Exponential data encoding for quantum supervised learning

Reliable quantum supervised learning of a multivariate function mapping depends on the expressivity of the corresponding quantum circuit and measurement resources. We introduce exponential-data-encoding strategies that are hardware-efficient and optimal among all nonentangling Pauli-encoded schemes, which is sufficient for a quantum circuit to express general functions having very broad Fourier frequency spectra using only exponentially few encoding gates. We show that such an encoding strategy not only reduces the quantum resources, but also exhibits practical resource advantage during training in contrast with known efficient classical strategies when polynomial-depth training circuits are also employed. When computation resources are constrained, we numerically demonstrate that even exponential-data-encoding circuits with single-layer training modules can generally express functions that lie outside the classically expressible region, thereby supporting the practical benefits of such a resource advantage. Finally, we illustrate the performance of exponential encoding in learning the potential-energy surface of the ethanol molecule and California's housing prices.

Figure 1

Schematic QML model used to express an $M$-variate bounded $f_{\mathrm{Q}}\left(\mathit{x}\right)$ [$|f_{\mathrm{Q}}\left(\mathit{x}\right)|\le 1$]. Each training unitary $W_l\left({\mathit{\theta }}_l\right)$ comprises $L$ layers of a single qubit (each carrying three independent training parameters) and the full nearest-neighbor cnot array. The unitary $V\left(\mathit{x}\right)$ encodes each element $x_m$ of a training datum $\mathit{x}$ onto an $N$-qubit Pauli-$Z$ gate with prechosen weights ${\beta }_{m1},...,{\beta }_{mN}$.

Figure 2

Visualization of the full convex space $C_{{\kappa }^M}$ of bounded ${\mathit{\kappa }}^M$-dimensional functions. The subspace $S_{\text{C/Q}}\subseteq C_{{\kappa }^M}$, and it has a dimension of at most $N_{\mathrm{tp}}$. (a) For a fixed $\mathrm{resrc}=O(\mathrm{polylog}\left({\kappa }^M\right))$, an optimal CFFLM generates $S_{\mathrm{C}}=C_{O(\mathrm{polylog}\left({\kappa }^M\right))}$. (b) With the same resrc constraint and dense exponential encoding, the QFFLM gives a subspace $S_{\mathrm{Q}}$ that can exceed the boundaries of $C_{O(\mathrm{polylog}\left({\kappa }^M\right))}$. Therefore, while a classically expressible $\mathit{c}$ strictly lies in $C_{O(\mathrm{polylog}\left({\kappa }^M\right))}$, QFFLMs can express $\mathit{c}\notin C_{O(\mathrm{polylog}\left({\kappa }^M\right))}$.

Figure 3

QFFLM ($L=1$) and CFFLM performances [of similar order in computational-resources ($\mathrm{resrc}$)] on learning two univariate functions $-1\le f\left(\mathit{x}\right)\le 1$ of period $[-\pi ,\pi )$ different characteristic $r$. (a) It is clear that the trained QFFLM expressivity for $r=0.05$ is higher than that for the CFFLM (inset). (b) For $r=1.6$, where the Fourier coefficients $c_n$ within and outside the classically expressible region are comparable, so are the expressivities of both models. The ${\mathcal{L}}_{\mathit{\theta }}$ graphs (averaged over five runs) in (c) reflect training accuracies for various $r$, where that for the CFFLM at $r=0.05$, in particular, saturates at a much larger value of around 0.08 than its QFFLM counterpart. The CFFLM expressivity is naturally high for $r=55.5$, where $f$ is classically expressible. QFFLM performs similarly for all regimes of $r$, with improved expressivities for larger $L$. Appendix pp5 further states the technical details behind these simulations.

Figure 4

(a) Learning the potential-energy surface with the naively (serial) reuploading [15] and exponentially encoded QFFLMs with otherwise identical training settings. Filled and unfilled markers refer to training and test losses. Training and testing shown, respectively, for (b) $N_{\mathrm{tp}}^{\mathrm{QFFLM}}=1134$ ($L=9$) and $N_{\mathrm{tp}}^{\mathrm{QFFLM}}=126L$ using (c) naive and (d) exponential models. All graphs are averaged over 10 runs, and all corresponding 95% confidence regions are shown.

Figure 5

Numerical results for the California Housing Prices dataset. Here, we set $N_{\mathrm{tp}}^{\mathrm{QFFLM}}=96L$ for (a) $L=1$ and (b) $L=3$. With increasing $L$, the gap between training and test losses of both types of encoding grows larger. All corresponding 95% confidence regions over 10 runs are shown.

Figure 6

Visualization of the frequency spectra. Elements of the spectrum $\mathrm{\Omega }$ are represented on the real line. Adding a $k\mathrm{th}$ $Z$-encoding gate with weight ${\beta }_k$ to $V\left(\mathit{x}\right)$ corresponding to the spectrum ${\mathrm{\Omega }}^{(k-1)}$ introduces new frequency elements, which are $\{\alpha \pm {\beta }_k|\alpha \in {\mathrm{\Omega }}^{(k-1)}\}$

Figure 7

Expressivity for $f_{\mathrm{step}}\left(\mathit{x}\right)$, where QFFLM predictions (red curve) are compared with the target function values (circular markers). In all panels (a)–(d), plots corresponding to the minimized mean-squared-error loss function are shown. We set $L=N+1$ for each trainable unitary $W_l$. QFFLM training is carried out with the Adam gradient optimizer of learning rate 0.05 for 500 iterative steps.

Figure 8

Quality of 2D image expression of the handwritten digit “2,” conveyed through the training accuracy in image regression. We consider $L=10$ layers for each trainable unitary $W_1$ and $W_2$, and we only vary the training-data encoding scheme. QFFLM training is carried out with the Adam gradient optimizer of learning rate 0.03 for 300 iterative steps. To compare the expressivity performances of QFFLMs with different encoded schemes, we choose the three-tuple of encoded weights $\mathit{\beta }=({\beta }_{11},{\beta }_{12},{\beta }_{13})=({\beta }_{21},{\beta }_{22},{\beta }_{23})$ as (a) (1,1,1), (b) (1,2,3), (c) (1,3,9), and (d) (1,7,49). Reconstructed images (c) and (d) are predictions of degree $d_{\mathrm{F}}=3^3=27$ bivariate Fourier series that are, respectively, dense and sparse in frequency spectra.

Figure 9

Circuit diagrams for training with the revised MD17 and California housing-price datasets. (a) Both circuits share a very similar structure that involves an initial trainable block $A$ and three classical-data encoding (or reuploading) blocks $B_1$, $B_2$, and $B_3$. (b) The specific details of $A$ and $B_l$ that depend on the task at hand differ by the number of encoding unitary operators (${\tilde{V}}_l$ or $V_l$ depending on which of the two tasks is executed) and trainable modules $W$, (c) where the latter involve $N=6$ and 8 qubits, respectively. The structure of $V_l$ follows that given in Fig. 1. In (c), the structure of ${\tilde{V}}_l\left({\mathit{\varphi }}_1\right)$ is explicitly shown as an example.

Figure 10

(a) Monte Carlo simulation of the boundary of $C_3$ and (b) the exact surface.

Figure 11

(a) General circuit for the demonstrations discussed in Appendix pp5, which deal with univariate target-function learning. The internal structures of the trainable modules in (b) and (c), which apply for Figs. 3 and 12, respectively.

Figure 12

Expressivity of QFFLMs with respect to the CFFLM and loss-function (${\mathcal{L}}_{\theta }$) minimization performances (averaged over five runs per plot marker). A monotonous drop in the saturated ${\mathcal{L}}_{\theta }$ values with increasing $L$ implies that the QFFLM expressivity improves as one uses deeper training modules. Note that all single-qubit rotation gates are still encoded with only two parameters.

Figure 13

Performances of CFFLMs and QFFLMs for $M=1$ and $\kappa =243$.