Emergent Recursive Kernels in Quantum Reservoir Computing

Abstract

Learning with a quantum reservoir can be far simpler than simulating its dynamics. We show that a quantum reservoir governed by a fixed evolution, consisting of an input coupling followed by an independently sampled Haar unitary, admits an exact recursive kernel in the large-dimension limit at fixed history length. Memory orthogonality and factorization of propagated overlaps reduce each kernel evaluation to a scalar recurrence, linear in history length and independent of reservoir size. We establish self-averaging around this limit and connect the kernel coefficients to out-of-time-ordered correlators. Our analysis of finite-dimensional effects shows that the computational simplicity of the Haar kernel does not come at the expense of expressive power. Moreover, we demonstrate how the kernel can be tuned for different learning tasks by adjusting properties such as memory retention and temporal interaction order. On the NARMA10 benchmark, the tuned kernel achieves lower prediction error than the tested Ising and Sachdev--Ye--Kitaev reservoirs.

Quantum reservoir computing can generate exponentially large feature spaces, but does its learning kernel require quantum simulation? This paper studies repeated qubit-input injection into a $D$-dimensional reservoir, followed by a Haar-random reservoir unitary sampled independently of a fixed input-coupling “writer” and reused at every step. The readout kernel is $k_L=D\operatorname{Tr}(\rho_L\rho'_L)-1$, corresponding to complete traceless-observable readout. For maximally mixed initialization and fixed history length $L$, the kernel approaches the scalar recurrence $\kappa_t=b(u_t,v_t)+a(u_t,v_t)\kappa_{t-1}$, with $\kappa_0=0$. Injection overlaps determine $b$, while normalized Kraus-operator overlaps determine $a$. Asymptotic freeness supplies memory orthogonality and propagation factorization; Haar-moment estimates establish bias $O_L(D^{-2})$ and root-mean-square error $O_L(D^{-1})$. Once local coefficients are known, evaluation costs $O(L)$, independently of $D$. Exact identities express injection coefficients as Pauli-averaged OTOCs. Finite-device linear readouts lie within the corresponding recursive feature space, although their required readout norms—and hence learning guarantees—can differ. A classical two-parameter extension controls memory retention and temporal interaction order. On bounded-input NARMA10, it yields lower mean test error than the tested Ising and complex-SYK reservoirs across four–nine total qubits at 1,024 training examples, and across 256–1,024 examples at nine qubits. The convergence theorem does not cover histories growing with $D$, arbitrary initial states, or writer–router dependence. Feature-space inclusion does not guarantee comparable sample efficiency. The benchmark comparison is exploratory, conditional on fixed parameter-candidate banks, and assumes ideal full-state readout.