the two-qubit gates that apply to the same qudit to reduce reduce the gate count.
We also need another lemma from quantum computation 53
Lemma C.2 (Phase Rotation). For any diagonal unitary matrix of size 2M x 2M (whose elements are in the form of ei0a), there exists a finite sequence of two-qubit gates with parameters {0k). where the structure of the sequence is fixed for all possible set of { and each two-qubit gate is an analytic function of {0x}. that can exactly represent the diagonal matrix.
Similar to Corollary C.I Lemma C.2 can also be extended to
Corollary C.2. For any diagonal unitary matrix of size IIn dn x II dn with dn a power of there always exists a finite sequence of two-qudit gates with parameters {0x}. where the structure of the sequence is fixed all possible set of {0x} } and each two- ww-qqudit gate is an analytic function of (8x) that can exactly represent the diagonal matrix.
We can analytically continue Corollary
C.2 to nonunitary diagonal matrices.
Corollary C.3. For any diagonal matrix of size IIn dn IIn dn with dn a power of 2 with diagonal elements {ayh, there exists a finite sequence of two-qudit gates with parameters {ax); where the structure of the sequence is fixed for all possible set of {0x}. that can exactly represent the diagonal matrix.
Proof. Consider Corollary since both the full unitary matrix and the sequence of two-qudit gates are finite, and are analytic functions of {6x}, } their analytic continuation must also be equal. Therefore, setting 2₂ 's be imaginary numbers concludes the proof.
Now, we are finally ready to prove the universality theorem.
Theorem C.1 (Universality of QuanTA), Let W be arbitrary matrix of shupe 2 x 2 For any collection of local dimensions {da} such that each dh is power of 2 and II dn = 2M, it is always possible to decompose W into a finite sequence of tensors (Tla),, where each tensor applies on two axes with local dimensions dimta) and dagas.
Proof. Let U, S and be the singular value decomposition of W. Since U and V are unitary matrices, it immediately follows from Corollary that they can be written as a finite sequence of tensors. In addition, since S is a diagonal matrix, Corollary C.3 shows that it can also be written as a finite sequence of tensors. Combining all tensors into the same QuanTA operator by applying the sequentially, we obtain the full Quan T'A decomposition of W.
L
Theorem C.2 (Rank representation). Let R r(T) be the rank of the full QuanTA operator, R(a) = =r(((a)) be the rank of individual tensors, d be the total dimension Tf d(a) dmmto) be the total dimension of the individual tensor T(a), and Nr be the total number of tensors. The
following inequality always holds
Σ d.((a)
d(a) - d(Ny 1) S R min (R
(C.2)
((a)
17

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Proof. The rank of the product of two matrices A and B of shape d x d satisfies the inequality r(A) +r(B)
r(AB) I min(r(A),r(B)) In QuanTA, each tensor can be viewed large matrix, where T( applied on the m and n axes, and identity matrix is applied to the rest of the axes. In this case, the rank of this operation is the same the rank of T(a) times the rank of the product of the identity matrices, which equals dR times concludes our proof.
Then, using the above inequality multiple
dimy
Theorem C.3 (Composition openness). There exists set {Ma} of matrices generated from fixed QuanTA structure and two matrices M1;Ma E S such that M1M2 g S.
Proof. We consider set of matrices & generated by the QuanTA structure that consists of one layer of single-qubit rotation gates followed by a layer of two-qubit CNOT gates [53) and then one layer of single-qubit rotation gates This is set of unitary matrices with entanglement generation determined by the number of layers of CNOT gates. Consider M2 E & according to quantum information theory, it is not possible to have M1 S. This is because M2 has two layers of CNOT gates which can generate more entanglement than any element M3 E that only contains one layer of CNOT gates.
D Datasets
Dataset
Task
Train
# Val
# #Test
Metric
Answer
DROP
Reading comprehension with discrete reasoming
1200
Fi-Score
Phruse
COMMONSENSE1 770K
54
Commonsense reasoning (Train)
170020
400
BoolQ
57
Commonsense reasoning (Test)
3270
Accuracy
Yes/No
PIQA
58
Commonsense reasoning (Test)
1838
Accuracy
Option
SIQA
59
Commonsense reasoning (Test)
508
Accuracy
Option
HellaSwag
60
Commonsense reasoning (Test)
10042
Accuracy
Option
WinoGrande
6
Commonsense reasoning (Test)
1267
Accuracy
Option
ARC-Easy
1
Commonsense reasoning (Test)
2376
Accuracy
Option
ARC-Challenge
Commonsense reasoning (Test)
1172
Accuracy
Option
OBQA 63
Commonsense reasoning (Test)
500
Accuracy
Option
MATH10K [4
Arithmetic reasoning (Train)
9519
400
AQuA 64
Arithmetic reasoning (Test)
254
Accuracy
Option
GSM8K
Arithmetic reasoning (Test)
1319
Accuracy
Number
MAWPS 66
Arithmetic reasoning (Test)
238
Accuracy
Number
SVAMP
Arithmetic reasoning (Test)
1000
Accuracy
Number
Table D.1: List of datasets used in this work.
In this section, describe the datasets used in this paper. In Table D.I the list of datasets used in this paper is listed.
For the DROP dataset 500 we subsample 2000 samples from the original train set as our train set, 800 samples from the train set as our validation set, and 1200 samples from the original validation set our test set, since the original dataset does not contain test set on Hugging Face. In addition, the Fi-score used to measure the closeness of the models' output compared to the ground truth since in general phrase.
For all commonsense reasoning tasks, we first fine-tune single model on the COMMONSENSE 70K dataset collected by [54], and evaluate the same model on eight different commonsense reasoning tasks [37 which we use the version provided by [54] The COMMONSENSEI 70K dataset is split
into train set of 1 170020 samples, and validation set of 400 samples. All of the commonsense
reasoning tasks are either Yes/No questions or multiple choice questions. these tasks, the model
asked to choose the best answer from all the options, and accuracy is used as the evaluation metric.
For all arithmetic reasoning tasks, we fine-tune a single model on the MАTIIОк dataset [54] and evaluate the same model on four different tasks 644 67]
We split the MATII dataset into a train
Note that the single-qube gates can be absorbed into the two-qubit gates, so this structure is within our QuanTA TA framework.
18

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set of 9519 samples, and a validation set of 400 samples. Similar to the commonsense reasoning tasks, we use the version the datasets provided by [34. In addition. in [54] there was found some data leak issues in some of the arithmetic datasets. Here, we only consider the datasets that are unaffected. In the arithmetic reasoning tasks, although the model is asked to generate the step-by-step solution for the final answer, only the final answer is parsed to measure the accuracy. For AQuA, we parse the output text to find the last character such that is one of the options. For the other three tasks which require numerical answers, simply parse the last number from the output text, and consider the answer to be correct it is the same as the ground truth for up to decimal places.
Hyperparameters and Experimental Details
In this section, describe the hyperparameter choices and the experimental details. All the experiments are conducted on NVIDIA A 100 GPUs with 80 GB memory. GPU count used in each experiment will be explained later. The code used produce the experiments is released GitHub at htttt:::///gt //git/ub..h...... com/ quanta- www--ffin-tttnnee /quntta. Our code is implemented using (54] and [68] as references.
E.1 Quan'TA parameterization
Although QuanTA supports decomposing the hidden dimension into an arbitrary number of axes N and wide selection of collections of tensors (T(a)) } as long as the tensors are compatible with the axes, in this work, focus on N 3, and 5, where exactly one tensor is applied each unique combination of axes, For example, there are 3 tensors when N = 3. 6 tensors when N 4, and 10 tensors when N = 5. Note that if N 2, there is only single tensor and this approach reduces to the full fine-tuning. Since these tensors are applied sequentially, and matrix multiplications in general don't t commute, the order of tensor application can also affect the result. In the case of N 3. the QuanTA layer is constructed as Fig. in the main paper. For = 4 and N = 5, we show the construction in Fig. E.4
Figure E.4: QuanTA architecture used in this work for N = 4 and N 5.
E.2 Experiments on DROP dataset
In Table E.2 we show the hyperparameters used for the DROP experiments. Only LoRA and QuanTA are applied the 13- and 70-billionparrmetter LLaMA2 models. For the [3-billion-paramere model or smaller, only a single A100 GPU is used for fine-tuning. And for the 70-billion parameter model, four A100 GPUs are used. For all experiments, the hyperparameters are only optimized on the 7-billion-paramete parameter LLaMA2 model, and applied directly on larger models. In addition, all the hyperparameters are optimized on the validation set, before evaluating the model on the test set. We further note that we choose the best checkpoint obtained during fine-tuning, in terms of the Fi-score on the validation set, as the final model to apply on the test set. Because of this, the number-of-epon -fe-occh parameter does not introduce significant effect to the final result, as long the training converges.
D-billioon-parameter
19

---

Therefore, this hyperparameter is chosen rather arbitrarily between and 6. We further note that the batch sizes reported here are the effective batch sizes, including the gradient accumulation steps.
E.3 Experiments commonsense datasets
We explain the details of the experiments on commonsense datasets. As mentioned before, in this experiment, we first fine-tune the model the joint COMMONSENSE 70K dataset and evaluate the same fine-tuned model on all downstream tasks. Similar to the drop dataset, we optimize the hyperparameters on the validation set that we created from the COMMONSENSE| dataset and choose the best checkpoint in terms of the validation accuracy to evaluate on the benchmarks. The hyperparameters are listed in Table E3
E.4 Experiments arithmetic datasets
We further explain the details of the experiments on arithmetic datasets. Similar to previous, we first fine-tune the model the joint МАтHHOK dataset and evaluate the fine-tuned model all downstream tasks and we optimize the hyperparameters on the validation set that we created from the MATHI0K dataset and choose the best checkpoint in terms of the validation accuracy to evaluate on the benchmarks. The hyperparameters are listed in Table E.4 Notice that we choose different set of module for LoRA match the experimental setup of [54]
20

---

Experiment
Hyperparameters
Values
FT
Batch Size
(2, 8) AdamW
Optimizer
Scheduler
Linear Scheduler (5e-6, le-5)
Learning Rate
Weight Decay
0
Dropout
0
Number of GPUs
1
Series Adapters
Batch Size
4
Optimizer
AdamW Linear Scheduler
Scheduler
Learning Rate
lc-4
Weight Decay
0
Dropout
0
Bottleneck
[64, 128]
Modules
Default
Number of GPUs
Parallel Adapters
Batch Size
4 A
Optimizer
Scheduler
Linear Scheduler
Learning Rate
le-4
Weight Decay
0
Dropout
0
Bottleneck
[64 .18] Default 1
Modules
Number of GPUs
LoRA
Batch Size
4 Adam W
Optimizer
Scheduler
Linear Scheduler
Learning Rate
{ |le-4, 2e-4, 5e.4, 6c-4)
Weight Decay Dropout
0
0
r
[4, 6, 8, 16, 24, 32, 48, 64, 96, 128]
16
Modules
(q-proj v_proj) (Same as Default) [1, 4]
Number of GPUs
QuanTTA
Batch Size
4 Adam W Linear Scheduler
Optimizer
Scheduler
Learning Rate
lc-4 0
Weight Decay Dropout N
0
[3, 5]
di d1-d2---- -d
18-8-4-44, 8-8-8-8, 16-8-8-4, 16-16444, 16-16-16, 16-8-8-5, 16-8-8-8]
Modules Number of GPUs
(q-proj v-proj) [1,
Table E.2: Hyperparameters used for DROP dataset for various fine-tuning methods. Curly brackets include the hyperparameter values tested during hyperparameter optimization, with the actual hyper- parameter(s) underscored. Square brackets include hyperparameter values for different experiments conducted in the main paper.
21

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Experiment
Hyperparameters
Values
FT
Batch Size
4
Optimizer
AdamW Linear Scheduler
Scheduler
Learning Rate
1e-5
Weight Decay
0
Dropout
0
Number of GPUs
1
QuanTA
Batch Size
4
Optimizer
AdamW
Scheduler
Linear Scheduler
Learning Rate
(5e-5, le-4)
Weight Decay
0
Dropout N
0
4
dj -d----.--00 -dN
[16-8-8-4, 16-8-8-5] (q-proj v-proj)
Modules Number of GPUs
Table E.3: Hyperparameters used for commonsense experiments. Curly brackets include the hy- perparameter values tested during hyperparameter optimization, with the actual hyperparameter(s) underscored. Square brackets include hyperparameter values for different experiments conducted in the main paper.
22

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Experiment
Hyperparameters
Values
FT
Batch Size
4
Optimizer
AdamW W Linear Scheduler
Scheduler
Learning Rate
le-5
Weight Decay
0
Dropout
0
Number of GPUs
1
LoRA
Batch Size
Optimizer
AdamW
Scheduler
Linear Scheduler
Learning Rate
1ee44. 3e-4)
Weight Decay Dropout
0
0
r
32
16
Modules
(q-proj k_proj v-proj up-proj down_proj)
Number of GPUs
I
QuanTA
Batch Size
Optimizer
AdamW
Scheduler
Linear Scheduler
Learning Rate
{ (1e-4. 3c-4) 0
Weight Decay Dropout N
0
di d1-d2-..--y Modules
(16-8-8-4, 16-16-4-4, 16-8-8-5, 16 16-16-5-4)
[(q_proj v_proj).
(q-proj k_proj v_proj up-proj down_proj))
Number of GPUs
Table E.4: Hyperparameters used for arithmetic experiments. Curly brackets include the hyperparam- eter values tested during hyperparameter optimization, with the actual hyperparameter(s) underscored. Square brackets include hyperparameter values for different experiments conducted in the main paper.
23

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Additional Benchmarking Results
In this section, include benchmarking results with additional fine-tuning methods and additional
datasets. In Table F.5 and F.6) we include additional comparisons to M [31, LoRETTA [13], and
KronA [23]. In Table F.7 we include additional results on five commonsense understanding tasks
from the GLUE benchmark
69
using RoBERTa model (701.
PEFT Method
# Params (%)
Fi Score (1)
FT
100%
59.4
Series Parallel
0.747%
58.8
0.747%
59.0
LoRA,=8
0.062%
54.0
LoRA,=32
0.249%
54.8
LoRA, 128
0.996%
56.2
MoRA,
0.062%
58.6
MoRA
0.249%
58.2
MoRA,=125
0.996%
58.9
LoRETTA
0.009%
48.6
LoRETTA,
0.083%
54.9
LoRETTA == 128
1.254%
59.1
KronAg4 64
0.008%
50.9
KronA256-16
0.062%
57.7
KronA102444
0.996%
58.5
QuanTA1668-8 (Ours)
0.041%
59.5
Quan 16-1661111 16
(Ours)
0.261%
59.6
Table F.5: Benchmark of various fine-tuning methods on the DROP dataset using LLaMA2 billion parameter model as the base model. Fine-tuning methods in addition the main paper are shown in italic font. In each case, we report the average of F1 score over 4 experiments with different random seeds. For LoRA, MoRA and LoRETTA, the subscript labels the rank; for KronA the subscript labels the sizes of the matrices; and for QuanTA, the subscript labels the dimensions of axes.
Accuracy (1)
PEFT Method
# Params (%)
BoolQ
PIQA
SIQA
HellaS.
WinoG.
ARC-e
ARC-c
OBQA
Avg.
LoRA!
0.70%
70.8
85.2
79.9
91.7
84.3
84.2
71.2
79.0
80.8
DORAt
0.35%
88.8
80.3
95.5
84.7
90.1
79.1
87.2
85.0
DoRAt
0.71%
74.6
89.3
79.9
95.5
85.6
90.5
80.4
85.8
85.2
LoRETTA
0.13%
74.3
87.5
80,9
94.5
86.7
92.1
81.5
85.8
85.4
KronA
0.052%
72.9
80.6
92.1
85.1
87.8
76.0
84.3
83.2
QuanTA (Ours)
0.035%
74.3
88.1
81.8
95.1
87.3
91.1
81.7
87.2
85.8
Table F.6: Benchmark on various commonsense reasoning tasks using LLaMA3 8 billion parameter model as the base model. Fine-tuning methods in additional the main paper are shown italic font. All results of models and PEFT methods labeled with *** are from [54]] and results with "t" are from (00
Accuracy (t)
PEFT Method
# Params (%)
SST-2
MRPC
CoLA
RTE
STS-B
LoRA
0.71%
94.01
91.48
62.08
74.51
90.48
TA (Ours)
0.62%
93.81
91.67
62.08
77.26
90.68
Table F.7: Benchmark on five natural language understanding tasks using RoBERTa model as the base model.
24

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G Systematical Way to Generate einsum Expressions
In the main paper, we show an example of how to implement QuanTA operation easily using einsum. Here, we show how to systematically generate the einsum expression more generally, For illustrative purposes, focus the case where there is exactly one tensor for every combination of two axes. First, we show how to generate the einsum expression for applying the QuanTA operator.
import itertools
import ppt_einsum as oe
quanta.spply.eeinmm..xpp.))))))
current__yybols_.nde list (range (N))
expr
for i in current_symbols sinie__dssmmbols_ddds
expr += (et_
for (diml, dim2) in itertools. combinations (range (-1, -N-1, -1), 2):
symbol_ind current_symbols_itt [dim1]
symbol_ind2 purrent__ymbole. indd [dim2]
symbol_ind3 symbol_ind1
symbol_ind4 symbol_ind2 +
expr +-
oe @et__yybollssyaao___ndee
oe get__eeebollleeeeee (symbol_ind3)
oe wew.get__ymbl... [ (eymbol_ind2)
oe no.gge_ssymbol oeg.ettsyybbollyyalll_o..n.ddddd (symbol_ind1)
surrent_symbols_iing ind [dim]]]]] [dim1]
symbol_ing ind3
current, Carrent_eymbolss.t inds [dim2] symbol_ing ind4
expr += "->..."
for in current serreent__ybols_steee _nnds: expr oe get_seebollleeee (i)
return expr
Then, applying the QuanTA operator to the hidden vector is as simple as
torch. einsum (quanta_ppply..ccomo
+T)
Similarly, it easy generate the einsum expression for obtaining the full QuanTA operator
import itertools import opt_einsum as oe
def quanta_op_einaum eeppeennnnrrre00000 (N):
current_symbole, inds list (range (NI)
expr
for in current_ seyrentt_ymbbls s__ inds:
expr oe gegg_ee_bbbll (i)
25

---

for (diml, dim2) in cow.:/iai.cmmmmiininnnno...o.o (range (-1, -N-1, -1), 2):
symbol_indi
vurrent_eymbols__iie _nnds [dimi]
symbol_ind ind2 current_eymbols. indd [dim2]
symbol_indh ind3
symbol_ind1
symbol_ind4
aymbol_ind2
expr
get__eymbol
(sy
get_symbollssyeee (symool_nno)))))
oe www.get_ymmolleeee (sy_bllmbb((yyabc___nnndd))
oe @sttesyybollsyyabll_oooo
eurrent__ymboll__ind [dim1]
symbol_inds
current, symbols,.l....... inds [dim2] symbol_inde _nd4
expr +
for i in setrent_symbols_nnde expr += oe oe. wew_gee__mmlll@emm (i)
return expr [1:]
and obtaining the full QuanTA operator is
full_T torch. einsum(quanta_ +T)
We note that the padding and truncation operators are omitted when the QuanTTA operator has different size than the original weight matrix. In addition, in our actual implementation, we use opt opt_eennsum library to optimize the contraction order, reducing the contraction cost.
H
Example Model Outputs
In this section, we provide some example output of QuanTA fine-tuned LLaMA model.
26

---

Task DROP
Model Output
Prompt:
Passage:
Hoping to rebound from their embarrassing home loss to the Lions, the Raiders flew to Invesco Field at Mile High for AFC West duel with the Denver Broncos. In the first quarter, Oakland trailed early Broncos Jay Cutler completed a 9-yard pass to Brandon Stokley for the only score of the period. In the second quarter, the Raiders got on the board with kicker Sebastian Janikouski getting a 38-yard field goal. However, Denver continued to pound away Cecil Sapp got a 4-yard run, while kicker Jason Elam got a 23-yard field goal. In the third quarter, Oakland began to come back as QB Josh McCown (who was a gametime decision heading into the game) completed 46-yard TD pass to WR Jerry Porter for the only quarter, the Raiders took the lead as Gerard Warren sacked Cutler in the end zone for a safety, while LB Thomas Howard returned an interception 44 yards for a touchdovn (followed by a successful two-point conversion pass from McCoun to WR Broncos tied overtime, Oakland managed to
of the period. In the fourth
Ronald Curry). However,
game up with
Elam's 20-yard field goal. make Denver go three-and-out on their first possession. 33-yard run by RB LaMont Jordan helped set up Janikowski for game- winning 52-yard field goal. Broncos head coach Mike Shanahan called timeout before the kick could begin.
Janikowski's 's second try hit off the very tip of the left goal post and was no good, giving Denver a chance to win the game. The Broncos won with Elam getting 23-yard field goal. With only did the Raiders fall 0-2, but they had
the loss,
lost 11-straight games (currently the NFL's longest losing streak) dating back to Week of the 2006 season.
Question: How many field goals did each kicker score in the first half?
Answer:
Output:
1
Table H.8: Examples of QuanTA trained LLaMA2-7B Outputs for the DROP dataset.
27

---

Task BoolQ
Model Output
Prompt:
Please answer following question true or false, question: - is ford escape a 4 wheel drive vehicle? Answer format: true/false
Highest probability choice:
Answer: the correct answer is true.
SIQA
Prompt:
Please choose the correct answer to the question: Carson took Lee's risk by going skydiving with him off of the plane. What will Lee want to after? Answer1: hug Carson Answer2: buy a ticket Answer3: kick Carson. Answer format:
answert/answerr//ansees
Highest probability choice:
Answer: the correct answer is answeri.
SIQA
Prompt:
Please choose the correct ending to complete the given sentence: Personal Care and Style: [header] How to make ice balls [title] Buy a package of water balloons. [step] This method is cheap, quick, and easy-perfect if you don't want to spend money specialty molds for making ice balls. All you'l '1l need few round water balloons (and, course, freezer. Ending1: [subateps] Uninflated balloons: this method requires 2 balls, 1 ice cream stick and water balloons in large bag (1 at time). Open the sides of your volcano and shake the tupperware from side to side few times. Ending2: [substeps] Of course, there is no telling water balloons will be successful; you just know that they will eventually explode on the first try. Plus, they're basically too small to correctly hang from your roof. Ending3: ) [substeps] Remember that, if you're using your ice balls cool your drinks, they'l1 need to able to fit inside your glasses. Thus, when it comes to water balloons, generally, the smaller, the better. Ending4:
and
[substeps] Use unopened water balloons, because they tend to be too full. Plastic ones dry and will harder to put the freezer. Answer format: http:///nninnn//nnnn///eeeeeee
Highest probability choice:
Ansver: the correct answer is ending3.
Table H.9; Examples of QuanTA trained LLaMA3-80 8B Outputs for selected commonsense datasets.
28

---

Task Model Output
MAWPS Prompt:
Bob will
for the weekend payment plans The first plan costs 57.98 for days plus
choose of
- cents per mile The second plan costs 45.98 98 for days plus
18 cents per mile How many miles does Bob need to drive for the 2 plans to cost the same ?
Answer:
Output:
Let's assume that Bob drives x miles. For the first plan, the cost would be: 57.98 + 0.13x For the second plan, the cost would be: 45.98 0.18x We want to find the value of such equal. So, we 57.98 + 0. 45.98 0.18x Simplifying this equation, we get: 0.05x = 12.00 Dividing both sides by 0.05, we get: x 240 Therefore, Bob needs to drive 240 miles for the two plans cost the same.
that the
up equation:
Table H.10: Examples of QuanTA trained LLaMA2-7B Outputs for selected arithmetic dataset.
29

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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: In the abstract and the introduction, we claimed our methods' quantum inspiration, high-rank representation, theoretical backup and strong empirical performances, and ease of implementation. All of these reflect the contributions and scope of this paper.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
is fine to include aspirational goals as motivation as long as is clear that these goals not attained by the paper.
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Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification:
We discuss the limitations just before the conclusion section.
Guidelines:
The answer NA means that the paper has limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create a separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (c.g.. independence assumptions, noiseless settings, model sellsspeciiiationn asymptotic approximations only holding locally). ). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-tex, system might not be used reliably to provide closed captions for online lectures because it fails handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and faimesss.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community, Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory Assumptions and Proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
30

---

Answer: [Yes]
Justification: All of the theoretical results presented in the main paper are backed up with proof' in the appendix.
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced in the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but if they appear in the supplemental material, the authors are encouraged to provide a short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental Result Reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that affects the main claims and/or conclusions of the paper (regardless whether the code and data are provided or not)?
Answer: [Yes]
Justification: In the appendix, we include the implementation details (including part of the code), dataset processing and evaluation details, and hyperparameter choices. The code is open-sourced later on GitHub.
Guidelines:
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Reviews:
[{"review_id": "LpNSsgFu26", "reviewer": "Reviewer_ddRu", "summary": "This work proposed a method, QuanTA, that leverages quantum-inspired techniques for efficient fine-tuning of large pre-trained language models. The authors show that it outperforms Low-Rank Adaptation (LoRA) in complex tasks and has significant improvements in common-sense and arithmetic reasoning.", "questions": "1. As quantum circuit can also be written in the form of a tensor network, what are the main advantages of the proposed method compared to other previous tensor-based PEFT？ such as FACT(https://arxiv.org/pdf/2212.03145) or KronA(https://arxiv.org/pdf/2212.10650) \n2. As the proposed method employs a quantum circuit to replace the low-rank adaptation,  there are various circuit architectures, which kind of circuit ansatz (architecture) is used here, arbitrary or specific? and what is the motivation for choosing such one?", "limitations": "no negative societal impact of their work", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "A novel idea to represent the weight updating by a quantum circuit inspired tensor adaptation.", "weaknesses": "the comparison of related tensor-based approaches is missing, it is unclear the outperformance of the proposed method over related previous works.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the valuable comments and suggestions! We address the raised weaknesses and questions below.\n\n**Weaknesses:**\n\n>the comparison of related tensor-based approaches is missing, it is unclear the outperformance of the proposed method over related previous works.\n\nWe thank the reviewers for the suggestion of comparing QuanTA to other tensor-based approaches. Here, we run additional experiments on tensor-based methods. In particular, we compared our method with KronA(https://arxiv.org/pdf/2212.10650) and LoRETTA(https://arxiv.org/pdf/2402.11417, another tensor-based method generalizing LoRA). Although we would like to include comparisons with FacT, we weren't able to perform the experiment due to the lack of open-source implementation.\n\nComparison of fine-tuning LLaMA2 7 billion models on DROP dataset using QuanTA, LoRA, KronA, and LoRETTA:\n\n| Method | Params (%) | F1 (↑) |\n|----|----|-----|\n| LoRA_r=8 | 0.062% | 54.0 |\n| LoRA_r=32 | 0.249% | 54.8 |\n| LoRA_r=128 | 0.996% | 56.2 |\n| KronA_64-64 | 0.008% | 50.9 |\n| KronA_256-16 | 0.062% | 57.7 |\n| KronA_1024-4 | 0.996% | 58.5 |\n| LoRETTA_r=8 | 0.009% | 48.6 |\n| LoRETTA_r=32 | 0.083% | 54.9 |\n| LoRETTA_r=128 | 1.254% | 59.1 |\n| **QuanTA_16-8-8-4** | 0.041% | 59.5 |\n| **QuanTA_16-16-16** | 0.261% | **59.6** |\n\nComparison of fine-tuning LLaMA3 8 billion model on various commonsense tasks using QuanTA, LoRA, KronA, and LoRETTA:\n\n| Method | Params (%) | BoolQ | PIQA | SIQA | HellaS. | WinoG. | ARC-e | ARC-c | OBQA | Avg. |\n|--------|------------|-------|------|------|---------|--------|-------|-------|------|------|\n| LoRA   | 0.70%      | 70.8  | 85.2 | 79.9 | 91.7    | 84.3   | 84.2  | 71.2  | 79.0 | 80.8 |\n| KronA  | 0.052%     | 72.9  | 87.1 | 80.6 | 92.1    | 85.1   | 87.8  | 76.0  | 84.3 | 83.2 |\n| LoRETTA| 0.13%      |**74.3**| 87.5 | 80.9 | 94.5    | 86.7   |**92.1**| 81.5  | 85.8 | 85.4 |\n| **QuanTA** | 0.035% |**74.3**|**88.1**|**81.8**|**95.1**|**87.3**| 91.1 |**81.7**|**87.2**|**85.8**|\n\nFrom the results above, we find that while these tensor-based approaches could be better than LoRA, they are not as good as our QuanTA method, in terms of both performance and parameter efficiency.\n\n**Questions:**\n\n>As quantum circuit can also be written in the form of a tensor network, what are the main advantages of the proposed method compared to other previous tensor-based PEFT？ such as FACT(https://arxiv.org/pdf/2212.03145) or KronA(https://arxiv.org/pdf/2212.10650)\n\nThanks for the question. This is a very good point. While a quantum circuit can be written as a form of tensor network, not all tensor networks have the same expressivity. Many existing tensor-based approaches still face low-rank limitations in certain ways. For example, FacT generalizes LoRA by viewing the weights over all layers jointly as a tensor to perform tensor decomposition. However, the proposed tensor methods (FacT-TT and Fact-TK) do not resolve the low-rank limitation. KronA, on the other hand, can parameterize full-rank matrices using the Kronecker product. However, it is still \"low-rank\" from a different perspective: in quantum circuit language, KronA parameterizes two single-qudit gates, which cannot introduce entanglements (correlations) between qudits. Introducing correlations with two-qudit gates is crucial for universal representation, which is lacking in KronA. Our QuanTA method is designed in a systematic way with a theoretical guarantee of its universality, rank representation, and composition openness (Theorem I to III). We also include the new benchmarks for KronA and LoRETTA above and demonstrate QuanTA achieves superior performance compared to other tensor-based methods. FacT experiment is omitted due to the previously mentioned reason.\n\n>As the proposed method employs a quantum circuit to replace the low-rank adaptation, there are various circuit architectures, which kind of circuit ansatz (architecture) is used here, arbitrary or specific? and what is the motivation for choosing such one?\n\nThanks for raising a very good point. In conventional quantum circuits (or quantum machine learning), people usually fix the two-qubit gates and only allow arbitrary single-qubit rotations. Different circuit architectures usually differ in the choice of two-qubit gates (CNOT or CZ) and the parameterization of single-qubit gates (X/Y/Z rotations). The architecture used in our work can be viewed as a generalization of conventional architectures in three ways: (1) we generalize unitary matrices to arbitrary matrices because classical computation does not have the unitarity constraint; (2) we generalize the circuit from qubits to \"qu*d*its\", meaning each dimension is generalized from \"2\" to arbitrary numbers; and (3) we allow the two-qudit gates to be fully parameterizable, instead of taking fixed forms. Because of the third generalization, the removal of single-qudit gates does not affect the expressivity, as they can be *merged* into the fully parameterizable two-qudit gates. Using this generalized architecture, we apply one two-qudit gate for each pair of qudits (axes or dimensions). This is one of the innovative designs of QuanTA that allows both full-rank matrix parameterization and entanglement generation between different qudits, while existing tensor-based methods fail to do so. In Appendix E1, we already described the architecture and gave examples for up to 5 qudits. Thanks for the reviewer's question, we further update this section to improve the clarity of the circuit architecture in the new version of our paper.\n\n>**Rating:** 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.\n\nWe are very grateful for the reviewer's valuable comments and questions for improving our work. We have supported new benchmarks and discussions to address the concerns. We would be very appreciative if the reviewer could consider raising the score.", "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": "yKfvbzEYXa", "reviewer": "Reviewer_GYap", "summary": "The authors address the issue that low-rank adaptation methods fail when applied to more complex tasks. They clearly present this motivation through experiments on two datasets of varying complexity. Drawing inspiration from quantum information science, the authors propose using the formalism from quantum circuits, which describe operations on quantum systems whose dimensions increase exponentially with the number of qubits.", "questions": "1. From a practical point of view, is there any preference regarding the decomposition of the output dimension? For example, should we favour smaller-dimensional axes? There are many possibilities for such a decomposition, so any tips that would help in exploring this hyperparameter subspace would be valuable.\n2. Why is the method limited to only two-axis gates? Multi-axis gates should be more expressive, correct? In quantum computing, we typically restrict ourselves to two-qubit gates due to the difficulty of implementing multi-qubit gates on real devices. However, this constraint does not apply here.\n3. I understand the motivation of this work, where the authors show the limitations of low-rank adaptation methods in the context of more complex reasoning tasks. Therefore, will the proposed method provide any benefits in terms of fine-tuning in the domain of computer vision?\n4. How large is the $\\alpha$ parameter? My understanding is that $\\alpha$ dictates the number of gates used in the circuit, and subsequently, the expressiveness of the final circuit is influenced by this value and the arrangement of these gates. Is this correct? If so, what is the relationship between the performance of the proposed method and the value of $\\alpha$?", "limitations": "yes", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "1. The paper is clearly motivated and well-written, with the only challenging aspect being the tensor formalism. However, the authors acknowledge this difficulty and frequently provide specific examples to aid understanding.\n2. Integrating quantum concepts in fine-tuning appears to be a novel approach and demonstrates promising results.", "weaknesses": "1. Please clarify the start and end limits in the summations.\n2. The authors justify their results by stating that adaptive methods based on low-rank decomposition perform poorly on more complex tasks. Why is LoRA the only baseline method considered? Why did the authors not compare their method with high-rank methods, such as MoRA or KronA?\n3. I appreciate that the authors devoted some attention to the complexity of the proposed methods, but it would be very beneficial if they also provided some evaluation of training time and GPU memory usage in comparison to other baselines.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are deeply appreciative of the reviewer's valuable comments and suggestions! We address the raised weaknesses and questions below. Due to the rebuttal length limitation, we address the weaknesses in this post and the questions in a new post.\n\n**Weaknesses:**\n\n>Please clarify the start and end limits in the summations.\n\nThanks a lot for the suggestion! For $i_m$, $j_m$ or $k_m$, the summation is from 1 to $d_m$, the number of dimensions of the $m$-th axis. For $\\alpha$, the summation (or product) is from $1$ to the number of matrices. We update the manuscript and include the limits to improve clarity.\n\n>The authors justify their results by stating that adaptive methods based on low-rank decomposition perform poorly on more complex tasks. Why is LoRA the only baseline method considered? Why did the authors not compare their method with high-rank methods, such as MoRA or KronA?\n\nThanks for the suggestion and we fully agree that it would be beneficial to include comparisons with other high-rank methods. We would like to note that MoRA is concurrent to our work and was posted on arXiv two days before the NeurIPS submission deadline, making it very difficult to include comparisons at the time of submission. In the tables below, we provide additional benchmarks with MoRA and KronA to further provide evidence of the effectiveness of QuanTA.\n\nComparison of fine-tuning LLaMA2 7 billion model on DROP dataset using QuanTA, LoRA, MoRA, and KronA:\n\n| PEFT Method                      | # Params (%) | F1 Score (↑) |\n|----------------------------------|--------------|--------------|\n| LoRA_r=8                         | 0.062%       | 54.0         |\n| LoRA_r=32                        | 0.249%       | 54.8         |\n| LoRA_r=128                       | 0.996%       | 56.2         |\n| MoRA_r=8                         | 0.062%       | 58.6         |\n| MoRA_r=32                        | 0.249%       | 58.2         |\n| MoRA_r=128                       | 0.996%       | 58.9         |\n| KronA_64-64                      | 0.008%       | 50.9         |\n| KronA_256-16                     | 0.062%       | 57.7         |\n| KronA_1024-4                     | 0.996%       | 58.5         |\n| **QuanTA_16-8-8-4 (Ours)**       | 0.041%       | 59.5         |\n| **QuanTA_16-16-16 (Ours)**       | 0.261%       | **59.6**     |\n\nComparison of fine-tuning LLaMA3 8 billion model on various commonsense tasks using QuanTA, LoRA, KronA:\n\n| PEFT Method          | # Params (%) | BoolQ | PIQA | SIQA | HellaS. | WinoG. | ARC-e | ARC-c | OBQA | Avg.  |\n|----------------------|--------------|-------|------|------|---------|--------|-------|-------|------|-------|\n| LoRA                 | 0.70%        |  70.8  | 85.2 | 79.9 | 91.7    | 84.3   | 84.2  | 71.2  | 79.0 | 80.8  |\n| KronA                | 0.052%       |  72.9  | 87.1 | 80.6 | 92.1    | 85.1   | 87.8  | 76.0  | 84.3 | 83.2  |\n| **QuanTA (Ours)**    | 0.035%       |  **74.3**  |**88.1**| **81.8** |  **95.1** | **87.3**   | **91.1**  | **81.7**  | **87.2** | **85.8** |\n\n\nFor the commonsense tasks, while we would like to compare QuanTA to both KronA and MoRA, we find MoRA can be sensitive to hyperparameters. Due to the limited computational resources and the large training set of COMMONSENSE170K, we were unable to find a good combination of hyperparameters for MoRA during the rebuttal period. Therefore we only include MoRA benchmarks on the DROP dataset. The new results demonstrate that while MoRA and KronA can improve significantly from base LoRA, QuanTA still achieves better performance while using fewer parameters. It further demonstrates QuanTA's effectiveness considering the concurrent development of MoRA.\n\nWe would like to further note that while KronA can parameterize full-rank matrices using the Kronecker product, it is still \"low-rank\" from a different perspective: in quantum circuit language, KronA parameterizes only single-qudit gates, which does not introduce entanglements (or correlations between qudits). Introducing correlations with two-qudit gates is crucial for universal representation, which is lacking in KronA. Our QuanTA method is designed systematically with a theoretical guarantee of its universality, rank representation, and composition openness (Theorem I to III).\n\n>I appreciate that the authors devoted some attention to the complexity of the proposed methods, but it would be very beneficial if they also provided some evaluation of training time and GPU memory usage in comparison to other baselines.\n\nThanks for the suggestion! To better understand the time and space complexity of our method, we run additional experiments and profile the average runtime per step and the maximum allocated GPU memory for training LLaMA 3 8 billion model on the COMMONSENSE170K dataset with a batch size of 4 (to fully saturates the GPU memory). The result is shown below.\n\n| PEFT Method          | Average Runtime per Step (second) | Maximum Allocated GPU Memory (GB) |\n|----------------------|------|-------|\n| LoRA                 | 0.76 | 76.8  |\n| MoRA                 | 0.79 | 78.5  |\n| KronA                | 0.73 | 76.4  |\n| **QuanTA (Ours)**    | 0.75 | 75.6  |\n\nWe find that while all methods have very similar runtime and memory allocation--potentially because most of the resources are devoted to running the base model--our method may still have a slight advantage. All the experiments are performed on A100 GPUs with 80GB VRAM. We would like to further note that the current implementation of QuanTA is based on a single ```einsum``` operation. Further optimizations using custom CUDA kernels could be possible.\n\nDue to the rebuttal length limitation, we address the questions in the next post.", "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": "SEEyjkN5JO", "reviewer": "Reviewer_Qk3g", "summary": "The paper proposes an efficient fine tuning method inspired from quantum circuits for large scale pretrained models. This is an alternative lorank approximation to weight updates (LoRA). The paper considers decomposing the dimension d of the square weight matrix as  d=d1xd2x...xdN) .The hidden vector in d dimensions can then be considered as a vector in the corresponding tensor product space of qudits. The action of the weight update is then considered as a sequence of quantum gates applied to these qudits.\nThe weighted layer is then defined as y = W_\\theta x = W_0 x + T_\\theta x - Sx where W_0 is the initial weight matrix, T_\\theta is trainable quantum circuit and S is fixed to be initial circuit for T_\\theta.\nThrough experiments on multiple datasets they are able to beat other fine tuning methods considering versions of Llama as the base model.\nThe paper also discusses the theoretical results to show the universality of tensor decomposition into tensors acting on atmost 2 axis. They have also derived lower and upper bounds on the rank of the overall tensor product operator in terms of ranks of individual tensors.\nThe work resolves the issue with low rank approximation which may not always be sufficient as shown through experiments.", "questions": "1. 1. The paper does not discuss the ordering over the pairs of qubits chosen. Also in general the universality theorem for tensor product might involve picking a set of pairs of qubits where a given pair might participate multiple times in the circuit. Is picking a pair of qubits only once always sufficient?", "limitations": "The authors have addressed their limitations and societal impact  well.", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The idea of using tensor decomposition for PEFT is original , novel and would find applications in the area of LLM fine tuning.\n2. The experimental results show promising results on multiple datasets including DROP, CommonSense Reasoning and Arithmetic Reasoning.\n3. The paper is backed by theoretical justification for universality of tensor product decomposition.\n4. The experiments also justify the need for higher rank approximations.", "weaknesses": "1. The paper does not discuss the ordering over the pairs of qubits chosen. Also in general the universality theorem for tensor product might involve picking a set of pairs of qubits where a given pair might participate multiple times in the circuit. Is picking a pair of qubits only once always sufficient?\n\n2.  A minor concern could also be that the dimension of of hidden state needs to be factorizable as product of small dimensions. This would typically not be a concern in practice since these are powers of 2, so would work perfectly in those scenarios.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are very appreciative of the reviewer's valuable comments and suggestions! We address the raised weaknesses and questions below.\n\n**Weaknesses:**\n\n>The paper does not discuss the ordering over the pairs of qubits chosen.\n\nThanks for the question. While the ordering of pairs in general should not significantly affect the results, the specific choices in this paper is generated with the following code:\n```\nitertools.combinations(range(-1, -N-1, -1), 2)\n```\nwith N being the number of axes. The details of the circuit architecture, including the ordering of pairs are already described in Appendix E1 and F. We update the paper to further improve the clarity of this point.\n\n>Also in general the universality theorem for tensor product might involve picking a set of pairs of qubits where a given pair might participate multiple times in the circuit. Is picking a pair of qubits only once always sufficient?\n\nThe reviewer is absolutely correct. Picking each pair of qudits only once does not necessarily satisfy the universality theorem. The universality theorem can be paraphrased as the following: given sufficient layers, QuanTA has the same representation as full fine-tuning. In practical cases, full fine-tuning is often not needed, either because the subsequent tasks are close to the pretraining data, or due to the small sample sizes of the subsequent tasks. Therefore, it is usually not necessary to have multiple gates for each pair of qudits. In addition, even with a single gate for each pair of qudits, QuanTA can already parameterize full-rank matrices and introduce correlations to any pair of qudits, which is not possible for existing methods such as LoRA or alike. While we find, empirically, that a single gate for each pair of qudits is sufficient for all the fine-tuning tasks we tried, additional gates can be employed when the task is significantly harder. This flexibility gives an additional tuning knob for the expressivity. Exploring how the performance scales as the number of gates on each pair of axes is also a very good theoretical question that involves further understanding of quantum information theory and will be a nice direction for future exploration.\n\n>A minor concern could also be that the dimension of of hidden state needs to be factorizable as product of small dimensions. This would typically not be a concern in practice since these are powers of 2, so would work perfectly in those scenarios.\n\nThanks for the comment. Yes, as the reviewer has noted, many existing LLMs have the hidden state chosen as a power of 2. However, we would like to note that, while the optimal design is not clear, QuanTA is flexible to be incorporated in LLMs with unfactorizable hidden dimensions. As already described in Appendix B, it is completely valid to either pad or truncate the hidden state to the nearest factorizable size in any layer before applying QuanTA. This also works in the inference stage, as the QuanTA matrices can also be padded or truncated analogously to merge with the original weight matrices to avoid any inference latency. We improve Appendix B to make this point clearer in the new version of the paper.\n\n**Questions:**\n>The paper does not discuss the ordering over the pairs of qubits chosen. Also in general the universality theorem for tensor product might involve picking a set of pairs of qubits where a given pair might participate multiple times in the circuit. Is picking a pair of qubits only once always sufficient?\n\nThanks for the reviewer's question! We provided the answers above in the weaknesses section.\n\n>**Rating: 8:** Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.\n\nWe are very grateful for the reviewer's comments and questions for improving our work. We have supported a new discussion to address the concerns. It would be deeply appreciated if the reviewer could consider raising the score.", "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": "66AN3eXuAW", "reviewer": "Reviewer_H46K", "summary": "This paper proposes a novel PEFT method that inspired by the Quantum Circuit. Their method is theoretically supported by the universality theorem and the rank representation theorem to achieve efficient high-rank adaptations on various downstream tasks. The QuanTA method surpasses LoRA, DoRA and even fine-tuning in some cases.", "questions": "Please refer to the weaknesses section.", "limitations": "Yes", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well-motivated and presents sufficient theoretical prove and justifications.\n2. The experiments are abundant and the model performs very well compared with other baselines with much less activated parameters\n3. A well-written paper", "weaknesses": "1. I'm curious about the performance of QuanTA on textual understanding (classification) tasks with BERT or RoBERTa. Afterall, all these PEFT models were initially applied to text classification tasks (LoRA, Adapter...)\n2. Do you have experiments that combine other PEFT methods with your QuanTA? Since the number of activated parameters of your method is much less than baselines. Can we further boost the model performance by adding your method to others?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are very appreciative of the reviewer's valuable comments and suggestions! We address the raised weaknesses below.\n\n**Weaknesses:**\n\n>I'm curious about the performance of QuanTA on textual understanding (classification) tasks with BERT or RoBERTa. Afterall, all these PEFT models were initially applied to text classification tasks (LoRA, Adapter...)\n\nThanks for the comments! In the original manuscript, we focus on LLaMA benchmarks because we believe such tasks are usually more challenging than tasks with BERT or RoBERTa. We note that one of the key questions that QuanTA addresses is high-rank fine-tuning of challenging tasks (which are non-trivial for existing methods), so that we do not focus on the textual understanding tasks that are usually considered to be simpler.  However, it is still very valuable to examine QuanTA in the context of text classification for completeness. In the table below, we compare QuanTA and LoRA (fine-tuning RoBERTa) on five natural language understanding tasks. While we would like to include more textual understanding benchmarks, some of the benchmarks have a large training set, and we were limited by our computational resources and the rebuttal timeline to include additional experiments. We note that while the results are less impressive than our experiments on LLaMA, QuanTA still achieves similar or better results than LoRA with fewer parameters. In addition, when fine-tuning BERT or RoBERTa, the output classifier layer usually contains most of the trainable parameters, making the parameter reduction advantage of QuanTA appear to be less significant. Even so, QuanTA still achieves better results with fewer parameters.\n\n\n| PEFT Method          | # Params (%) | SST-2 | MRPC | CoLA | RTE | STS-B |\n|----------------------|--------------|-------|------|------|---------|--------|\n| LoRA                 | 0.71%        |  **94.01**  | 91.48 | **62.08** | 74.51    | 90.48   |\n| **QuanTA (Ours)**    | 0.62%       | 93.81 |**91.67**| **62.08** |**77.26**| **90.68**   |\n\n\n>Do you have experiments that combine other PEFT methods with your QuanTA? Since the number of activated parameters of your method is much less than baselines. Can we further boost the model performance by adding your method to others?\n\nThanks for the comment and this is a very good point. While we did not have the time to explore such possibilities, it could be very advantageous to combine QuanTA with other PEFT methods. For example, it would be possible to do Q-QuanTA to further reduce the cost by quantizing the QuanTA weights. In addition, the current QuanTA weights are mostly \"square\". It would also be interesting to further combine QuanTA and LoRA, making low-rank parameterizations of QuanTA weights. Many other modifications or variants of LoRA could also be adapted to QuanTA to further improve our results. It will be a good future direction to further boost the model performance by integrating QuanTA and other methods.\n\n**Questions:**\n\n>Please refer to the weaknesses section.\n\n>**Rating: 7:** Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.\n\n\nWe are very grateful for the reviewer's comments and questions for improving our work. We have supported new benchmarks and discussions to address the concerns. It would be greatly appreciated if the reviewer could consider raising the score.", "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:
We thank all the reviewers for their valuable comments! We appreciate all reviewers for noting the novelty of our work as well as our strong theory and experiment results.

Below, we summarize the major concerns of the reviewers and our responses.

>In general, reviewers agree that our results are already impressive, although some reviewers are interested in seeing additional benchmarks: (Reviewer ddRU) QuanTA vs other tensor-based fine-tuning methods; (Reviewer GYap) QuanTA vs other high-rank fine-tuning methods; and (Reviewer H46K) QuanTA's performance on textual understanding tasks.

To address the reviewer's concern, we perform new experiments comparing QuanTA with KronA (tensor-based and high-rank), LoRETTA (tensor-based), and MoRA (high-rank) and show that although these methods can achieve better performance than LoRA, they are not as good as QuanTA in terms of both performance and parameter efficiency. In addition, we compare QuanTA and LoRA on text understanding tasks by fine-tuning the RoBERTA base model. We show that QuanTA is still advantageous in this application.

>Reviewer GYap acknowledges our theoretical discussions on the space and time complexity of QuanTA, but is interested in the actual runtime and memory cost for the experiments.

To answer this question, we include new results profiling the runtime per iteration as well as the memory usage during training.

>While reviewers in general agree that our paper is clear and well-written, especially considering the complexity of tensor operations, reviewers have some questions requiring further clarifications of certain parts of our paper.

In our response below, we discuss all the points in detail and we update the new version of our paper to further clarify these points.

>Reviewers are also interested in further explorations of QuanTA, such as additional experimentations of circuit architecture, QuanTA's performance in computer vision tasks, and combining other PEFT methods with QuanTA.

We thank all the reviewers for these great suggestions and believe it would be important to explore them in subsequent works.

We would like to further highlight the novelty and significance of our work: (1) as a quantum-informed algorithm, our work bridges quantum computation and machine learning, opening up more opportunities for quantum-inspired ideas or even real quantum hardware realization to improve LLMs (2) our method allows high-rank fine-tuning that is extremely parameter efficient with no inference overhead, achieving state-of-the-art results in multiple benchmarks.

Meta-review: The paper presents a novel fine-tuning method inspired by quantum circuit-like reparameterization, which is clearly motivated and well-written. The authors have done a commendable job addressing the inherent complexity of tensor formalism, offering specific examples that enhance the reader's understanding. The integration of quantum concepts into the fine-tuning process appears to be an innovative approach, yielding promising results that contribute positively to the field.

There are several key areas that require attention for the camera-ready version. It is crucial to include all the additional results provided during the rebuttal phase, including the additional baselines, runtime analysis, and the various discussion points. These elements are essential for a comprehensive evaluation of the proposed method. Additionally, the discussion on the "composition openness" of QuanTA compared to LoRA is somewhat misleading, as LoRA adapters are not composed directly with each other but with a nonlinear operator in between. It is recommended to revise this section for clarity. Overall, with these revisions, the paper is strong and merits acceptance.
Decision: Accept (poster)