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Q13 / Computational complexity / conjecture

The singlet conjecture

For an exchange-symmetric interaction, does subtracting a triplet projector ever increase the complexity of its positive-weight local-Hamiltonian problem?

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The conjecture

Marwaha–Sud, Conjecture 3: for an exchange-symmetric two-qubit interaction KK and a normalized triplet state ∣τ⟩\lvert\tau\rangle, let K′=K−∣τ⟩⟨τ∣.K^{\prime}=K-\lvert\tau\rangle\langle\tau\rvert. The positive-weight {K′}+\{K^{\prime}\}^{+}-Hamiltonian promise problem is conjectured to be at most as hard as its {K}+\{K\}^{+} counterpart.

Scope & assumptions

'Symmetric' means exchange symmetry, not invariance under all collective SU(2)\mathrm{SU}(2) rotations. Treat a specific reduction notion as an additional formalization.

Why this is interesting

This is a proposed ordering of complexity by local symmetry structure, beyond the paper's proved phase classification.

A place to start

Study one edge of the paper's 20-arrangement diagram and ask for an explicit polynomial reduction.

Related work

These works provide context and established results. They need not be the origin of this exact formulation.

  1. A complexity phase transition at the EPR Hamiltonian ↗

    Kunal Marwaha, James Sud · 2026

    Read the relevance note
  2. The complexity of antiferromagnetic interactions and 2D lattices ↗

    Stephen Piddock, Ashley Montanaro · 2017

    Read the relevance note

Related questions

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