Noisy quantum kernel machines

In the noisy intermediate-scale quantum era, an important goal is the conception of implementable algorithms that exploit the rich dynamics of quantum systems and the high dimensionality of the underlying Hilbert spaces to perform tasks while prescinding from noise-proof physical systems. An emerging class of quantum learning machines is that based on the paradigm of quantum kernels. Here, we study how dissipation and decoherence affect their performance. We address this issue by investigating the expressivity and the generalization capacity of these models within the framework of kernel theory. We introduce and study the effective kernel rank, a figure of merit that quantifies the number of independent features a noisy quantum kernel is able to extract from input data. Moreover, we derive an upper bound on the generalization error of the model that involves the average purity of the encoded states. Thereby we show that decoherence and dissipation can be seen as an implicit regularization for quantum kernel machines. As an illustrative example, we report exact finite-size simulations of machines based on chains of driven-dissipative quantum spins to perform a classification task, where the input data are encoded into the driving fields and the quantum physical system is fixed. We determine how the performance of noisy kernel machines scales with the number of nodes (chain sites) as a function of decoherence and examine the effect of imperfect measurements.

Figure 1

Scheme of a noisy quantum kernel machine. An element $\mathit{x}$ of the input space $\mathcal{X}$ is encoded into a density matrix $\widehat{\rho }\left(\mathit{x}\right)$ obtained by evolving in time a fixed initial state described by the density matrix ${\widehat{\rho }}_0$ [see Fig. 2 for a specific example of the encoding process described by the evolution map $\mathcal{M}\left(\mathit{x}\right)]$. The measured features are represented by a vector of observables $\mathit{\varphi }\left(\mathit{x}\right)$ (with an added 1 corresponding to the unity operator as the first element to create an offset term) that belongs to the feature space $\mathcal{F}$. The trial function is obtained by applying a linear transformation to the feature vector (depending nonlinearly on $\mathit{x}$) with a vector of weights $\mathit{w}$ that is optimized via the training procedure described in the main text.

Figure 2

Schematic representation of the encoding procedure for the MNIST classification task. The input grayscale image, of original size $N_p\times N_p$, with $N_p=28$, is first downsampled to a size of $N_p^{\prime }\times N_p^{\prime }$ (or $N_p^{\prime 2}\times 1$ when viewed as a column vector), with $N_p^{\prime }=8$, and linearly transformed by a fixed $N_p^{\prime 2}\times M$ random projection filter $\mathit{W}$ to yield the vectors ${\mathit{x}}^{\prime }$ containing $M=10$ random-projection features. Those features are normalized by three times the standard deviation over the set of all features for all images in the training set, and we denote $\mathit{x}$ the normalized vectors representing the images. The vector $\mathit{x}$ is then encoded into a sequence of driving pulses $\xi \left(t\right)$, where the amplitude of the $i\mathrm{th}$ pulse (at time $t_i$) is proportional to the input's $i\mathrm{th}$ component $x_i$. Finally, the pulses are used to drive a spin chain (initially prepared in the state ${\widehat{\rho }}_0$), where the driving amplitude at site $j$ is $F_j\left(t\right)={\eta }_j\xi \left(t\right)$ with ${\eta }_j$ a random site-dependent scale factor. We define the state of the spin chain immediately after the driving sequence to be the encoded state, represented by its density matrix $\widehat{\rho }\left(\mathit{x}\right)$.

Figure 3

Equivalent circuit of the encoding procedure for the MNIST classification task. If the driving pulses are sharp enough, the encoding process of Fig. 2 can be equivalently seen as a quantum circuit, where the $i\mathrm{th}$ driving pulse on site $j$ is effectively a single-qubit $X$-rotation gate $R^X\left({\eta }_j{x}_i\right)$, and the pulses at different times are separated by the gate $\mathcal{G}$ generated by the free dynamics of the spin system in the absence of the drive. Note that the entire process between ${\widehat{\rho }}_0$ and $\widehat{\rho }\left(x\right)$ serves as the dynamical map $\mathcal{M}\left(x\right)$ shown in Fig. 1.

Figure 4

Time dynamics of the average entanglement negativity $\overline{\mathcal{N}\left(\widehat{\rho }\right)}$ (a) and the average von Neumann entropy $\overline{S\left(\widehat{\rho }\right)}$ (b) in presence of pure dephasing with different values of the corresponding rates $\gamma$. At the initial time $t=0$ the system is in the pure state ${\widehat{\rho }}_0$ defined in the text. Note that the driving sequence finishes at the time $Jt\simeq 5$ indicated by the vertical dotted lines. The time is expressed in units of $1/J$ where $J$ is the average value of the spin coupling. We define $\overline{\mathcal{N}\left(\widehat{\rho }\right)}$ as the average over all the sites of the negativities associated to the system partitions having the form $\left\{\right\{\mathrm{site}i\},\{\mathrm{site}j\mid j\ne i\left\}\right\}$. This quantity is averaged over 20 inputs $\mathit{x}\in \mathcal{X}$, over five disordered configurations of spin couplings, and for a chain of $N=5$ spins. The filled areas correspond to a one standard deviation confidence interval.

Figure 5

(a) Training error $\mathcal{R}$ (dashed lines) and testing error ${\mathcal{R}}^{*}$ (solid lines and markers) as a function of the regularization parameter $\lambda$ for a chain with $N=7$ spins in the presence of pure dephasing for different values of the corresponding rate $\gamma$ (different markers in the legend) in units of the average spin coupling $J$. Measurements are assumed to be ideal. (b) Same as panel (a) with an extra random Gaussian noise of width $\sigma ={10}^{-3}$ added to the observable expectation values to account for imperfect measurements. For each value of $\lambda$ the corresponding errors are averaged over 15 disordered configurations, and the error bars are bootstrap estimates of the standard deviation for the estimated mean values. We use ten bootstrap sets, each consisting of 15 samples randomly drawn with replacement from the original set of 15 disorder realizations. (c) Minimal testing error as a function of the number of spins $N$ for different values of the dephasing rate $\gamma$. For each disorder configuration, the regularization parameter $\lambda$ is chosen to minimize the testing error and the resulting minimum is averaged over the disorder. The error bars are derived using the same bootstrap procedure. Number of disorder configurations: 50 for $N=2$ to 5 spins, 25 for $N=6$, 15 for $N=7$, and 5 for $N=8$. (d) Same as panel (c) with an extra random Gaussian noise of width $\sigma ={10}^{-3}$ added to the observable expectation values to account for imperfect measurements.

Figure 6

(a) Kernel effective rank for the full tomography decoding as a function of the number of spins $N$ and for different values of the dephasing rate $\gamma$. We have used the empirical representation of the kernel matrix on the training set. The results are averaged over the same numbers of disorder realizations as for Fig. 5. (b) Kernel empirical spectrum for the full tomography decoding, $N=6$ spins, and different values of the dephasing rate $\gamma$. The markers correspond to the optimal generalization parameter $\lambda$. The curves have been obtained via the kernel empirical representation on the training set. Results are averaged over 25 disorder configurations. The filled area corresponds to twice the estimated standard error on the averaged value, using the same bootstrap method as for Fig. 5.

Figure 7

(a) Minimal testing error as a function of the number of repetitions $N_{\mathrm{rep}}$ of all the local spin measurements for $N=6$ spins and different values of the dephasing noise $\gamma$. The regularization parameter value is chosen to minimize the testing error. The horizontal dashed lines represent the errors obtained using a full tomography decoding for the corresponding dephasing rates. The results are averaged over 50 disorder realizations. (b) Difference between the testing error of the time multiplexing decoding and the one for full tomography as a function of the number $N_{\mathrm{meas}}=3N{N}_{\mathrm{rep}}$ of local measurements performed for $N_{\mathrm{rep}}=50$, dephasing rate $\gamma /J=0.01$, and different values of the number $N$ of spins. For each realization of the disorder, the regularization parameter value is chosen to minimize the testing error. The results are averaged over 25 disorder realizations.

Figure 8

Equivalent circuit of the encoding with the information bottleneck removed. Instead of injecting a single random projection feature $x_i$ per time step as represented in Fig. 3, where a total number of $M=N_{\mathrm{pulse}}$ random projection features are fed into the kernel machine, here we inject $N$ random projection features in each time step, with a total number of $M=N\times N_{\mathrm{pulse}}$ random projection features injected by the end of the encoding sequence.

Figure 9

Optimal testing accuracy as a function of the number of driving pulses, for the encoding presented in Fig. 8, $N=6$ spins, and $\gamma /J=0.01$. The regularization parameter is chosen to maximize the testing accuracy. Note that for this encoding the number of features $M$ yielded by the preprocessing is $M=N\times N_{\mathrm{pulse}}$. The results are shown for ten realizations of the disorder on both the preprocessing and the system parameters. The maximal accuracy obtained is $94.5\%$ for $N_{\mathrm{pulse}}=3$. The boxes extend from the first to the third quartile of the distributions, the middle line indicates the median, and the whiskers indicate the extreme values of the distributions.