Robust data encodings for quantum classifiers

Data representation is crucial for the success of machine-learning models. In the context of quantum machine learning with near-term quantum computers, equally important considerations of how to efficiently input (encode) data and effectively deal with noise arise. In this paper, we study data encodings for binary quantum classification and investigate their properties both with and without noise. For the common classifier we consider, we show that encodings determine the classes of learnable decision boundaries as well as the set of points which retain the same classification in the presence of noise. After defining the notion of a robust data encoding, we prove several results on robustness for different channels, discuss the existence of robust encodings, and prove a lower bound on the number of robust points in terms of fidelities between noisy and noiseless states. Numerical results for several example implementations are provided to reinforce our findings.

Figure 1

A common architecture for a binary quantum classifier that we study in this paper. The general circuit structure is shown in (a) and the structure for a single qubit is highlighted in (b). In both, a feature vector $\mathit{x}$ is encoded into a quantum state ${\rho }_{\mathit{x}}$ via a “state preparation” unitary $S_{\mathit{x}}$. The encoded state ${\rho }_{\mathit{x}}$ then evolves to $U{\rho }_{\mathit{x}}{U}^{†}=:{\tilde{\rho }}_{\mathit{x}}$ where $U\left(\mathit{\alpha }\right)$ is a unitary Ansatz with trainable parameters $\mathit{\alpha }$. A single qubit of the evolved state ${\tilde{\rho }}_{\mathit{x}}$ is measured to yield a predicted label $\widehat{y}$ for the vector $\mathit{x}$.

Figure 2

A visual representation of data encoding (7) for a single qubit. On the left is shown a set of randomly generated points ${\{{\mathit{x}}_i,y_i\}}_{i=1}^M$ normalized to lie within the unit square, $({\mathit{x}}_i={[x_{1,i},x_{2,i}]}^T)$, separated by a true decision boundary shown by the dashed black line. A data encoding maps each ${\mathit{x}}_i\in {\mathbb{R}}^2$ to a point on the Bloch sphere ${\rho }_{{\mathit{x}}_i}\in {\mathbb{C}}^2$, here using the dense angle encoding (14). The dashed black line on the Bloch sphere shows the initial decision boundary of the quantum classifier. During the training phase, unitary parameters are adjusted to rotate the dashed black line to correctly classify as many training points as possible. Different data encodings lead to different learnable decision boundaries and different robustness properties, as discussed in the main text.

Figure 3

Cartoon illustration of robust points for a single qubit classifier. In panel (a), input training data points ${\mathit{x}}_i$ with classes $y_i\in \{\text{yellow},\text{blue}\}$ are mapped into quantum states ${\rho }_{{\mathit{x}}_i}$ according to some encoding function $E$. The dashed line through the Bloch sphere indicates the initial decision boundary. Points with a green outline are classified correctly, while points with a red outline are misclassified. In (b), data points are processed by the QNN with optimal unitary parameters (after minimizing a cost function to find such parameters). For clarity, we keep data points fixed and adjust the location of the decision boundary, which is now rotated to correctly classify more points (fewer points with red outlines). In (c), a noise process $\mathcal{E}$ occurs which shifts the location of the final processed points (or location of decision boundary), causing some points to be misclassified. The set of points which maintain the same classification in (b) and (c) are the robust points. Example points ${\mathit{x}}_1$ and ${\mathit{x}}_2$ are correctly classified in (a) and (b) then misclassified in (c) due to the noise. Example point ${\mathit{x}}_3$ is incorrectly classified in (a), correctly classified in (b) after propagating through the QNN, and remains correctly classified in (c).

Figure 4

Examples of learnable decision boundaries for a single qubit classifier with the (a) dense angle encoding, (b) amplitude encoding, and (c) superdense angle encoding where $\theta =\pi$ and $\varphi =2\pi$. Colors denote class labels. The QNN used here consisted of an arbitrary single qubit rotation (see Fig. 13) with random parameters.

Figure 5

Partial robustness for the dense angle encoding. The data set consists of 500 points in the unit square separated by a vertical decision boundary and we use a train-test split of $80\%$. Red crosses and green circles indicate decision boundary location. Orange circles and blue crosses indicate points labeled as each class. Misclassified points have opposite color border. Panel (a) shows the classifier test accuracy after optimizing the unitary without noise. Panel (b) shows the reduced accuracy after amplitude damping noise of strength $p=0.4$ is added. The robust set is at the far left and far right of the unit square, explicitly shown in panel (c). Here, blue circles indicates the robust set and black crosses indicates its complement.

Figure 6

Partial robustness for the amplitude encoding. The data set consists of 500 points in the unit square separated by a vertical decision boundary, and we use a train-test split of $80\%$. Panel (a) shows classifier test accuracy after optimizing the unitary without noise. Panel (b) shows the reduced accuracy after adding amplitude damping noise with strength $p=0.4$. The robust set is shown explicitly in panel (c) where a blue circle indicates a robust point and a black cross indicates a misclassified point. Panel (d) is the same as (b) but with decreased strength $p=0.2$ of the amplitude damping channel. Test accuracy reduces from $81\%$ to $60\%$ in this case. Panel (e) shows the robust set for (d).

Figure 7

Cartoon illustration of the encoding learning algorithm with a single qubit classifier. In (a), a preset encoding with no knowledge of the noise misclassifies a large number of points. In (b), the encoding learning algorithm detects misclassifications and tries to adjust points to achieve more robustness, attempting to encode into the robust set for the channel.

Figure 8

Minimum cost achieved (vertical axis) from applying the encoding learning algorithm to three example data sets (each plot) using three different encodings (horizontal axis). The blue crosses [$\times$] show the minimum cost achieved when training over only unitary parameters without any noise present (ideal). The green triangles [$▴$] show the same case with the addition of amplitude damping noise of strength $p=0.3$. The orange circles [$•$] show minimum cost after applying the encoding learning algorithm. The dense angle encoding (DAE) is seen to perform well on all data sets and is capable of adapting well to noise, even outperforming the ideal case without noise and fixed encoding. The superdense angle encoding (SDAE) does not perform well on any shown data set since the generated decision boundary is highly nonlinear and cannot correctly classify more than half the data set. The generalized amplitude encoding (GAE) performs well on the diagonal boundary, since it generates a suitable decision boundary.

Figure 9

Learnability versus robustness on the “vertical” data set using the parameterized dense angle encoding. The horizontal and vertical axes show the encoding hyperparameters $\varphi$ and $\theta$, respectively. Panels (a) and (b) show the classifier accuracy while panel (c) shows the proportion of robust points. Panel (a) shows accuracy without noise as a function of encoding parameters. Panel (b) shows accuracy with the ad- dition of an amplitude damping channel of strength $p=0.3$. Panel (c) shows $\delta$ robustness for different parameter values. As expected, the robust set is largest when all points are encoded into the zero state, i.e., $\theta =0$. This leads to all points labeled 1 being misclassified, with a resulting accuracy of approximately $50\%$. The orange [$\mathbf{\text{−}}$] filled line indicates optimal $\theta$ parameters in each panel. From (a) to (b), the $\theta$ parameters corresponding to highest accuracy are shifted towards the robust points (i.e., toward $\theta =0$) in (c). See Appendix pp3 for the optimal parameters found in each case.

Figure 10

Top row: Fidelity between noisy and noiseless states for different encodings on the Iris data set. Comparing average fidelity over every encoded state in data set, versus fidelity of noisy and noiseless encoded mixed states. From left to right, the noise models are bit-flip noise, amplitude damping noise, dephasing noise, and global depolarizing noise, with strengths varied across the horizontal axis. Bottom row: Upper bounds on partial robustness in terms of fidelity using the bounds (72) and (73). For each plot (in both rows), three different data encodings are considered—curves corresponds to three product qubit encodings: dense angle encoding, amplitude encoding, and superdense angle.

Figure 11

Misclassification percentage as a result of (b) measurement noise as a function of probabilities $\{p_{00},p_{11}\}$. If either $p_{00}$ or $p_{11}$ is less than $1/2$, then half the correctly classified points will be misclassified, with the probability increasing as expected, with the number of misclassified points increasing as the of-diagonal terms, $p_{01},p_{10}\to 1$, as expected from Theorem t33. By classified correctly, in this context, we mean the fraction of points which are classified the same with and without noise.

Figure 12

Three single qubit (two dimensional) data sets which we use. (a) vertical, (b) diagonal, and (c) the moons data set from scikit-learn [78] rotated by ${90}^{\circ }$ with a noise level of 0.05. 20% of each set is test data, indicated by the points circled with the opposite color. We chose the latter two due to the fact that the moons and vertical data sets can be well classified by the dense angle encoding, while the diagonal data set can be well classified by the amplitude encoding, which can be seen by studying the decision boundaries generated in Fig. 4.

Figure 13

Circuit diagrams for the QNN Ansätze we use to obtain numerical results. (a) Ansatz for the single qubit classifier. Here, $R_z^{\alpha }$ and $R_y^{\beta }$ denote rotations around the $z$ axis and $y$ axis with angles $\alpha ,\beta$, respectively, of the Bloch sphere. We note that any element of $U\left(2\right)$ can be represented by this Ansatz [68]. (b) Ansatz for the two qubit classifier, with 12 parameters, ${\left\{{\alpha }_k\right\}}_{k=1}^{12}$. The first six parameters, ${\alpha }_1,\cdots ,{\alpha }_6$, are contained in the first two single qubit unitaries, and ${\alpha }_{10},\cdots ,{\alpha }_{12}$ are the parameters of the final single qubit gate. The parameters of the intermediate rotations are defined by $\{{\gamma }_1,{\gamma }_2,{\gamma }_3\}:=\{2{\alpha }_7+\pi ,2{\alpha }_8,2{\alpha }_9\}$. This decomposition can realize any two qubit unitary [87] up to global phase with the addition of a single qubit rotation on the bottom qubit at the end of the circuit. We omit this rotation since we only measure the first qubit for classification. As such, we reduce the number of trainable parameters ($\mathit{\alpha }$) from 15 to 12.

Figure 14

Misclassification percentage as a result of Pauli noise with strengths $\{p_X,p_Y,p_Z=0\}$ As expected, classification is robust for $p_X+p_Y<1/2$ based on Theorem t16, and a sharp transition occurs when this constraint is violated to give maximal misclassification. By classified correctly, in this context, we mean the fraction of points which are classified the same with and without noise.