Reservoir Computing with Feedback for Quantum State Identification

Identifying quantum states through measurements is a complex challenge encountered in many quantum applications. Given ample time and multiple measurements, quantum tomography is frequently used to pinpoint the state. However, real-world qubit readouts are often constrained by the information acquired from specific read-out setups. While neural networks have been used to address quantum state identification, their creation and operation come at a significant computational expense. Reservoir computing offers a potential solution by substituting a naturally occurring nonlinear dynamical system for the neural network. This switch reduces training costs while preserving functional versatility.



Echo state networks (ESNs) are a subset of reservoir computers. In ESNs, the internal reservoir state undergoes a linear transformation at each timestep, which is both fixed and randomly chosen. Subsequently, this state is subject to a nonlinear transformation. The reservoir dynamics then guide the system towards a sequence of states dictated by the input signal. This sequence can then be tailored to a linear function, encapsulating the characteristics of the desired target system. A potential drawback is that the fixed reservoir states might not offer the complexity needed for specific problems. While directly altering (training) the ESN would reintroduce the computational burden, an indirect modification can be achieved by redirecting some output as input. This feedback loop can potentially influence the internal reservoir state, yielding enhanced ESN outputs suitable for a broader array of challenges.



We suggest deploying an ESN, with output feedback, to interpret quantum state readouts from partial or incomplete measurements. Given the intrinsic quantum nature of the problem, our proposal leans towards a quantum solution: a quantum ESN. This approach seems optimal, as replicating quantum states on classical computing systems is notoriously computationally intensive. Thus, a quantum-based ESN should be more efficient for quantum state readouts. By feeding the measurements from a disordered quantum state, we aim to calibrate our feedback-augmented ESN to approximate the original state closely.