Tensor Networks: The Classical Competitor
Tensor-network methods are the classical benchmark quantum simulation must beat; they already solve many correlated-electron problems Kronos cares about.
Classical methods that mimic quantum states
Any honest quantum-advantage claim must beat the best classical method, and for correlated quantum systems that method is often a tensor network. Tensor networks (matrix product states / DMRG in 1D, PEPS in 2D) represent quantum states with a controlled amount of entanglement, capturing exactly the physics that makes VQE and Hubbard-model problems hard, without a quantum computer.
# Matrix product state (MPS) with bond dimension chi:
|psi> = sum A^{s1}_{a1} A^{s2}_{a1 a2} ... A^{sN}_{a_{N-1}} |s1..sN>
# cost ~ O(N * chi^3); chi controls captured entanglement
# exact when chi >= 2^{N/2}; efficient when entanglement is bounded
# DMRG variationally optimizes the A tensors for the ground stateWhere classical wins today
- One-dimensional and quasi-1D correlated systems: DMRG is essentially exact and fast.
- Moderate 2D systems: PEPS and related methods reach sizes VQE cannot.
- Molecular active spaces: DMRG and coupled cluster set the accuracy bar for first-wall chemistry.
Where they strain, and why quantum matters long-term
Tensor networks fail when entanglement grows beyond what a tractable bond dimension chi can hold: large two-dimensional systems at strong coupling, and real-time dynamics where entanglement grows linearly in time. The cost scales steeply in chi, so these cases eventually become intractable, exactly the regime a fault-tolerant quantum computer would own.
Kronos uses tensor networks two ways: as the classical production tool for correlated-electron questions we can answer now, and as the mandatory baseline every quantum result is measured against. A quantum method earns trust only by matching tensor networks where they are reliable and then extending past where they break. This is the operational meaning of 'no assumed quantum advantage'; see the honest assessment.