The processing of temporal data, crucial for applications like real-time inference and online learning, has been revolutionized by classical machine learning algorithms such as recurrent neural networks and transformers. Physical neural networks (PNNs), employing physical systems for computation, offer a promising avenue for online data processing. Physical reservoir computing, a subset of PNNs, uses a linear projector on the physical system's state, enabling fast convex optimization and efficient temporal learning. Quantum systems, with their exponentially scaling Hilbert space, offer significant potential for scalable and resource-efficient machine learning. However, current noisy intermediate-scale quantum (NISQ) hardware faces challenges in handling temporal data. Quantum Sampling Noise (QSN) and optimization barren plateaus hinder QML's ability to train and infer on high-dimensional data. Furthermore, finite coherence times and measurement backaction (information scrambling) limit the processing of long data streams. This paper introduces a novel approach to address these limitations.

Existing research explores physical reservoir computing using various physical systems, including quantum systems. While quantum systems hold immense promise for more efficient machine learning, their practical application is limited by several factors. Quantum sampling noise (QSN) introduces uncertainty, reducing the accuracy of both training and inference, even on fault-tolerant hardware. The optimization landscape for quantum systems often presents barren plateaus, making training exponentially difficult. When dealing with long data streams, finite coherence times and information scrambling due to repeated measurements severely restrict the length of data that can be processed effectively. The concept of fading memory in classical systems requires a strict condition for a physical system to maintain a persistent temporal memory on long data streams, which is usually analysed using Volterra series theory. Prior work in quantum reservoir computing has utilized reset operations implicitly, but the crucial role of deterministic reset in establishing persistent memory has not been adequately highlighted.

This paper presents a Quantum Volterra Theory (QVT) for analyzing the memory capabilities of quantum systems subjected to repeated measurements. Based on this theory, the authors propose NISQRC, an algorithm that uses mid-circuit measurements and deterministic reset operations to enable inference on arbitrarily long temporal data, irrespective of coherence time limitations. The algorithm involves dividing the quantum system into memory and readout qubits. After each encoding step (injecting data via a parameterized quantum channel), only the readout qubits are measured, and then deterministically reset. The memory qubits retain information from previous inputs. The proposed method efficiently samples from deep circuits under partial measurements, avoiding the exponential complexity associated with traditional approaches. The output features are constructed as probabilities of measurement outcomes. Simulations and experiments are conducted using a 7-qubit quantum processor to demonstrate the algorithm's capabilities in channel equalization, a task relevant to communication systems. The QVT provides a method for characterizing the system's memory timescale, enabling optimized encoding design for specific machine learning tasks. Simulations include noise modeling to accurately reflect experimental conditions.

The key findings of the paper demonstrate that NISQRC successfully addresses several limitations in quantum machine learning on temporal data: 1. **Overcoming Coherence Time Limitations:** NISQRC allows for inference on signals far exceeding the coherence times of individual qubits. Simulations and experiments show successful recovery of signals 500 times longer than qubit lifetimes. This is achieved by balancing the encoding step length with the rate of information extraction via mid-circuit measurements. 2. **Mitigation of Quantum Sampling Noise (QSN):** The algorithm's performance is robust to QSN, with simulations accurately reflecting experimental results even with a large number of shots. 3. **Addressing Information Scrambling:** The deterministic reset protocol effectively prevents information scrambling and thermalization, even in the presence of QSN. This ensures that the system retains persistent memory and avoids becoming an "amnesiac" reservoir. 4. **Efficient Sampling from Deep Circuits:** The proposed method efficiently samples from deep circuits with repeated partial measurements, avoiding the exponential scaling in computation associated with traditional methods. 5. **Experimental Validation:** Experiments on a 7-qubit quantum processor using a circuit-based ansatz demonstrate successful channel equalization on a signal lasting 117 µs, significantly longer than the qubit lifetimes (63 µs - 164 µs for T1 and 9 µs - 231 µs for T2). The results align closely with simulations that assume infinite coherence times. 6. **Importance of Reset and Connectivity:** Experiments also highlight the critical role of the deterministic reset operation in maintaining persistent memory. Removing the reset leads to severely degraded performance. Similarly, reduced connectivity between qubits reduces the system's expressiveness, impacting its ability to effectively learn the channel equalization task. The Jacobian rank is identified as an indicator of this expressiveness. 7. **Quantum Volterra Theory (QVT):** The developed QVT framework provides a powerful tool for analyzing the memory properties of quantum systems under repeated measurements, guiding design choices for encoding and measurement protocols.

NISQRC's success in handling long temporal data on NISQ hardware significantly advances the field of quantum machine learning. The algorithm's robustness to QSN and coherence time limitations demonstrates its practicality for real-world applications. The QVT provides a theoretical framework for understanding and optimizing quantum reservoir computing, enabling the design of systems with tailored memory properties. The finding that connectivity plays a crucial role in the system's expressiveness highlights the importance of hardware architecture in quantum machine learning. The observed agreement between simulations and experimental results validates the accuracy of the noise model used in simulations and further underscores the potential of NISQRC for broader applications. Future work can focus on optimizing the algorithm for different quantum architectures and exploring more complex machine learning tasks.

This paper presents NISQRC, a novel algorithm that overcomes key limitations in quantum machine learning for temporal data. Through simulations and experiments, NISQRC demonstrates the ability to process arbitrarily long signals, unconstrained by qubit coherence times. The developed Quantum Volterra Theory provides a powerful tool for designing and optimizing quantum reservoir computing systems. Future research directions include applying NISQRC to more challenging tasks and exploring its capabilities on larger and more advanced quantum computers.

The current implementation of NISQRC faces some limitations. The experiments were conducted on a 7-qubit processor, limiting the complexity of the tasks that can be addressed. Technical constraints associated with mid-circuit measurements on the utilized IBM Quantum platform, such as non-contiguous data collection and potential parameter drifts, introduced some discrepancies between experimental and simulation results. The relatively simple channel equalization task was used for demonstration, and future research should explore more complex tasks. Furthermore, the optimal separation of memory and readout qubits might depend on specific tasks and warrants further investigation.