Quantum locality preserving projection algorithm

Abstract

In this paper, we propose a quantum algorithm to efficiently perform locality preserving projection (LPP) for dimensionality reduction, named quantum locality preserving projection (QLPP). We present QLPP in 4 steps and give their corresponding quantum circuits. Furthermore, we give an improved algorithm to construct the density matrix proportional to the Hermitian/non-Hermitian chain product composed of multiple Hermitian/non-Hermitian matrices in QLPP. The analysis results show that the computation complexity of QLLP is $O(d\kappa _\mathrm{eff}^{10}\sqrt{m}\mathrm{polylog}(m)/\varepsilon ^{7})$ for $m\gg n$, where $\kappa _\mathrm{eff}$ is a predefined condition number, m is the number of training data with dimension n, d is the dimension of reduced dimension feature space, and $\varepsilon$ denotes the tolerance error. It is polynomial speedup in m compared to classical LPP algorithm, whose complexity is $O (m^2n)$.

Appendix A: proof of theorem

First, we extend matrices $\mathbf{B} _{1}, \mathbf{B} _{2},...\mathbf{B} _{k},$ to matrices $\mathbf{B} _{1}', \mathbf{B} _{2}',...\mathbf{B} _{k}',$. The expanded matrices are shown below,

$$\begin{aligned} (\mathbf{B} _{j}')_{2^{q-1}r_{k}\times 2^{q-1}r_{k}}= \begin{pmatrix} (\mathbf{B} _{j})_{r_{j}\times r_{j-1}} &{} {\varvec{0}}_{r_{j}\times (2^{q-1}r_{k}-r_{j-1})}\\ {\varvec{0}}_{(2^{q-1}r_{k}-r_{j})\times r_{j-1}} &{} {\varvec{0}}_{(2^{q-1}r_{k}-r_{j})\times (2^{q-1}r_{k}-r_{j-1})} \end{pmatrix}, \end{aligned}$$

(57)

where $j=1,2,...k$, $r_{j}$ represents the number of rows of matrix $\mathbf{B} _{j}$, $2^{q-1}r_{k}\ge \max (r_{0}, r_{1}, ..., r_{k})$, q is a positive integer and $r_0=r_1$.

Then, we extend matrices $\mathbf{B} _{1}', \mathbf{B} _{2}',...\mathbf{B} _{k}'$ to matrices $\mathbf{B} _{1}'', \mathbf{B} _{2}'',...\mathbf{B} _{k}''$. The expanded matrices are shown as follows.

$$\begin{aligned} \begin{aligned}&\mathbf{B} _{1}''= \begin{pmatrix} {\varvec{0}}_{2^{q-1}r_{k}\times 2^{q-1}r_{k}} &{} {\varvec{0}}_{2^{q-1}r_{k}\times 2^{q-1}r_{k}}\\ {\varvec{0}}_{2^{q-1}r_{k}\times 2^{q-1}r_{k}} &{} \mathbf{B} _{1}' \end{pmatrix},\\&\mathbf{B} _{2j}''= \begin{pmatrix} {\varvec{0}}_{2^{q-1}r_{k}\times 2^{q-1}r_{k}} &{} \mathbf{B} _{2j}'\\ (\mathbf{B} _{2j}')^{\dagger } &{} {\varvec{0}}_{2^{q-1}r_{k}\times 2^{q-1}r_{k}} \end{pmatrix},\\&\mathbf{B} _{2j+1}''= \begin{pmatrix} {\varvec{0}}_{2^{q-1}r_{k}\times 2^{q-1}r_{k}} &{} (\mathbf{B} _{2j+1}')^{\dagger }\\ \mathbf{B} _{2j+1}' &{} {\varvec{0}}_{2^{q-1}r_{k}\times 2^{q-1}r_{k}} \end{pmatrix}, \end{aligned} \end{aligned}$$

(58)

where $j=1, 2 ,...[\frac{k+1}{2} ]$, $[\frac{k+1}{2} ]$ is the largest integer less than $\frac{k+1}{2}$ and $2j+1\le k$.

When k is even, we have

$$\begin{aligned} \begin{aligned} \mathbf{B} '&= \left[ \mathbf{B} _{k}''... \mathbf{B} _{2}''(\mathbf{B} _{1}'')^{\frac{1}{2} }\right] \left[ \mathbf{B} _{k}''... \mathbf{B} _{2}''(\mathbf{B} _{1}'')^{\frac{1}{2} }\right] ^{\dagger }\\&= \begin{pmatrix} (\mathbf{B} )_{r_{k}\times r_{k}} &{} {\varvec{0}}_{r_{k}\times (2^{q}r_k-r_{k})}\\ {\varvec{0}}_{(2^{q}r_k-r_{k})\times r_{k}} &{} {\varvec{0}}_{(2^{q}r_k-r_{k})\times (2^{q}r_k-r_{k})} \end{pmatrix}\\&= \begin{pmatrix} 1 &{} {\varvec{0}}_{1\times (2^{q}-1)}\\ {\varvec{0}}_{(2^{q}-1)\times 1} &{} {\varvec{0}}_{(2^{q}-1)\times (2^{q}-1)} \end{pmatrix} \otimes \mathbf{B} , \end{aligned} \end{aligned}$$

(59)

otherwise

$$\begin{aligned} \mathbf{B} '= \left[ \mathbf{B} _{k}''... \mathbf{B} _{2}''(\mathbf{B} _{1}'')^{\frac{1}{2} }\right] \left[ \mathbf{B} _{k}''... \mathbf{B} _{2}''(\mathbf{B} _{1}'')^{\frac{1}{2} }\right] ^{\dagger }= \begin{pmatrix} {\varvec{0}}_{(2^{q}-1)\times (2^{q}-1)}&{} {\varvec{0}}_{(2^{q}-1)\times 1}\\ {\varvec{0}}_{1\times (2^{q}-1)} &{} 1 \end{pmatrix} \otimes \mathbf{B} , \end{aligned}$$

(60)

where B denotes $\left[ \mathbf{B} _{k}... \mathbf{B} _{2}(\mathbf{B} _{1})^{\frac{1}{2}}\right] \left[ \mathbf{B} _{k}... \mathbf{B} _{2}(\mathbf{B} _{1})^{\frac{1}{2}}\right] ^{\dagger }$.

Since $\mathbf{B} _{1}'', \mathbf{B} _{2}'',...\mathbf{B} _{k}''$ are Hermitian matrices, we can obtain the density matrix proportional to $\mathbf{B} '$ by using the improved algorithm. Specifically, we use $\mathbf{A} _{1}=\mathbf{B} _{1}''$, $f_{1}(\mathbf{A} _{1})= \mathbf{A} _{1}^{\frac{1}{2} }$ and $\mathbf{A} _{j}=\mathbf{B} _{j}''$, $f_{j}(\mathbf{A} _{j})= \mathbf{A} _{j}$ for $j=2, 3,..k$. When k is odd, the density matrix ${\varvec{\rho }}_\mathbf{B '}$ proportional to $\mathbf{B} '$ is

$$\begin{aligned} {\varvec{\rho }}_\mathbf{B '}=\frac{1}{T} \begin{pmatrix} {\varvec{0}}_{(2^{q}-1)\times (2^{q}-1)} &{} {\varvec{0}}_{(2^{q}-1)\times 1}\\ {\varvec{0}}_{1\times (2^{q}-1)} &{} 1 \end{pmatrix} \otimes \mathbf{B} =\frac{1}{T} (\left| 1 \right\rangle \left\langle 1\right| )^{\otimes q} \otimes \mathbf{B} , \end{aligned}$$

(61)

otherwise

$$\begin{aligned} {\varvec{\rho }}_\mathbf{B '}=\frac{1}{T} \begin{pmatrix} 1 &{} {\varvec{0}}_{1\times (2^{q}-1)}\\ {\varvec{0}}_{(2^{q}-1)\times 1} &{} {\varvec{0}}_{(2^{q}-1)\times (2^{q}-1)} \end{pmatrix} \otimes \mathbf{B} =\frac{1}{T} (\left| 0 \right\rangle \left\langle 0\right| )^{\otimes q} \otimes \mathbf{B} , \end{aligned}$$

(62)

where $\frac{1}{T}$ is a constant for normalization.

Removing $(\left| 1 \right\rangle \left\langle 1\right| )^{\otimes q}$ from Eq. (61) or removing $(\left| 0 \right\rangle \left\langle 0\right| )^{\otimes q}$ from Eq. (62), one can obtain the density matrix proportional to $\left[ \mathbf{B} _{k}... \mathbf{B} _{2}{} \mathbf{B} _{1}\right] \left[ \mathbf{B} _{k}... \mathbf{B} _{2}{} \mathbf{B} _{1}\right] ^{\dagger }$.