Daniel D. Scherer

Nico Meyer, Daniel D. Scherer, Axel Plinge et al.

Quantum machine learning implemented by variational quantum circuits (VQCs) is considered a promising concept for the noisy intermediate-scale quantum computing era. Focusing on applications in quantum reinforcement learning, we propose a specific action decoding procedure for a quantum policy gradient approach. We introduce a novel quality measure that enables us to optimize the classical post-processing required for action selection, inspired by local and global quantum measurements. The resulting algorithm demonstrates a significant performance improvement in several benchmark environments. With this technique, we successfully execute a full training routine on a 5-qubit hardware device. Our method introduces only negligible classical overhead and has the potential to improve VQC-based algorithms beyond the field of quantum reinforcement learning.

Nico Meyer, Christian Ufrecht, Maniraman Periyasamy et al.

Quantum reinforcement learning is an emerging field at the intersection of quantum computing and machine learning. While we intend to provide a broad overview of the literature on quantum reinforcement learning - our interpretation of this term will be clarified below - we put particular emphasis on recent developments. With a focus on already available noisy intermediate-scale quantum devices, these include variational quantum circuits acting as function approximators in an otherwise classical reinforcement learning setting. In addition, we survey quantum reinforcement learning algorithms based on future fault-tolerant hardware, some of which come with a provable quantum advantage. We provide both a birds-eye-view of the field, as well as summaries and reviews for selected parts of the literature.

Nico Meyer, Daniel D. Scherer, Axel Plinge et al.

Reinforcement learning is a growing field in AI with a lot of potential. Intelligent behavior is learned automatically through trial and error in interaction with the environment. However, this learning process is often costly. Using variational quantum circuits as function approximators potentially can reduce this cost. In order to implement this, we propose the quantum natural policy gradient (QNPG) algorithm -- a second-order gradient-based routine that takes advantage of an efficient approximation of the quantum Fisher information matrix. We experimentally demonstrate that QNPG outperforms first-order based training on Contextual Bandits environments regarding convergence speed and stability and moreover reduces the sample complexity. Furthermore, we provide evidence for the practical feasibility of our approach by training on a 12-qubit hardware device.

Marco Wiedmann, Marc Hölle, Maniraman Periyasamy et al.

VQA have attracted a lot of attention from the quantum computing community for the last few years. Their hybrid quantum-classical nature with relatively shallow quantum circuits makes them a promising platform for demonstrating the capabilities of NISQ devices. Although the classical machine learning community focuses on gradient-based parameter optimization, finding near-exact gradients for VQC with the parameter-shift rule introduces a large sampling overhead. Therefore, gradient-free optimizers have gained popularity in quantum machine learning circles. Among the most promising candidates is the SPSA algorithm, due to its low computational cost and inherent noise resilience. We introduce a novel approach that uses the approximated gradient from SPSA in combination with state-of-the-art gradient-based classical optimizers. We demonstrate numerically that this outperforms both standard SPSA and the parameter-shift rule in terms of convergence rate and absolute error in simple regression tasks. The improvement of our novel approach over SPSA with stochastic gradient decent is even amplified when shot- and hardware-noise are taken into account. We also demonstrate that error mitigation does not significantly affect our results.

Maniraman Periyasamy, Nico Meyer, Christian Ufrecht et al.

The data representation in a machine-learning model strongly influences its performance. This becomes even more important for quantum machine learning models implemented on noisy intermediate scale quantum (NISQ) devices. Encoding high dimensional data into a quantum circuit for a NISQ device without any loss of information is not trivial and brings a lot of challenges. While simple encoding schemes (like single qubit rotational gates to encode high dimensional data) often lead to information loss within the circuit, complex encoding schemes with entanglement and data re-uploading lead to an increase in the encoding gate count. This is not well-suited for NISQ devices. This work proposes 'incremental data-uploading', a novel encoding pattern for high dimensional data that tackles these challenges. We spread the encoding gates for the feature vector of a given data point throughout the quantum circuit with parameterized gates in between them. This encoding pattern results in a better representation of data in the quantum circuit with a minimal pre-processing requirement. We show the efficiency of our encoding pattern on a classification task using the MNIST and Fashion-MNIST datasets, and compare different encoding methods via classification accuracy and the effective dimension of the model.

Nico Meyer, Christopher Mutschler, Dominik Seuß et al.

Logical operations are essential for quantum computation within quantum error-correcting codes. However, discovering their physical realizations is challenging, especially for non-additive codes that lack a stabilizer description. We present a general learning-based framework that, given only an encoding circuit, constructs physical implementations of logical operations while enforcing structural properties such as transversality or shallow depth. Our approach is validated by rediscovering known logical operations of standard stabilizer codes. We then extend it to a co-design procedure, dubbed variational early fault-tolerant quantum computing (VarEFTQC), which tailors non-additive encodings to a given noise model and enforces desired logical gate sets, such as transversal IQP-type families or low-depth universal sets. A software library implements the complete learning pipeline, including loss-function variants, ansatz families, and optimization routines. Together, these results position VarEFTQC as a practical tool for discovering hardware-adapted logical gadgets for early fault-tolerant quantum computing.

Maniraman Periyasamy, Marc Hölle, Marco Wiedmann et al.

Deep reinforcement learning (DRL) often requires a large number of data and environment interactions, making the training process time-consuming. This challenge is further exacerbated in the case of batch RL, where the agent is trained solely on a pre-collected dataset without environment interactions. Recent advancements in quantum computing suggest that quantum models might require less data for training compared to classical methods. In this paper, we investigate this potential advantage by proposing a batch RL algorithm that utilizes VQC as function approximators within the discrete batch-constraint deep Q-learning (BCQ) algorithm. Additionally, we introduce a novel data re-uploading scheme by cyclically shifting the order of input variables in the data encoding layers. We evaluate the efficiency of our algorithm on the OpenAI CartPole environment and compare its performance to the classical neural network-based discrete BCQ.

Nico Meyer, Christopher Mutschler, Dominik Seuß et al.

Concatenating quantum error correction codes scales error correction capability by driving logical error rates down double-exponentially across levels. However, the noise structure shifts under concatenation, making it hard to choose an optimal code sequence. We automate this choice by estimating the effective noise channel after each level and selecting the next code accordingly. In particular, we use learning-based methods to tailor small, non-additive encoders when the noise exhibits sufficient structure, then switch to standard codes once the noise is nearly uniform. In simulations, this level-wise adaptation achieves a target logical error rate with far fewer qubits than concatenating stabilizer codes alone--reducing qubit counts by up to two orders of magnitude for strongly structured noise. Therefore, this hybrid, learning-based strategy offers a promising tool for early fault-tolerant quantum computing.

Maniraman Periyasamy, Axel Plinge, Christopher Mutschler et al.

The study of variational quantum algorithms (VQCs) has received significant attention from the quantum computing community in recent years. These hybrid algorithms, utilizing both classical and quantum components, are well-suited for noisy intermediate-scale quantum devices. Though estimating exact gradients using the parameter-shift rule to optimize the VQCs is realizable in NISQ devices, they do not scale well for larger problem sizes. The computational complexity, in terms of the number of circuit evaluations required for gradient estimation by the parameter-shift rule, scales linearly with the number of parameters in VQCs. On the other hand, techniques that approximate the gradients of the VQCs, such as the simultaneous perturbation stochastic approximation (SPSA), do not scale with the number of parameters but struggle with instability and often attain suboptimal solutions. In this work, we introduce a novel gradient estimation approach called Guided-SPSA, which meaningfully combines the parameter-shift rule and SPSA-based gradient approximation. The Guided-SPSA results in a 15% to 25% reduction in the number of circuit evaluations required during training for a similar or better optimality of the solution found compared to the parameter-shift rule. The Guided-SPSA outperforms standard SPSA in all scenarios and outperforms the parameter-shift rule in scenarios such as suboptimal initialization of the parameters. We demonstrate numerically the performance of Guided-SPSA on different paradigms of quantum machine learning, such as regression, classification, and reinforcement learning.

Nico Meyer, Christian Ufrecht, Maniraman Periyasamy et al.

Quantum computer simulation software is an integral tool for the research efforts in the quantum computing community. An important aspect is the efficiency of respective frameworks, especially for training variational quantum algorithms. Focusing on the widely used Qiskit software environment, we develop the qiskit-torch-module. It improves runtime performance by two orders of magnitude over comparable libraries, while facilitating low-overhead integration with existing codebases. Moreover, the framework provides advanced tools for integrating quantum neural networks with PyTorch. The pipeline is tailored for single-machine compute systems, which constitute a widely employed setup in day-to-day research efforts.

Marco Wiedmann, Maniraman Periyasamy, Daniel D. Scherer

VQC can be understood through the lens of Fourier analysis. It is already well-known that the function space represented by any circuit architecture can be described through a truncated Fourier sum. We show that the spectrum available to that truncated Fourier sum is not entirely determined by the encoding gates of the circuit, since the variational part of the circuit can constrain certain coefficients to zero, effectively removing that frequency from the spectrum. To the best of our knowledge, we give the first description of the functional dependence of the Fourier coefficients on the variational parameters as trigonometric polynomials. This allows us to provide an algorithm which computes the exact spectrum of any given circuit and the corresponding Fourier coefficients. Finally, we demonstrate that by comparing the Fourier transform of the dataset to the available spectra, it is possible to predict which VQC out of a given list of choices will be able to best fit the data.

Nico Meyer, Jakob Murauer, Alexander Popov et al.

Reinforcement learning is a powerful framework aiming to determine optimal behavior in highly complex decision-making scenarios. This objective can be achieved using policy iteration, which requires to solve a typically large linear system of equations. We propose the variational quantum policy iteration (VarQPI) algorithm, realizing this step with a NISQ-compatible quantum-enhanced subroutine. Its scalability is supported by an analysis of the structure of generic reinforcement learning environments, laying the foundation for potential quantum advantage with utility-scale quantum computers. Furthermore, we introduce the warm-start initialization variant (WS-VarQPI) that significantly reduces resource overhead. The algorithm solves a large FrozenLake environment with an underlying 256x256-dimensional linear system, indicating its practical robustness.

Nico Meyer, Christian Ufrecht, George Yammine et al.

Benchmarking and establishing proper statistical validation metrics for reinforcement learning (RL) remain ongoing challenges, where no consensus has been established yet. The emergence of quantum computing and its potential applications in quantum reinforcement learning (QRL) further complicate benchmarking efforts. To enable valid performance comparisons and to streamline current research in this area, we propose a novel benchmarking methodology, which is based on a statistical estimator for sample complexity and a definition of statistical outperformance. Furthermore, considering QRL, our methodology casts doubt on some previous claims regarding its superiority. We conducted experiments on a novel benchmarking environment with flexible levels of complexity. While we still identify possible advantages, our findings are more nuanced overall. We discuss the potential limitations of these results and explore their implications for empirical research on quantum advantage in QRL.

Nico Meyer, Julian Berberich, Christopher Mutschler et al.

Quantum machine learning leverages quantum computing to enhance accuracy and reduce model complexity compared to classical approaches, promising significant advancements in various fields. Within this domain, quantum reinforcement learning has garnered attention, often realized using variational quantum circuits to approximate the policy function. This paper addresses the robustness and generalization of quantum reinforcement learning by combining principles from quantum computing and control theory. Leveraging recent results on robust quantum machine learning, we utilize Lipschitz bounds to propose a regularized version of a quantum policy gradient approach, named the RegQPG algorithm. We show that training with RegQPG improves the robustness and generalization of the resulting policies. Furthermore, we introduce an algorithmic variant that incorporates curriculum learning, which minimizes failures during training. Our findings are validated through numerical experiments, demonstrating the practical benefits of our approach.

Nico Meyer, Martin Röhn, Jakob Murauer et al.

Linear systems of equations can be found in various mathematical domains, as well as in the field of machine learning. By employing noisy intermediate-scale quantum devices, variational solvers promise to accelerate finding solutions for large systems. Although there is a wealth of theoretical research on these algorithms, only fragmentary implementations exist. To fill this gap, we have developed the variational-lse-solver framework, which realizes existing approaches in literature, and introduces several enhancements. The user-friendly interface is designed for researchers that work at the abstraction level of identifying and developing end-to-end applications.

Nico Meyer, Christopher Mutschler, Andreas Maier et al.

Quantum error correction is crucial for protecting quantum information against decoherence. Traditional codes like the surface code require substantial overhead, making them impractical for near-term, early fault-tolerant devices. We propose a novel objective function for tailoring error correction codes to specific noise structures by maximizing the distinguishability between quantum states after a noise channel, ensuring efficient recovery operations. We formalize this concept with the distinguishability loss function, serving as a machine learning objective to discover resource-efficient encoding circuits optimized for given noise characteristics. We implement this methodology using variational techniques, termed variational quantum error correction (VarQEC). Our approach yields codes with desirable theoretical and practical properties and outperforms standard codes in various scenarios. We also provide proof-of-concept demonstrations on IBM and IQM hardware devices, highlighting the practical relevance of our procedure.

Alexander Mattick, Maniraman Periyasamy, Christian Ufrecht et al.

Quantum computing is currently strongly limited by the impact of noise, in particular introduced by the application of two-qubit gates. For this reason, reducing the number of two-qubit gates is of paramount importance on noisy intermediate-scale quantum hardware. To advance towards more reliable quantum computing, we introduce a framework based on ZX calculus, graph-neural networks and reinforcement learning for quantum circuit optimization. By combining reinforcement learning and tree search, our method addresses the challenge of selecting optimal sequences of ZX calculus rewrite rules. Instead of relying on existing heuristic rules for minimizing circuits, our method trains a novel reinforcement learning policy that directly operates on ZX-graphs, therefore allowing us to search through the space of all possible circuit transformations to find a circuit significantly minimizing the number of CNOT gates. This way we can scale beyond hard-coded rules towards discovering arbitrary optimization rules. We demonstrate our method's competetiveness with state-of-the-art circuit optimizers and generalization capabilities on large sets of diverse random circuits.

Maniraman Periyasamy, Christian Ufrecht, Daniel D. Scherer et al.

Whether QML can offer a transformative advantage remains an open question. The severe constraints of NISQ hardware, particularly in circuit depth and connectivity, hinder both the validation of quantum advantage and the empirical investigation of major obstacles like barren plateaus. Circuit cutting techniques have emerged as a strategy to execute larger quantum circuits on smaller, less connected hardware by dividing them into subcircuits. However, this partitioning increases the number of samples needed to estimate the expectation value accurately through classical post-processing compared to estimating it directly from the full circuit. This work introduces a novel regularization term into the QML optimization process, directly penalizing the overhead associated with sampling. We demonstrate that this approach enables the optimizer to balance the advantages of gate cutting against the optimization of the typical ML cost function. Specifically, it navigates the trade-off between minimizing the cutting overhead and maintaining the overall accuracy of the QML model, paving the way to study larger complex problems in pursuit of quantum advantage.

Maja Franz, Lucas Wolf, Maniraman Periyasamy et al.

Deep Reinforcement Learning (RL) has considerably advanced over the past decade. At the same time, state-of-the-art RL algorithms require a large computational budget in terms of training time to converge. Recent work has started to approach this problem through the lens of quantum computing, which promises theoretical speed-ups for several traditionally hard tasks. In this work, we examine a class of hybrid quantum-classical RL algorithms that we collectively refer to as variational quantum deep Q-networks (VQ-DQN). We show that VQ-DQN approaches are subject to instabilities that cause the learned policy to diverge, study the extent to which this afflicts reproduciblity of established results based on classical simulation, and perform systematic experiments to identify potential explanations for the observed instabilities. Additionally, and in contrast to most existing work on quantum reinforcement learning, we execute RL algorithms on an actual quantum processing unit (an IBM Quantum Device) and investigate differences in behaviour between simulated and physical quantum systems that suffer from implementation deficiencies. Our experiments show that, contrary to opposite claims in the literature, it cannot be conclusively decided if known quantum approaches, even if simulated without physical imperfections, can provide an advantage as compared to classical approaches. Finally, we provide a robust, universal and well-tested implementation of VQ-DQN as a reproducible testbed for future experiments.