Quantum algorithms for machine learning

Soutenance de thése de

Alessandro Luongo

20 Novembre 2020

Supervisor: Iordanis Kerenidis

Co-supervisor: Frédéric Magniez.

aluongo@irif.fr

This thesis addresses this question:

This thesis addresses this question:

The end of Moore's law?

The QML toolkit

• Query access to matrices

• Quantum linear algebra

• Distance estimations

• Singular Value Estimation

• Tomography (of pure states)

• Hamiltonian simulation

• Amplitude estimation and amplification

• Singular value transformations

• Polynomial approximations

• ...

Quantum access to a matrix

Quantum query:

\( |i\rangle |0 \rangle \mapsto |i\rangle |x_i\rangle \)

Where  \(|x_i\rangle=\frac{1}{\|x_i\|_2} x_i = \frac{1}{\|x_i\|_2} \sum_i (x_i)_j |j\rangle \)

\( X \in \mathbb{R}^{\textcolor{red}{n \times d}} = [x_1, \dots, x_n]^T \)    \( x_i \in \mathbb{R}^d\)

• Classical preprocessing time: \( O(\textcolor{red}{nd} \log nd) \)

• Classical space: \( O\left( \textcolor{red}{nd} \log nd \right) \)

• Query time: \( O(\textcolor{red}{\log nd}) \)

• Quantum space: \( O \left(\textcolor{red}{ \log nd }\right) \)

Where  \(|x_i\rangle=\frac{1}{\|x_i\|_2} x_i = \frac{1}{\|x_i\|_2} \sum_i (x_i)_j |j\rangle \)

Quantum query:

\( \frac{1}{\sqrt{n}}\sum_i |i\rangle |0 \rangle \mapsto \frac{1}{\sqrt{n}}\sum_i |i\rangle |x_i\rangle \)

\( X \in \mathbb{R}^{\textcolor{red}{n \times d}} = [x_1, \dots, x_n]^T \)    \( x_i \in \mathbb{R}^d\)

Quantum access to a matrix

Quantum linear systems of equations

Given:

• quantum sparse access \(A \in \R^{n \times n}\),

• and a vector \(x \in \R^n\)

The HHL algorithm produces a state \(|z\rangle\) such that

\( \kappa(A) = \frac{\sigma_1(A)}{\sigma_n(A)} \) and \( s = \text{row's sparsity} \)

Harrow, Aram, Avinatan Hassidim, Seth Lloyd - Physical review letters - 2009 

Quantum linear systems of equations

The HHL algorithm produces a state \(|z\rangle\) such that

\( \kappa(A) = \frac{\sigma_1(A)}{\sigma_n(A)} \)

Harrow, Aram, Avinatan Hassidim, Seth Lloyd - Physical review letters - 2009 

Given:

• quantum access \(A \in \R^{n \times n}\),

• and a vector \(x \in \R^n\)

Quantum singular value transformations

Given matrix \(A \in \R^{n \times m}\),  and a vector \(x \in \R^m\)

It is possible to produce a state \(|z\rangle\) s.t.

Iordanis Kerenidis, Anupam Prakash - 8th Innovations in Theoretical Computer Science Conference - 2017.

András Gilyén, et al.  - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing -  2019.

Guang Hao Low, Isaac L. Chuang -  Physical review letters  - 2017.

\(\mu(A) = \min\left(\|A\|_F, \sqrt{\max_{i \in [n]} \|a_i\|_{2p}^{2p} \max_{i \in [d]} \|a_{*i}\|_{2(1-p)}^{(1-p)} }\right) \)

Distance estimation subroutines

• Euclidean distance [QMEANS]

• Quadratic forms [Thesis]

• Distance induced by \(A\) [QEM]

\(|i,j\rangle \mapsto |i,j ,\|x_i- x_j\|_2\rangle  \)

\(|i,j\rangle \mapsto |i,j ,d_A(x_i,  x_j) \rangle  \)

\(|i,j\rangle \mapsto |i,j ,x_i^TA^\textcolor{orange}{-1}x_j \rangle  \)

\( \widetilde{O}(\frac{1 }{\epsilon} ) \)

\( \widetilde{O}(\frac{\mu(A)}{\epsilon} ) \)

\( \widetilde{O}(\frac{\mu(A)\textcolor{orange}{\kappa(A)}}{\epsilon} ) \)

\( X \in \mathbb{R}^{n \times d} = [x_1, \dots, x_n]^T \)    \( x_i \in \mathbb{R}^d\)

We can produce an estimate \(\overline{x}\) of \(|x\rangle\) such that \(\|\overline{x}\| = 1\) and

Tomography of pure states

 Iordanis Kerenidis, Anupam Prakash -  ACM Transactions on Quantum Computing, 2020.

Iordanis Kerenidis, Anupam Prakash, Jonas Landman -  International Conference on Learning Representations, 2019.

samples

samples

SVE of product of matrices

Theorem: Assume to have quantum access to \(A, B \in \mathbb{R}^{n \times n} \).

There is an algorithm that performs the mapping

\( \sum_i \alpha_i |i\rangle \mapsto \sum_i \alpha_i|i, \sigma_i \rangle \)

where \(\sigma_i \) is the i-th singular value of \(AB\) in time

\(O\left(\frac{(\kappa(A)+\kappa(B))( \mu(A)+\mu(B))}{\epsilon}\right) \)

[QSFA]

Supervised

Learning

• \(X \in \mathbb{R}^{n \times d} \) dataset

• \( L \in [K]^{n} \) labels

(classification)

Quantum supervised learning

• Quantum slow feature analysis for dimensionality reduction

• Quantum Frobenius distance classifier: a simple NISQ classifier

• Simulation on MNIST dataset of handwritten digits

• Extension to other generalized eigenvalue problems in ML

[QSFA, Thesis]

https://cmp.felk.cvut.cz/cmp/software/stprtool

Slow Feature Analysis

\(Y \in \mathbb{R}^{n \times \textcolor{orange}{K}} \)

\( y_i =\left[ w_1^Tx_i, \dots, w_K^Tx_i\right] \)

\(L \in [K]^{n}  \) (labels)

Slow Feature Analysis

\(Y \in \mathbb{R}^{n \times \textcolor{orange}{K}} \)

\( y_i =\left[ w_1^Tx_i, \dots, w_K^Tx_i\right] \)

\(L \in [K]^{n}  \) (labels)

Quantum Slow Feature Analysis

Theorem:

• Quantum access to \(X\), derivative matrix \(\dot{X}\)

• Let \( \epsilon, \theta, \delta, \eta >0\)

     There are quantum algorithms to get:

• Map dataset in the slow feature space \( |\overline{Y}\rangle \)  in time: $$ \tilde{O}\left(  \frac{ ( \kappa(X) + \kappa(\dot{X})) ( \mu({X})+ \mu(\dot{X}) ) }{\delta\theta}  \gamma_{K-1} \right) $$

• Find \( \{ w_j\}_{j=0}^{K}\) in time:    $$O\left(d^{1.5}\sqrt{K}\frac{\kappa(X)\kappa(\dot X X)(\mu(X) + \mu(\dot{X}))}{\epsilon^2}\right)$$

[QSFA]

QFDC: classification in slow feature space

[QSFA]

We get high accuracy with a fast quantum classifier

Classification of handwritten digits of MNIST dataset

Ex: polynomial expansion

of degree 2:

\([x_1, x_2, x_3] \mapsto [x_1^2, x_1x_2 \dots x_3^3 ]\)

We get high accuracy with a fast quantum classifier

Malware detection via DGA classification

[Thesis]

• Using SFA in a classification problem improves its accuracy

• QSFA can process old datasets in new ways!

SFA as instance of a more general problem

The GEP (Genrealized Eigenvalue Problem) is defined as:

• ICA Independent Component Analysis

• G-IBM Gaussian Information Bottleneck Method

• CCA Canonical Correlation Analysis

• SC (some) Spectral Clustering  [1]

• PLS Partial Least Squares

• LE Laplacian Eigenmaps

• FLD Fisheral Linear Discriminant

• SFA Slow Feature Analysis

• KPCA Kernel Principal Component Analysis

\( AW = BW\Lambda \)

[1] Iordanis Kerenidis, Jonas Landman - Quantum spectral clustering. arXiv2007.00280. (2020)

Quantum supervised learning

• Quantum slow feature analysis for dimensionality reduction

• Simulation on MNIST dataset of handwritten digits

• Quantum Frobenius distance classifier: a simple NISQ classifier

• Extension to other generalized eigenvalue problems in ML

[QSFA, Thesis]

Unsupervised

Learning

• \(X \in \mathbb{R}^{n \times d} \)

(clustering)

Quantum unsupervised

learning

• q-means for clustering (quantum version of k-means)

• Quantum Expectation-Maximization

• Simulation on VoxForge dataset for speaker recognition

The k-means algorithm:

\( t \leftarrow 0 \)

Step 1:

• Compute distance for all points \(x_i \) and centroid \(\mu_j^{t}, \)  $$ d(x_i, \mu_j^{t}) $$

•  Assign points to closest cluster: $$ l(x_i) = \argmin_{c \in [K]}{ d(x_i, \mu_j^{t})}$$

 Step 2:

• Compute the barycenter:  $$\mu_j^{t+1} = \frac{1}{|C_j|}\sum_{i \in C_j}^{} x_i $$

t \(\leftarrow \)t+1

q-means

\( t \leftarrow 0 \)

Step 1:

• Compute distance for all points \(v_i \) and centroid \(\mu_j^{t}, \)  $$|i,j\rangle \mapsto  |i,j,d(v_i, \mu_j^{t})\rangle $$

•  Generate characteristic vector of a cluster: $$ |\chi_j \rangle = \frac{1}{|C_j|}\sum_{i \in C_j} |i \rangle$$

Step 2:

• Use quantum linear algebra to build $$|\mu_j^{t+1}\rangle = \frac{1}{|C_j|}\sum_{i \in C_j}^{} |v_i\rangle $$

• Perform tomograph on \( |\mu_j^{t+1}\rangle \)

• Build quantum access to \( \mu_j\).

\( t \leftarrow t+1 \)

q-means

\( t \leftarrow 0 \)

Expectation:

• Compute distance for all points \(v_i \) and centroid \(\mu_j^{t}, \)  $$|i,j\rangle \mapsto  |i,j,d(v_i, \mu_j^{t})\rangle $$

•  Generate characteristic vector of a cluster: $$ |\chi_j \rangle = \frac{1}{|C_j|}\sum_{i \in C_j} |i \rangle$$

Maximization:

• Use quantum linear algebra to build $$|\mu_j^{t+1}\rangle = \frac{1}{|C_j|}\sum_{i \in C_j}^{} |v_i\rangle $$

• Perform tomograph on \( |\mu_j^{t+1}\rangle \)

• Build quantum access to \( \mu_j\).

\( t \leftarrow t+1 \)

Gaussian mixture models

\( k \) labels

Multinomial distribution

\( [\theta, \mu_1, \dots, \mu_k, \Sigma_1, \dots, \Sigma_k ] \)

\( \gamma^* =   \text{argmax} \prod_i \sum_{j \in [k]} \theta_j p(x_i|\mu_j,\Sigma_j) \)

Gaussian distribution

\(_\gamma\)

•     \(\|\theta-\overline{\theta}\|<\delta_\theta \)

• \(\|\mu_j - \overline{\mu_j}\| < \delta_\mu \)

• \(\|\Sigma_j - \overline{\Sigma_j}\| < \delta_\mu\sqrt{\eta}\)

Maximum Likelihood Estimation

Expectation-Maximization

                      Repeat

                      \(t = 0\)

• Expectation   \[ r_{ij}^t \leftarrow \frac{\theta^t_j N(v_i; \mu^t_j, \Sigma^t_j )}{\sum_{l=1}^k \theta^t_l N(v_i; \mu^t_l, \Sigma^t_l)}\]

                         \( t \leftarrow t+1 \)

                      Until  \( | \ell(\gamma^{t-1};V) - \ell(\gamma^t;V) | < \tau \)

• Maximization

• Update the parameters \(\theta, \mu, \Sigma \) using the responsibilities \(r_{ij} \)

Quantum Expectation-Maximization for

Gaussian mixture models

           Theorem: Given quantum access to a matrix \( X \in \mathbb{R}^{n \times d} \) there is a quantum EM algorithm that fits a GMM in time:

[QEM]

Classical: \( O(d^2 k n) \)

We get high accuracy with a fast quantum classifier

• Classical ML accuracy: 169/170

• Quantum ML accuracy: 167/170

• Max element of \( \Sigma_j^{-1}\) set to \(5\)  via \(\kappa = \frac{1}{\lambda_\tau} \)

Speaker recognition problem on VoxForge dataset

Quantum unsupervised

learning

• q-means for clustering (quantum version of k-means)

• Quantum Expectation-Maximization

• Simulation on VoxForge dataset for speaker recognition

Quantum Spectral Sums

\(S_f(A) = \sum_i^n f(\lambda_i) \)

\(A \in \mathbb{R}^{n \times n} \) SPD

\( f : \mathbb{R} \mapsto \mathbb{R}  \)

Quantum algorithms for log-determinant

\(  S_{\log(x)}(A) =\log\det(A) = \sum_i^n \log(\lambda_i) \)

[QSS]

Tyler's M-estimator

• Data from sub-Gaussian distributions.

• Robust to outliers

• Valid for data \( X^{n \times d}\) with \( n,d \mapsto \infty \)

[Thesis, AQSS]

In many cases \(C = X^TX\) is not a "good" sample covariance matrix

Tyler's M-estimator

[Thesis, AQSS]

Runtime:

\[ \tilde O\left(\textcolor{red}{d^2}\frac{\mu(X)\kappa(\Sigma_k)\mu(\Sigma_k)}{\epsilon^3}\gamma\right) \]

\( \Gamma_{k+1} = \sum_{i=1}^n \frac{x_ix_i^T}{ x_i^T \Gamma_k^{-1}x_i}/ Tr[\sum_{i=1}^n \frac{x_ix_i^T}{x_i^T \Gamma_k^{-1}x_i}]\)

Stopping condition: log-likelihood with a log-determinant

Classical:

\( O\left( d^2n \right) \)

Other spectral sums and applications (not in thesis)

• Schatten p-norm             \(O(2^{p/2}\mu(A)(p+\kappa(A))\sqrt{n}/\epsilon) \)

• Von Neumann entropy       \(O(\mu(A)\kappa(A)n /\epsilon )\)

• Trace of Inverse              \(O(\mu^2 \kappa(A)^2/\epsilon) \)

[QSS]

Thanks

Elham Kashefi
Iordanis Kerenidis
Frédéric Magniez
Filippo Miatto
Simon Perdrix
Simone Severini

Conclusions and outlook

• We have a corpus of algorithms with provable speedups.

  • Simple to extend current algorithms to more powerful models.

• Quantum algorithms seems to work promisingly well in ML:

  • \(\kappa(A) \), \(\mu(A), s, \eta, \epsilon \),

• QML might allow solving new or  existing problems:

  • better, faster, cheaper, or a combination.

• Artificial Intelligence might be promising to explore

  • Smaller QRAM?

• We should work directly on state-of-the-art ML algorithm:

  • Interpretable, explainable, fair, robust, privacy-preserving ML.

Thanks for your time, there is never enough.

Evolution of Mutual Information between layers while training a QNN

The dropout technique for avoiding barren plateaus

Rebecca Erbanni - Master's thesis

Alexander Singh - IRIF internship

Counting triangles

\[ \Delta(G) = \frac{1}{6} Tr[A^3] \]

• Create block encoding of \(B=A^{1.5}\)

• Estimate \(Tr[B^TB] \)

Van Apeldoorn, Joran, et al. - 58th Annual Symposium on Foundations of Computer Science - 2017.

Canonical Correlation Analysis

Input: \( \{ (x_i, y_i) \}_{i=0}^n\) where \( x_i \in \mathbb{R}^{d_1}\) and \(y_i \in \mathbb{R}^{d_2}\) i.e. matrices \(X,Y\)

Classical:

• Step1: Solve GEP \[ \Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX}w_x = \lambda^2\Sigma_{XX}w_x \]

• Step 2: Find \(w_y \) by

  \( w_y = \frac{\Sigma_{YY}^{-1}\Sigma_{YX}w_x}{\lambda} \)

CCA model: find \(w_x, w_y\) such that

\( {w_x, w_y} =\arg\max_{w_x, w_y} cos((Xw_x, Yw_y)) \)

Quantum:

\( \Sigma_{XX}\Sigma_{XY}^{-1}\Sigma_{YY} = U\Sigma V^T \)

\( W_x = \Sigma_{XX}^{-1/2}U \)

 \( W_y = \Sigma_{YY}^{-1/2}V \).

Quantum algorithms for model checking

A state-space exploration approach

• Formalize our software as an automaton \(A_P\).

• For a temporal property \(f\) we build the automaton \(A_{\neg f}\).

• Solve the emptiness problem of the language:  \[L(A_P\times A_{\neg f}) = \emptyset\].

Software \(\mapsto\) specification LTL

\(\mapsto\) Büchi automata \(\mapsto\)  \(\omega\)-language

Idea: use quantum DFS!

Dürr, Christoph, et al. - SIAM Journal on Computing 35.6 (2006)

Theorem: The emptiness problem for \(\omega\)-languages is decidable!