Quantum Federated Learning Based on Bures--Uhlmann Geometry for Heterogeneous Noisy Clients

Haruki Emori; Masaki Uchihara; Yuuki Tokunaga

quant-phAug 28, 2026

Quantum Federated Learning Based on Bures--Uhlmann Geometry for Heterogeneous Noisy Clients

Authors: Haruki Emori, Masaki Uchihara, Yuuki Tokunaga

Organizations: Graduate School of Information Science and Technology, Hokkaido University, Kita 14, Nishi 9, Kita-ku, Sapporo, Hokkaido 060-0814, Japan · RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan · Department of Diabetes, Endocrinology, and Metabolism, National Center for Global Health and Medicine, Japan Institute for Health Security, 1-21-1 Toyama, Shinjuku-ku, Tokyo 162-8655, Japan

Abstract

Quantum federated learning enables collaborative model training across quantum devices without sharing raw data, and it faces the data and hardware heterogeneity inherent to noisy quantum devices. Utilizing the quantum geometric tensor is a natural remedy, yet pure-state approaches and diagonal approximations discard the correlations that encode parameter incompatibility. To address this, we extend the parameter-space geometry to the mixed states that noisy clients actually prepare. The real part of the resulting mixed-state geometric tensor is the Bures metric, which measures how fast the physical state changes under parameter variation, and the imaginary part is the mean Uhlmann curvature, which quantifies the incompatibility of estimating multiple parameters simultaneously. Accordingly, we employ the Bures metric as a local preconditioner and use the mean Uhlmann curvature to develop an achievable-precision aggregation rule that dynamically down-weights unreliable clients. Furthermore, we establish theoretical guarantees by proving a convergence theorem and a variance-dominance proposition. Empirical evaluations on a trapped-ion quantum emulator demonstrate that the proposed method maintains high accuracy across diverse device-heterogeneity conditions and outperforms standard federated averaging, whose accuracy degrades under strong noise.

Figures & tables

Panel of Figure 1 — Figure 1: One layer of the shared variational quantum classifier, drawn for three qubits, with the experiments using $n=6$ and $L=3$ . Each layer re-uploads the data through $R_{y}(x_{j})$ (green), applies a trainable single-qubit block $R_{z}R_{y}R_{z}$ (blue), and a linear CNOT entangler (pink). After $L$ layers the Pauli observables ( 23 ) are measured in three settings ( $Z$ , $X$ , $Y$ ) and fed to the shared readout head ( 24 ). The three QFL methods use this same circuit and the same head, and differ only in the local update and the aggregation.

Panel	Regime	Noisy shots $M_{k}$	Noisy rate $p_{k}$	$\varrho_{\rm noisy}/\varrho_{\rm good}$
(a)	few shots, strong noise	$\{100,200,300\}$	$\{0.4,0.5,0.6\}$	$1.1\times 10^{-3}$
(b)	more shots, strong noise	$\{500,600,700\}$	$\{0.4,0.5,0.6\}$	$5.6\times 10^{-3}$
(c)	few shots, weak noise	$\{100,200,300\}$	$\{0.1,0.2,0.3\}$	$1.8\times 10^{-2}$
(d)	more shots, weak noise	$\{500,600,700\}$	$\{0.1,0.2,0.3\}$	$9.2\times 10^{-2}$

Table 1: The four device-heterogeneity conditions. The reliable clients always have

M_{k}=1000

shots and

p_{k}=0

, and the entries below are the shot budgets and depolarizing rates drawn for the unreliable clients. The last column is the closed-form weight ratio

\varrho_{\rm noisy}/\varrho_{\rm good}

of Eq. ( 30 ) for the most heavily discounted client of each condition, that is, the client with the highest depolarizing rate and the smallest shot budget. Panels (a)–(d) of Fig. 2 correspond to these rows.

Figure 2: Test accuracy against the communication round on MNIST-10 (

n=6

qubits,

L=3

layers,

K=9

clients), evaluated on the reimei-E trapped-ion emulator with

4000

shots per circuit and averaged over three seeds, with error bars the cross-seed standard error. Panels (a)–(d) are the four device-heterogeneity conditions of Table 1 , namely (a) few shots and strong noise, (b) more shots and strong noise, (c) few shots and weak noise, (d) more shots and weak noise. QFedQGT (solid, red) attains the highest round-averaged accuracy in every panel. The two baselines are closest to it in the weak-noise panels (c), (d) and fall further behind in the strong-noise panels (a), (b). The top and bottom rows differ only in the noise rate, and the left and right columns only in the shot budget, and the vertical separation is the larger of the two, as Eq. ( 28 ) predicts.

Condition	Method	Final	Mean	Std	Worst
(a) few shots,	QFedAvg	$0.803\pm 0.009$	$0.774$	$0.040$	$0.703$
strong noise	QFedFisher	$0.783\pm 0.007$	$0.738$	$\mathbf{0.039}$	$0.667$
	QFedQGT	$\mathbf{0.870\pm 0.023}$	$\mathbf{0.814}$	$0.044$	$\mathbf{0.753}$
(b) more shots,	QFedAvg	$0.810\pm 0.017$	$0.779$	$0.050$	$0.670$
strong noise	QFedFisher	$0.807\pm 0.003$	$0.751$	$0.065$	$0.610$
	QFedQGT	$\mathbf{0.843\pm 0.009}$	$\mathbf{0.815}$	$\mathbf{0.049}$	$\mathbf{0.733}$

Table 2: Accuracy on the reimei-E emulator (

4000

shots, averaged over three seeds) for all four device-heterogeneity conditions of Table 1 . “Final” is the round-

60

accuracy with its cross-seed standard error, “Mean” the average over the

23

evaluated rounds, “Std” the standard deviation across those rounds (a round-to-round stability measure, smaller is better), and “Worst” the lowest accuracy attained at any evaluated round. Best in each block in bold, with ties both bolded.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Method	noisy	reliable	ratio	weight jitter
QFedAvg	$0.120$	$0.100$	$1.20$	$0.000$
QFedFisher	$0.129$	$0.088$	$1.46$	$0.044$
QFedQGT	$0.010$	$0.238$	$0.042$	$0.000$

Appendix

Table 3: Mean aggregation weight given to the noisy and to the reliable clients in condition (a), and the mean per-client round-to-round standard deviation of the weights. A ratio below one indicates down-weighting of the noisy clients.

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Quantum Federated Learning (QFL) offers a promising framework to train quantum models across distributed clients while keeping data strictly local. Due to its simplicity and low communication overhead, Federated Averaging (FedAvg) is the standard aggregation choice in QFL literature. However, deploying QFL on practical hardware exposes a severe double-drift phenomenon: the global model is simultaneously derailed by client drift from non-IID data and hardware bias from noisy quantum gradient estimates. In this work, we first analyze the convergence of FedAvg under these realistic conditions, mathematically demonstrating that quantum hardware bias creates a persistent error floor that standard averaging cannot correct. To overcome this limitation, we propose Q-ANCHOR, a quantum-aware federated aggregation architecture that anchors server updates with zero-noise extrapolation while applying stateful client correction to suppress both client drift and hardware-induced bias. Our convergence theory proves that Q-ANCHOR successfully mitigates classical client drift while actively reducing the hardware-bias floor. Experimental results demonstrate that Q-ANCHOR achieves significantly more stable training than conventional FL baselines.

Abstract

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