Machine learning via relativity-inspired quantum dynamics

We present a machine-learning scheme based on the relativistic dynamics of a quantum system, namely a quantum detector inside a cavity resonator. An equivalent analog model can be realized for example in a circuit QED platform subject to properly modulated driving fields. We consider a reservoir-computing scheme where the input data are embedded in the modulation of the system (equivalent to the acceleration of the relativistic object) and the output data are obtained by linear combinations of measured observables. As an illustrative example, we have simulated such a relativistic quantum machine for a challenging classification task, showing a very large enhancement of the accuracy in the relativistic regime. Using kernel-machine theory, we show that in the relativistic regime the task-independent expressivity is dramatically magnified with respect to the Newtonian regime.

Figure 1

Scheme of the relativistic reservoir-computing protocol. (a) Each point $\mathit{x}=(x_1,x_2)$ of the dataset is linearly mapped to acceleration values $(a_1,a_2)$ according to Eq. (5). (b) The acceleration values are used to construct a piecewise constant acceleration profile $a\left(\tau \right)$. (c) The quantum detector, initially at rest in the cavity prepared in a single-mode coherent state, undergoes noninertial motion with proper acceleration $a\left(\tau \right)$. (d) Analog circuit QED system where the analogous proper acceleration is controlled by modulated driving fields. (e) Observables of the detector are measured at different times giving the feature vector $\mathit{X}$ and the affine trial function $\widehat{f}={\mathit{w}}^T\mathit{X}+b$. (f) The classification result is predicted by $\mathrm{sgn}\left[\widehat{f}(x_1,x_2)\right]$.

Figure 2

Figures of merit of the relativistic reservoir-computing protocol. (a) Light and dark histograms correspond to testing samples belonging to different spirals of the dataset. Parameters: $a_0=3, T=2$, and $m=4$. (b) Same quantity plotted for the Newtonian model with same parameters. (c) The empirical kernel spectrum computed for the relativistic (solid line) and Newtonian (dashed line) models with same parameters. The first 40 nonzero eigenvalues ${\gamma }_l$ are plotted in descending order. (d) Inaccuracy of the relativistic (triangles) and Newtonian (squares) models evaluated on both the training (solid lines) and testing (dashed lines) set, as a function of the acceleration time $T$. Parameters: $a_0=1$ and $m=4$. (e) Same quantities plotted as a function of the base acceleration $a_0$, for $T=2$ and $m=4$. (f) Same quantities plotted as a function of the number of repetitions $m$, for $a_0=2$ and $T=2$. Quantities are expressed in natural units, where the scale is fixed by the proper frequency of the atom $\mathrm{\Omega }$.

Figure 3

The phase modulations ${\theta }_{\pm }\left(\tau \right)$ (left panel) and the corresponding rates ${\dot{\theta }}_{\pm }\left(\tau \right)$ that provide the desired simulation of the accelerated motion, where $\mathrm{\Omega }=1\mathrm{MHz}$. Note that modulation rates are in the regime of ${\dot{\theta }}_{\pm }\left(\tau \right)\lesssim 10\mathrm{\Omega }=10\mathrm{MHz}$.

Figure 4

Performances of the reservoir-computing model considered in the main text, but with a qubit replacing the harmonic oscillator. Same parameters as Figs. 2 and 2 in the main text, respectively, showing very similar results.