{"@context":"https://schema.org","@type":"DataCatalog","name":"Tensor Network Optimization Atlas — claim ledger","url":"https://research.mahastrategies.com/atlas/tensor-networks/claims.json","isPartOf":"https://research.mahastrategies.com/atlas/tensor-networks#atlas","version":"0.1.0","dateModified":"2026-07-29","license":"CC BY 4.0","statusVocabulary":[{"id":"established","label":"Established result","definition":"A published theorem, standard construction, or reproduced numerical result. The label does not imply the method is practical at every system size, nor that its cost is acceptable for a given application."},{"id":"active-research","label":"Active research","definition":"An open scientific or engineering question with no field-wide resolution. Evidence exists on more than one side, or the comparison needed to settle it has not been run."},{"id":"conjecture","label":"Conjecture","definition":"A proposed correspondence or interpretation that is argued structurally rather than derived, and which has published objections or unmet consistency conditions. It is recorded because it drives research, not because it is settled."}],"boundary":"Each record carries a status label and a limitations field. Both are part of the claim and must not be dropped when reused.","claims":[{"@type":"Claim","@id":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-001#claim","id":"tn-001","slug":"mps-cost-is-set-by-bond-dimension","status":"established","claim":"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.","explanation":"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.","limitations":"This is a statement about representational capacity, not a guarantee that any particular state of interest is reachable at a practical D.","conceptIds":["mps","bond-dimension","tensor-network"],"sourceIds":["schollwoeck-2010","orus-2013"],"benchmarkIds":[],"reviewDate":"2026-07-29","canonicalUrl":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-001"},{"@type":"Claim","@id":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-002#claim","id":"tn-002","slug":"area-law-explains-when-mps-works","status":"established","claim":"The one-dimensional area law proves that ground states of gapped one-dimensional local Hamiltonians have entanglement bounded independently of system size.","explanation":"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.","limitations":"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.","conceptIds":["area-law","mps","bond-dimension"],"sourceIds":["hastings-2007"],"benchmarkIds":[],"reviewDate":"2026-07-29","canonicalUrl":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-002"},{"@type":"Claim","@id":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-003#claim","id":"tn-003","slug":"dmrg-is-mps-optimization","status":"established","claim":"DMRG is understood as variational optimization over the manifold of matrix product states.","explanation":"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.","limitations":"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.","conceptIds":["dmrg","mps"],"sourceIds":["white-1992","schollwoeck-2010"],"benchmarkIds":[],"reviewDate":"2026-07-29","canonicalUrl":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-003"},{"@type":"Claim","@id":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-004#claim","id":"tn-004","slug":"truncation-error-must-be-reported","status":"established","claim":"Tensor-network accuracy is controlled by singular-value truncation, and the discarded weight is the quantity that makes a result interpretable.","explanation":"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.","limitations":"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.","conceptIds":["truncation","bond-dimension"],"sourceIds":["schollwoeck-2010","vidal-2003"],"benchmarkIds":[],"reviewDate":"2026-07-29","canonicalUrl":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-004"},{"@type":"Claim","@id":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-005#claim","id":"tn-005","slug":"ads-mera-is-a-conjecture-not-a-derivation","status":"conjecture","claim":"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.","explanation":"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.","limitations":"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.","conceptIds":["holographic-conjecture","mera","scale-invariance"],"sourceIds":["swingle-2009","bao-2015"],"benchmarkIds":[],"reviewDate":"2026-07-29","canonicalUrl":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-005"},{"@type":"Claim","@id":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-006#claim","id":"tn-006","slug":"mera-disentanglers-reach-critical-scaling","status":"established","claim":"MERA introduces disentanglers before coarse-graining, allowing it to reproduce the entanglement scaling of critical systems that a tree tensor network cannot.","explanation":"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.","limitations":"The cost prefactor in the bond dimension is high, and the higher-dimensional generalization is substantially harder than the one-dimensional construction.","conceptIds":["mera","disentangler","scale-invariance"],"sourceIds":["vidal-2005","vidal-2006"],"benchmarkIds":[],"reviewDate":"2026-07-29","canonicalUrl":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-006"},{"@type":"Claim","@id":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-007#claim","id":"tn-007","slug":"qubo-and-ising-are-equivalent-encodings","status":"established","claim":"Quadratic unconstrained binary optimization problems map exactly onto Ising spin models, and explicit Ising formulations exist for a catalogue of NP-hard problems.","explanation":"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.","limitations":"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.","conceptIds":["qubo-ising","penalty-term"],"sourceIds":["lucas-2013"],"benchmarkIds":[],"reviewDate":"2026-07-29","canonicalUrl":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-007"},{"@type":"Claim","@id":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-008#claim","id":"tn-008","slug":"contraction-cost-tracks-network-structure","status":"established","claim":"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.","explanation":"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.","limitations":"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.","conceptIds":["contraction-complexity","tensor-network"],"sourceIds":["markov-shi-2005"],"benchmarkIds":[],"reviewDate":"2026-07-29","canonicalUrl":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-008"},{"@type":"Claim","@id":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-009#claim","id":"tn-009","slug":"peps-contraction-is-hard","status":"established","claim":"PEPS extend tensor networks beyond one dimension, but exact contraction of PEPS is computationally hard.","explanation":"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.","limitations":"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.","conceptIds":["peps","contraction-complexity"],"sourceIds":["verstraete-cirac-2004","schuch-2006"],"benchmarkIds":[],"reviewDate":"2026-07-29","canonicalUrl":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-009"},{"@type":"Claim","@id":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-010#claim","id":"tn-010","slug":"entanglement-growth-bounds-time-evolution","status":"established","claim":"Tensor-network time evolution is limited by entanglement growth, which forces the bond dimension upward as the simulated time increases.","explanation":"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.","limitations":"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.","conceptIds":["bond-dimension","truncation","area-law"],"sourceIds":["vidal-2003","schollwoeck-2010"],"benchmarkIds":[],"reviewDate":"2026-07-29","canonicalUrl":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-010"},{"@type":"Claim","@id":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-011#claim","id":"tn-011","slug":"classical-contraction-narrowed-a-sampling-claim","status":"established","claim":"Classical tensor-network contraction has been reported to solve the Sycamore random-circuit sampling problem, narrowing the advantage originally claimed for that specific task.","explanation":"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.","limitations":"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.","conceptIds":["contraction-complexity","quantum-inspired-optimization"],"sourceIds":["pan-2021"],"benchmarkIds":["bench-sycamore-sampling"],"reviewDate":"2026-07-29","canonicalUrl":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-011"},{"@type":"Claim","@id":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-012#claim","id":"tn-012","slug":"tensor-network-portfolio-work-is-narrow","status":"active-research","claim":"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.","explanation":"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.","limitations":"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.","conceptIds":["quantum-inspired-optimization","qubo-ising"],"sourceIds":["mugel-2020"],"benchmarkIds":["bench-dynamic-portfolio"],"reviewDate":"2026-07-29","canonicalUrl":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-012"},{"@type":"Claim","@id":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-013#claim","id":"tn-013","slug":"qaoa-advantage-is-unresolved","status":"active-research","claim":"Whether QAOA delivers a practical advantage over strong classical baselines on real optimization instances remains unresolved.","explanation":"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.","limitations":"This records an open question. It is neither a claim that QAOA will fail nor that classical methods are permanently ahead.","conceptIds":["qaoa","qubo-ising","quantum-inspired-optimization"],"sourceIds":["farhi-2014","lucas-2013"],"benchmarkIds":[],"reviewDate":"2026-07-29","canonicalUrl":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-013"},{"@type":"Claim","@id":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-014#claim","id":"tn-014","slug":"no-general-classical-advantage-is-recorded-here","status":"active-research","claim":"No source in this atlas establishes that classical tensor-network contraction generally outperforms quantum hardware on industrial optimization workloads.","explanation":"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.","limitations":"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.","conceptIds":["quantum-inspired-optimization","contraction-complexity","qaoa"],"sourceIds":["pan-2021","mugel-2020","farhi-2014"],"benchmarkIds":["bench-sycamore-sampling","bench-dynamic-portfolio"],"reviewDate":"2026-07-29","canonicalUrl":"https://research.mahastrategies.com/atlas/tensor-networks/claims/tn-014"}],"benchmarks":[{"id":"bench-sycamore-sampling","task":"Sampling from the output distribution of the Sycamore random quantum circuits.","classicalMethod":"Tensor-network contraction with an optimized contraction order, run on classical hardware.","comparedAgainst":"The superconducting-processor demonstration that originally framed this task as beyond practical classical reach.","reportedResult":"The cited work reports solving the Sycamore sampling problem classically, substantially narrowing the gap the original demonstration claimed for this task.","doesNotEstablish":"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.","sourceId":"pan-2021","sourceUrl":"https://research.mahastrategies.com/atlas/tensor-networks/sources/pan-2021"},{"id":"bench-dynamic-portfolio","task":"Dynamic portfolio optimization on real market datasets.","classicalMethod":"Quantum-inspired tensor-network optimization.","comparedAgainst":"Quantum processors applied to the same problem instances.","reportedResult":"The cited work reports applying both tensor-network methods and quantum processors to dynamic portfolio optimization with real datasets.","doesNotEstablish":"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.","sourceId":"mugel-2020","sourceUrl":"https://research.mahastrategies.com/atlas/tensor-networks/sources/mugel-2020"}]}