Tensor Network Optimization Atlas

MPS ground-state search (DMRG)

Bond dimension · D

Rank retained on each virtual bond; sets the variational manifold the sweep optimizes over.

An MPS of bond dimension D carries at most log(D) entanglement entropy across any single cut.

Truncation

Singular value decomposition at each bond, keeping the D largest singular values.

Error measure: Discarded weight — the summed squares of the singular values thrown away.. Discarded weight is the quantity to report. A converged sweep with a large discarded weight is converged to the wrong state, and the sweep itself will not say so.

Canonical form

Left/right canonical gauge fixed by successive QR or SVD, so the effective problem at each site is well conditioned.

Cost scaling

Polynomial in D and linear in site count N for a fixed sweep count; the D-dependence dominates.

MPS time evolution

Bond dimension · D

Same rank budget, but now re-truncated after every applied gate rather than optimized once.

Unchanged at log(D) per cut — the ceiling is a property of the ansatz, not of what it is used for.

Truncation

SVD truncation after each two-site gate application.

Error measure: Accumulated discarded weight across the gate sequence.. Errors compound over time steps; a per-step truncation that looks harmless can dominate the result by the end of the evolution.

Canonical form

Canonical form restored between gate applications so each truncation is taken with respect to the correct reduced state.

Cost scaling

Polynomial in D per gate, multiplied by the number of gates.

Scale-invariant MERA

Layer structure

One (u, w) pair reused at every layer, giving a renormalization-group transformation with no preferred length scale.

Entanglement scaling

Reproduces the logarithmic entanglement scaling characteristic of critical one-dimensional systems, which a plain tree tensor network cannot.

Cost scaling

Polynomial in the bond dimension; the layer count grows logarithmically in system size rather than linearly.

Encoding

Substituting x = (1 + s) / 2 with x in {0, 1} and s in {-1, +1} converts a quadratic binary objective into an Ising energy with couplings J, local fields h, and a constant offset. The transformation is exact and invertible.

Constraint handling

Hard constraints are added as penalty terms weighted by a multiplier large enough that violating a constraint costs more than any feasible gain. The multiplier widens the energy scale, which worsens conditioning — a correctness-preserving step that makes the numerics harder.

Tensor construction

Each spin becomes an index; each coupling becomes a tensor on the edge joining its two spins. The problem graph becomes the network graph, so the problem topology is the contraction topology.

Extracted quantity

Contracting the network evaluates a partition function or, in the zero-temperature limit, a minimum-energy configuration.

Encoding

Explicit Ising formulations exist for a catalogue of NP-hard problems including partitioning, covering, colouring, and Hamiltonian-cycle families.

Constraint handling

Each formulation states its own penalty structure and the multiplier scaling needed for the ground state to encode a feasible solution.

Tensor construction

The interaction graph of the resulting spin model determines the tensor network; a dense coupling matrix produces a dense network with high treewidth.

Extracted quantity

Ground-state configuration, or low-energy configurations sampled from the model.

Benchmark record · bench-sycamore-sampling

Sampling from the output distribution of the Sycamore random quantum circuits.

Classical method

Tensor-network contraction with an optimized contraction order, run on classical hardware.

Compared against

The superconducting-processor demonstration that originally framed this task as beyond practical classical reach.

Reported result

The cited work reports solving the Sycamore sampling problem classically, substantially narrowing the gap the original demonstration claimed for this task.

Source · arXiv:2111.03011

Benchmark record · bench-dynamic-portfolio

Dynamic portfolio optimization on real market datasets.

Classical method

Quantum-inspired tensor-network optimization.

Compared against

Quantum processors applied to the same problem instances.

Reported result

The cited work reports applying both tensor-network methods and quantum processors to dynamic portfolio optimization with real datasets.

Source · arXiv:2007.00017

A matrix product state represents a many-body state with a cost controlled by its bond dimension, which bounds the entanglement it can carry across any cut.

The bond dimension D fixes the variational manifold. Because an MPS of bond dimension D carries at most log(D) entanglement entropy across a cut, the representation is efficient precisely when the target state's entanglement is bounded.

The one-dimensional area law proves that ground states of gapped one-dimensional local Hamiltonians have entanglement bounded independently of system size.

This is the structural result explaining why DMRG succeeds in one dimension: the entanglement a correct answer must carry does not grow with the chain, so a fixed bond dimension can suffice.

DMRG is understood as variational optimization over the manifold of matrix product states.

The density-matrix renormalization group was formulated before the tensor-network language existed; the later reformulation showed the states it produces are matrix product states and its sweeps are variational updates on them.

Tensor-network accuracy is controlled by singular-value truncation, and the discarded weight is the quantity that makes a result interpretable.

Each bond is reduced to its D largest singular values; the discarded tail is a measurable error. A result reported without its bond dimension and discarded weight cannot be assessed.

The reading of a MERA network as a discretized holographic geometry is a proposed correspondence, not a derived one, and published work argues its consistency conditions are in tension.

The proposal observes that the extra layer direction of a MERA behaves like a radial bulk coordinate and that entanglement entropy has a geometric reading in the network. Subsequent work set out conditions such a correspondence would have to satisfy and argued they conflict.

MERA introduces disentanglers before coarse-graining, allowing it to reproduce the entanglement scaling of critical systems that a tree tensor network cannot.

A unitary applied across block boundaries removes short-range entanglement that would otherwise have to be carried upward through the isometries, so the network reaches logarithmic entanglement scaling.

Quadratic unconstrained binary optimization problems map exactly onto Ising spin models, and explicit Ising formulations exist for a catalogue of NP-hard problems.

The substitution between binary variables and ±1 spins is exact and invertible, and published formulations give the couplings, fields, and penalty structures for a range of NP-hard families.

The cost of contracting a tensor network is governed by the contraction order and by structural properties of the network graph such as its treewidth.

Quantum circuits and spin models alike can be expressed as tensor networks, and the feasibility of evaluating them depends on graph structure rather than on the arithmetic of any single contraction step.

PEPS extend tensor networks beyond one dimension, but exact contraction of PEPS is computationally hard.

The lattice generalization of MPS gives the right entanglement structure for two-dimensional systems, and hardness results for contracting it are why higher-dimensional work uses approximate contraction schemes.

Tensor-network time evolution is limited by entanglement growth, which forces the bond dimension upward as the simulated time increases.

Generic dynamics generate entanglement across cuts, and holding a fixed accuracy then requires a bond dimension that grows with it. This is the entanglement barrier.

Classical tensor-network contraction has been reported to solve the Sycamore random-circuit sampling problem, narrowing the advantage originally claimed for that specific task.

The result is the clearest published case of tensor-network methods closing a gap that hardware had been said to open, and it is evidence that classical baselines move.

Tensor-network methods have been applied to dynamic portfolio optimization on real datasets alongside quantum processors, but the reported work does not establish a general advantage.

The cited study applies both approaches to the same problem family, which is what makes it relevant. Its scope is a specific formulation on specific datasets.

Whether QAOA delivers a practical advantage over strong classical baselines on real optimization instances remains unresolved.

QAOA is the standard gate-based approach to QUBO and Ising problems and the usual comparison point for quantum-inspired classical methods. Settling the question requires end-to-end comparisons at matched accuracy and cost.

No source in this atlas establishes that classical tensor-network contraction generally outperforms quantum hardware on industrial optimization workloads.

This claim exists to state an absence rather than leave it to be inferred from silence. The strongest results recorded here are a specific sampling task reproduced classically and a specific portfolio-optimization study. Neither is a throughput comparison across portfolio construction, supply-chain logistics, and molecular simulation, and this atlas publishes no performance figures it cannot attribute to a cited source.