Optimal quantum learning in proximity to universality

TL;DR

It is demonstrated that the learnability and scalability of the reservoir can be continuously controlled by the parameter $p, allowing us to navigate from classically tractable to maximally expressive quantum dynamics.

Abstract

The boundary between classically simulable and computationally superior quantum systems is fundamental to identifying true quantum advantage. We investigate this within the framework of quantum reservoir computing by introducing a tunable $N$-qubit random circuit model, where a fraction $p$ of Clifford gates are probabilistically substituted with nonstabilizing conditional-$\hat{T}$ gates. We establish a direct correspondence between the reservoir's performance on temporal processing tasks and its entanglement spectrum statistics and long-range nonstabilizer resource content. To assess scalability, we study the scaling of the anti-flatness of states in the large-$N$ limit at a fixed circuit depth ratio $d/N \sim \mathcal{O}(1)$. This is taken as a witness to concentration of measures, a known impediment to learning in thermalizing systems. We demonstrate that the learnability and scalability of the reservoir can be continuously controlled by the parameter $p$, allowing us to navigate from classically tractable to maximally expressive quantum dynamics. These architecture-agnostic results offer a general strategy for designing powerful and trainable quantum machine learning systems and clarify the physical resources underpinning quantum computational advantage.

Figures (5)

• Figure 1: A probabilistic quantum reservoir computer.(a) The circuit’s qubits are split into memory $M$ and readout $R$ subsets, and evolution is captured by an effective contractive channel with the unitary update $\hat{U}_n\equiv\hat{U}_{\rm res}\,\bigotimes_{j\in M} \hat{R}^{Y}_j(\theta_n)$, followed by (idealized) measure-and-reset of readout qubits. Classical inputs, $\theta_n$, are encoded as local rotations $R^Y(\theta):=\exp [-i\hat{Y}\theta/2]$, and reservoir is iterated for $n$ steps. Each brick is either a non-Clifford $\hat{CT}$ with probability $p$ or a random two-qubit Clifford gate with $1-p$. (b) Learnability is characterized via relative distance to quantum universality. The probability $p^{\star}(d)$ indicates the depth-dependent onset of chaotic-integrable crossover from the perspective of entanglement-spectrum. In an optimal $(p,d)$ window, a linear readout on the model’s observables accurately reconstructs nonlinear, time-dependent functionals of the input history.
• Figure 2: Properties of entanglement.(a) Entanglement dynamics starting from a random product-state for $N=14$. (b) Same vs the rescaled depth $(1-p)d/N$. (c) Mean level spacing ratio $\langle r\rangle$ for $d/N=2$ and (inset) $d/N=1$. Dotted lines indicate the quantum and classical limits, with $\langle r_{\mathrm{Q}}\rangle\approx0.6$ and $\langle r_{\mathrm{C}}\rangle\approx0.39$, respectively. Plotted for various $N$ by starting from the initial state $\left|0\right\rangle\!\left\langle0\right|^{\otimes N}$. (d) Relative entropy of distributions for $N=20$. $p^{\star}$ denotes the onset of the crossover to integrable (classical) regime for $d/N=2$. The results are averaged over $200-600$ independent realizations.
• Figure 3: Relative gap of mutual magic.$\Delta\mathcal{I}$ as a function of $p$ at depth $d/N=2$. $p^{\sharp}$ denotes the point after which, for sufficiency large systems, MM becomes relatively submaximal. The results are averaged over $400$ independent realizations.
• Figure 4: Linear memory and nonlinear learnability.(a) Linear memory as function of delay $\tau$ for various $p$ with $(N,d/N)=(10,2)$. Inset shows the mean memory $\overline{\mathcal{C}}$ over the interval $1\leq\tau\leq12$. (b) Nonlinear learnability as function of $p$ for $d/N=1,2$, with $(\tau,N)=(10,10)$. Each input sequence has $\approx2000$ steps; we discard the first $\approx 500$ (washout) and use the rest for training and testing via supervised linear regression on standardized features (reservoir observables). Each data point is obtained by averaging the performance metrics over $50-100$ random configurations and input sequences.
• Figure 5: Scaling of anti-flatness.(a) Size-scaling of anti-flatness and linear fits, plotted in logarithmic scale for various $p$. (b) The decay slope $\alpha(p)$, extracted from a linear fit to $\log_2(\mathcal{F})\!\sim -\alpha N$. The results are averaged over $200$ realizations. On the right axis we have replotted the nonlinear memory for the same depth $d=2N$ to match the performance to the obtained points $p^{\sharp}$ and $p^{\star}$.