The algorithm

Successive Randomized Compression (SRC) (Camaño et al., 2026) computes a compressed approximation of a product of tensor trains, such as $A\,|\psi\rangle$ for an MPO $A$ and an MPS $|\psi\rangle$, without ever forming the exact product. The exact product of an MPO with bond dimension $\chi_A$ and an MPS with bond dimension $\chi_\psi$ has bond dimension $\chi_A \chi_\psi$; compressing it afterwards, as in the usual contract-then-truncate approach, pays for that intermediate size. SRC targets the output bond dimension $\chi_{\text{out}}$ directly, or an oversampled sketch size $\chi_{\text{os}}$ followed by a final SVD truncation to $\chi_{\text{out}}$.

Randomized range findingCopy Anchor Link

The building block is the randomized range finder (Halko et al., 2011): to approximate the column space of a matrix $M$ with rank about $k$, multiply it by a random Gaussian matrix $\Omega$ with $k$ columns and orthonormalise the result, $Q = \operatorname{orth}(M\Omega)$. Then $M \approx Q Q^\dagger M$ with high probability, and the cost is dominated by the product $M\Omega$.

SRC applies this idea one site at a time along the chain, with the matrix $M$ being an unfolding of the (never formed) product network.

Sketching and truncationCopy Anchor Link

By default, $\chi_{\text{os}} = \chi_{\text{out}}$ and only the first two sweeps run: the output is right-canonical, with bond dimension at most $\chi_{\text{out}}$. An oversample factor sets $\chi_{\text{os}} = \lceil \text{oversample}\,\chi_{\text{out}}\rceil$, and oversample=None the paper's $\max(\lceil 1.5\chi_{\text{out}}\rceil, \chi_{\text{out}} + 10)$; the third sweep then truncates to $\chi_{\text{out}}$.

1. Left to right: sketching. At every site the open physical legs are contracted with a Gaussian tensor $\Omega_j$ of sketch size $\chi_{\text{os}}$. Accumulating these from the left gives the sketched environments $C_j$, each of size $\chi_{\text{os}}$ times the product of the input bond dimensions. The same sketch index is shared by every site, which makes the sketch a Khatri-Rao product rather than a dense Gaussian matrix of the full Hilbert space.

2. Right to left: orthonormalisation. Starting from the last site, the environment $C_j$ is contracted with the site tensors and the projected right environment $S$, and a QR decomposition of the result gives the new site tensor as an isometry. The isometry is then folded into $S$ for the next site. The first site absorbs the remaining weight.

3. Left to right: exact truncation (oversampling only). The intermediate is already right-canonical, so no re-orthogonalisation is needed. An SVD of each centre tensor gives the Schmidt values at that cut. Keep at most $\chi_{\text{out}}$ values, apply the relative cutoff, and absorb the remaining weight into the next site. The output is left-canonical, with the norm on the last site. The discarded weights add in quadrature, giving the usual TT-SVD quasi-optimality bound for this intermediate.

CostCopy Anchor Link

For an MPO with bond dimension $\chi_A$ applied to an MPS with bond dimension $\chi_\psi$, the dominant per-site cost grows like $\chi_{\text{os}}^2 \chi_A \chi_\psi$, compared with the $(\chi_A\chi_\psi)^3$ of an exact contraction followed by an SVD-based truncation. The same kernel handles MPO-MPO products and plain compression; for a stack of trains, the product of all their bond dimensions replaces $\chi_A\chi_\psi$.

Being randomized, SRC is not optimal: its error is somewhat above the best rank-$\chi_{\text{out}}$ truncation, with a gap that shrinks quickly as the spectrum decays. The benchmarks quantify this on concrete cases.