Block MPS Forward/Reverse Sweep

Author: TheoryLand

This demo isolates the one-site block-sweep primitive that sits underneath matrix-product-state algorithms. In modern MPS language, a sweep step contracts a site tensor with an incoming environment block, factorizes the result, and truncates by singular value; that local move is the core update behind DMRG-style methods and their descendants [1, 2].

The immediate target here is narrower than a full TEMPO calculation. We use an identity MPO and rank-preserving local embeddings so that the exact internal bond rank should stay fixed through both the forward and reverse passes. That makes the figure a controlled algorithmic check before reintroducing the nontrivial influence-functional MPO needed for imaginary-time TEMPO benchmarks of the Ohmic spin-boson problem [3].

The sliders vary chain length and exact block bond dimension, while the plotted traces show the kept bond dimension and per-step wall time through a complete forward/reverse sweep. In this stripped-down setting, any unexpected rank growth or loss is a bug in the sweep primitive rather than a feature of the physics model.

Method

We label the incoming carry block at site $j$ and symmetry sector $\lambda$ by $C_j^{(\lambda)}$ , and the local MPS tensor by $A_j[s,\alpha,\beta]$ . Because the demo MPO is chosen to be the identity in both physical and sector indices,

W_j[s,n,\lambda,\rho]=\delta_{sn}\,\delta_{\lambda\rho},

the forward update reduces to a sectorwise contraction of the carry with the local site tensor [2]:

\Theta_j^{(\lambda)}[(s,a),\beta] = \sum_\alpha C_j^{(\lambda)}[a,\alpha] A_j[s,\alpha,\beta].

Each sector block is then factorized independently with a thin SVD,

\Theta_j^{(\lambda)} = U_j^{(\lambda)}\, \Sigma_j^{(\lambda)}\, \left(V_j^{(\lambda)}\right)^\dagger.

but the truncation rule is applied globally rather than sector by sector: all singular values from all sectors are pooled, ranked together, and only the largest retained values are kept. The emitted objects are

M_j^{(\lambda)}[s,a,b] = U_j^{(\lambda)}[(s,a),b], \qquad C_{j+1}^{(\lambda)}[b,\beta] = \left(\Sigma V^\dagger\right)_j^{(\lambda)}[b,\beta].

so the forward recurrence is

\{C_j^{(\lambda)}\}+A_j \;\longrightarrow\; \{M_j^{(\lambda)}\}+\{C_{j+1}^{(\lambda)}\}.

For the reverse sweep, the compressed block site is contracted from the right,

\widetilde{\Theta}_j^{(\lambda)}[(s,c),a] = \sum_b R_j^{(\lambda)}[c,b]\, M_j^{(\lambda)}[s,a,b].

and the same sectorwise SVD plus pooled singular-value truncation is applied again. This is the same structural idea as a one-site MPS sweep, but written in the block form needed later for influence-functional tensor networks and TEMPO-style history compression [1, 3].

In this demo each local site is a rank-preserving embedding and each MPO site is the identity, so the exact internal bond rank should stay fixed through both sweep directions. That is precisely the behavior we want to verify before moving on to the full imaginary-time TEMPO benchmark.

References

[1] Steven R. White. “Density matrix formulation for quantum renormalization groups,” Physical Review Letters 69 (19), 2863--2866 (1992). DOI: 10.1103/PhysRevLett.69.2863.
[2] Ulrich Schollwoeck. “The density-matrix renormalization group in the age of matrix product states,” Annals of Physics 326 (1), 96--192 (2011). DOI: 10.1016/j.aop.2010.09.012.
[3] A. Strathearn, P. Kirton, D. Kilda, J. Keeling, and B. W. Lovett. “Efficient non-Markovian quantum dynamics using time-evolving matrix product operators,” Nature Communications 9, 3322 (2018). DOI: 10.1038/s41467-018-05617-3.