Quantum machine learning with subspace states

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 is the quantum subspace state, defined as $|Col(X)\langle = \sum_{S\subset [n], |S|=d} det(X_{S}) |ket{S}\langle$ for matrices $X \in \mathbb{R}^{n \times d}$ s]uch that $X^{T} X = I_{d}$, that encodes $d$-dimensional subspaces of $\mathbb{R}^{n}$ and for which 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.