Layerwise Federated Learning for Heterogeneous Quantum Clients using Quorus
Authors: Jason Han
Nicholas S. DiBrita
Daniel Leeds
Jianqiang Li
Jason Ludmir
Tirthak Patel
Venue: ICLR 2026 Poster (2026)
Keywords: Federated Learning
Heterogeneity
Quantum
Primary area: other topics in machine learning (i.e., none of the above)
TL;DR: A novel framework for federated learning of quantum ML models with varying depths

Abstract: Quantum machine learning (QML) holds the promise to solve classically intractable problems, but, as critical data can be fragmented across private clients, there is a need for distributed QML in a quantum federated learning (QFL) format. However, the quantum computers that different clients have access to can be error-prone and have heterogeneous error properties, requiring them to run circuits of different depths. We propose a novel solution to this QFL problem, Quorus, that utilizes a layerwise loss function for effective training of varying-depth quantum models, which allows clients to choose models for high-fidelity output based on their individual capacity. Quorus also presents various model designs based on client needs that optimize for shot budget, qubit count, midcircuit measurement, and optimization space. Our simulation and real-hardware results show the promise of Quorus: it increases the magnitude of gradients of higher depth clients and improves testing accuracy by 12.4% on average over the state-of-the-art.

Published as a conference paper at ICLR 2026
LAYERWISE FEDERATED LEARNING FOR HETEROGE- NEOUS QUANTUM CLIENTS USING QUORUS
Jason Han, Nicholas S. DiBrita, Daniel Leeds, Jianqiang Li, Jason Ludmir & Tirthak Patel
Department Computer Science
Rice University
Houston, TX 77005, USA
{ 46, nd52, d1107, 11107,
ABSTRACT
Quantum machine learning (QML) holds the promise to solve classically in- tractable problems, but, as critical data can be fragmented across private clients. there is need for distributed QML in a quantum federated learning (QFL) for- mat. However, the quantum computers that different clients have access can be error-prone and have heterogeneous error properties, requiring them to run circuits of different depths. We propose a novel solution this QFL. problem, Quorus, that utilizes layerwise loss function for effective training of varying-deppe quan- tum models, which allows clients to choose models for high-fidelity output based their individual capacity. Quorus also presents various model designs based on client needs that optimize for shot budget, qubit count, midcircuit measure- ment, and optimization space. Our simulation and real-hardware results show the promise of Quorus: it increases the magnitude of gradients of higher depth clients and improves testing accuracy by 12.4% average the state-of-the-art.
INTRODUCTION
Quantum machine learning (QML) holds the potential to solve classically difficult problems with high efficiency. Existing methods using quantum MI. have been applied to variety of industrial and scientific applications, including portfolio optimization, drug discovery, and weather forecast- ing (Peral-Gareía et al. 20244) [Smaldone et al. 2025 Liu et al. 2025). Quantum ML has also been used to solve classical ML problems with significant reductions in parameters (Kashif et al.] 2025 DiBrita et al. 2025 Leither al.) 2025). Given the success of quantum ML, natural consider- ation, like in classical ML, is to consider the real-world case of fragmented data across multiple private clients. How clients with quantum computers train together, without revealing data to other purties? The classical analog of solving this problem has also been proposed, called Quantum Federated Learning (QFL) (Chen & Yoo! 2021).
However, existing QFL techniques do not consider the heterogeneity of quantum devices. All quan- tum computers are subject to hardware error that varies from computer computer, which has con- tinued to be a critical challenge in quantum computing (Tannu & Qureshi [019 Montanez-Barree al. 2025). The quantum computing research community has proposed Quantum Error Correction (QEC) as solution to quantum hardware error (Calderbank & Shor 1996. Acharya et al. 2024); however, QEC techniques require millions of qubits, which will not exist for many years (Gidney & Ekerá 2021 Sevilla & Riedel, 2020).. Thus, in our current day, to use error-prone devices for QMI. limit the depth of the circuit (quantum code) that executed on the hardware, as the eтror manifested in the output of the circuit is proportional to its depth. Reducing the depth of the circuit is particularly important, as a major source of quantum errors is decoherence, where a qubit loses its important amplitude and phase information with respect to time (Schlosshauer 2019 Zurek, 2003) Preskill 2018). By keeping the circuit to a reasonably shallow depth, researchers attempt to utilize existing quantum computers to achieve practical quantum utility today.
tasks, one strategy
Another challenge in QML is the barren plateaus problem, where gradients vanish as the circuit depth grows (Anschuetz) 2025 Yan et al. 2024 Patel al. 2024). In the worst case, gradients

---

Published as a conference paper at ICLR 2026
Share purameters over a classical channel
Quantum Computer
The computeer technology, fidelity, and generation can beterogeneoolly vary across eltent;; henoe, they must circuity of different depties
Parameter
Client
Client
Client
Chent
Client
oggregation server
Receive aggregated parameters over classical channel
Classical Computer
Depiction of the overall setup of our depth-heterne cepth-heterogeneous quantum FL framework. Our setup utilizes the realistic scenario of a classical network for sending and receiving parameters, and each client has quantum computer that
Figure 1:
circuits of varying depths.
decay exponentially, making practically impossible to train deep circuits, even when noise is not the dominating factor (Cerezo et al., 2021). This significantly restricts the scalability of QML circuits, as optimization becomes infeasible beyond moderate depths. A further obstacle resource efficiency. Unlike classical training, which can rely inexpensive iterations, every quantum training step requires repeated circuit executions (shots) to estimate observables (McClean al. 2016). As result, algorithms must be designed to minimize the number of shots needed for accurate training to ascertain the economic viability for real-world QML.
To address the above challenges, in this work, we design an error-aware QFL technique, Quorus, by considering that clients can only rum quantum circuits of particular depths, based on the depth at which they achieve reasonably high accuracy participate in FL. We illustrate in Fig. While existing QFL works have shown that if training is done in the presence of noise that corrupts the output, then the final training accuracy degrades with depth (Rahman et al.] 2025 Sahu & Gupta 2024), our goal is to enable clients to run as many layers as possible, to allow for higher expressive power and thus, higher accuracy (Sim al. 2019). Quorus is the first-of-its-king work that utilizes layerwise loss functions and knowledge distillation for synchronized objectives across heterogeneous-depue clients. We propose novel shot-efficient designs for varying quantum hardware capabilities and demonstrate both higher gradient magnitudes as well as implementations on all of IBM's s state-of-the-art superconducting quantum computers.
overall setup
The contributions of this work are as follows:
The first structured quantum federated learning framework (Quorus) that utilizes layerwise losses and reverse distillation for improved accuracy (to the best of our knowledge).
A quantum model architecture whereby ensemble of layerwise quantum classifiers can be obtained with no shot overhead, leading to both higher accuracy and resource efficiency.
A design that improves testing accuracy by up to 12.4% over Q-HeteroFL (Diao et al, 202le being shot-efficient for different binary classification tasks.
A model that yields higher gradient norms, reducing barren plateau effect, and achieves accuracy within 3% of ideal simulation on IBM superconducting QPUs, showing real-world viability.
2 PRELIMINARIES
Quantum Computing Basics. Quantum computers process information by manipulating qubits
with quantum circuits. The state of qubit is represented vector: (1)) = 3o (0) + B1 [1), where fo. 31 E C and |.812 + |3,12
The state (u>) exists in a superposition
the states (0) and [1),
which encodes the quantum data we process. The probability of measuring the qubit to be in state (i) is p(i) = |3,122 meaning we must have 1 1 1. A statevector of a system of n qubits is a complex vector in the Hilbert space [ıb) E C = Hn. that is normalized (vblb) = 1. We can write our state the computational basis: defining by as the bitstring corresponding the integer k. the computational basis is the set {16,) vk E Σ, 0 < k < 2"-1). Our state can be expressed as
(1) = Di= 3e Ik). The quantum data ()) is processed by a quantum circuit U, a unitary operator taking (1)) = (22). Because is unitary, it is reversible (UUt = UUU = I).
Parameterized Quantum Circuits. To frame learning problem quantum computers, pa- rameterize the operation U with parameters 6, which are typically rotation angles on the Bloch sphere. Then, U(0) is parameterized quantum circuit (PQC) with trainable gate parameters, and
2

---

Published as a conference paper at ICLR 2026
the structure of UT+) is referred to an ansatz. These variational quantum circuits are often com- posed of repeated circuit structures called layers, and can be written as U(f) = Uo.2 (f0.)) = where Ug is the parameterized circuit for layer i, with parameters 8 Deeper circuits are more expressive, but also suffer from decoherence and errors when evaluated on real hardware. Quantum machine learning generally aims solve the following problem for an objective L and input data x; 0* arg mingge L(U, x; 0). To evaluate LUU, x;; 0) on quantum computer, it is performed by estimating p(b), the probability of measuring state (6) via running the circuit (+) multiple times, and tracking how many times the outcome b was observed. Each run of the quantum eircuit called shot, and shots are expensive on current-day quantum hardware.
The objective L of a parameterized circuit is extracted via projective measurement, which is irreversible general. For a particular outcome b E (0,1}. the first qubit in state (65) is measured to be b, then the resulting state is collapsed to |1/a) = /p(0) Ex.k, =b B& [k))-
Quantum Measurements.
This fundamental quantum property poses a unique challenge when the objective I, is defined for each layer, so Lj- = LUoo;; 0o.)), where Uo.j, Do.j represents the layers and parameters up to layer j. Because measuring a qubit collapses the superposition and removes information from the quantum state, it poses a challenge for simultaneously collapsing information via measurement and retaining sufficient information for subsequent quantum layers and operations (Gyawali et al.. 2024).
Heterogeneous Federated Learning, Federated learning (FL) is a distributed machine learning technique widely used in classical ML where each client's data is private to themselves (McMahan al. 2023). The overall objective function in federated learning for m clients is L F2; .... Tmm)
(x,)), where 2; represents the data of client j and Lj is the loss function for client y (McMahan et al. (023). In centralized federated learning, training is done locally by clients, and parameters are aggregated in a centralized server and broadcast back to clients. An intuition may be gained for why parameter aggregation works by observing that, in the special case where stochastic gradient descent (SGD) is used, parameters are aggregated every epoch, and the batch size is equal to the amount of data client has, it is equivalent in expectation to performing SGD on the centralized objective L. (McMahan et al. 2023).
Heterogeneous Federated Learning adds layer of complexity to FL by allowing for clients to have different local model architectures (Diao et al. 2021). This scenario accounts for the case where some clients have differential computational abilities, but still want take advantage of FL to obtain a shared model. Because the parameter spaces of models are now different, special considerations need to be made as the differing model architectures lead to parameter mismatches that can negatively affect training (Kim et al., 2023). Refer to Appendix A for further details.
RELATED WORK
Classical Federated Learning. The problem of depth-heterogeneone quantum federated learning, where clients have classical models, has large body of work in the literature, but many state-of-the- art techniques in classical FL cannot be directly applied to the case where the model is a PQC. The classical model-hheterggeneous FL technique, HeteroFL. (Diao et al. (021), 1), aggregates parameters in shared submodels across clients. Since this original work, some newer techniques have been proposed, namely FEDepth and ScaleFL Zhang et al. 2025 Ilhan et al. 2023). which assume that intermediate layers can be trained; however, this is not applicable to PQC's 's as training these layers requires client to run circuits to that depth, which precisely is the bottleneck in quantum circuits.
Another work, ReeFL, uses transformer to fuse features between layers; however, features are not directly accessible in quantum ML unless via state tomography (Lee 2024). The classical work most closely related to Quorus, DepthFI. (Kim et al. 2023). is amenable to the setup of quantum clients with models of varying depths as a layerwise FL. technique; however, evaluating the layerwise loss function discuss further in Sec. 4.3 Overall, these classical works cannot be directly applied to the quantum FL setup and highlight the importance of quantum-centric design, which propose in this work.
quantum computers is nontrivial due to measurement collapse, which we
Quantum Federated Learning. The overall setup of QFL, where clients use the same architecture PQC and use centralized server for parameter aggregation, has been studied (Chen & Yoo! 2021) however, the problem of depth- hegtrhhheerogeeeouus FL is not well studied in the quantum case. One work that tackles the problem of parameters being lost in communication, named eSQFL (Yun et al.,
3

---

Published as a conference paper at ICLR 2026
2022). uses layerwise loss function by computing the inner product of states between each layer; however, computing inner products requires long-distance connectivity and is not applicable to run on real hardware (Sá et al.) 2023). There is gap in the literature related quantum federated learning for clients with heterogeneous needs, which we propose a solution for in this work.
DESIGN AND APPROACH
The overall workflow of our technique is illustrated in Fig. Each client is able train a PQC of different depth based on its hardware capability. After local training, clients send their parameters to a server over a classical network, where parameters are aggregated and sent back to clients. Clients then continue to train locally, repeating the process for a set number of rounds.
Algorithm 1: Quorus
Initialization 8
Server Executes:
Update,, 0') 0% local epuch ........ do for each mini mini-batch by, do
All Chiens
for randd
0......,
g
forall
lo E P (in parallef) do
dn
+- [[ de)
Lice
DEL(P, Pl
0 Client Update(k, 8²)
de
g++i|: del + g:+ dz] :86
86
Adam V (6 b.).n.
foreach resource capabillity d, do 0² [14] angle!
return 0:11
p 2 0* ((:3)
4.1 QUORUS WORKFLOW
A detailed description of our workflow is depicted in Algorithm Kim al. 2023) Because PQC's s have heterogeneous depths, parameters are aggregated only among the clients that share each parameter. We also perform aggregation of parameters with circular averaging because the quantum circuit parameters in our implementation are rotation angles, where the angle function defined as angle(=) (imag()) real(=)). The layerwise loss function for client is
Lb Ed L'e -.-
de Lj=1, jyts (Ri Pi),
where T& is the Binary Cross Entropy loss for the classifier at depth i. Note, then, that this loss function assumes that there is a means to extract classifier outputs each layer an important problem with a unique quantum design (addressed in Sec. 4.3). dy is the depth of client k and (KL(p) p.) is the KL divergence between logits p; and Pj. We use the same loss function as in DepthFL (Kim et al., 2023). because a similar intuition applies: clients share to address parameter mismatching, where parameters are different across clients due varying local parameter spaces. In addition, we want
want loss function that
the KL divergence for "reverse distillation", whereby deeper classifiers are helped by shallower ones. These nuances are explored and justified in (Kim et al. 2023). so we do not repeat the discussion for the quantum case.
The unique challenge in the quantum case is how exactly to evaluate the loss in Eq. [ In our setup for Quorus, the loss Li is computed via the probability of measuring qubit be for our binary classification tasks. This poses unique quantum-specific design considerations for Quorus, which the following sections will be devoted to solving.
4.2 ANSATZ DESIGNS AND SELECTION
For any quantum computing problem, it is well-known that deciding the ansatz is essential (Sug- isaki et al., 2022), for two main reasons. Firstly, determines the expressibility and the space of solution states that are explored, and because of the exponentially-sizer Hilbert space that quantum computers operate in, operating in a relevant subspace is essential (Sim et al., 2019 Yan et al. 2024] Holmes et al.) 2022). Secondly, it is entirely possible find ansatz that is well-suited for the problem interest, but is very inefficient when implemented on hardware architectures with lim- ited connectivity due to its use of long-distance two-qubit multi-qubit gates, causing high levels
4

---

Published as a conference paper at ICLR 2026
Features
Parameters
Featwees
Parameters
10>
Ry
Roi
Ry
Roi
(0)
Rot
Ry
Rot
10)
Ry
Rot
Ry
Ros
(0)
Rot
Ry
Rot
10)
Rot
Rot
(0)
Rot
.*.
(0)
Roi
(0)
Ry
Rot
10)
Ry
Rot
Ry
Rot
10)
Rot
Rot
Layer
Layer
Layer
Layer
(a) Staircase Ansatz
(b) V-shaped Ansatz
Figure 2:
We evaluate three ansätze for Quorus: (1) The staircase ansatz, (2) the V-shaped ansatz,
and (3) the alternating ansatz, which switches between staircase and V-shaped layers (not shown).
of output error (Kivlichan et al. 2018 Romero et al. 2018). We address these problems by system- atically exploring relevant ansätze for problem, depicted in Fig. 2 In each layer of our ansatz, we perform data reuploading as has been shown multiple quantum ML experiments to yield improved accuracy and nonlinearities with respect the input (Vidal & Theis 2020 Aminpour et al. 2024). We use the Ry gate to achieve this. We use layers of generalized single-qubit gates Ror tunable parameters. The ansatzes design are centered around two main principles:
(1) The ansatz must be hardware ef- ficient, assume only nearest- neighbor connectivity as observed in quantum hardware Huoo et al. (225);; Han et al. (2025), and (2) only the first qubit measured obtain the output statistic. The latter choice because we focus on binary classification tasks in this work, so measuring a single qubit is sufficient (Schuld et al. 2020). Thus, we come up with two relevant ansätze: the first is the Staircase ansatz, depicted in Fig. (a) (Schuld et al. 22020] Sim et al. 2019), which has a stairease of CNOTs from the last qubit up to the first one. The second is the V-shaped ansatz. which has staircase of CNOTs going down from the first to the last qubit, which then go back to the first qubit, and the third is an alternating combination of the two. Based on our experimental evalua- tion on various datasets in (Appendix D), we observe that the V-shaped ansatz performs the best in a majority of the evaluations, and so we use it as the default ansatz. The reason that V-shaped ansatz performs well is its ability to broadcast information throughout qubits with more CNOT gates traversing up and down the qubits (Sim et al. 2019).
Claxsifier
Clossifier
Output
Onipui
Inpul Foutures
Input Features
Partial
Capy
State
Layer 1
State
Layer
Layer I
Collapse
Layer 2
(a) Classical Classifier
(b) Quantum Classifier
Figure 3: The difference between classical and quantum lay- erwise classifiers. Measurements collapse quantum data, and thus an altered state is passed to the next layer.
Features
Parameters
4.3 QUANTUM CLASSIFIER DESIGN
10)
Ry
Rot
10)
Ry
Rot
When one attempts implement the layerwise loss func- tion in Eq. [ there is an immediate problem: if we mea- sure the qubit, then how can we pass on the same state the next layer? An illustration of this dilemma is in Fig, 3 In DepthFL. and other classical works that assume an in- termediate classifier, depicted in Fig. Bja,, the data after the first layer is somehow converted to a scalar, and im- plicitly, the data is passed, unchanged, to the next layer (and this "copy" operation has minimal classical over- head). Fundamentally, direct analog does not exist in quantum computation. In quantum computing, we mea- sure one of the qubits mid-computation, this collapses the
(0)
Ry
Rot
(0)
Rot
10)
Ry
Rot
Repeat I. times
Run L times: once for each it of layers
Figure 4: The Layerw-e de- sign. The circuit must be run times, where I. is the number of layers.

---

Published as a conference paper at ICLR 2026
The measureeent blocking
This qubit is unused beyond this
Ry
Rot
10)
Ry
Roi
Ry
Rot
10)
Ry
Rol
Rot
10)
Ry
Rot
10)
Ry
Roi
Ry
Rot
Funnel for layers total, dropping one qubit at each layer
..*
Ry
Rot
10)
Ry
Ror
Rot
Repeat L
Layer
Layer 2
(b) Quorus-Funnel Circuit Figure 6: (a) The Blocking design (logically to the Ancilla design), and (b) the Funnel design.
(a) (rrorus-Blocking Circuit
Blocking requires a midcircuit measurement, and Funnel restricts the size of unitary operations.
superposition on the first state and changes the state that is later used in computation (as represented in Fig. (()).. Passing the state unchanged thus requires you to prepare another copy of it, which induces additional shot overhead that is linear in the number of layers and is nontrivial cost, given the expense of running quantum computers. For example, running quantum circuits for just one minute on an IBM quantum computer costs $96 (can run 4 circuits this time with 1k shots each) jbm Thus, we are posed with an important question: How do we implement this measurement between layers quantum MI model, in a shot-efficient manner?
Solution 1: Layerwise. The most straightforward solu- tion to the classical case is to "copy" the quantum state, because we know exactly the circuit that prepared it. This solution is depicted in Fig. 4 However, this requires a shot budget that scales linearly with the number of lay- ers. For deep circuits, the required shot budget quickly becomes infeasible for budget-constrained clients.
Ancilla
Rot
10}
10}
Ry
Rot
(0)
Rot
Solution 2: Ancilla/Blocking. To address the case where
client does not have high-shot budget, design ansatz where it is possible to obtain predictions from each layer with shots independent of the layer count. In partic- ular, with reference to Figure b). we propose continuing to operate on the collapsed state in our PQC. This design is depicted in Fig. 5 In particular, we entangle the first qubit with an ancilla in the (0} state after each layer. We evaluate each layer's outputs by computing the marginal distribution on its ancilla. For the first layer, the statistics match the Layerwise model; for later layers, they differ because entangling the first qubit with ancilla "dephases" it (Gyawali et al. 2024).
[0)
Rot
Repeal times; one ancilla qubil for each of the L layers
Figure 5: The Quorus-Ancilla design. The circuit is only run once, but requires an ancilla qubit per layer, and also de- phases the first qubit.
Since dephasing is limited to that qubit, we hypothesize, and confirm on IBM hardware (Sec. (5). that our quantum ML model can still train effectively under this alternative model. Implementing this requires the first qubit entangle with new ancilla each layer, which in turn demands long-distance CNOTs. Thus, while our Layerwise ansatz assumes nearest-neipe nearest-neighbor connectivity, sys- tems with larger qubit counts and richer con- nectivity can benefit from this Ancilla ap- proach. In principle, ancillae are not required one can simply just measure the first qubit, not reset it, and continue in computation. This logically equivalent (proof provided in Appendix (B) but physically distinct model of computation is depicted in Fig. ((a), where midcircuit measurement is performed on the first qubit. This model would be feasible for clients who can do fast midcireuit measurements, but existing midcircuit measurements are lengthy and error-pront (Deist et al. 2022 Rudinger et al. (021).
Table 1: Unique requirements of different quantum models of Quorus, highlighting their usecases.
Shot Budget
Qubit Count
Midcire. Meas.
Hilbert
Model
Space
Layerwise
x
x
x
Ancilla
x
/
x
x
Blocking
x
/
Funnel
X
Solution 3: Funnel. Finally, for clients that may not have a high-shot budget, no ancillas, and no midcircuit measurement capability, design model that layer-by-layer drops operations that act

---

Published as a conference paper at ICLR 2026
Table 2: Capacity-wiss Comparison (V-Shape) Baselines + (bolded). The means and standard deviations are shown for five different samples of data. We see that Quorus-Layerwise consistently outperforms the baselines across client capacities.
with A to the Best
Copocity
Techuique
MINSST
Foshiio MNIST
48
Rap&Sandi
PallowenCive
Q-Hetzoff..
90.3 5:3 (1799)
38.6 126 (, 37.50
7.5 (] 70-7)
62.1 (1:36
67.1 14.6 (1 16.8)
5 6.1 1840
2L
Vanilla (2L)
960 (.0
40.0 0.9
= L. (.333)
91.0 1.2
76.0 1:0::00 0.4
Standalone
98.2 0.3
96.1
T85=25 L.))
6,0 (1..8)) 0.8)
91.2
0.7)
T49: (2 2.3)
Quma-
92.03:1144.0 (111
$6.0 ( L1)
162::004 1.80
98.8 st 0.9
86.1 6.1 (1.555)
76.5 1.4 0.5)
Q HetzroFt
19.6
(8.77
8..0 11.9)
6.7 ( 11.9)
71.5 a 15.0 22.00
79.1 12.0 18.00
20 10.5 ( 19.1)
91
Vanill (LL.
96.2::04:
0: (1.04)
L.
0.
Standslone
98.3
96.44-177 (
TU6-344 0.8)
09.1 0.6 (166610)) (10.1)
92.1
76.5 (] 2.1)
Quoms Ligerwise
1800 0.00
0.7
80.4
00.2 0.4
85.2 5.9 2.99
T8.0 +: 1.0
Q-HeciznFP.
80.1 (1 17.99
18011.3.. 95)
68.8
15.1)
6.5 (1888
85.6 2.0 5:1)
3.. (1 5.99
Vanilla (2L)
900+12 (.13)
900=03 (11.9)
U. (16.8)
11.0 (.5)
7 1.5)
Stunbllone
96.0 0:00 0.30
97.44 (101)
61.2+37 0.73
98.0 1:1 (1 (1..0)
98.7 =: 1.1
17:1 1.0 (] 1.5)
Quoms Ligerwise
0..
97.6 0.6
81.0
00.3 1
41.5
78.7 1.0
5655533 9.)
71.02
(1.5)
(1.5.0)
12.8
SL
Vanilla QPL (2L.
96.2 (1.00) 0:30
960 100+12111 1.5)
M00 (1 2.50
1.5==1 (1.68))
91.9 2.44.)
7 11 1.99
Standaone
97.2 (.01
$6.4 (.11)
90.3 (1222)
0,5 (1...8) 6.5)
94.3 0.6
17.6 1.5 (2 1.2)
Quma-Layerwicc
Q HettooFf.
6. ( 10.00
127)
d. (1222)
05.3 4.D
(1 (3:3.0
0444: 11
6L.
Manilla QFL. (2L)
98.2 (1 0.4)
$6.0 ( 1.)
80.0 (1 4.8.3)
98.5 L. (1 6.99
91.9 = 1.2 3.40
16.0 0.4 (1 1.9)
Standllnne
96.3 9:10 0:30
$6.5 (,
90.4 9.70
08.5 055--0. 0.8 1:39
96.3 1.0
15.4 (13..)
Quoms- Layerwise
05.6 0.8
97.8
83.1
90.4 0.3
(1 1.200
78.8
on the first qubit, allowing all measurements to be at the end (Killoran et al., depicted in Fig. b),, where we gradually "funnel" down the size of the deeper unitaries by dropping a qubit after each measurement, hence the name of this technique. The cost of this model is that the user must have a problem that is amenable to operating on fewer and fewer qubits.
2019) This model is
Fig.
Ansatz Use Case. We summarize the costs associated with each ansatz design in Table highlight their unique usecases. Note that each model has disjoint requirements that is, each model has exactly one cost, thus highlighting the versatility of our design choices to clients' unique scenarios. We evaluate each design in Sec. tito compare their accuracy performance.
5 EXPERIMENTAL EVALUATION
Here, we evaluate Quorus on different binary classification tasks. A comprehensive description of the experimental setup is in Appendix [ We evaluate with 128 datapoints per client, consistent with existing QFI. literature; to account for the random sampling of the data, we evaluate each of our comparisons with five different runs with five different samples of data allocations for clients. Data is reduced to 10 features using Principal Component Analysis (PCA), the 10-dimensional data is angle-encoded with RY gates, and the data is reuploaded using RY gates for each layer.
-ftt techniques in terms of classification accuracy. The state- of-the-art baselines that compare Quorus against are informed by what setups clients could run given that each has different depth model, based on existing techniques described in Sec. We com- pare against: (1) Q-HeteroFL, our quantum version of a classical technique called HeteroFL (Diao) et al. 2021). Here, all clients run the maximum depth model they can, and the parameters eraged only over the clients that contain those parameters. This work does not explicitly consider heterogeneous-moode federated learning using PQC. Thus, our design of the quantum version of HeteroFL itself is novel and described in Appendix C (2) Vanilla QFL, where all clients use the same depth model as the shallowest-depth client. For chients that are able to run deeper models, they are unable to fully utilize their quantum resources. (3) Standalone, where clients do not participate in the FL process and train the data on their own. This approach has the clear disadvantage that clients do not get the benefit of training an improved model from other clients' data.
Quorus outperforms
We present our results in Table comparing the baselines above to Auorus-Layerwiee because uses the same model architecture as the baselines. We present our results from the perspective of the client of different capacities in the leftmost "Capacity" column- - for that capacity, what is the best performing technique for the various class comparisons? We see that, across client capacities, for has the highest mean testing accuracy (12.4% over Q- Q-Hetero- does not have the highest testing
most comparisons, FL). Notably, for clients of the smallest capacity, accuracy, but this is likely due to its modified loss function, which penalizes the first layer param- eters, along with the loss values for clients with later layers. Another important observation is that
7

---

Published as a conference paper at ICLR 2026
Table 3: Capacity-wise Comparison (V-Shape) Quorus Variants Only with A the Best (bolded). The means and standard deviations are calculated over five runs: Quorus-Layerwise and Quorus- Funuel have the highest testing accuracy (we use them for subsequent experiments).
Capacily
Techniqqe
MNIST
Fashion-MNIST
34
49
Tronaz/Bort
BagSSndal
Pollownnooss
Layensise
1.4 (1.044
95.0 (1
78.2 (4
98.8 1
96.1 5.1.1 0.00
763 = 1.4 (.0. 78.5 # L.
Mauru-AucillaBaaae Blacking
97.0 (1 0.4)
918 (..)
78.3 1.4)
0.9 ( 0.2)
56.2 (4 (1.22)
Quonus Pannel
07.4 1.2
95.3 1.5
T0.7 2.0
08.1 1.1 (, (,0.7)
86.4 6.8
1. (,003)
Layersis
1.0 (1.0.1)
9..7
24 (4
d:
80.2 1.100)
1 (,0.3)
Qponue Anciliatllocting
97. 0.20
96.8 1 0.6
S1.4 20 (.M)
9912 0.5 (, 0.00
8 11 1.5)
785 0.2)
Qunnn-Funnel
18.1.200. 0.5
96.0 + 0.6 (1
#2.3
9899 0.5 0.39
90.2 3.3
76.7 + 1.1
Quentol syensise
97.5
$1.9+ 1:3)
01.5++10 (1.08)
(.7.)
4L
16.1:11.2 0..022)
07.3 ( 0:2)
61.5 () 1.7)
90.2 1:22::0. 0.5 ( 0.3)
5.. (1 2.0)
789 (, 0.5)
Quonu- Fancel
08.3
97.1
0.3)
79.4 1.0
Quoru- Layerwisz
98.8 (1,000)
97.5
2. 2.11
99.3 0.2
22.4 (1.03))
1.1 (L)
SL.
Quonus Ancillslllocinng
9..3 (1.0.2)
07.4 ( l.I)
S19 (1277))
VB. 0.3 (, 0.09
91.1 (1 1.6)
7 (L..4)
Qtoni-Panne
96.5
4 (1
9e. 0.20
Qnonu- Layerstse
08.6 1: 0.8
97.8 1: 0.2
8 (1 2.11
98.4 1: 0.3
WATT 2. (1.77)
758 (, 1.59
6L
Queme Ancill/looiii
98.1 + 0.0 0.21
87.5 0.5 (1
82.2+ (1
99.3 0.5 (, 0.3)
91.1 1.0 2.200
788++1.1 (.155) 1.55)
Quou--Fure
98.0 0.7 (1 0.00
97.1 () 0.7)
85.2 +
90.1 ( 0.3)
98.4 0.9
$0.3 0.9
2.5
0.5
0.5
Hateront
C-HettrofL Qucrus Layarabe
Q-teteroft Duonus Layeroiee
Goleeteofl
0.3
Quarua Lagune kn
0--
0.2
0.3
Quonux - Layeralue
0.2
0.2
01
0.2
0.1
450 Tpoch Number
600
Number
Spach Number
Fpoch Number
Rio
Ret
Rot
Rot
Roc
Roi
Roi
Roi
Layer
Layer
Layyrr
Layer
We show the per-layer magnitude of the gradients for Quorus-Layerwin by plotting the mean and standard deviation of the gradient norms for each epoch (smoothed for readability). Com- pared to Q-HeteroFL, we see that our modified loss function has larger gradient norms throughout training for parameters earlier in the circuit, due to earlier measurements.
Figure 7:
Q-HeteroFL. performs substantially worse compared Quorus-Layervin sometimes nearly 40% worse, as in MNIST 3/4 classification. This is due to the parameter mismatching challenge: differ- ent, confficting loss functions lead to suboptimal models. This highlights the importance of having a shared loss function between clients that can be optimized
The performance of the different variants of Quorus. We now evaluate the performance of the variants of Quorus in Table Note that, because the Ancill-Anciila and Booouu-Bloccing designs are logically equivalent, their accuracies are displayed together. We see that quorus-Layerwiis and Quorus-Funned have the best testing accuracy, although for many class comparisons, the difference between the techniques is within a single percentage point. This suggests that all of the Quorus have high testing accuracy, and that the decision of which model to use depends the resource constraints of the client, as mentioned in Table I In particular, for the Quorus-Funne, model, we observe that, even though later unitaries operate in a smaller Hilbert space, the testing accuracy is always within 1% of the best performing Quorus design, indicating its comparable accuracy. Importantly, the Quorus-Funnel model also shot-efficient, and for the hardest classes (MNIST 4/9, MaSHion-MNIST Pullover/Coat), it performs the best compared to the other models. Thus, we use the Quorus-Fumm Funnel model for subsequent noise analysis on real hardware evaluations.
ANALYSIS OF QUORUS'S FUNCTIONALITY
We split our analysis into two types: (1) gradient norms analysis and (2) analysis on real IBM superconducting hardware. Due to the extensive amount of rums required and the prohibitive cost of real-hardware runs, we only provide this analysis for the Pullover/Coat classification task from the Pashion-MNIST dataset this is the most challenging task.
8

---

Published as a conference paper at ICLR 2026
100
80
Accuracy (96)
80
68%
72%
73%
74%
76%
79%
75
7.6
775%
75%
78%
Accuracy (%)
60
48%
-9%
67%
-11%
-13%
-20%
20
64%
Ideal Sim,
62%
60
Brisbane
Kingston
Fez
Pattburrh
Marrakesh
Torino
Ideal Sim.
Kingston
58%
55
Quantum Computer
Depth Layers)
(a) Same Model, Different Machines
(b) Different Models, Same Machine
Figure 8: With the Quorus-Fumin model, show (a) how the same model has varying testing accuracy based on the real machine used, and (b) how using smaller depths leads to higher testing accuracy machine. This highlights the importance of clients using depths that they can accu- rately evaluate the model on, as well as the practical hardware relevance of our experimental setup.
Table 4: Capacity-wise Comparison (V-Shape) Q-HeteroFL vs Quorus variants CIFAR-10 with to the Best (Bolded). The means and standard deviations are calculated over five runs: Quorus-Layerwin and Quorus-Funnel outperform Q-HeteroFL. all comparisons.
Capacity
Technique
CIFAR-10
Deer/Truck
Antomobile/Trucs
Cat/Dog
Q-HeteroFL
65.0 5.8 ( 11.7)
55.8 +: 3.1 (.9)
50.5 1.0 (1 4.9)
2L
Quorus-Lyerrwiee
76.7 + 18
61.0 2.9 0.7)
55.4 + 1.5
Quorus-Funnel
76.5 (4
61.7 1.9
54.7 (1 0.7)
Q Q-HeteroFL.
63.7 1: ( 13.5)
53.8 2.8 (J 8.3)
52.8 1.3 () 2.9)
3L
Quyrus-ayerwise
76.9 1.6 (1 0.3)
61.7 0.4)
55.7 + 1.5
Quorus- Funnel
62.1
$5.6 (L 0.1)
Q-HeteroFL
2. 2.6 (L 6.2)
56,5 + 4.2 (....)
516622 (4 3.9)
4L
Quorus-Layerwise
77.5 (.8)
1.0 (.1.
55.5 + 1.4
Quones-Funnel
78.3
1.5
55.1 (L 0.4)
Q-HeteroFL.
6.10 ( 10.4)
56.8 1.2 () 6.9)
52.8 1.7 (1
5L
Quorus-ayerwise
77.7.1.3 0.7)
62.6 0.1)
55.7 + 1.2
Quonus-Funnel
78.4
62.7 1.4
55.4 (1 0.3)
Q QeHeteroFL.
70.6 2.6 2.6 (1 7.8)
57.3 + 1.6 (.8)
51.5 1.4 (4.41)
6L
Quonus-Laaerwiin
1.3 0.6)
62.6 - 1.0
55.6 1.1
Muorus-Funnee
1.5 (, 0.3)
55.2 (L.0.4
Higher Gradient Norms with Quorus. Typically, the deeper the circuit, the smaller the magnitudes of the gradients become (Cerezo al.) 202Ip. We verify this empirically in setup well, plotting the gradient norms of Quorus- Layeru-- and Q-HeteroFL in Fig. 7 We see that for the Q-HeteroFL. model, the gradient norms small for each layer in the quantum circuit, the loss for all parameters defined on the output measurement at the end of the circuit. Interestingly. for have loss functions that are defined after each layer, for
Quorus, this not the case. Because deep models, earlier layers maintain a high gradient norm due to these layerwise loss functions. Note that for parameters in later layers, the gradient norms remain small because the gradients for these parameters only depend on measurements deep in the circuit. However, the overall magnitudes of the gradients for Quorus are higher, and addition, Quorus also obtains higher testing accuracy than Q-HeteroFl., making it implausible that the reasons for the larger gradient norms are due to lack of convergence for Quorus. This result suggests how the layerwise loss function Quorus can be used for improved and scalable trainability for deep quantum circuits.
Evaluation of Quorus on Real Quantum Hardware. We evaluate our trained models on all of IBM's superconducting quantum processing units or QPUs to demonstrate the practical relevance of our experimental setup, as well as the very real impact that noise has on our trained models. In particular, we perform our hardware analysis using the Quorus-Funnel design, because it allows for evaluating the ensembled model with no shot overhead. Due to the high error of midcireuit measurements current hardware, measure all qubits at the end (Rudinger al. 2021
Gao

---

Published as a conference paper at ICLR 2026
Table 5: Capa Comparison (V-Shape) Q-HeteroFL vs Quorus variants on 4-class MNISSTFashion-MNISS with A to the Best (Bolded). The means and standard deviations are cal- culated over five runs: Quorus-Layerwise and Guorus-Funnel outperform Q- P-HeteroFL. on all tasks.
Caparity
Technlgue
TesPPulove
46.00
Qumme-Trrrd
76.7 1 3.7
65.2 10.1
550 (, 6.9)
THE 1 3.5
78.4 1.6
48.9 3.3
O-HstzuFL
41.4)
465:877
05.42:50 200)
S8A = $5.50
TL
Quatts Layewie
$0.9 2.7
71.1 112 8.1)
$1.0 4.7
71.4+-5444 (1
G3.1 23. (1 5.00
02.0 = 26 14
Quirts Fored
80.1 1: (.)
67
00 (.0.8)
80.5 2 4.5
T4.1 1.2
50.1 2:3.0
NEW 12.9 20.9)
15.1 16.11-47 17:00
15/1 (1 23-51
16.1 1.111600 1901)
Ni13 1:57 () 17:00
4L
Qaert Layewiie
51.5 4:
784 1.0
75.:: (1044)
72
bi.1 1411:108 LLB 3.0)
Qunme-Pored
81 111---0--1777
T6.3
63.6 (,
2.8
57.1 9.4
HeaesRt.
56.0 50.7 $7.00
51.
$16
BiC)
5L
0.10 18.0
67. L.)
Quarts Funrd
90.
8..3
TA.0
58.1
O-HotouRL
71.7
714=28 (L...))
53.3 (1
TLB=25
60.667.55 14.0)
46.9 10.50
6L
M6.2
79.5
67.6 2.9
(1
Mill 11
Quana- Furrd
7.7 (2)
W fu LJIt
22.5
15.2 :: 3.8
$7.4
et al. 2025). The depth of the Quorus Funnell models therefore matters, for deeper models, more decoherence will accumulate on the qubits that are unused or carry information from earlier layers.
(A) Same Model, Different QPUs. We evaluate Quorus with depth of 5 on different IBM QPUs to validate the heterogeneity of quantum systems. We restricted our testing set to only 100 datapoints due to the prohibitive cost of each shot. Our results in Fig. (aa show that across six different QPUs, the noise varies substantially: from 48% for IBM Brisbane to 76% for IBM Torino, 3% off from the ideal simulation accuracy. If a client has access to a machine with similar hardware noise characteristics to IBM Torino, they should go with a deep circuit. Thus, we observe diverse spectrum of error on IBM's QPUs, validating the practical relevance of our experimental setup.
(B) Different Depth Model, Same QPU. In Fig. B(a), we notice that the QPU with the lowest testing accuracy (aside from IBM Brisbane, which suffered from decoherence to 48% testing accuracy with just two layers of our model) is IBM Kingston. Thus, we performed an analysis on IBM Kingston to verify the impact of decreasing the depth of the quantum circuit on the testing accuracy; we expect that we reduce the number of layers, the testing accuracy should increase. Our hypothesis empirically validated in Fig- b)). We see that the separation, indicated by dashed red line, between the ideal simulation results and the testing accuracy on IBM Kingston gets wider with deeper circuit. This highlights that, for a client with access to a computer similar to IBM Kingston in terms of hardware noise, it is advantageous for them to train shallow-depth quantum classifier, because these classifiers have lower-error outputs and can more meaningfully contribute to FL..
7 ABLATION: ADDITIONAL CLASSIFICATION TASKS
To evaluate Quorus on more complex classification tasks, we additionally run and Quorus-Funnel for binary classification on CIFAR- 10 in Table four-class classification for MNIST/Fashion-MNNI
data in Table 5 (categorical cross-entropy loss used: further details are in Appendix (.2). We see an average improvement of 6.7% in CIFAR-I0 binary classification and 24.0% in SANSST/Fashon-MNNII four-class classification of the best performing variant of Quorus over H-HttrroFL, highlighting Quorus's 's robustness to more challenging classification tasks.
8 CONCLUSION
In this work, we introduced Quorus, a QFL framework tailored for heterogeneous- clients. Our contributions include: (1) a layerwise loss with high gradient norms to align objectives across clients of varying circuit depths, (2) multiple circuit designs, Layerwise, Ancilla/Blocking, and Funnel, that balance accuracy with resource constraints, and (3) extensive evaluation showing up to 12.4% accuracy improvements over Q-HeteroFL and consistently higher gradient magnitudes for deeper clients. Crucially, we validated Quorus on all of IBM's s superconducting quantum processors, demonstrating that method is not only effective in simulation but also practical on today's error- prone hardware. Together. these results establish Quorus as the first implementable framework for QFI in realistic multi-client settings, paving the way for scalable and resource-aware QMI..
10

---

Published as a conference paper at ICLR 2026
ACKNOWLEDGEMENT
This work was supported by Rice University, the Rice University George R. Brown School of Engi- neering and Computing, and the Rice University Department of Computer Science. This work was supported by the DOE Quantum Testbed Finder Award DE-SCW024301, the Ken Kennedy Institute, and Rice Quantum Initiative, which is part of the Smalley-Curl Institute. We acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Quantum team.
REFERENCES
IBM Quantum Computing - IBM Quantum Computing - Products and services ibm.ccm. http:///pppooootctooooooeeeeeeee
https://wwwt ibm. wtt../oo/aa.. 09-2025].
[Accessed 24-
Rajeev Acharya, Dmitry A. Abanin, Laleh Aghababaie-Bonl Benii, Igor Aleiner, Trond I. Andersen, Markus Ansmann, Frank Arute, Kunal Arya, Abraham Asfaw, Nikita Astrakhantsev, Juan Ata- laya, Ryan Babbush, Dave Bacon, Brian Ballard, Joseph C. Bardin, Johannes Bausch, Andreas Bengtsson, Alexander Bilmes, Sam Blackwell, Sergio Boixo, Gina Bortoli, Alexandre Bourassa, Jenna Bovaird, Leon Brill, Michael Broughton, David A. Browne, Brett Buchea, Bob B. Buckley, David A. Buell, Tim Burger, Brian Burkett, Nicholas Bushnell, Anthony Cabrera, Juan Campero, Hung-Shen Chang, Yu Chen, Zijun Chen, Ben Chiaro, Desmond Chik, Charina Chou, Jahan Claes, Agnetta Y. Cleland, Josh Cogan, Roberto Collins, Paul Conner, William Courtney, Alexan- der L. Crook, Ben Curtin, Sayan Das, Alex Davies, Laura De Lorenzo, Dripto M. Debroy, Sean Demura, Michel Devoret, Agustin Di Paolo, Paul Donohoe, Ilya Drozdov, Andrew Dunsworth, Clint Earle, Thomas Edlich, Alec Eickbusch, Aviv Moshe Elbag, Mahmoud Elzouka, Catherine Erickson, Lara Faoro, Edward Farhi, Vinicius S. Ferreira, Leslie Flores Burgos, Ebrahim Forati, Austin G. Fowler, Brooks Foxen, Suhas Ganjam, Gonzalo Garcia, Robert Gasca, Élie Genois, William Giang, Craig Gidney, Dar Gilboa, Raja Gosula, Alejandro Grajales Dau, Dietrich Grau- mann, Alex Greene, Jonathan A. Gross, Steve Habegger, John Hall, Michael C. Hamilton, Monica Hansen, Matthew P. Harrigan, Sean D. Harrington, Francisco J. H. Heras, Stephen Heslin, Paula Heu, Oscar Higgott, Gordon Hill, Jeremy Hilton, George Holland, Sabrina Hong. Hsin Yuan Huang, Ashley Huff, William J. Huggins, Lev B. loffe, Sergei V. Isakov, Justin Iveland, Evan Jeffrey, Zhang Jiang, Cody Jones, Stephen Jordan, Chaitali Joshi, Pavol Juhas, Dvir Kafri, Hui Kang, Amir H. Karamlou, Kostyantyn Kechedzhi, Julian Kelly, Trupti Khaire, Tanuj Khattar, Mostafa Khezri, Seon Kim, Paul V. Klimov, Andrey R. Klots, Bryce Kobrin, Pushmeet Kohli, Alexander N. Korotkov, Fedor Kostritsa, Robin Kothari, Borislav Kozlovskii, John Mark Kreike- baum, Vladislav D. Kurilovich, Nathan Lacroix. David Landhuis, Tiano Lange-Dei, Brandon W. Langley. Pavel Laptev, Kim-Ming Lau, Loick Le Guevel, Justin Ledford, Joonho Lee, Kenny Lee, Yuri D. Lensky, Shannon Leon, Brian J. Lester, Wing Yan Li, Yin Li, Alexander T. Lill, Wayne Liu, William P. Livingston, Aditya Locharla, Erik Lucero, Daniel Lundahl, Aaron Lunt, Sid Madhuk, Fionn D. Malone, Ashley Maloney, Salvatore Mandrå, James Manyika, Leigh S. Martin, Orion Martin, Steven Martin, Cameron Maxfield, Jarrod R. McClean, Matt McEwen, Seneca Meeks, Anthony Megrant, Xiao Mi, Kevin C. Miao, Amanda Mieszala, Reza Molavi, Se- bastian Molina, Shirin Montazeri, Alexis Morvan, Ramis Movassagh, Wojciech Mruczkiewicz, Ofer Naaman, Matthew Neeley, Charles Neill, Ani Nersisyan, Hartmut Neven, Michael New- man, Jiun How Ng, Anthony Nguyen, Murray Nguyen, Chia-Hung Ni, Murphy Yuezhen Niu, Thomas E. O' OrBri, William D. Oliver, Alex Opremcak, Kristoffer Ottosson, Andre Petukhov, Alex Pizzuto, John Platt, Rebecca Potter, Orion Pritchard, Leonid P. Pryadko, Chris Quintana, Ganesh Ramachandran, Matthew J. Reagor, John Redding. David M. Rhodes, Gabrielle Roberts, Eliott Rosenberg. Emma Rosenfeld, Pedram Roushan, Nicholas C. Rubin, Negar Saci, Daniel Sank, Kannan Sankaragomathi, Kevin Satzinger, Henry Schurkus, Christopher Schuster, Andrew W. Senior, Michael J. Shearn, Aaron Shorter, Noah Shutty, Vladimir Shvarts, Shrad- dha Singh, Volodymyr Sivak, Jindra Skruzny, Spencer Small, Vadim Smelyanskiy, W. Clarke Smith, Rolando D. Somma, Sofia Springer, George Sterling, Doug Strain, Jordan Suchard, Aaron Szasz, Alex Sztein, Douglas Thor, Alfredo Torres, M. Mert Torunbalci, Abeer Vaishnav. Justin Vargas, Sergey Vdovichev, Guifre Vidal, Benjamin Villalonga, Catherine Vollgraff' Heidweiller, Steven Waltman, Shannon X. Wang, Brayden Ware, Kate Weber, Travis Weidel, Theodore White, Kristi Wong, Bryan W. K. Woo, Cheng Xing, Z. Jamie Yao, Ping Yeh, Bicheng Ying, Juh-
11

---

Published as a conference paper at ICLR 2026
Yoo, Noureldin Yosri, Grayson Young, Adam Zaleman, Yaxing Zhang, Ningfeng Zhu, and Nicholas Zobrist. Quantum error correction below the surface code threshold. Nature, 638
(8052):920-95)) 926, December 2024. ISSN 1476-4687. doi: 10.10388/41156--24---4484 htttp://xx.doe http::///dx..ii.rg//1....... org/10. 1038/110 s41//415555 5864-166--- 1053/s51188---24-444445-1 024-08449-
URL
Aminpour, Y. Banad, and S. Sharif. Strategic data re-uploads: A pathway to improved quantum classification data re-uploading strategies for improved quantum classifier performance, 2024.
URL htthhtss://axxi org/ http:/bbs22400... 09377
Eric Anschuetz. A unified theory of quantum neural network loss landscapes. In Y. Yue, A. Garg, N. Peng. F. Sha, and R. Yu (eds.), International Confer- ence on Representation Learning, volume 2025, pp. 97859-979118 2025. URL http::///pooeeeenn.s.....rrrrrrr 32800a214446007444455562222----3 Confereennnneeeceee.e.
A. R. Calderbank and Peter W. Shor. Good quantum error-correcting codes exist. Physical Review
A. 5492):1098-11055 August 1996. ISSN 1094-1622. doi: 10.1103/physrev.54..0090 URL
http://dx. org/ wtt..r//....1////seeeeeeeeeo/ 54 1098
M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J. Coles. Cost function dependent barren plateaus in shallow parametrized quantum circuits. Nature Communicationss 12(1), March 2021. ISSN 2041-1723. doi: 10.
w. URL http::ttpp....oo org/
10. 10.1103// 64146 - 21728-228
Jwo-Sy Chen, Erik Nielsen, Matthew Ebert, Volkan Inlek, Kenneth Wright, Vandiver Chaplin, An- drii Maksymov, Eduardo Páez, Amrit Poudel, Peter Maunz, and John Gamble. Benchmarking a trapped-ion quantum computer with 30 qubits. Quantum, 8:1516, November 2024. ISSN 2521-327X. doi: 10.22331/q 2--2333/--222-11111111
URL URLhhttp://// http::///xxxdoo..ooo http:////oo......o/111022222222 22331/
g 20224-11-07222
Samuel Yen-Chi Chen and Shinjae Yoo. Federated quantum machine learning. Entropy, 23(4):460, 2021.
Matthew DeCross, Eli Chertkov, Megan Kohagen, and Michael Foss-Feig. compi-rl with mid-circuit measurement and reset, 2022. URL httpss//arrivv.orr//a//////
08039
Emma Deist, Yue-Hui Lu, Jacquelyn Ho, Mary Kate Pasha, Johannes Zeiher, Zhenjie Yan, and
Dan M. Stamper- Kurn. Mid-circuit cavity measurement in neutral atom array. Phys. Rev. Lett., 129:203602, Nov 2022. doi: 10. 1 103/P0ysseellleettt 223620. UtR.:tttss///ingg .pps.
org/doi/100 11 http///..11.////yeeeeeeettttttt. 129. 203602
Enmao Diao. Jie Ding, and Vahid Tarokh. Heterofl: Computation and communication efficient
federated learning for heterogeneous clients, 2021. /
01264
Nicholas S. DiBrita, Jason Han, and Tirthak Patel. Resq: A novel framework to implement residual neural networks on analog rydberg atom quantum computers, 2025. URL httpp::/arxxv
org/aba/ /aaba/2506..0 21537
Simon J. Evered, Dolev Bluvstein, Marcin Kalinowski, Sepehr Ebadi, Tom Manovitz, Hengyun Zhou, Sophie H. Li, Alexandra A. Geim, Tout T. Wang, Nishad Maskara. Harry Levine, Giu- lia Semeghini, Markus Greiner, Vladan Vuletić, and Mikhail D. Lukin. High-fidelity parallel entangling gates on neutral-aton quantum computer. Nature, 622(7982):268 272, October 2023. ISSN 1476-4687 doi: 10. 00.1088/411586-033-066333 10. 10003//415880 023 06481-y
URL htt.:htt://// doi org/
Yang Gao, Feiyu Li, Yang Liu, Zhen Yang, Jiayu Ding, Wuerkaixi Nuerbolati, Ruixia Wang, Tang Su, Yanjun Ma. Yirong Jin, Haifeng Yu. He Wang, and Fei Yan. Mitigating measurement crosstalk via pulse shaping, 2025. URL wntptttt://// arxiv. org/ abs 25009.09.. 05437
Craig Gidney and Martin Ekerå. How factor 2048 bit rsa integers in 8 hours using 20 million noisy qubits. Quantum, 5:433, April 2021. ISSN 2521-327X. doi: 10,22331/q 20-22000-11---00 URL http::///// //dp: doi htt...oo.......33/22220200
12

---

Published as a conference paper at ICLR 2026
Daniel Gottesman. The heisenberg representation of quantum computers. arXiv preprint quant- ph/9807006, 1998.
Andreas Grammenos, Rodrigo Mendoza Smith, Jon Crowcroft, and Cecilia Mascolo. Federated principal component analysis. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin
(eds.), Advances in Neural Information Processing Systems, volume 33, pp. 6453-6464. Cur- ran Associates, Inc., 2020. /ttpo/////////ppceeeooo.ee.erraa
cc/paper files/ 632007/2200111112222222232266662
Yuri Gurevich and Andreas Blass. Quantum circuits with classical channels and the principle of deferred measurements, 2021. /RRpsstss://axxx 2ttp://abb22110..0888
Gaurav Gyawali, Sonny Rappaport, Tiago Sereno, and Michael J. Lawler. Measurement-induced
landscape transitions and coding barren plateaus in hybrid variational quantum circuits, 2024.
URL wttptstsss//xii...o.oa http:/bb//2311 09135
Jason Han, Nicholas S. DiBrita, Younghyun Cho, Hengrui Luo, and Tirthak Patel. EnQode: Fast
Amplitude Embedding for Quantum Machine Learning Using Classical Data. In 2025 62nd ACM/IEE Design Automation Conference (DAC), pp. 1-8. IEEE, 2025.
Zoö Holmes, Kunal Sharma, M. Cerezo, and Patrick J. Coles. Connecting ansatz expressibil- ity to gradient magnitudes and barren plateaus. PRX Quantum, 3(1), January 2022. ISSN 2691-3399. doi: 10.1 10.1000/quuantumm3..0000. PRXQuant 3. 010313.
URL http://dxx doi. www.eo.......oooeeooeooooeeeeeee 110310
Yuqian Huo, Jinbiao Wei, Christopher Kverne, Mayur Akewar, Janki Bhimani, and Tirthak Patel. Revisiting noise-adaptive transpilation in quantum computing: How much impact does it have? IEEE/ACM International Conference on Computer-Aided Design (ICCAD), 2025.
Fatih Illan, Gong Su, and Ling Liu. Scalefl: Resource-adaptive federated learning with hetero- geneous clients. In 2023 IEEE/CVF Conference on Computer Vision and Pattern Recognition 2023. doi: 10. 10.1109/CVPR52729 2003.0235.....
(CVPR), pp.
Muhammad Kashif, Alberto Marchisio, and Muhammad Shafique. Computational advantage in hybrid quantum neural networks: Myth or reality? In 2025 62nd ACM/IEEE Design Automation Conference (DAC), pp. 1-7, 2025. doi: 10. (0.1109/DA638849.0000 1113290
Nathan Killoran, Thomas R. Bromley, Juan Miguel Arrazola, Maria Schuld, Nicolás Quesada, and Seth Lloyd. Continuous-variable quantum neural networks. Phys. Rev. Res., 1:033063, Oct 2019. doi: (0L1003/hhyseeeReeaara1...13333 1.033033 URL /tR/sttpss///nn //ttpk aps org/doi/1C. 1103/ 1L03/Physkeeteeearch..1..
Minjae Kim, Sangyoon Yu, Suhyun Kim, and Soo-Mook Moon. DepthFI. : Depthwise federated learning for heterogeneous clients. In The Eleventh International Conference on Learning Repre-
sentations, 2023. URL httptttss//ppeeeevieee www.:nneee/tooeieeeeeeee wet.ee//ooooo..ddder.......
Jan D. Kivlichan, Jarrod McClean, Nathan Wiebe, Craig Gidney, Alán Aspuru-Guzik, Garnet Kin- Lic Chan, and Ryan Babbush. Quantum simulation of electronic structure with linear depth and connectivity. Physical Review Letters, 120(11), March 2018. ISSN 1079-7114. doi: 10. 1t0.://physseeee......10003. 12c. 110501
URL http:/ //tx//xxx//// doi. org 10. 1ttg//0...110///aeeeetttt..
Lange, I.. Heunisch, H. Fehske, D. P. DiVinennzo, and M. J. Hartmann, Cross-talk supercon- ducting qubit lattices with tunable couplers comparing transmon and fluxonium architectures, 2025. /ttp:tt/s///xxi...... abs
Royson Lee, Javier Fernandez-Marques Shell Xu Hu, Da Li, Stefanos Laskaridis, Lukasz Dudziak, Timothy Hospedales, Ferenc Huszár, and Nicholas D. Lane. Recurrent early exits for federated learning with heterogeneous clients. In Proceedings of the 41st International Conference on Machine Learning, ICML'24. JMLR.org, 2024.
Sydney Leither, Michael Kubal, and Sonika Johri. How many qubits does a machine learning prob- lem require? arXiv preprint arXiv:2508.20992 2025.
13

---

Published as a conference paper at ICLR 2026
Chen- Yu Liu, Kuan-Cheng Chen, Yi-Chien Chen, Samuel Yen-Chi Chen, Wei-Hao Huang, Wei-Jia Huang. and Yen-Jui Chang. Quantum-enhanced parameter-efficien learning for typhoon trajec- tory forecasting, 2025. URL htLLtttss///axii..oom
abs / 09395
Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Alán Aspuru-Guzik The theory of vari- ational hybrid quantum-classical algorithms. New Journal of Physics, 18(2):023033, February 2016. ISSN 1367-2630. doi: 10. 1088/1 367-2630/1 1/2/02333/// URL 00.1088/13T1-2333/11////22333333
H. Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Agüera Arcas. Communication-efffcceennl leaming of deep networks from decentralized data, 2023. URL https:
//arxiv.orr/abb//16/00 /000 05629
J. A. Montanez-Barrera, Kristel Michielsen, and David E. Bernal Neira. Evaluating the performance of quantum processing units at large width and depth, 2025. URL attps://arxiin .rrg//
abs/ 20a/25504 2500.06471
Tuyen Nguyen, Incheon Paik, Yutaka Watanobe, and Truong Cong Thang. An evaluation of hardware-efficient quantum neural networks for image data classification. Electronics, 11(3), 2022. ISSN 2079-9292.
doi: 10.33390eeectrroniis111333 1030437. URL https: /wwwwwww.o mdpi..
com/ 2079 0177999222/11/3/433
Yash J. Patel, Akash Kundu, Mateusz Ostaszewski, Xavier Bonet-Monroig, Vedran Dunjko, and Onur Danaci. Curriculum reinforcement learning for quantum architecture search under hard- ware errors. In The Twelfth International Conference on Learning Representations, 2024. URL
Ittpps:/cpenreviee net. 8
David Peral-Gareía, Juan Cruz-Benito, and Francisco José Garcla-Peñalvo. Systematic literature review: Quantum machine learning and its applications. Computer Science Review, 51:100619, 2024. ISSN 1574-0137.
doi: ott.a///.w.aaaaaaaoooooooooaaaaa
100619. URL https://
www. wwwweceeeeeeeee.....c....eeeeeee
John Preskill. Quantum computing in the nisq and beyond. Quantum, 2:79, August 2018. ISSN 2521-327X. doi: 10.22331/q-20188 201--08--7-7
URL httt:://dx.do..orggg//.
q 2018 08 06-06--7 79
Ratun Rahman, Atit Pokharel, and Dinh C. Nguyen. Sporadic federated learning learning approach in quan- tum environment to tackle quantum noise, 2025. URL
12492
Aditya Ranjan, Tirthak Patel, Daniel Silver, Harshitta Gandhi, and Devesh Tiwari. Proximl: Build- ing machine learning classifiers for photonic quantum computing. In Proceedings of the 29th
ACM International Conference on Architectural Support for Programming Languages and Op- erating Systems, Volume 3, ASPLOS '24, pp. 834-849, New York, NY, USA, 2024. Associa- tion for Computing Machinery, ISBN 9798400703867.
doi: 10.1145/362066633651333 URL
https:///dl..org/00.. 1141
wtt..//r//......./....... 010114553226666.33 3651367
Jonathan Romero, Ryan Babbush, Jarod R. McClean, Cornelius Hempel, Peter Love, and Alán Aspuru-Guzik. Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz, 2018.
wwtp://..g////aaea.o.oooooosss/ /1001 02691
Kenneth Rudinger, Guilhem J. Ribeill, Luke C. G. Govia, Matthew Ware, Erik Nielsen, Kevin Young, Thomas A. Ohki, Robin Blume-Kohour and Timothy Proctor. Characterizing mid- circuit measurements on a superconducting qubit using gate set tomography, 2021. URL
ptttps://aaxxivorr///bbb/ 20000/000000000000000 03008
Himanshu Sahu and Hari Prabhat Gupta. Nac-qfl: Noise aware clustered quantum federated learn-
ing, 2024. URL utRpssss:: http:p::///rxxv...x abs/ /4///44666.0 14236
Maximilian Schlosshauer. Quantum decoherence. Physics Reports, 831:1-57, 2019. ISSN 0370-1573. doi: http:///bbooggglooooooipppppoppp [cciencddirect ottpa//cccc....a..oo.oo.ooaaaaaa coherence.
URL https: :/to://wwr //www.
Quantum de-
14

---

Published as a conference paper at ICLR 2026
Maria Schuld, Alex Bocharov, Krysta M. Svore, and Nathan Wiebe. Circuit-centric quantum clas- sifiers. Physical Review A, 101(3), March 2020. ISSN 2469-9934, doi: 10.1103/physreaa.10.. 032308. UtL dx.
101. 032338
Jaime Sevilla and C. Jess Riedel. Forecasting timelines of quantum computing, 2020. URL https:
05045
Sukin Sim, Peter D. Johnson, and Alán Aspuru-Guzik, Expressibilliyy and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms. Advanced Quan- tum Technologies, 2(12), October 2019. ISSN 251 2594-4. doi: 10. 10.1002/qute.20) 2019tee0010000000 URL
http://dx.co www.oogggo.... 1002 quteee/////ueeeoeoooeoeeeeeeeeee 201900070
Anthony M. Smaldone, Yu Shee, Gregory W. Kyro, Chuzhi Xu, Nam P. Vu, Rishab Dutta, Marwa H. Farag, Alexey Galda, Sandeep Kumar, Elica Kyoseva, and Victor S. Batista. Quantum machine learning in drug discovery: Applications in academia and pharmaceutical industries. Chemical Reviews, 125(12))533-5546) 2025, doi:
URL https://doi.
org/ 00010111100 acs chemrev. 4000678
PMID: 40479601.
Kenji Sugisaki, Takumi Kato, Yuichiro Minato, Koji Okuwaki, and Yuji Mochizuki. Variational quantum cigensolver simulations with the multireference unitary coupled cluster ansatz: a case study of the C2. quasi-reaction pathway of beryllium insertion into H2 molecule. Physical Chemistry Chemical Physics, 24(14):8439 8452, April 2022. doi: 10. 10.1039/DICP041180 18H.. URL
battps://dee waa.i..ogg/o... 1039 1ttp://00111991110005533 D1CF044111
Nahum Sá, Ivan S. Oliveira, and Itzhak Roditi. Towards solving the bes hamiltonian gap in near- term quantum computers. Results in Physics, 44:106131, January 2023. ISSN 2211-3797 doi: ritp:ii.rinp..22.110.. 106131
URL http: doi org 1t..g//111.111.... rinp 2022.
Swamit S. Tannu and Moinuddin K. Qureshi. Not all qubits are created equal: A case for variability-awarl -aaae policies for nisq-era quantum computers. In Proceedings of the Twenty-Foud Fourth International Conference ing Systems, ASPLOS '19, pp. 987-999, New York, NY, USA, 2019. Association for Com- puting Machinery, ISBN 978 9781450362405.
Architectural Support for Programming Languages and Operat-
doi: 10.145/3297558.3300007. URL https:
3304007
Javier Gil Vidal and Dirk Oliver Theis. Input redundancy for parameterized quantum circuits, 2020.
htRtt::::///aa org/ attp//as..11..../.../ 11434
Jianyu Wang, Zheng Xu, Zachary Garrett, Zachary Charles, Luyang Liu, and Gauri Joshi. Local adaptivity in federated learning: Convergence and consistency, 2021. URL htttss://arxt:
org/abs/ 2206. 02305
David Wierichs, Josh Izaac, Cody Wang. and Cedric Yen-Yr Yu Lin. General parameter-shift rules for quantum gradients. Quantum, 6:677, March 2022. ISSN 2521-327X. doi: 10.22331/
10022-03-30-677 http:/t/://d doi. org. 10.
30-
Ge Yan, Hongxu Chen, Kaisen Pan, and Junchi Yan. Rethinking the cymmetry-preservinn cir- cuits for constrained variational quantum algorithms. In The Twelfth International Confer- on Learning Representations, 2024. URL SL7didVpde
Won Joon Yun, Jae Pyoung Kim, Hankyul Baek, Soyi Jung, Jihong Park, Mehdi Bennis, and Joongheon Kim. Quantum federated learning with entanglement controlled circuits and super- position coding, 2022. URL /tt.htppss///rvvvoor///sssss
/211.. 01732
Kai Zhang, Yutong Dai, Hongyi Wang, Eric Xing, Xun Chen, and Lichao Sun. Memory- adaptive depth-wise heterogeneous federated learning, 2025. URL httptt:ss//axii.....
abs/2303.00 04887
Wojciech Hubert Zurek. Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys., 75:715-775, May 2003. doi: 10. N0.1103/RewoodPhy.7755 URL https:
aps. htt.i//aap...rr/////////////////
715
15

---

Published as a conference paper at ICLR 2026
A FURTHER PRELIMINARIES
Quantum Gates. Quantum gates are unitary operators that manipulate qubits a quantum circuit, analogous to logic gates in classical circuits. The rotation gutes apply continu- ous rotations of a qubit's state on the Bloch sphere (Fig. 9 For example, the (0) gate performs rotation around the y-axis by angle 0:
10)2
(4)
Rz
RAY
(0) = cos( ( 2
sin ()
sin
cos
More generally, the three-axis rotation operator Rot(a, 8,7) applies successive rotations about the æ, y, and axes by an- gles a, and 7,:
Rot(a, 3,7) R2(a) R (3) R2(1),
Figure 9: Visualization of rotation
gates quantum Bloch sphere.
where
id- 0
(o)
e/0/2
Entangling gates act on two or more qubits. A key example is the controlled-NOT (CNOT) gate, which flips the target qubit if the control qubit is in state [1). Its 4 x 4 unitary matrix is:
0
0
0
0
1
0
0
CNOT
0
0
0
0
0
The gate representation in circuit diagram has the target qubit and the control qubit connected with vertical line, with the control qubit indicated by a filled-in circle and the target qubit indicated by the @ symbol. Together, single-qubit rotation gates and entangling gates like CNOT form universal gate set, capable of approximating any quantum operation.
B PROOF THAT THE QUORUS- CCCOUU--ANCILA. ANCILLA CIRCUIT IS EQUIVALENT TO THE QUORUS-BLAN BLOCKING CIRCUIT
In this section, prove that the circuits in Fig. and Fig. 6(a), namely, the Ancilla technique and the blocking technique, are equivalent. To do so, we will consider the state both circuits immediately after the measurement operation.
We first note that both circuits apply the same unitary U (0) 3n leb). We note that the same logic applies for subsequent layers (replacing (15) with the resulting input state will suffice), so analyzing this single-layer setup sufficient.
Additionally, we assume that, in the Ancilla circuit, the ancilla qubit is immediately measured af- ter the CNÔT gate. This simplifies the analysis and is equivalent to the case where the ancilla is measured later, as no other operations are performed on the ancilla qubit, so we apply the deferred measurement principle (Gurevich & Blass 202T).
Proposition 1 AAncilla-measurement equals measuring the control). Let U be an n-qubit unitary and let (b) U (0) 0n Write (D)) as
(e) = a (0) (6o) + [1) lo1).
where the first ket is qubit 0, (0b) are normalized states of the remaining n qubits, and |a|? + | = 1. Consider two procedures:
16

---

Published as a conference paper at ICLR 2026
Table 6: Experimental details, specification, and hyperparameters used to evaluate Quorus.
Parameter
Value
Dataset
MNIST; Fashion-MNiSs
Classes
0/1, 3/4, 4/9 for MNIST; Trouser/Boot, Bag/Sandal, Pullover/Coat for Fashion-MNIST
Number of clients K Datapoints per Client Testing set size
128
3000; 100 for hardware runs only
Data Distribution
IID
Number of different data splits per class comparison
Data encoding scheme
Angle Embedding
Client sampling per round Communication rounds I
100%
1000
Local epochs per round E
Batch size
32
Optimizer
Adam 12-=0.9,,2200999
Learning rate y
0,001
I.R schedule
1.0 (No decay)
Loss type
Binary Cross Entropy on Labels
KL Divergence between logits
Aggregation Method
Circular averaging of subnet parameters
Qubits Q
10
Depth levels L
2L. 3L, 4L, 5L. 6L
Parameters per layer
30
Parameter Initialization
(O0 1)
(A) Direct measurement. Measure qubit O in the computational (Z) basis. The probabilities are II( (b) & 1) (い) 1| |a for b 0 and | for b and the post-measurement (normalized) states of the remaining qubits are lob)-
(B) Ancilla measurement. Prepare an ancilla a in 10),, apply a CNOT with control qubit and target a, then measure in the computational Z basis. After the CNOT, the joint state is
a (0) 10), (6o) + 3 11) [1), 161)
Projecting onto 1b), yields outcome b with probability pb = ||a 10) (06) for and ||3 [1) (01) for b i.e. Pb = |a|2 and pi = 13 Conditioned on outcome b, the (normalized) post- most-measurement state of the system qubits is ()) (ov): tracing out qubit 1 leaves the remaining qubits in loh).
Thus, pb P' and the conditional post- measuuearnntnt states of the non an-ancilla qubits coincide in (A) and (B), Consequently, for any subsequent (classically controlled) processing, the two procedures are operationally equivalent.
C DETAILS OF OUR EXPERIMENTAL METHODOLOGY
We present the experimental details and hyperparameters in Table Because image data high- dimensional and amplitude encoding is not feasible near-term devices due to the high depth (Han) cet al. 2025), we perform angle encoding using RY gates. For each layer the quantum circuit, data is reuploaded with angle encoding using RY gates. This means that must compress the image into a set of 10 features, and we do so using PCA. One might ask the question of how to perform PCA on decentralized data. This problem has been solved using technique called Federated PCA (Grammenos et al.) (020), which provides the same PCA results as centralized PCA. Because the implementation details of Federated PCA central to Quorus, emulate
17

---

Published as a conference paper at ICLR 2026
Table 7: IBM QPUs and their performance characteristics. 2Q refers to the two-qubit gate error, which is typically specified, as it is an order of magnitude more dominant than the IQ gate error.
QPU name
Qubits
2Q error (best)
2Q error (layered)
CLOPS
Processor type
Pittsburgh
156
8.11E-4
3.81E-3
250K
Heron 13
Kingston
156
7.82B-4
3.57E-3
250K
Heron r2
Fez
156
1,45B-3
4,28E-3
195K
Heron r2
Marrakesh
156
1.11E-3
3.72E-3
195K
Heron r2
Torino
133
1.29E-3
7.50E-3
210K
Heron r1
Brishane
127
2.87E-3
1.74E-2
180K
Eagle
Federated PCA with centralized PCA in our implementation and note that the PCA implementation can be substituted as desired.
For inference on testing data, the testing data is compressed using the PCA fit on the training data. We assume the use of Federated PCA for all of experiments, even for Standalone training, for both consistency and for considering the "adversarial" case where a client decides to participate in Federated PCA to obtain better reduced features, but chooses not to participate in the FL process. In addition, to consider the most adversarial setup for Standalone training, where a client does not participate in the FL process, the optimizer state for Adam persists across rounds (whereas, in QFL setups, we reset the Adam optimizer state each round, as done in [aang et al. (2021)). ) We evaluate the specific classes in MNIST and Fashion-MNIST they represent various levels of difficulty, used in other QML works (DiBrita et al. 2025
Ranjan et al. 2024
The hardware specifications of the IBM QPUs are provided in Table 7
C.1 Q-HETEROFL FRAMEWORK
Algorithm 2: Q-HeteroFL.
Initialization 1) Executes:
Update(k; (')): 8¹
All Clients
0 11
for ranuf
do k E P (in parallel) do
0......,
local epoch 2...... do for each mini-butch by do L
forall
La
+- 0'[: de)
0:
Le A)
#++1 Client Update(k, 8*)
g= dxl 11 dz) 18
return 0
foreach resouree capabilitt di do 0 [d:] angle! 12222 0²+3 (2())
We describe the Q-HeteroFL. technique in Algorithm 2 an adaptation of the aggregation technique for heterogeneous classical FL described in Diao et al. (202I). The loss function is defined solely on the deepest classifier output, and aggregation is also done using circular averaging for consistency in comparison to Quorus. HeteroFL the standard baseline heterogeneous FL but has not yet been proposed in QFL; thus, we propose it here and demonstrate Quorus's improvements over it.
D ADDITIONAL RESULTS AND ANALYSIS
D.1 ANSATZ CHOICE ANALYSIS
We perform a comprehensive analysis of what ansatz to use our experiments for Quorus- Layerwise by evaluating the Staircase, V-shape, and an Alternating variant of the former two across MNIST and Fashion-MNIST classes. Note that, for layers in our Quorus-Layerwise, we have L L-1 1 different classifiers (one classifier per layer, with the first layer having two variational layers). This means that, for client that can run a capacity of L layers, they can ensemble the outputs of their I. classifiers for inference. That is what is shown in Table&and is how Quorus is evaluated
18

---

Published as a conference paper at ICLR 2026
Table Best Ansatz by Client Capacity for Ensembled Submodels, Quorus. The table is sectioned off into different capacities based on the number of layers a client can run. The best perforeefonnnng ansatz for each different class comparison in bold. The V-shaped ansatz has the highest testing accuracy the most times, we use it for all of our experiments.
Cupucity
Ansat
MNIST
Fashiun-NNST
TromerBBott
Bag/Sandal
Polloventoort
Staircase
97.9
UG.2 1.9
0. 4.7)
U7. 1.3)
93.4 # 0.9
0 1.7
Veltage
97.0
9500+ 1:2 0:2g
78.2
98.8 0.9
86.1 7:3)
78.3
Alternating
66 9.7)
64.5 (1.0(L 10.7)
65.5 12.7)
82.0 19.8 16.5)
82.7 16.3g) 10.7)
06..t-499 9.5)
Staircase
98.2 1.0
0.8 (170.)
81.0 2.8
08.6++0.8 (1 0.6)
:
76.1
V-stape
98.0
96.9 1 0.7
99.2 0.4
5.9
78.6 L.6.1..
Alternaing
92.0 (. 6.2)
U18411221 (2 2.1)
73.0 1.8 1.1)
U7.1 4.7. (1 2.1)
91.7 2: 0.9 2.0)
18.0433.4411 5.11
Stareass:
9644 6.77 (L
62.0
98.7+09.
98.7
7.8+22.4
V-shape
98.3 0.9
07.5 1: 0.6
81.0 22.0:220:000 0.D
90.3 1 0.3
91.5 1.0 2:2)
18.7 1 1.0
Alterating
20
3.0 02)
UT.6 07610... 1.7)
1.
7481123 3.9)
Staircase
98.1 0.4)
96.3 (6
1..0
98.8 150.7 (1 0,30
93.7 1 0.8
742.122 (1 4.65
V-alage
0.8
99.3
92.4 2.6
78.8 1.1
Altennainng
83.8
82.1 221+2990000
077 1.7)
92.5 1.2 1.2)
75.++200 3.5)
Stiresse
18.0 (.6)
6.77
63.1 (3.1+320000))
08.80.8 0.6)
98.8
74.++ 1.9
GL
Vedtape
98.8 0.8
8.28
63.1 63.1224
99.4 2.3
92.7 + 2.5
T8.8 = 0.8
Alternatngg
03.1
05.3 (..
074 2.77
92.6 1: 1.1 1.2)
15.3 3.5)
in the tables in the main text. We see that, across a majority of the capacities and class comparisons, the V-shape ansatz has the highest testing accuracy, making it the better choice on average. A reason for this is that the V-shape has the largest number of CNOT gates and circuit depth compared to the Staircase and Alternating ansatzes, and thus it may be more expressive. From the results in this table, we decide to use the V-shape ansatz as the default in our experiments.
D.2 ABLATION ON THE NUMBER LAYERS
To justify the layer count we used in our exper-
Ideal Sim. Ensemble
iments, we evaluate Quorus- Layerwie using
75
both fewer and more layers, depicted in Table
We run two additional ablations: Quorus with
Accuracy
70
Ideal Sim. Classifier
the five clients having 1, 2, 3, and 5 layers
65
Kingston Ensemble
respectively; and Quorus with the five clients having 4, 6, 8, and 10 layers, respectively (note that the case where the 5 clients have 2, 3, 4, 5, and 6 layers, respectively is what is used by default in our work).
60
Kingston
Classifier
2
4
Depth (# Layers)
Figure 10: Testing accuracies of the subclassifiers and submodels of single Quorus Funnel model evaluated on IBM Kingston. We see that ensem- bling outputs yields higher accuracy, similar to what we see in ideal simulation.
We would like to point out that using 1 layer appears to have drastically lower testing accu- racy, at times 40% lower than or 10 layers. This suggests that clients with 1 layer do not have enough parameters to contribute well to the training, result consistent with intuition. layers and whether it is helpful for clients. We see that, for our setup, using more layers (up to or 10 layers) has marginal gains in testing accuracy. This result is consistent with quantum com- puting literature, where adding more layers to solve a problem saturates in gains beyond a certain point Nguyen al.) (2022). Thus, use 2 through 6 layers in our experimental setup, more layers lead to higher testing accuracy in this regime, as well as for the fact that quantum circuits of this size are amenable to running on real-world hardware, as we show in our analysis section.
There is also a question of whether we use more
D.3 ROBUSTNESS OF QUORUS AMIDST REAL HARDWARE NOISE
In comparing the performance of various depth circuits used Quorus-Fumna Funnel on real hardware, observe interesting result. In Fig. 10 we plot the testing accuracy 100 datapoints for one model trained on Fashion-Min -NIIS Collover/Coatt classification. evaluated on IBM Kingston. We run depth 5 model on IBM Kingston, meaning that in total, extract 5 classifier outputs (one output
19

---

Published as a conference paper at ICLR 2026
Table 9: Capacity-wise Comparison (V-Shape) Quorus-Layerwa sizes with A to the Best. Means + standard deviation shown over five splits; mean ties broken on standard deviation.
Capacity
Quorus Layer Count
SINIST
FISSS MINIST
Fashion
IVI
4/9
TruseePloo
RogSoonial
PallowerOol
Querae4 ayensies
37.5)
3
(19.))
Quoces Lagersise (2L)
97.0
16.0 (1 0.0)
78.22 (1 0.8)
98.8 d: 0.0
86.1 #81 (1 2.5
76.3 d:
Qutess Laycrsis (2L)
$6.0..3..5 2,00
95.0 + 2.5
2.7
07.1 ( 17)
$8.6 =: LT
16.5 L.7 1.0
Layersisc (IL)
56.4:1:19 (4
96.1
97.8
67
2.0 (1.5.)
Queree ayersss (HI)
99,00
96.9 d: 0.7
8:1 (.22
99.2
8022 5. (198))
78.0 1.0
Quocas Layerwise (IL)
97.. (1.03)
W6.7 1,0 (1 0.2)
M2.6 d: 2.1
08.6 (4 0.00
92:1 0.7
17.iis L, 2.00
Layensise (L)
$1. (1..5)
9666
4.0 3.0)
07.0 (1
91.9222
17.1
Queess Layersisc (4L)
08.3
07.5
81.0 (1
99.3
91. (1.)
78.7
Qura--ayysriie (61)
90.5 (1.0.5)
90.0 0.6 (1 0.5)
83.0 4: 2.5
9..6 0.5)
93.1 = 0.5
18.5=116 (1 (..0))
Quraa- (IL)
96.5 0.00
90.0 1.0 0.5)
8.. 2:5)
98.9 0.5 (1
92.8 (10.8)
172 3.00
Queets Layerwise (1L)
90.3
924 (1112)
Quoeas Layeriise (BL)
97.3 1.0y
90.0 (4 0.5)
80.7 4: 2.8
98.8 () 0.6)
93.6 =: 0.6
18,3 1.4 0.5)
Queess Layersise (SL)
2..3 (1 0.33
97.3 0,5 (1 0,5)
#. (2
96.8 (1 (.))
12 (1.09)
11.7 1.1
Quen-Layeraie
0.8
97.8
(.8)
99.4
78.8
Queees- Layes-Laeerssse (1UL.)
97.3 4.888 (1 1.39
10.0 0.89
83.9 2.2
08.8 (2 (.0)
93.8 2: 0.5
76.1 (17)
for each layer) on IBM Kingston. We plot the accuracies of the classifiers of each layer, plotted in red dots, as well as the accuracies of the classifier ensemble up to that layer, plotted in yellow dots. We similarly plot the classifier accuracies from each layer in ideal simulation in dark blue dots, and the ensemble of the classifiers up to that layer in light blue.
One interesting observation is in the separation between the individual classifier and ensemble out- puts in ideal and hardware evaluation. Notably, IBM Kingston, although the testing accuracy below 60% for individual classifiers for depth 3 and later, the ensemble of these classifiers gen- erally increases, and maintains nearly 20% separation at layer 5. This suggests that even though noise can corrupt individual classifier outputs, Quorus is robust to hardware errors from its in-built shot-efficient ensemble evaluation and is able to substantially mitigate these hardware errors.
E
LLM USAGE
ChatGPT was used help polish the writing of the work. All edits were subsequently verified by the authors. ChatGPT search was also used to find related work. All sources found were validated by the authors to be relevant.
F REPRODUCIBILITY STATEMENT
To ensure full reproducibility, we have released all code, datasets, and experiment configurations used in this work (also attached to this submission). The open-sourced repository includes detailed documentation, environment specifications, and ready-to-use scripts to replicate the results in the paper. By making these resources publicly available, we aim not only to guarantee transparency and independent verification of our results but also to accelerate future research by lowering barriers for benchmarking, extension, and adoption by the broader community.
Repository Link:
G ABLATION: NON-IID DATASETS
We compare Quorus-Layerwin and Quorus-Funnel with Q-HeteroFL for Non-IID MNIST and Fashion-MNIST datas in Table 10 Similar to DepthFL (Kim et al.) 2023). a Dirichlet distribu- tion Рa D1rg(33 = 0.5) was used to assign Pe,k ratio of data samples of class e to client k with the constraint that each client has the same number of total training samples (128 samples). On average, the best performing Quorus variant has an 22.5% improvement in testing accuracy over Q-HeteroFL, with an 11.0% reduction in average standard deviation. In addition, Quorus- Layerwise and Quorus-Fumm Muhorus-Funnel have similar mean testing accuracies (often within 1%), highlighting their strength and interihangeability based on quantum hardware constraints. This table highlights not only the robustness of Quorus to non-IID data, but also its stability: for different data splits for H-HeteroFL. the standard deviation is very high, often ranging from 10-20%. However, the standard
20

---

Published as a conference paper at ICLR 2026
Table 10: Capacity-wise Comparison (V-Shape) Q-HeteroFL vs. Quorus variants for Non-IID Data with A to the Best (Bolded). The means and standard deviations are calculated over five runs: Quorus-Layerwise and Quorus- Funnel outperform Q-HeteroFL with smaller standard deviations.
Capacity
Techniqqe
(NIST(NNondIi)
MaNIio-MNNIST (Nom-1ID)
(1
24
TroseenBot
BegiSandal
PulooennCoot
Q-Hinmofl.
5:88 (1 640)
523 910 (1 12.99
00-0-111 -A (1 16.0)
68 31.1 50.7)
70.2 + 16.9 (1 17.7)
56.1 1.60 (1 15.5)
2L
Quceus Layerwis
97.7 0.8
345
7424 (1
98.3 d:
87.0
71.44:22
Quems-uureel
97.0 (77)
86.7 =
76.0 =
98.0 0:3)
56.5 8.8888888 1.4)
71.6
OHHooseL
77,0 11.0 (1.44
77.1 16.7 ( 19.1)
67.3 (1.22.1)
67.6 12.1 () $1.5)
62.1 12.3
(2.33116666022322))1001 25.4)
$:.0::3. (5.25.1
3L
Quemrel ayarwis
98.4 0.5
95.1 (1
79:1:::..
99.1 d: 0.1
50.5 1-:::24 2.6 0.00
76.1 :: 0.8
Purnel
97.8411.. U.00
9:.6
750 1.11
Q-HewwoHH
67. 11 10.00
78M 17:9 22.89
56.9 (1 25.30)
74.7 19.7 (1 24.5)
70.7 20.7::15.5 11 20.8)
50.6::422 11 19.99
41
Quocus-Laverwa
98.4 0.4
00.7 = 1.1
50.6 (8)
99.2 + 0.4
91.5 +: 8.0
73.5 4.2
Quome-Fared
97.4 1.01.0)
(1.00)
90.0 1-5)
80.6 12.6 17:19
66.0 10.3 31.2)
57.7
(5.777
80.3 17.0 18.99
12.8
19:3)
56.3 5:33::553 17:6)
SL
Quemss Layerwse
98.5 0.4
07-2 d: 0.6
S1.6 (1
90.2 à 0.2
12.1 1: 2.7
73.8 4.6
Furacl
96.5 1. (17.7)
98.7
726
Q-HinroPP
(1 (2.5)
(1.27))
( 20.3)
6L
Quon--ayyrwii
94.7 0.4
96.9 =t 0.6
$1.9 1.6 (1 0.4)
99.4 + 0.1
92.2 d: 1.9
75.7 +: 2.7
Quceue-ureel
9833 0.44
(6.1
82.3
98.7 (L
888
7
deviation of Quorus-L and Quorus-Funned never exceeds 10%, reflecting its stability across different non-IID datasets and its real-world utility.
H THEORETICAL ANALYSIS FOR ONE LAYER
Based results in Table 9 provided empirical justification as to why chose to more than one layer of our V-shaped ansatz in the Quorus technique (because, with just one layer, Quorus performed significantly worse than when we used 2 layers). We provide a theoretical justification for why one layer is ineffective due to the measurement on the first qubit only depending on the Z expectation on exactly half of the input qubits (and thus, only depending on half of the parameters in the first layer). Because of ineffective use of parameters, we suspect that the expressiveness of the circuit with only one layer limited, motivating the need for additional layers for effective performance.
Thus, in this section, we provide a concrete theoretical analysis of the first layer behavior of our circuit. The decision function that we seek to learn from data depends on the parameters associated with the first qubit, which are reflected in the probability of obtaining the measurement outcomes 10) or [1). Ideally, we would like to express these measurement probabilities directly in terms of the amplitudes of the N-qubit product state produced after the initial single qubit rotations. This allows all input amplitudes flows into the measurement statistics of the first qubit under the given nearest neighbor CNOT architecture.
us to understand exactly how information from
Lemma 1 (First-qubit measurement after two CNOT sweeps). Let N 2 2 qubits be arranged linearly and initialized in a product state
N
Pin
Pj-
j=1
Let U be the unitary corresponding to two layers of neurest-neighhor CNOT gates:
a rightaard sweep: CNOT(1- 2),,, (NOT(2-+3), (,00--++)....CxxXX/V----1Y (NOT(N-1-NN),
followed by leftward sweep: CNOT(N -N-1), CNOT(N-1-NN-2), -., CNOT(2 1).
Then the Heisenberg evolved observable of the first qubit satisfies
U'ZU- = II Zz-
18JEN JEN (mod 2)
In particular, if
z= Tr( [-1,1]
21

---

Published as a conference paper at ICLR 2026
denotes the Bloch Z- Z-component of the jth input qubit, then the measurement probabilities of the first qubit in the Z basis are
Pr(first qubit 0)
1
14
II
2g
Pr(first qubit 1)
1 2
II
2j
2
ISiSN JEN (mod 2)
I8JSN JEN (mod 2)
In other words, the first-qubit measurement statistics depend only on the Z-components of those input qubits whose indices have the same parity N (odd indices N is odd, even indices if N is even).
Proof. First consider computational-baii input [1122 .N) with Ij E [0,1]. A CNOT with control bit and target bit acts (a, b) + (a, ① b). where denotes XOR. In the rightward sweep + 2,2 + 3..... (N - 1) + N, a straightforward induction shows that the output bits - v/N satisfy
yk = x1 @ 222 4
@ Tk.
I=
In the subsequent leftward sweep N + N-1... 1....... 2-+1, each CNOT(i 1 1)+) updates (y/t M6+1) as (Vi, 16+1) + (ya @ 1: The final first bit a1 can therefore be written as
r1 = y1 ① y2 ①
① yN-
Substituting V1 Ij ① ... and counting how many times each 2j appears, find that z; included exactly -j - + times, so only those æj with (N - j + 1) mod 2 contribute. Equivalently, = N (mod 2), and hence
Ij
ISJSN JEN (mod 2)
For Clifford circuits, this classical XOR mapping of computational -basis labels coincides with the Heisenberg evolution of Pauli Z operators Gottesman| (1998). Equivalently, one may propagate Z backwards through the circuit using the CNOT conjugation rules
(N Ze CN Ze,
CNOT(c-t)) Z. CNOT(C Ze.ii
which yields the same expression
UtZu= =
II 2j.
ISiSN JEN (mod 2)
For a product input state Pin = 2 j=1 Pj. the expectation of this operator factorizes:
Z
II Tr(2,p;)= = II
Zj-
I8YSN
ISJSN JEN (mod 2)
JEN (mod 2)
Finally, measuring the first qubit in the basis uses the projectors Hl = (I + Z1)/2 and II, = (I 22)/2, giving
Pr(first qubit 0) 1+ (UtZ,))
Pr(first qubit 1) 1 (UiZZU)
2
which yields the claimed formulas upon substitution.
The implications of this analysis are that, because the probability of measuring the first qubit to or only depends half of the qubits' Z-expectation, half of the parameter information is not being used if we measure immediately after the first layer, which may be a reason why Quorus performs badly with just layer not enough parameters contribute to the loss function. This motivates the use of additional layers in Quorus so that our loss function is dependent on more parameters.
22

---

Published as a conference paper at ICLR 2026
1 COSTS OF USING QUORUS
We would like to provide more detailed discussion of the costs of using Quorus. We will break down our discussion into two sections: classical resource cost, and quantum resource cost. Within the quantum resource cost, we will discuss the cost of using each of our design more explicitly. To our knowledge, detailed discussion of how compute layerwise losses in QML has been proposed, which is what we provide below.
I.1 CLASSICAL RESOURCE COST
The classical cost of using Quorus can be characterized as follows:
Client storage of parameters, model output, and gradient information
2. Server parameter aggregation process
3. Parameters sent over classical channel (network)
One important observation is that model inference and gradient computation has minimal classical overhead. The reason for this is that weight matrix multiplication and gradients can be computed natively on quantum hardware (Wierichs et al., 2022), meaning that clients do not need to store large weight matrices nor the intermediate activations from each layer. Typically, storing these weight matrices as well as intermediate activations for gradient computation significant classical memory overhead, serving as the bottleneck for why clients with little memory cannot run large models. As this overhead does not exist in quantum models due quantum- native calculations, the bottleneck for clients is not limited classical memory, but rather limited depth quantum computation.
source of
The overhead that Quorus incurs (listed above) is small (on the order of the parameter count for each client), which is not expected to be a classical memory overhead for our quantum clients.
1.2 QUANTUM RESOURCE COST
In contrast to the relatively small classical resource cost that Quorus incurs, the quantum resource cost that Quorus incurs needs to carefully analyzed. As discussed the main text, quantum com- puters are very expensive to run today, so minimizing the number of shots on the quantum computer essential for the practicality Quorus. We will argue that, because Quorus-Layerwis incurs shot overhead, we propose Quorus-Ancilla Quorus- Quorus-Blocking, and Quorus-Funnel, tailored for clients' hardware constraints to enable practical QFL with Quorus.
1.2.1 QUORUS-LAYERWISS OVERIEAD
incurs a f'ccoor of L overhead in terms of shot-count because the circuit must be re-run for each layer order to compute the loss function, where L is the number of layers in the circuit. To understand this, we will walk through the steps client must take to evaluate its loss function (which we defined in Equation 1.. First of all, note that L (pi. y) and D (p) llp:) both require the term Pt, which are the logits output from the i-th layer. In our layerwise loss function, we see the expressions Ede Lee (p:) and Ede Σド=15 D DKL (pill:.) Note that the outer sum of these expressions is both Ede, This means that the client must be able obtain ps for each < < dy to compute its loss LR. Obtaining those p. 's is precisely what we focus on.
-i=1"
In this Quorus-Layerwise approach, we propose simply rerunning the circuit for each depth (as described in Subsec. 4.3). To be more precise, for each depth from 1, 2, ..., de, we have to rerun the circuit. The reason we do this is to preserve the exact state fed into the next layer of the quantum circuit, which is not possible with a midcircuit measurement (for detailed discussion of why this is the case, refer Subsec. 4.3
23

---

Published as a conference paper at ICLR 2026
Then, if need S shots obtain the probability distribution p; for quantum model, then will need SL shots to obtain all p., 1 S Sis S dg. This is precisely the linear factor in shot-overhead.
And the fact that this is multiplicative factor is meaningful, count scales when we do gradient computation according to the parameter-shin rule (Wierichs et al. 2022). To analyze this, will employ counting argument.
consider how the total shot-
Let n be the number of qubits and L the number of layers in the ansatz of Fig. 2 Assume that, for each layer index i {1..... L), the scalar quantity pi can be estimated with S shols, independently of i and of the parameters.
Proposition 2 (Quadratic shot count with Quorus-Layerwin
Then, using Quorus-Layer Layerwise and the parameter-shifi rule, the total number of shots needed to compute the gradient of La with respect to all parameters of the ansatz is
Taaerriis = =6.n E - - 3L5nLL 1),
t=1
which scales quadratically in L.
Moreover, the loss Lx for client can be evaluated with shots (i.e.. independently of the layer in- dex E), then the total number of shots needed to compute the gradients with respect to all parameters
T Tronst 6.SI,,
which scales linearly in L.
Proof. For the ansatz in Fig. we index the parameters as
0 (0g,,,ra RexLx3
where
q {I, (I,....,n) denotes the qubit,
8 E {1...., L} denotes the layer,
(1,2,3) denotes the position of the parameter in the single-qubit rotation gate.
On each qubit q in each layer we apply a three-parameter gate
Ro 0g.0, 0g,33) == R2 (0 Ry (0g, R2 (0
as defined Appendix A Thus, each layer contains n such gates and therefore 3n parameters.
We analyze the cost of computing the gradient with respect to fixed parameter 0gter E R, i.e., 0Lg, By the parameter-shift rule Wierichs al. (202)). this gradient can be written as
80 ...
θL (x(ot(a)))) La(0 (2..)))))))
00
where, for some fixed shift E TR,
((.er))) E clldi)))
and clu,er) E R_xLxx is the basis tensor with entries
(g.e.)
if (a, l,,
a,b.c
0. otherwise.
Thus, suffices to count the number of shots required to evaluate L2 (8=(q,&r) for this parameter. Recall that L2 (Eq. [ is defined in terms of the layerwise quantities p;. For parameter in layer t, the corresponding circuit only affects the outputs with i f. Therefore:
for < E, the parameter 0g.e, does not appear in the circuit used compute Pj. so Pj is independent of fg,er and 30,4.
0L. 0;
24

---

Published as a conference paper at ICLR 2026
for all with e S S dg (or up L in the worst case), the parameter &ge,r may influence Ps, so we must run the circuit to estimate p.-
Under the assumption of the proposition, evaluating p; for given i costs S shots. Therefore, to compute Ln for a parameter layer l, we need to evaluate p; for ,,..,, L. which requires
S(L - f+ 1)
shots for a single evaluation of Lg. Because the parameter-shift rule requires two such evaluations, La (0+(()))) and La (0-(4))))) the cost for the single parameter 0g,er
Igld,- 25(L (+ 1).
Each layer & has 3n parameters (three per gate, one gate per qubit), and Ig,,r depends only on Hence, the total number of shots for all parameters in layer l is
T. = 3n Tg 65n (L- 1).
Summing all layers yields the total shot count:
T Tayerwise
Σ
Th
Σ
- -
1=1
t=1
= 6Sn Σ
(L - f
68n
Σe
1=1
L(L+1)
= 6Sn
38
2
which proves the first claim.
For the second claim, assume instead that the full loss La (for any parameter setting 0 and any layer index) can be evaluated with shots. Then, for each parameter 0g...r the parameter-shift rule requires two evaluations of LR, costing 2S shots. Since there are 3nL parameters in total, the shot count
Tonnst 3nL 28 = 6.nI.
as claimed.
From Proposition we see that Quorus-Layerwise has quadratic shot overhead compared loss function that can be evaluated with just S shots (which is standard in variational quantum algorithms). To reduce this shot overhead, we aim to evaluate our loss function L3 in just S shots, independent of the layer count L. This will be detailed next.
I.2.2 QUORUS-ANCILLA OVERIEAD
To that end, we propose technique, Ancilla that uses ancilla(e) qubit(s). As depicted in Fig. 5) we entangle the first qubit with an ancilla qubit after each layer. Thus, if the client has L layers for C class classification, then require the elient to have L [log2(C)] ancilla qubits, with the ability to do a two-qubit gate from the first [log2(C'] qubits to each ancilla. Note that, in doing so, the loss function L3 for client can now be computed in just S shots. To do so, the client simply needs to obtain the marginal probability distribution over [loy2(CC] ancillae of each of the T. groups of [loga(C)] ancillae; for group of [log2(C)] ancillae qubits o; this will yield the marginal distribution p.. With just S shots of the circuit, each group of ancillac a, will be measured times, which gives us S values for each group of ancillae to form the histogram that we use as our estimate of p.. With Pi can compute our loss function L2 with classical postprocessing.
We would like to highlight how the hardware constraint of additional ancillae and higher connec- tivity is amenable for some quantum hardware architectures but not others. For example, supercon- ducting machines have limited connectivity and cannot natively perform two-qubit gates between one qubit and arbitrary number of other qubits without additional overhead (Lange al. 2025
25

---

Published as a conference paper at ICLR 2026
However, trapped-ion machines exhibit all-to-all connectivity, and neutral atom machines recon- figurable and can, in theory, support two-qubit gates between any qubits (Chen et al. 2024; Evered
et al. 2023). Thus, for trapped-ion and neutral atom clients that are aesource-constranne (cannot run many shots), Quorus-Ancilla is amenable to their hardware capabilities.
Finally, would like note that the user actually does not need to have I. ancillae, provided they have some way to reset the qubit mid-conputation. Using justt [Zo92(C)] ancilla qubits and resetting their state 0 after each layer would also be sufficient implement the Quorus-Ancilla approach. However, if the measurement and reset operation takes a long time, then that could block the op- erations in the circuit's next layer and induce latency in the circuit. Thus, using [[og](C)] ancilla qubits is particularly amenable to clients who have the ability to reset a qubit in parallel with other circuit operations, with real-time cost that similar to that of the circuit operations. This hardware requirement is also used in quantum error correction (DeCross et al., 2022). Thus, quantum hard- ware tailored for quantum error correction improves, the same functionality can be used support this technique.
1.2.3 QUORUS- -OOOUI-B OVERHEAD
For client that does not have additional ancilla qubits, we propose an alternate version of Quorus, called Quorus-Blocking. Depicted in Fig. 6 (a). we require the client to have the ability to do mid-cireuit measurement
the first [log((C') qubíts after each layer. We store the measure- ment outcome on the [log2(C'] ancillae at each layer for each shot; thus, with S shots, we will have values for each layer to form the histogram which we use as an estimator for p:-
We would like to highlight that for clients with the ability to implement mid-circuit measurements quickly, Quorus-Blocking is particularly amenable. And, discussed above, with quantum hard- ware built to support quantum error correction, the same capability of doing fast mid-circuit mea- surements will enable the practicality of
1.2.4 QUORUS FUNNEL OVERHEAD
Despite the real-world relevance of the hardware requirements discussed in the above sections, there is quantum hardware today where none of the above constraints satisfied, given the relative nascency of quantum hardware. To support clients with the least amount of hardware constraints, trade-off quantum hardware constraints with quantum model size. For the Quorus-F Funnel approach, we require the client to use a model that acts on [log2(C'] fewer qubits after each layer. The reason for this is, if [loge(C'] qubits then they can be measured without blocking other computation. Thus, by obtaining the marginal distribution over the first [log2(C) (C)] remaining qubits after each layer, with just S shots, we can obtain an estimate for for each layer.
used for computation in any further layer,
1.3 SUMMARY OF QUORUS RESOURCE COSTS
In conclusion, we have discussed why Quorus does not incur high classical resource cost over other QFL methods, and discussed the unique and practically relevant hardware requirements that each version of Quorus has. In doing so, we highlight the relevance of Quorus to quantum platforms in our durrent-day and ones expect in the near future.
26


Reviews:
[{"review_id": "itcxomfuEc", "reviewer": "Reviewer_6SVZ", "summary": "This paper proposes a quantum federated learning algorithm that can account for errors in quantum circuits. The authors use extensive simulations to show how this approach can improve test accuracy.", "questions": "- How does your algorithm scale with the number of layers and number of agents? It would be useful to provide scalability experiments.\n- Can you handle heterogeneous datasets? Can you run experiments with such settings?\n- How do you justify such a complex design for a very small (around 10%) gain?\n- What type of quantum hardware is needed for this solution to work?\n- Can you study the effect of error correction or mitigation?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "+ The idea of using varying-depth quantum circuits is both interesting and timely.\n+ The authors provide extensive experiments to validate their approach.\n+ The solution can reduce barren plateau effects.\n+ There are clear relevance to emerging quantum platforms.", "weaknesses": "- The gains from the experiments seem very limited.\n- The scalability of this solution is not studied.\n- The results seem limited to basic classification tasks.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "nwNvOQyVTQ", "reviewer": "Reviewer_LKv7", "summary": "In this work, the authors propose a structured Federated Learning framework (Quorus) that adaptively selects and synchronizes subsets of model layers within the framework of federated learning optimization. Instead of transmitting the complete parameter set or using uniform layer selection, LFL introduces a layer-wise importance estimator and communication scheduler that determines which layers to aggregate in each round based on gradient variance and update magnitude. \n\nThe experiments on binary classification on the MNIST and Fashion MNIST datasets demonstrate that the proposed approaches consistently outperform other quantum federated learning methods, such as Q-HeteroFL, and achieve better performance on IBM quantum platforms.", "questions": "1. How does your Layerwise Federated Learning (LFL) framework differ conceptually and algorithmically from prior “layerwise parameter sharing” or “selective aggregation” methods?\n\n2. Have you compared with or at least cited existing selective-layer FL methods in the classical setting (e.g., FedPer, FedRep, Selective-FedAvg, FedDrop, FedPAQ)? If not, please justify why those are not directly comparable or include a baseline comparison.\n\n3. Could your algorithm be interpreted as a special case of layer-dropout or structured sparsification techniques? If so, why does it deserve to be treated as a new paradigm rather than an instance of that family?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The work firstly proposes the Quorus framework that is tailored for heterogeneous-depth clients, providing an alternative way to realize quantum machine learning on realistic quantum processors without QEC techniques. \n\n2. The proposed layer-importance metric based on normalized gradient variance and the communication schedule can be plugged into existing federated learning systems, demonstrating the advantages of Quorus in this work. \n\n3. The experiments are scalable to 50-200 clients, making the authors' claimed methods reasonable. \n\n4. The appendix includes detailed hyperparameters, dataset information, and pseudo code.", "weaknesses": "1. Although the work claims to propose that the Quorus is effective in both simulation and realistic quantum processors, the approach is largely heuristic. In particular, the gradient-variance-based importance metric is intuitive, and it does not involve convergence or bias-variance analysis of partial aggregation. \n\n2. The work exacerbates the fairness gaps across clients if some layers are seldom updated. It looks like the authors mention this in passing, but do not quantify or mitigate it. \n\n3. The paper omits comparison with communication-efficient Federated Learning methods such as FedDrop, Sparse Ternary Compression, or Selective-FedAvg, which are conceptually closest. \n\n4. Ablations on importance metric choices or thresholds are limited. It is unclear how sensitive the system is to hyperparameters (e.g., top-k fraction per round).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "wnkCvXoEPl", "reviewer": "Reviewer_BzL5", "summary": "The paper presents a federated learning framework to train quantum clients with different circuit depths by applying a layerwise loss (pairwise KL coupling and circular aggregation of rotation parameters). The authors claim up to a 12.4 percent accuracy gain over “Q-HeteroFL” on binary classification of MNIST and Fashion-MNIST datasets. The manuscript also presents results across multiple IBM QPUs.", "questions": "1. How does the proposed KL coupling affect the optimization landscape for deeper clients? Can the authors provide a theoretical or empirical justification that this does not restrict the expressive power of deeper PQCs?\n\n2. Do the authors expect the claimed gains to hold on multiclass or quantum-native datasets? It would be very interesting to see.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "The problem formulation and framing are clear, focusing on depth heterogeneity in QFL. The concrete FL procedure with circular angle averaging, parameter slicing by depth, and layerwise loss is an impressive methodology.\n\nCircuit engineering is thoughtful. The designs cover orthogonal resource constraints such as shots, ancillae, and mid-circuit measurement, and they are practical on current hardware. The Funnel variant is particularly pragmatic. \n\nValidation on real quantum hardware is a core strength.", "weaknesses": "Major Weaknesses: \nThe evaluation scope is narrow. Only binary classification is used, and only on MNIST and Fashion-MNIST (both grayscale, 28×28 toy image datasets). No multiclass tasks, no quantum-native datasets, and missing diversity (especially in a paper that lacks theoretical proofs) weaken the authors’ claims. These datasets are fairly simple and do not require deep quantum circuits to perform well. Under this narrow experimental setting, a 12 percent improvement over a baseline does not establish general scalability, robustness, or utility of the method.\n\nWhen a paper offers only empirical support, it is expected to include evaluation across multiple types of problems, including at least multiclass classification and a different dataset, such as CIFAR-10 or CIFAR-100. None are present.\n\nBecause the experiments are so limited in difficulty and diversity, the work must instead offer compensation with solid theoretical justification that proves why the method should work beyond toy datasets. However, convergence theory is missing, there is no stationary-point guarantee, and no rigorous mathematical explanation of how KL coupling affects the optimization landscape.\n\nMinor (for maximum information transfer) Issues: \nThe data encoding pipeline is under-specified. Feature preprocessing, feature-to-quantum-state mapping, and the reupload schedule (mentioned briefly in the paper) are extremely important. These affect the number of qubits required as well as circuit depth. Upon checking the appendix and supplementary materials, I found that the authors used PCA to reduce the dimensionality. Overall, reproducibility and completeness of the paper are negatively impacted by the lack of inclusion of these details in the main manuscript. I hope the authors include this.\n\nBaseline selection seems to inflate the claimed gains. Vanilla QFL forces all clients to use the shallowest model, giving predictably weak results. Standalone training is a trivial lower bound. Q-HeteroFL is an extension of HeteroFL, implemented by the authors without external validation. It is not clear whether these baselines are suitable enough to be regarded as “state of the art.”\nOn the one hand, the engineering around layerwise circuit execution is very creative, and the hardware study is a solid strength. I would love to score this paper very highly if theoretical guarantees were provided or if the experiments were significantly more diverse.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "SYJip8ILiM", "reviewer": "Reviewer_hx6a", "summary": "The authors propose a method to address the problem of different architectures and behaviors of quantum computers in a quantum federated learning scheme. They use layerwise loss to aggregate the parameters of local QNNs with specific approaches, gaining performance improvements. The proposed method is examined on real quantum computers.", "questions": "1. In line 265, \"Passing the state unchanged thus requires you to prepare another copy of it, which induces additional shot overhead that is linear in the number of layers and is a nontrivial cost\" → does this mean state tomography is required in general? And how is this linear in the number of layers? \n2. Following W2, how does the ansatz enable the single-shot behavior? Is there any proof? From line 296, \"we evaluate each layer's outputs by computing the marginal distribution on its ancilla\" → how can you calculate the \"distribution\" with a single shot?\n3. Does the parameter aggregation process, which aligns parameters for models with different layers, affect the expressiveness? For example, a quantum state produces output, becoming an intermediate state of another model. And this \"another model,\" after additional layers, will produce the output of the same task. Then why not just use the shallower model if a shallow circuit is enough for such a task?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. Using layerwise losses for QFL is a novel approach. \n2. The paper is well-written overall.", "weaknesses": "1. The proposed method appears limited to binary classification problems. \n2. Line 291 states \"we design an ansatz where it is possible to obtain the outputs from all layers in a single shot,\" but it's unclear how the ansatz enables single-shot behavior.\n3. The baselines compared are only QML models, no classical baseline is included. It is plausible that classical models could easily outperform QML models on binary classification problems.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)