Quantum Speedups in Regret Analysis of Infinite Horizon Average-Reward Markov Decision Processes

We thank the reviewer for the insightful comments and valuable suggestions. Below we carefully address each of your concerns:

Bound on epochs $E$ in Equation (84)

We appreciate your careful review. Indeed, your observation is correct that the bound on the number of epochs $E$ as $E \leq poly(SA\log T)$ is essential. This bound follows directly from Proposition 18 in Appendix C.2 of [1], explicitly mentioned right after equation (86) in our manuscript. We will further clarify and explicitly reference this in the revised version for clarity.

Missing references in classical regret minimization

Thank you for suggesting these essential references [2] and [3]. We agree that these works provide essential context and relevance to classical regret minimization. We will incorporate them explicitly in the revised manuscript to improve completeness.

Definition and example of quantum experiment in RL

We clarify that a "quantum experiment" in reinforcement learning corresponds exactly to performing a state-action transition and obtaining the next state encoded as a quantum state $|\psi\rangle$. Specifically, given classical state-action pairs $(s,a)$, our method obtains quantum samples by leveraging a quantum oracle that prepares the superposition state $|\psi\rangle = \sum_{s'} \sqrt{P(s'|s,a)}|s'\rangle$, representing the probability distribution of transitioning to the next state. Each quantum sample preparation constitutes precisely one quantum experiment in our RL setting.

Interaction protocol between learner and environment

We agree with the reviewer on their interpretation. Our learner performs one quantum experiment at each step. Specifically, after observing a classical state-action pair, the agent queries the quantum oracle, performing a quantum experiment to obtain the next state as a quantum superposition. Thus, at each interaction step, one trajectory step is sampled quantum-mechanically rather than classically.

We will clearly describe this interaction protocol in our revision to eliminate any confusion.

Technical novelties in the algorithms and proofs

We respectfully highlight our technical and conceptual novelties:

• Our paper provides the first quantum RL framework for infinite-horizon average-reward MDPs, a notably more challenging setting compared to finite-episode or discounted problems. The quantum infinite-horizon setting introduces complexities (e.g., convergence analysis without episodic resets, the inability to directly apply classical martingale techniques and the quantum measurement-collapse issue) not encountered in classical analyses or in finite-horizon analyses.

• To handle the critical quantum measurement-collapse issue (absent in classical RL), we developed a novel momentum-based updating scheme (Eq. 21–25). In addition, unlike [1], our analytical approach avoids classical martingale-based arguments and introduces entirely novel analytical techniques (Lemma 2–3, Eqs. 34–38).

• Thus, our proofs, while building upon foundational results such as [1] and [4], involve substantial technical novelties tailored specifically to quantum infinite-horizon RL, beyond straightforward combination of known results.

We sincerely believe these novel contributions significantly enrich the theoretical landscape of quantum RL.

We hope these clarifications address your concerns, highlighting our work's relevance, novelty, and contribution clearly. We kindly request reconsideration of your evaluation in light of these clarifications.

References:

[1] Near-optimal Regret Bounds for Reinforcement Learning, 2008.

[2] Reinforcement Learning for Infinite-Horizon Average-Reward Linear MDPs via Approximation by Discounted-Reward MDPs, 2024.

[3] Model-free Reinforcement Learning in Infinite-horizon Average-reward Markov Decision Processes, 2020.

[4] Near-optimal quantum algorithms for multivariate mean estimation, 2022

We sincerely thank the reviewer for their thoughtful review. We address each point carefully below, hoping to clarify our contributions clearly.

Writing style intended for ML community

Indeed, our intended audience is the ML/RL community, which may not be deeply familiar with quantum computing. To address this, our manuscript provides comprehensive background in Sections 1–2, including:

• Definitions 1–3 (page 2-3): clearly defining classical random variables, quantum experiments, and quantum evaluation oracles.
• Lemma 1 (pages 3–4): clearly explaining the quantum mean estimation lemma critical for our quantum speedups.
• Appendix B explicitly details Quantum Amplitude Amplification (QAA), foundational to quantum mean estimation.

These definitions and explanations are designed specifically for researchers familiar primarily with classical RL.

How quantum computing improves statistical complexity

We appreciate your insightful hypotheses, all of which correctly capture our quantum RL framework:

• Implicit statistical structure: Quantum computations inherently encode probability distributions within quantum states' amplitudes, adding implicit side information absent in classical samples.

• Computational speedup vs. statistical barriers:
  Our framework leverages quantum computation’s unique statistical properties (beyond computational speedup alone), enabling quantum mean estimation with significantly reduced sample complexity.

• More information per sample:
  Quantum states carry richer statistical information due to superposition, fundamentally enhancing each quantum sample’s informational content.

Quantum computing fundamentally accelerates mean estimation via Quantum Amplitude Estimation (QAE) [1]. By encoding probability distributions directly into quantum amplitudes, QAE estimates expectations through Grover-like iterations [2], achieving quadratic improvement in the required number of samples compared to classical methods. Thus, our quantum approach translates quadratic mean-estimation improvements directly into regret improvements from classical $\tilde{O}(\sqrt{T})$ to quantum $\tilde{O}(1)$.

Concrete illustrative example

Consider an infinite horizon RL scenario for dynamic resource allocation in large-scale communication networks, a classical RL application. At each time step, the RL agent decides resource allocations (bandwidth, routing paths, etc.) based on observed network states, including current node congestion and usage levels. Classically, accurately estimating next-state transition probabilities (e.g., likelihood of congestion patterns given certain actions) typically requires extensive repeated sampling and interaction with the network, causing delays and inefficient resource utilization.

In our quantum setting, imagine a scenario where a quantum oracle—consistent with standard quantum oracles extensively studied in quantum optimization and Grover algorithms [2]—encodes the entire probability distribution of next network states into quantum states as a superposition. Such quantum oracles could realistically be implemented via quantum sensors or quantum-assisted sampling techniques that inherently exploit quantum mechanical effects (e.g., interference, coherence). By using Quantum Amplitude Estimation (QAE), our quantum RL agent directly accesses these encoded probability distributions, rapidly and accurately estimating expected state transitions using significantly fewer interactions compared to classical sampling.

This quantum-enhanced method fundamentally reduces statistical complexity and significantly accelerates policy learning, leading to improved real-time decisions in critical scenarios (e.g., minimizing congestion, maximizing throughput). Such quantum-assisted techniques align with emerging research trends in quantum technology, where quantum-based sensors and simulators increasingly enable access to rich statistical data through quantum sampling.

Regret reporting clarification

Thank you for highlighting this point. Our regret bound is explicitly reported as $\tilde{O}(1)$, where $\tilde{O}$ hides logarithmic factors in $T$. As defined in footnote 1 on page 1, the growth is at most logarithmic in $T$.

We hope this clarifies our work clearly and kindly ask for reconsideration of your evaluation.

References:

[1] Quantum Sub-Gaussian Mean Estimator, 2021.

[2] A fast quantum mechanical algorithm for database search, 1996.

We thank the reviewer for their positive feedback and valuable suggestions. We address each point carefully and explicitly below:

Comparison with [1]

We clarify that the problem studied in [1] is fundamentally different from our setting:

• Episodic vs. Infinite Horizon:
  Reference [1] focuses on episodic RL with a finite horizon, whereas our work investigates infinite-horizon average-reward RL. Infinite-horizon RL is generally considered a more challenging setting because the optimal policy needs to sustain performance indefinitely without episodic resets, requiring different theoretical frameworks, convergence analyses, and stability considerations.

• Technical Differences:
  The techniques in [1] depend heavily on episode-wise decomposition, and their quantum acceleration leverages lazy updating with episodic resets. Conversely, our infinite-horizon scenario necessitates a novel momentum-based updating framework (Eqs. 21–25) to avoid repeated quantum measurement collapse. Unlike the episodic case, infinite-horizon settings cannot directly leverage the lazy episodic updating; thus, our work introduces entirely novel analytical tools (Lemma 2–3, Eqs. 34–38).

Our contribution is the first to achieve quantum RL speedup in infinite-horizon average-reward settings, filling a significant theoretical gap distinct from [1].

Scalability and tabular approach

We fully acknowledge the limitation identified by the reviewer regarding the reliance on a tabular approach. Indeed, scalability to large state-action spaces is challenging, and addressing this limitation is an important direction for future work. We explicitly mention this limitation and highlight that extending our quantum RL framework beyond tabular methods (e.g., using function approximation methods or linear mixture models) is an essential next step, deserving further exploration.

Conceptual and technical novelty compared to classical methods

We appreciate the reviewer highlighting this critical point. We emphasize our conceptual and technical novelties:

• Our work is the first to provide a quantum RL framework and regret analysis for infinite-horizon average-reward settings, a distinct and more challenging problem than episodic RL.
• Unlike classical methods, our work uses quantum amplitude estimation (QAE) techniques that fundamentally enhance statistical complexity, achieving exponential improvement ($\tilde{O}(1)$ regret) over classical $\tilde{O}(\sqrt{T})$.
• The momentum-based updating scheme (Eq. 21–25) developed particularly to avoid quantum state collapse is novel and uniquely necessary for infinite-horizon RL. The absence of martingale techniques—ubiquitous in classical regret analysis—further differentiates our analytical approach (Lemma 2–3, Eqs. 34–38).

Our contribution not only establishes theoretical quantum RL foundations in an unexplored domain but also provides novel tools essential for future research in quantum RL.

We sincerely hope this addresses your concerns and clearly clarifies our contributions and novelties. We kindly ask you to reconsider and possibly improve your evaluation in light of these clarifications.

References:

[1] Provably Efficient Exploration in Quantum Reinforcement Learning with Logarithmic Worst-Case Regret, 2024.