Fig. 2: Quantum Volterra Theory (QVT) analysis for (M + R)-qubit reservoir.
From: Overcoming the coherence time barrier in quantum machine learning on temporal data
a First and second order Volterra kernels in a (2 + 1)-qubit QRC, which vanish at large n1 and n2 due to finite memory nM. b Fixed-point of memory subsystem \({\hat{\rho }}_{{{{\rm{FP}}}}}^{{\mathsf{M}}}\) with reset (top) and without reset (bottom), starting from an arbitrary initial state (center). Without reset, the fixed point is always the trivial fully-mixed state and Volterra kernels vanish. The top panel shows the distribution of the 4M = 256 eigenvalues of \({{{{\mathcal{P}}}}}_{0}\) in a (4 + 2)-qubit QRC, where red dots correspond to the static unit eigenvalue λ1 = 1. The remaining eigenvalues λα≥2 (blue) evolve with evolution time τ, leading to a variable memory time. The bottom panel shows the resulting memory time nM as a function of the evolution duration τ. c Memory time nM as a function of qubit lifetimes T1 = γ−1, in terms of the evolution duration τ in a (4 + 2)-qubit QRC. Provided T1 ≫ τ, \({n}_{{{{\rm{M}}}}}\to {n}_{{{{\rm{M}}}}}^{0}\), so that the QRC memory is mostly dominated by its lossless dynamical map and not by T1 in this regime.
