  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 $\Pr[S]= det(X_{S}X_{S}^{T})$ 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 $O(d^{3})$ operations \cite{derezinski2019minimax}. The second is
a quantum singular value estimation algorithm for compound matrices
$\mathcal{A}^{k}$, 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 $\mathbb{R}^{n}$. 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 $\Gamma(x) = \sum_{i} x_{i} Z^{\otimes
(i-1)} \otimes X \otimes I^{n-i}$ with $O(\log n)$ depth circuits that we term
Clifford loaders.
