Context-Aware Error Mitigation Orchestration for Hybrid Quantum Reinforcement Learning on NISQ Systems

Bisma Majid; Shabir Ahmed Sofi; Mir Mohammad Yousuf

cs.LGOct 1, 2026

Context-Aware Error Mitigation Orchestration for Hybrid Quantum Reinforcement Learning on NISQ Systems

Authors: Bisma Majid, Shabir Ahmed Sofi, Mir Mohammad Yousuf

Organizations: Department of Information Technology, National Institute of Technology Srinagar, Hazratbal, Srinagar, 190006, Jammu and Kashmir, India

Abstract

Quantum Reinforcement Learning (QRL) integrates reinforcement learning with parameterized quantum circuits and is a promising approach to combinatorial optimization. On Noisy Intermediate-Scale Quantum (NISQ) devices, however, decoherence, gate imperfections, and measurement errors reduce policy quality and make learning less reliable. Existing error mitigation techniques are generally applied as fixed corrections that do not adapt to changing noise conditions or to the evolving state of training. This work presents Adaptive Policy-Guided Error Mitigation (APGEM) as a context-aware orchestration layer of the hybrid quantum-classical training loop that dynamically selects the most suitable mitigation strategy during QRL training. APGEM evaluates Zero-Noise Extrapolation (ZNE), Probabilistic Error Cancellation (PEC), Clifford Data Regression (CDR), and Readout Error Mitigation (REM) using policy-level indicators, including quantum-state fidelity, policy entropy, cumulative reward, and approximation ratio, and integrates the selected strategy directly into the reinforcement learning loop. The framework is evaluated on the Capacitated Vehicle Routing Problem (CVRP), a representative NP-hard problem in urban logistics, under a range of NISQ noise models and noise levels. APGEM consistently outperforms conventional static mitigation methods, reaches approximately 94% of the utility of an oracle strategy, maintains higher quantum-state fidelity as noise increases, and produces more stable learning behaviour throughout training. Ablation studies show that the framework learns context-aware mitigation policies that adapt to different noise environments and circuit execution conditions. These findings demonstrate that integrating adaptive error mitigation into the learning process substantially improves the robustness and reliability of QRL on NISQ hardware.

Figures & tables

Ref(s)	Focus of Prior Work	Identified Gap	Contribution of our Work
[ 23 ] , [ 24 ]	Surveys of quantum computing in logistics and supply chains; discuss QAOA, annealing, and hybrid methods.	Do not address noise-resilience or adaptive error mitigation in NISQ settings.	Propose APGEM-QRL , embedding adaptive noise mitigation within a reinforcement learning framework.
[ 25 ] , [ 26 ]	Quantum-inspired algorithms for logistics optimization; scalable heuristics on classical hardware.	Lack of true quantum implementation and no adaptive error management.	Implement noisy quantum circuits with feedback-driven error mitigation beyond static QIC baselines.
[ 27 ]	Application of QAOA/annealing to logistics problems (TSP, VRP) via QUBO/Ising formulations.	Assumes fixed noise or applies static mitigation; no policy-aware adaptivity.	Introduce policy-guided mitigation that adjusts dynamically to circuit depth, reward dynamics, and noise type.
[ 28 ]	Cognitive/learning-inspired quantum models for route optimization and decision-making.	Conceptual focus; lacks robustness analysis under noisy quantum hardware.	Extend to QRL with adaptive mitigation , ensuring reliable convergence under real-world noise.
[ 29 ] [ 30 ]	Systems perspective on integrating quantum methods into logistics workflows and enterprise KPIs.	Limited demonstrations on realistic urban road networks; no dynamic noise control.	Evaluate on standard CVRPLIB benchmark instances under diverse noise models, reporting best-known-referenced metrics with statistical testing.
[ 26 ]	Data-driven adaptive decision-making in logistics using quantum-inspired techniques.	Adaptivity applied only to classical decision processes, not to quantum error mitigation.	Make error mitigation adaptive , guided by policy feedback and noise diagnostics.

Table 1: Summary of prior works, identified gaps, and contributions of our Work.

Figure 1: Block diagram of the Adaptive Policy-Guided Error Mitigation (APGEM) framework. Quantum circuits generate policies degraded by noise, which are corrected through an adaptive selection of mitigation strategies (ZNE, PEC, CDR, REM). The corrected policy guides QRL updates for solving the CVRP.

Figure 2: Block diagram of the proposed methodology. The workflow begins with VRP instance generation, followed by reinforcement learning environment interaction through a variational quantum circuit. Execution under noise is adaptively corrected by the APGEM controller, with fidelity and entropy metrics guiding policy optimization. The training loop produces results compared against a classical baseline.

Metric	Definition / Formula	Interpretation
Distribution Fidelity ( $F$ )	$F(p,q)=\big(\sum_{i}\sqrt{p_{i}q_{i}}\big)^{2}$	Classical (Bhattacharyya) fidelity between mitigated and ideal output distributions; higher is better.
Normalized Entropy ( $H$ )	$H(p)=-\tfrac{1}{n_{q}}\sum_{i}p_{i}\log_{2}p_{i}$	Shannon entropy of the measured distribution, normalized to $[0,1]$ ; lower indicates a sharper, less noise-dominated distribution.
Cumulative Reward ( $R$ )	$R=\sum_{t}r_{t}$	Accumulated shaped reward over an episode; higher corresponds to lower-cost feasible routing.
Routing Cost ( $C$ )	Total travel distance of all routes (with depot returns)	Domain-specific solution quality on the feasibility-guaranteed decode; lower is better.
Approximation Ratio ( $AR$ )	$AR_{\text{BKS}}=\dfrac{C_{\text{quantum}}}{C_{\text{BKS}}}$ (and vs. best classical)	Cost relative to best-known (primary) and to the best classical baseline; values near $1$ are near-optimal.
Mitigation Gain ( $\Delta M$ )	$M_{\text{mitigated}}-M_{\text{unmitigated}}$	Execution-quality improvement from mitigation at matched conditions.

Table 2: Summary of performance metrics used in evaluating APGEM.

Figure 3: Quantum policy ansatz for the CVRP–QRL framework. The first layer applies

R_{y}(\theta_{i})

rotations to encode environment-state features into the quantum register. This is followed by a linear entanglement layer of controlled-

Z

gates to capture correlations across qubits. Two variational layers are then stacked, each consisting of trainable

R_{y}(\phi^{(k)}_{i})

rotations and entangling operations, allowing expressive state transformations. The final measurement distribution guides action selection within the reinforcement learning loop.

Figure 4: Circuit-level illustration of the adaptive error mitigation module integrated into the CVRP–QRL framework.

Component	Version
Python	3.10.11
NumPy	2.2.6
SciPy	1.14.1
pandas	2.2.3
Qiskit	1.4.0
Qiskit Aer	0.16.1

Table 3: Software environment used for all reported results.

Method	Mean AR	95% CI	Avg rank	Feasible	Wilcoxon vs OR-Tools
OR-Tools	1.043	[1.026, 1.062]	1.53	29/30	(reference)
Clarke–Wright	1.048	[1.038, 1.059]	1.77	30/30	$p=0.50$ ( $\delta=-0.19$ )
Simulated Annealing	1.163	[1.128, 1.206]	2.97	30/30	$p=4.6\times 10^{-5}$ ( $\delta=-0.72$ )
Tabu Search	1.276	[1.218, 1.340]	4.10	30/30	$p=4.6\times 10^{-5}$ ( $\delta=-0.82$ )
Nearest Neighbour	1.392	[1.351, 1.435]	5.30	30/30	$p=3.8\times 10^{-6}$ ( $\delta=-0.92$ )
Genetic Algorithm	1.430	[1.326, 1.550]	5.33	30/30	$p=3.8\times 10^{-6}$ ( $\delta=-0.87$ )

Table 4: Classical baselines over 30 CVRPLIB instances. Approximation ratio to best-known (lower is better),

95\%

confidence interval, average Friedman rank, number of instances solved feasibly, and Wilcoxon signed-rank

p

-value against the top method (OR-Tools) with Cliff’s delta.

Figure 5: Mean approximation ratio to best-known with bootstrap

95\%

confidence intervals over 30 CVRPLIB instances. OR-Tools and Clarke–Wright are the strongest and statistically indistinguishable.

Figure 6: Friedman average ranks (1 = best) with the Nemenyi critical difference (

\text{CD}=1.38

). OR-Tools and Clarke–Wright group together and dominate the metaheuristics.

Quantity	Value
Frozen-policy cost (mean)	629.3
Trained-policy cost (mean)	610.8
Improvement over frozen	2.08%
Beats frozen policy	yes
Feasibility rate	1.00

Table 5: Policy learning on P-n16-k8 (three seeds, REINFORCE with parameter-shift gradients). Route cost of the frozen policy versus the trained policy, and feasibility rate.

Figure 7: Route cost versus policy update on P-n16-k8 (mean over three seeds). The REINFORCE learner descends below the frozen-policy baseline; the best classical cost is shown for reference.

Instance	Customers	Qubits	Depth	AR to BKS	Total shots	Runtime/ep (s)
P-n16-k8	15	4	10	1.162	607,232	0.58
A-n32-k5	31	5	12	2.597	946,176	1.14
E-n51-k5	50	6	14	3.132	1,174,016	2.29
M-n101-k10	100	7	16	3.352	2,977,792	14.88

Table 6: Scalability of the quantum policy across instance sizes (contextual bandit controller). Approximation ratio to best-known, qubits, circuit depth, total shots, and mean runtime per episode.

Figure 8: Quantum policy versus classical baselines on A-n32-k5 (best-known

=784

, dashed line). Classical solvers, led by OR-Tools at the optimum, remain ahead of the quantum policy at this scale.

Figure 9: Scalability across CVRPLIB instances: approximation ratio to best-known, mean episode runtime, and total shots as the customer count grows from 15 to 100.

Technique	Fidelity	Entropy
CDR	0.835	0.824
PEC	0.829	0.735
ZNE	0.816	0.768
REM	0.769	0.856
No Mitigation	0.711	0.908

Table 7: Per-technique distribution fidelity and normalized entropy on P-n16-k8.

Figure 10: Per-technique distribution fidelity and normalized entropy. Mitigation raises fidelity from

0.711

(unmitigated) to

0.83

(PEC, CDR).

Technique	Shots/ episode	Overhead	Projected HW (s)
No Mitigation	10,240	1.00 $\times$	81.9
PEC	20,480	2.00 $\times$	163.8
ZNE	30,720	3.00 $\times$	245.8
REM	51,200	5.00 $\times$	409.6
CDR	92,160	9.00 $\times$	737.3
APGEM (contextual bandit)	40,499	3.96 $\times$	324.0

Table 8: Sampling overhead per technique on P-n16-k8, relative to unmitigated execution, with per-episode shot counts and a projected cloud-hardware time at

0.4

ms per shot.

Figure 11: Sampling overhead factor per technique relative to unmitigated execution. The adaptive controllers sit between ZNE and REM by mixing techniques according to context.

Noise	No Mitigation	PEC	ZNE	REM	CDR
0.01	0.65	0.09	0.12	0.11	0.03
0.03	0.12	0.22	0.14	0.49	0.03
0.05	0.02	0.74	0.17	0.05	0.01
0.08	0.03	0.19	0.73	0.04	0.02
0.12	0.04	0.15	0.71	0.04	0.06
0.18	0.10	0.09	0.54	0.08	0.18

Table 9: APGEM technique-selection share by depolarizing noise level (2600 decisions). Each technique dominates a different regime.

Figure 12: APGEM technique selection versus depolarizing noise. The dominant technique transitions from no mitigation to readout mitigation, PEC, ZNE and CDR as noise increases.

Figure 13: APGEM technique selection against unmitigated fidelity, entropy and circuit depth, showing that selection is context-dependent.

Figure 14: Windowed selection share over training. All techniques retain nonzero share; the policy does not collapse to a single technique.

Method	Mean reward	Avg rank	Wilcoxon vs APGEM
Oracle	0.639	–	(upper bound)
APGEM	0.602	2.23	(reference)
ZNE	0.585	2.09	$p=0.57$ ( $\delta=-0.01$ )
PEC	0.569	3.33	$p=1.8\times 10^{-4}$ ( $\delta=0.07$ )
CDR	0.522	4.53	$p=6.2\times 10^{-5}$ ( $\delta=0.31$ )
REM	0.511	4.69	$p=4.3\times 10^{-10}$ ( $\delta=0.20$ )

Table 10: Mean cost-aware reward over 60 held-out conditions and Wilcoxon signed-rank of APGEM against each fixed technique, with Cliff’s delta. The oracle is the expected-utility upper bound.

Figure 15: Mean cost-aware reward across regimes. APGEM exceeds every fixed technique and approaches the expected-utility oracle.

Figure 16: Corrected regret. Left: per-condition expected regret (policy) versus the irreducible noise floor. Right: the raw cumulative regret separated into its noise-floor and expected-regret components.

Figure 17: APGEM reward against the per-condition oracle (left) and cumulative regret (right) over training.

Controller	Mean fidelity	Circuit evals	Best AR
APGEM (Contextual Bandit)	0.820	5550	1.20
APGEM (Bayesian)	0.816	5533	1.16
APGEM (Multi-Objective)	0.799	5214	1.16
Original (Epsilon-greedy)	0.820	5752	1.20

Table 11: APGEM controller comparison on P-n16-k8: mean fidelity, total circuit evaluations, and best approximation ratio.

Noise level	0.01	0.05	0.08	0.10
APGEM	0.966	0.901	0.817	0.794
No mitigation	0.975	0.820	0.712	0.684

Table 12: Mean distribution fidelity versus depolarizing noise level, APGEM against no mitigation (P-n16-k8).

Figure 18: Noise robustness: APGEM sustains higher distribution fidelity than unmitigated execution as noise grows.

Episodes	Best AR	Mean AR	Mean fidelity	Total shots	ZNE share
100	1.200	1.234	0.824	4,659,712	97.5%
500	1.162	1.215	0.823	23,022,080	99.1%
1000	1.162	1.221	0.822	46,104,064	99.5%

Table 13: APGEM metrics versus training horizon on P-n16-k8.

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APGEM: Adaptive Policy-Guided Error Mitigation for Quantum Reinforcement Learning on a Real-World CVRP Case Study

Shabir Ahmad Sofi, Bisma Majid, Mir Mohammad Yousuf

Department of Information Technology NIT Srinagar J&K, India

Variational Quantum Circuits Q-Learning Vehicle Routing Problem Mitigation

Open paper AskPDF arXiv

Quantum Reinforcement Learning (QRL) represents policies as variational quantum circuits (VQCs), making it attractive for combinatorial optimization such as the Capacitated Vehicle Routing Problem (CVRP). On noisy intermediate-scale quantum (NISQ) hardware, however, decoherence degrades fidelity and destabilizes learning, and conventional error mitigation is applied statically without regard to the learning context. We introduce Adaptive Policy-Guided Error Mitigation (APGEM), a controller that selects among Zero-Noise Extrapolation (ZNE), Probabilistic Error Cancellation (PEC), Clifford Data Regression (CDR), and Readout Error Mitigation (REM) online, driven by a fidelity, entropy, and cost aware utility function and an epsilon-greedy rule over temporal-difference Q-scores. We evaluate on a realistic urban-logistics testbed, a Delhi-based CVRP over real landmarks with geodesic inter-node costs, exercised across five noise families and four severity levels. On this instance, the QRL agent outperforms constructive heuristics and approaches metaheuristics, while mitigation restores approximation ratios from 0.84-0.87 to 0.92-0.94 under high noise. The controller shifts from a CDR-dominated regime under short training horizons to a balanced deployment across all four techniques under longer horizons, indicating genuine regime-dependent selection. These preliminary results position adaptive, learning-aware mitigation as a practical route to noise-resilient QRL.

Abstract

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