Feedback Connections in Quantum Reservoir Computing with Mid-Circuit Measurements

TL;DR

This work investigates a mid-circuit, feedback-enabled quantum reservoir computing (QRC) architecture designed to operate within qubit coherence time, bridging restart-based and continuous protocols. By incorporating input and feedback gates into a minimal two-qubit scheme and resetting after each step, the approach isolates the impact of feedback on memory and prediction, while leveraging Haar-random unitaries to induce richness in dynamics. Training uses a straightforward linear readout learned via the Moore-Penrose pseudoinverse, and performance is evaluated on both classical (Mackey-Glass) and quantum (Ising-chain) time series, showing significant memory retention with appropriate feedback strength and shot counts. Hardware demonstrations on IBM QPU back the viability of mid-circuit feedback in near-term devices, and analysis of ESP across models indicates that the proposed scheme largely preserves fading memory, supporting its potential for scalable quantum temporal processing.

Abstract

Existing approaches to quantum reservoir computing can be broadly categorized into restart-based and continuous protocols. Restart-based methods require reinitializing the quantum circuit for each time step, while continuous protocols use mid-circuit measurements to enable uninterrupted information processing. A gap exists between these two paradigms: while restart-based methods naturally have high execution times due to the need for circuit reinitialization, they can employ novel feedback connections to enhance performance. In contrast, continuous methods have significantly faster execution times but typically lack such feedback mechanisms. In this work, we investigate a novel quantum reservoir computing scheme that integrates feedback connections, which can operate within the coherence time of a qubit. We demonstrate our architecture using a minimal example and evaluate memory capacity and predictive capabilities. We show that the correlation coefficient for the short-term memory task on past inputs is nonzero, indicating that feedback connections can effectively operate during continuous processing to allow the model to remember past inputs.

Figures (6)

• Figure 1: QRC protocols: Each architecture is designed to process an input sequence ${s_k}$. The systems are initialized in the state $\rho_0$. A unitary operator $U(\cdot)$, constructed from parameterized Pauli rotations, governs the state evolution. Each $U(\cdot)$ represents the processing of one time step (cycle). (a) Graphical representation of a conventional restart-based QRC scheme. For each timestep, the architecture must be restarted and run from the beginning. This results in a quadratic runtime. (b) A rewind protocol that does not restart from the beginning, but only processes the last $\tau$ time steps. (c) A feedback-controlled QRC protocol. Only one timestep is processed at a time, but the expectation value from the previous trial is stored in a buffer. (d) A mid-circuit measurement architecture that continuously monitors the quantum state. No restart is required. This architecture can be combined with weak measurements. (e) A feedback-driven QRC architecture with mid-circuit measurements. Feedback is incorporated by feeding the measurement strings $\textbf{m}_k$ back into the reservoir, a process that can be done at runtime. The expectation values $\langle Z \rangle_k$ are then computed in the post-processing step after all the circuits have been executed.
• Figure 2: Minimal two-qubit unitary circuit. The circuit processes the input $s_k$ via a rotation $R(a_{\mathrm{in}}\,s_k)$ and then incorporates feedback via rotations $R(a_{\mathrm{fb}}\,m_{k-1}^0)$ and $R(a_{\mathrm{fb}}\,m_{k-1}^1)$, where $\{m_{k-1}^0, m_{k-1}^1\}$ are the previous measurement results. The structure of $R(\cdot)$ is explained in Eq. \ref{['eq1']}. A final Haar-random unitary $U_{\text{Haar}}$ is applied at the end.
• Figure 3: Evaluation of short term memory and predictive capabilities: All values are averaged over 128 Haar-random unitaries. Since there is no single 'privileged' choice of unitary, our primary interest is in the average performance of the protocol over these random instances. Therefore we choose to indicate one standard deviation of the mean with the error bars to show how much this mean would vary if we resampled the Haar measure many times. (a) The left panel shows the short-term memory (STM) capacity for several different values of $a_{fb}$, with $\tau$ ranging from $0$ to $-3$. The right panel provides a more detailed view for $a_{fb} = 1.3$, plotting all values of $\tau$. (b) Here, we plot the NMSE for the quantum 1D Ising chain, showing the prediction capabilities for $\tau = 1$ over different $a_{fb}$ values. The right panel shows the prediction capabilities for different $\tau$ values for selected $a_{fb}$ values. (c) shows the same analysis for the Mackey-Glass task. $5k$ shots were used for the right panels in both (b) and (c).
• Figure 4: Echo state property for different architectures: The top figures show the first component of the reservoir state over time for five randomly initialized runs, while the bottom figures show the absolute difference between these five reservoir states. (a) Classical ESN model. (b) Feedback-driven QRC regime feedbackdriven. (c) Mid-circuit measurements QRC mid_circuit_measurements. (d) Proposed feedback-based QRC with mid-circuit measurements and post-measurement resets. (e) Proposed feedback-based QRC with mid-circuit measurements and continuous operation (no resets).
• Figure 5: QPU Test: NMSE for various $\tau$ using (a) Ising chain and (b) Mackey-Glass sequences.