Unfortunately you can't. With any orthogonal factorization (e.g. QR, LQ, or SVD) you have the problem that because some or all of the columns of the orthogonal matrix have to span a particular subspace, and because the remaining columns have to form an orthogonal basis for the complement to this subspace, and because these spaces can be completely arbitrary, the orthogonal matrix won't be sparse unless you happen to be very lucky.

There are "sparse QR" methods that effectively represent Q as a product of Givens rotations rather than storing Q explicitly. The Givens rotations are transformations that are extremely sparse/structured matrices.

Unfortunately you can't. With any orthogonal factorization (e.g. QR, LQ, or SVD) you have the problem that because some of the columns of the orthogonal matrix have to span a particular subspace, and because the remaining columns have to form an orthogonal basis for the complement to this subspace, and because these spaces can be completely arbitrary, the orthogonal matrix won't be sparse unless you happen to be very lucky.

There are "sparse QR" methods that effectively represent Q as a product of Givens rotations rather than storing Q explicitly. The Givens rotations are transformations that are extremely sparse/structured matrices.