# Tensor Network Optimization Atlas A source-led, machine-readable research library for matrix product states, MERA, and tensor-network contraction as a classical method for QUBO/Ising optimization and quantum-circuit simulation. Version: 0.1.0 Evidence cutoff: 2026-07-29 Canonical URL: https://research.mahastrategies.com/atlas/tensor-networks Claims JSON: https://research.mahastrategies.com/atlas/tensor-networks/claims.json Sources JSON: https://research.mahastrategies.com/atlas/tensor-networks/sources.json Methodology: https://research.mahastrategies.com/atlas/tensor-networks/methodology ## Boundary This edition publishes source-bounded claims only. It excludes performance figures without a resolvable cited source, vendor rankings, and investment guidance. It does not assert that classical tensor-network contraction generally outperforms quantum hardware on industrial optimization workloads — see tn-014. The AdS/MERA correspondence is recorded as a conjecture with its published objection, and no computational claim here depends on it. ## Claims - tn-001 [established]: 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. - tn-002 [established]: The one-dimensional area law proves that ground states of gapped one-dimensional local Hamiltonians have entanglement bounded independently of system size. - tn-003 [established]: DMRG is understood as variational optimization over the manifold of matrix product states. - tn-004 [established]: Tensor-network accuracy is controlled by singular-value truncation, and the discarded weight is the quantity that makes a result interpretable. - tn-005 [conjecture]: 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. - tn-006 [established]: MERA introduces disentanglers before coarse-graining, allowing it to reproduce the entanglement scaling of critical systems that a tree tensor network cannot. - tn-007 [established]: Quadratic unconstrained binary optimization problems map exactly onto Ising spin models, and explicit Ising formulations exist for a catalogue of NP-hard problems. - tn-008 [established]: 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. - tn-009 [established]: PEPS extend tensor networks beyond one dimension, but exact contraction of PEPS is computationally hard. - tn-010 [established]: Tensor-network time evolution is limited by entanglement growth, which forces the bond dimension upward as the simulated time increases. - tn-011 [established]: Classical tensor-network contraction has been reported to solve the Sycamore random-circuit sampling problem, narrowing the advantage originally claimed for that specific task. - tn-012 [active-research]: 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. - tn-013 [active-research]: Whether QAOA delivers a practical advantage over strong classical baselines on real optimization instances remains unresolved. - tn-014 [active-research]: No source in this atlas establishes that classical tensor-network contraction generally outperforms quantum hardware on industrial optimization workloads. ## Claim limitations - tn-001: This is a statement about representational capacity, not a guarantee that any particular state of interest is reachable at a practical D. - tn-002: The proof covers gapped 1D local Hamiltonians. It does not transfer wholesale to higher dimensions or to gapless systems, and it says nothing about states reached by long-time evolution. - tn-003: The reformulation is a change of description, not of results. It does not by itself extend DMRG's reach beyond the regimes where it already worked. - tn-004: Small per-step discarded weight does not bound global error in general. Truncations accumulate, and a sweep can converge cleanly onto a state that the truncation put out of reach. - tn-005: This atlas records the conjecture and the objection; it does not adjudicate them. Critically, no numerical use of MPS, MERA, or PEPS as an optimization method depends on the correspondence being true — the algorithms predate it and stand on their own analysis. Treating the holographic reading as a warrant for a computational claim is a category error. - tn-006: The cost prefactor in the bond dimension is high, and the higher-dimensional generalization is substantially harder than the one-dimensional construction. - tn-007: An exact encoding does not make a problem easy. Penalty multipliers widen the energy scale and worsen conditioning, and the hardness of the original problem survives the change of variables intact. - tn-008: Favourable structure is a property of the instance. Hardware acceleration improves the constant factor; it does not change the complexity class, and a dense problem graph produces a network with no good contraction order to find. - tn-009: Approximate schemes carry errors that are not generally certified, so a two-dimensional tensor-network result requires more care in interpretation than a one-dimensional one. - tn-010: The barrier is a property of the physics being simulated, not of a given implementation, so it is not removed by better engineering. Specific non-generic dynamics can evade it. - tn-011: It concerns one benchmark sampling task, not an application workload, and it does not generalize to a claim that classical tensor networks outperform quantum hardware broadly. It also does not settle circuits at other depths or sizes. - tn-012: A single application study does not transfer to supply-chain logistics or molecular simulation, and it does not license the claim that tensor networks are the better production method for portfolio construction. Instance size, cost model, and solution-quality criteria all bound what such a result means. - tn-013: This records an open question. It is neither a claim that QAOA will fail nor that classical methods are permanently ahead. - tn-014: The absence of a general result is not evidence that classical methods are inferior, nor that they are superior. It records that the comparison the commercial framing asserts has not been established by the sources here. A future benchmark could change this claim; a vendor figure without a resolvable source could not. ## Benchmark records - bench-sycamore-sampling: Sampling from the output distribution of the Sycamore random quantum circuits. reported: The cited work reports solving the Sycamore sampling problem classically, substantially narrowing the gap the original demonstration claimed for this task. does not establish: It does not show classical tensor networks outperform quantum hardware in general, and it is a benchmark sampling task rather than an application workload. It also does not settle later circuits at different depths or sizes. source: pan-2021 - bench-dynamic-portfolio: Dynamic portfolio optimization on real market datasets. reported: The cited work reports applying both tensor-network methods and quantum processors to dynamic portfolio optimization with real datasets. does not establish: It does not establish a general throughput advantage over quantum annealers or gate-based hardware, does not extend to supply-chain logistics or molecular simulation, and does not license the claim that tensor networks are the better production method for portfolio construction. Instance sizes, cost models, and solution-quality criteria all bound what a result like this transfers to. source: mugel-2020 ## Sources - Density matrix formulation for quantum renormalization groups (1992) — DOI:10.1103/PhysRevLett.69.2863 — https://doi.org/10.1103/PhysRevLett.69.2863 - Efficient classical simulation of slightly entangled quantum computations (2003) — arXiv:quant-ph/0301063 — https://arxiv.org/abs/quant-ph/0301063 - Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions (2004) — arXiv:cond-mat/0407066 — https://arxiv.org/abs/cond-mat/0407066 - Entanglement renormalization (2005) — arXiv:cond-mat/0512165 — https://arxiv.org/abs/cond-mat/0512165 - A class of quantum many-body states that can be efficiently simulated (2006) — arXiv:quant-ph/0610099 — https://arxiv.org/abs/quant-ph/0610099 - The computational complexity of PEPS (2006) — arXiv:quant-ph/0611050 — https://arxiv.org/abs/quant-ph/0611050 - Simulating quantum computation by contracting tensor networks (2005) — arXiv:quant-ph/0511069 — https://arxiv.org/abs/quant-ph/0511069 - An Area Law for One Dimensional Quantum Systems (2007) — arXiv:0705.2024 — https://arxiv.org/abs/0705.2024 - Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems (2008) — arXiv:0907.2796 — https://arxiv.org/abs/0907.2796 - Entanglement Renormalization and Holography (2009) — arXiv:0905.1317 — https://arxiv.org/abs/0905.1317 - The density-matrix renormalization group in the age of matrix product states (2010) — arXiv:1008.3477 — https://arxiv.org/abs/1008.3477 - A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States (2013) — arXiv:1306.2164 — https://arxiv.org/abs/1306.2164 - Ising formulations of many NP problems (2013) — arXiv:1302.5843 — https://arxiv.org/abs/1302.5843 - A Quantum Approximate Optimization Algorithm (2014) — arXiv:1411.4028 — https://arxiv.org/abs/1411.4028 - Consistency Conditions for an AdS/MERA Correspondence (2015) — arXiv:1504.06632 — https://arxiv.org/abs/1504.06632 - Dynamic Portfolio Optimization with Real Datasets Using Quantum Processors and Quantum-Inspired Tensor Networks (2020) — arXiv:2007.00017 — https://arxiv.org/abs/2007.00017 - Solving the sampling problem of the Sycamore quantum circuits (2021) — arXiv:2111.03011 — https://arxiv.org/abs/2111.03011