Accelerating Quantum Reinforcement Learning with a Quantum Natural Policy Gradient Based Approach

Thank you for your constructive feedback and insightful comments. Below we clarify and address the points raised in your review:

Regarding Novelty:

We appreciate the reviewer’s concern regarding the novelty of our work. To clarify, our paper is indeed the first in quantum parameterized reinforcement learning to provide a rigorous sample complexity analysis with concrete bounds that surpass classical asymptotic limits. To achieve this, we introduce a novel quantum algorithm incorporating truncated NPG estimators (Algorithm 1), along with a detailed and novel theoretical analysis of the bias-variance tradeoff (specifically Lemma 5 and Lemma 6). This thorough analytical framework constitutes a significant step forward compared to previous quantum reinforcement learning literature.

Regarding Integration with raw-PQC:

The reviewer’s suggestion on integrating our QNPG approach with raw-PQC is indeed interesting. We note, however, that the policy parameterizations of QNPG and raw-PQC differ fundamentally: QNPG utilizes classical parameterization, whereas raw-PQC explicitly leverages quantum circuit parameterization. Despite this difference, exploring integration or hybrid approaches between these two setups could be an intriguing direction for future research, which we will mention in the future work directions.

Regarding Formatting Issues:

We carefully re-examined our LaTeX source file following the reviewer's comment on the line spacing discrepancies. Upon checking, we find no use of any command that reduces the line spacing. We believe this formatting inconsistency is likely due to LaTeX's inherent compilation behavior. One thing we see is that $\hat{g}_\rho$ is in text, and seems to be taking more space than the text, and thus the space seems to be less due to in-line Math text.

Regarding Essential References:

We sincerely thank the reviewer for highlighting the additional references. These valuable references will be explicitly cited and discussed in the final version of our related work section. The mentioned RL paper does not study sample complexity guarantee. Further, will describe the literature on Quantum Multi-Armed Bandits where speedup is shown.

Thank you once again for your insightful feedback, which significantly strengthens our manuscript.

Thank you for your detailed feedback and constructive comments. We address your concerns and questions clearly below:

Response to Weaknesses 1, 4, and Question 1:

Practicality of Quantum Oracle Implementations and Experimental Validation:
We acknowledge the reviewer’s valid concerns about the real-world implementation of quantum oracles and experimental validations. Our current work is primarily theoretical, and extensive numerical experiments and implementations are planned as future work. Importantly, the quantum oracles required by our algorithm can be implemented with polynomial cost on quantum computers as shown in [1]-[3], posing no conceptual barriers. However, classically simulating these oracles is NP-hard. We can ofcourse simulate for a very small scale, but that will not help validate in a larger example. Future studies will aim to implement and validate our methods on quantum simulators or quantum hardware.

Response to Weakness 2:

Given additional page in the final version, we will work on adding explanations. Further, we appreciate the reviewer’s suggestion regarding the complexity of Lemma 6's proof. While we have provided detailed explanations and derivations for each steps of the proof in the Appendix, we agree that further clarification could help readers that are unfamiliar with the specific concepts involved understand better. To clarify, we summarize the high-level idea of the proof as follows, which we will incorporate into the final draft:

"The proof of Lemma 6 proceeds by first expressing the error between the estimated gradient and the ideal gradient as a recursive update from the inner-loop iteration. It then demonstrates that each update contracts the error exponentially, meaning the error shrinks by a fixed multiplicative factor with each iteration. Subsequently, the proof carefully bounds the additional errors introduced by truncation bias and estimator variance, leading to explicit residual terms. Finally, these contraction and residual terms are combined through a recursive argument to establish the overall error guarantees."

If you can provide some specific comments for confusing symbols, we would appreciate that so that we can revise them in the final version.

Response to Weakness 3:

Comparison with Existing Quantum RL Algorithms:
We clarify that, to our knowledge, there are currently no quantum reinforcement learning algorithms with provable guarantees for general parameterization setting in the existing literature. Thus, our approach introduces significant novelty in providing theoretical foundations and complexity improvements as compared to classical and prior quantum algorithms.

Response to Question 2:

Validity of Score Function Assumptions:
We note as we stated after the assumption, regarding Lipschitz continuity and smoothness of the score function, this assumption is common in reinforcement learning literature, and can indeed be verified for commonly used parameterized policies, such as softmax policies [4] and Gaussian policies, where the action is modeled as a Gaussian distribution, with parameters (mean and standard deviation) learned from the environment [5]. The assumption is necessary since without this, we would have unbounded gradients.

Thank you again for your thoughtful comments. We hope these clarifications enhance the clarity and impact of our contributions.

References

[1] Implementing Grover oracles for quantum key search on AES and LowMC, 2019

[2] Automatic Generation of Grover Quantum Oracles for Arbitrary Data Structures, 2021

[3] Verified Compilation of Quantum Oracles, 2022

[4] On the Theory of Policy Gradient Methods: Optimality, Approximation, and Distribution Shift, 2021

[5] An Improved Analysis of (Variance-Reduced) Policy Gradient and Natural Policy Gradient Methods, 2020

Thank you for your positive feedback and valuable insights. We address your primary concerns clearly below:

Clarification on the "Model-Free" Nature of QNPG:

We appreciate the insightful discussion about whether our Quantum Natural Policy Gradient (QNPG) algorithm should be considered "model-free". Indeed, we agree with the reviewer that we explicitly assume the existence of a quantum transition oracle. However, our use of the term "model-free" aligns with the standard distinctions within reinforcement learning literature:

• Classical model-based reinforcement learning algorithms explicitly attempt to learn or estimate the transition dynamics (state-action probabilities) of the environment from sampled data. This learning process typically introduces a complexity that scales directly with the size of the state-action space $|S||A|$, making it computationally expensive for large environments.

• In contrast, our QNPG algorithm does not explicitly attempt to estimate the environment's transition probabilities. Instead, we directly use the provided quantum transition oracle to sample trajectories and estimate gradients without explicitly estimating or storing transition dynamics. Consequently, the sample complexity for parameterized model-free methods (like our proposed QNPG) does not scale with $|S||A|$. Instead, it depends primarily on mixing times and related properties, reflecting a fundamentally different complexity structure from traditional model-based methods.

We note that the use of quantum transition oracle is like the use of generative models in classical decision making, and thus unless the transition dynamics is learnt, the algorithm is denoted model-free. This important distinction will be further emphasized in the final version of the manuscript.

Regarding Numerical Results:

We acknowledge the reviewer's point regarding numerical experimentation. The current focus of our work is theoretical, and comprehensive numerical experiments are planned for future studies. We clarify that while implementing quantum oracles in actual quantum computers is feasible with polynomial complexity and no fundamental conceptual barriers, simulating these oracles classically is NP-hard.

We greatly appreciate your constructive comments and hope these clarifications clearly address your points.

We appreciate the reviewer’s thoughtful comments and valuable feedback. We address the main points and questions raised as follows:

Response to Weakness 1 and Questions 1-3:

Why Quantum Computing Accelerates Mean Estimation (Lemma 1):
Quantum computing fundamentally accelerates mean estimation through the Quantum Amplitude Estimation (QAE) algorithm [1]. Quantum states inherently encode probability distributions directly into their amplitudes, allowing QAE to estimate probabilities and expectations using methods similar to Grover iterations [2]. This results in a quadratic speedup in terms of accuracy and the required number of queries or experiments compared to classical sampling methods.

Quantum Acceleration in Convergence Analysis (Lemma 5):
We agree that Lemma 5 is central to the speedup presented in our paper. Specifically, quantum computing provides variance-reduced estimators for $F_h$ and $g_h$. Compared with classical NPG, our quantum estimators achieve quadratically reduced variance. However, this reduction introduces a bias-variance trade-off absent from classical NPG, which traditionally provides unbiased gradient estimators.

Key Novelties:
We thus sum up our key novelties. The introduction of biased estimators in our quantum NPG algorithm is due to the use of truncated estimators for the Fisher information and policy gradients, which are necessary for compatibility with quantum RL oracles (Equations 16-17). This truncation introduces a bias absent in classical estimators. Specifically:

• Classical NPG methods rely on stochastic algorithms that sample the NPG gradient from trajectories of random length, typically following a geometric distribution. However, we cannot achieve quantum superpositions of trajectories with random lengths due to fundamental quantum computational constraints. Consequently, we adapt the classical algorithm to deterministically truncate trajectory lengths.
• This deterministic truncation changes the classical unbiased estimation approach into one that inherently introduces bias. Therefore, we modify the methods of estimating the Fisher information and gradient estimators, replacing geometric distribution sampling with deterministic truncation.
• To rigorously address the bias introduced by truncation, we significantly adjust the analytical framework. Specifically, Lemma 4 quantifies complexities associated with obtaining quantum superpositions, and Theorem 1 explicitly bounds the biases and variances resulting from truncation. Lemmas 5 and 6 subsequently utilize these bounds, meticulously handling bias accumulation and variance reduction through modified analytical techniques and recursive error bounding methods.

These methodological adaptations and theoretical analyses represent substantial novelty over classical NPG algorithms and are crucial to achieving our main theoretical results.

Response to Weakness 2 and Question 4:

Numerical Experiments:
We acknowledge the reviewer's concern regarding the absence of numerical results. The primary objective of this paper is theoretical, and thus detailed numerical experimentation is intended as future work. While implementing quantum oracles on an actual quantum device would be of polynomial complexity and presents no conceptual challenges, simulating these oracles classically is NP-hard. We can ofcourse simulate for a very small scale, but that will not help validate in a larger example.

Thank you again for your valuable comments. We hope these clarifications clearly articulate the novelty and significance of our contributions.

References

[1] Quantum Sub-Gaussian Mean Estimator, 2021

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