Quantum machine learning with subspace states

Abstract: We introduce a new approach for quantum linear algebra based on quantum subspace states and present three new quantum machine learning algorithms. The first is a quantum determinant sampling algorithm that samples from the distribution $\Pr[S]= det(X_{S}X_{S}<sup>{T})$ for $|S|=d$ using $O(nd)$ gates and with circuit depth $O(d\log n)$. The state of art classical algorithm for the task requires $O(d<sup>{3})$ operations \cite{derezinski2019minimax}. The second is a quantum singular value estimation algorithm for compound matrices $\mathcal{A}<sup>{k}$, the speedup for this algorithm is potentially exponential. It decomposes a $\binom{n}{k}$ dimensional vector of order-$k$ correlations into a linear combination of subspace states corresponding to $k$-tuples of singular vectors of $A$. The third algorithm reduces exponentially the depth of circuits used in quantum topological data analysis from $O(n)$ to $O(\log n)$. Our basic tool are quantum subspace states, defined as $$|Col(X)\rangle = \sum_{S\subset [n], |S|=d} det(X_{S}) |S\rangle$$ for matrices $X \in \mathbb{R}^{n \times d}$ such that $X^{T} X = I_{d}$, that encode $d$-dimensional subspaces of $\mathbb{R}<sup>{n}$. We develop two efficient state preparation techniques, the first using Givens circuits uses the representation of a subspace as a sequence of Givens rotations, while the second uses efficient implementations of unitaries $$\Gamma(x) = \sum_{i} x_{i} Z^{\otimes (i-1)} \otimes X \otimes I<sup>{n-i}$$ with $O(\log n)$ depth circuits that we term Clifford loaders.