# 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 [S]= det(X_SX_S^T) for |S|=d using O(nd) gates and with circuit depth O(dlog n). The state of art classical algorithm for the task requires O(d^3) operations <cit.>. The second is a quantum singular value estimation algorithm for compound matrices š^k, the speedup for this algorithm is potentially exponential. It decomposes a nk 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)ā© = ā_Sā [n], |S|=d det(X_S) |Sā© for matrices X āā^n Ć d such that X^T X = I_d, that encode d-dimensional subspaces of ā^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 Ī(x) = ā_i x_i Z^ā (i-1)ā X ā I^n-i with O(log n) depth circuits that we term Clifford loaders.

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