Quantum average neighborhood margin maximization for feature extraction

Abstract

Average neighborhood margin maximization (ANMM) is a local supervised metric learning approach, which aims to find projection directions where the local class discriminability is maximized. Furthermore, it has no assumption of class distribution and works well on small sample size. In this work, we address this problem in the quantum setting and present a quantum ANMM algorithm for linear feature extraction. More specifically, a quantum algorithm is designed to construct scatterness and compactness matrices in the quantum state form. Then a quantum algorithm is presented to obtain the features of the testing sample set in the quantum state form. The time complexity analysis of the quantum ANMM algorithm shows that our algorithm may achieve an exponential speedup on the dimension of the sample points $D$, and a quadratic speedup on the numbers of training samples $N$ and testing samples $M$ compared to the classical counterpart under certain conditions.

Appendices

Appendix A: quantum comparison circuit diagram

Here, we design a quantum comparison circuit used in this article, as shown in Fig. 1. Assume there are two quantum states to be compared, $\left|a\right.\rangle =|{a}_{l-1}{a}_{l-2}\cdots {a}_{0}\rangle$ and $\left|b\right.\rangle =|{b}_{l-1}{b}_{l-2}\cdots {b}_{0}\rangle$, after passing this quantum comparison circuit, if $a\ge b$, we can obtain $|0\rangle$ in the lower most register, otherwise, this quantum state will be flipped to $|1\rangle$.

Fig. 1

Quantum comparison circuit

Appendix B: Proof of Eq. (22)

According to the Hadamard Lemma [38], we can get

$${\mathrm{e}}^{-\mathrm{i}\rho \Delta t}\sigma {\mathrm{e}}^{\mathrm{i}\rho \Delta t}=\sigma -i\Delta t\left[\rho ,\sigma \right]-\frac{\Delta{t}^{2}}{2}\left[\rho ,\left[\rho ,\sigma \right]\right]+\cdots .$$

(38)

As for ${\mathrm{Tr}}_{p}\left\{{\mathrm{e}}^{-\mathrm{i}Sw\Delta t}\left(\sigma \otimes \rho \right){e}^{iSw\Delta t}\right\}$, suppose $\sigma ={\sum }_{i}{\sigma }_{i}|{\sigma }_{i}\rangle \langle {\sigma }_{i}|$ and $\rho ={\sum }_{j}{\rho }_{j}|{\rho }_{j}\rangle \langle {\rho }_{j}|$, the derivation is as follows:

$$\begin{array}{l}{\mathrm{Tr}}_{p}\left\{{\mathrm{e}}^{-\mathrm{i}Sw\Delta t}\left(\sigma \otimes \rho \right){e}^{iSw\Delta t}\right\}\\ ={\mathrm{Tr}}_{p}\left\{\left(\sigma \otimes \rho \right)+\frac{{\left(-\mathrm{i}Sw\Delta t\right)}^{1}}{1!}\left(\sigma \otimes \rho \right)+\left(\sigma \otimes \rho \right)\frac{{\left(\mathrm{i}Sw\Delta t\right)}^{1}}{1!}+\mathcal{O}\left(\Delta{t}^{2}\right)\right\}\\ ={\mathrm{Tr}}_{p}\left\{{\sum }_{ij}{\sigma }_{i}{\rho }_{j}|{\sigma }_{i}\rangle \langle {\sigma }_{i}||{\rho }_{j}\rangle \langle {\rho }_{j}|-\mathrm{i}Sw\Delta t{\sum }_{ij}{\sigma }_{i}{\rho }_{j}|{\sigma }_{i}\rangle \langle {\sigma }_{i}||{\rho }_{j}\rangle \langle {\rho }_{j}|+{\sum }_{ij}{\sigma }_{i}{\rho }_{j}|{\sigma }_{i}\rangle \langle {\sigma }_{i}||{\rho }_{j}\rangle \langle {\rho }_{j}|\left(\mathrm{i}Sw\Delta t\right)+\mathcal{O}\left(\Delta{t}^{2}\right)\right\}\\ ={\mathrm{Tr}}_{p}\left\{{\sum }_{ij}{\sigma }_{i}{\rho }_{j}|{\sigma }_{i}\rangle \langle {\sigma }_{i}||{\rho }_{j}\rangle \langle {\rho }_{j}|-\mathrm{i}\Delta t{\sum }_{ij}{\sigma }_{i}{\rho }_{j}|{\rho }_{i}\rangle \langle {\sigma }_{i}||{\sigma }_{j}\rangle \langle {\rho }_{j}|+{\sum }_{ij}{\sigma }_{i}{\rho }_{j}|{\sigma }_{i}\rangle \langle {\rho }_{i}||{\rho }_{j}\rangle \langle {\sigma }_{j}|\left(\mathrm{i}\Delta t\right)+\mathcal{O}\left(\Delta{t}^{2}\right)\right\}\\ ={\sum }_{i}|{\sigma }_{i}\rangle \langle {\sigma }_{i}|-\mathrm{i}\Delta t{\sum }_{ij}{\sigma }_{i}{\rho }_{j}|{\rho }_{i}\rangle \langle {\sigma }_{i}|\langle {\rho }_{j}||{\sigma }_{j}\rangle +{\sum }_{ij}{\sigma }_{i}{\rho }_{j}|{\sigma }_{i}\rangle \langle {\rho }_{i}|\langle {\sigma }_{j}||{\rho }_{j}\rangle \left(\mathrm{i}\Delta t\right)+\mathcal{O}\left(\Delta{t}^{2}\right)\\ =\sigma -\mathrm{i}\Delta t\left[\rho ,\sigma \right]+\mathcal{O}\left(\Delta{t}^{2}\right).\end{array}$$

(39)

Appendix C: Proof of Eq. (23)

$$\begin{array}{c}{\mathrm{Tr}}_{p}\left\{\left({\sum }_{n}|n\Delta t\rangle \langle n\Delta t|\otimes \prod_{i=1}^{n}{\mathrm{e}}^{-\mathrm{i}S{w}_{i}\Delta t}\right)\left({\sum }_{n}|n\Delta t\rangle \langle n\Delta t|\otimes \sigma {\left(\otimes \rho \right)}^{n}\right)\left({\sum }_{n}|n\Delta t\rangle \langle n\Delta t|\otimes \prod_{i=1}^{n}{\mathrm{e}}^{\mathrm{i}S{w}_{i}\Delta t}\right)\right\}\\ ={\mathrm{Tr}}_{p}\left\{{\sum }_{n}|n\Delta t\rangle \langle n\Delta t|\otimes \prod_{i=}^{n}{\mathrm{e}}^{-\mathrm{i}S{w}_{i}\Delta t}\left(\sigma {\left(\otimes \rho \right)}^{n}\right)\prod_{i=1}^{n}{\mathrm{e}}^{\mathrm{i}S{w}_{i}\Delta t}\right\}\\ ={\mathrm{Tr}}_{p}\left\{{\sum }_{n}|n\Delta t\rangle \langle n\Delta t|\otimes \left(\prod_{i=2}^{n}{\mathrm{e}}^{-\mathrm{i}S{w}_{i}\Delta t}\left(\begin{array}{c}\sigma \otimes \rho -\mathrm{i}S{w}_{1}\Delta t\left(\sigma \otimes \rho \right)\\ +\left(\sigma \otimes \rho \right)\mathrm{i}S{w}_{1}\Delta t\end{array}\right){\left(\otimes \rho \right)}^{n-1}\prod_{i=2}^{n}{\mathrm{e}}^{\mathrm{i}S{w}_{i}\Delta t}+\mathcal{O}\left(\Delta{t}^{2}\right)\right)\right\}\\ ={\mathrm{Tr}}_{p}\left\{{\sum }_{n}|n\Delta t\rangle \langle n\Delta t|\otimes \left(\prod_{i=3}^{n}{\mathrm{e}}^{-\mathrm{i}S{w}_{i}\Delta t}\left(\begin{array}{c}\sigma \otimes \rho \otimes \rho \\ -\mathrm{i}S{w}_{2}\Delta t\left(\sigma \otimes \rho \otimes \rho \right)\\ +\left(\sigma \otimes \rho \otimes \rho \right)\mathrm{i}S{w}_{2}\Delta t\\ -\mathrm{i}S{w}_{1}\Delta t\left(\sigma \otimes \rho \otimes \rho \right)\\ +\left(\sigma \otimes \rho \otimes \rho \right)\mathrm{i}S{w}_{1}\Delta t\end{array}\right){\left(\otimes \rho \right)}^{n-2}\prod_{i=3}^{n}{\mathrm{e}}^{\mathrm{i}S{w}_{i}\Delta t}+\mathcal{O}\left(\Delta{t}^{2}\right)\right)\right\}\\ ={\mathrm{Tr}}_{p}\left\{{\sum }_{n}|n\Delta t\rangle \langle n\Delta t|\otimes \left(\sigma {\left(\otimes \rho \right)}^{n}-\mathrm{i}\Delta t{\sum }_{n}S{w}_{i}\left(\sigma {\left(\otimes \rho \right)}^{n}\right)-\left(\sigma {\left(\otimes \rho \right)}^{n}\right)S{w}_{i}+\mathcal{O}\left(\Delta{t}^{2}\right)\right)\right\}\\ ={\sum }_{n}|n\Delta t\rangle \langle n\Delta t|\otimes \left(\sigma -\mathrm{i}\Delta tn\left[\rho ,\sigma \right]+\mathcal{O}\left(\Delta{t}^{2}\right)\right)\\ \approx {\sum }_{n}|n\Delta t\rangle \langle n\Delta t|\otimes {\mathrm{e}}^{-\mathrm{i}\rho n\Delta t}\sigma {\mathrm{e}}^{\mathrm{i}\rho n\Delta t},\end{array}$$

(40)