The quantum PCP conjecture
Is estimating the ground state energy of a local Hamiltonian to constant relative precision still QMA-hard?
Problem statement
Conjecture: there is a constant such that it is QMA-hard to decide, given a local Hamiltonian on qubits with terms, whether the ground state energy is below or above .
Equivalently, the constraint-satisfaction version: approximating the maximum fraction of satisfiable local terms to within a constant is QMA-hard.
Lean 4
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Background
The classical PCP theorem states that approximating MAX-3SAT to within a constant is NP-hard. Kitaev's circuit-to-Hamiltonian construction gives the quantum analogue of the Cook–Levin theorem, with an inverse-polynomial promise gap. Amplifying that gap to a constant is the open step, and the classical amplification techniques — parallel repetition, gap amplification by expander walks — have no known quantum counterpart.
A positive answer would imply that entanglement survives at constant temperature in some systems, contradicting a widespread physical intuition, and would rule out efficient classical approximation schemes for ground-state energies. A refutation would give a general approximation algorithm and would be equally significant.
Known partial results
- The NLTS theorem (Anshu, Breuckmann, Nirkhe, 2022) established a necessary consequence: there are local Hamiltonians whose low-energy states all require circuits of superlogarithmic depth. This removed the main obstacle people had hoped would refute the conjecture.
- Quantum PCP is known for the "quantum PCP of proximity" and in the setting of games (via MIP* = RE, which is a different model with entangled provers).
- Dinur's gap amplification does not survive quantisation: the quantum analogue of the union bound over assignments fails.
Recorded progress
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Discussion
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