An almost sudden jump in quantum complexity

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The Quantum Satisfiability problem (QSAT) is the generalization of the canonical NP-complete problem - Boolean Satisfiability. (k, s)-QSAT is the following variant of the problem: given a set of projectors of rank 1, acting non-trivially on k qubits out of n qubits, such that each qubit appears in at most s projectors, decide whether there exists a quantum state in the null space of all the projectors. Let f *(k) be the maximal integer s such that every (k, s)-QSAT instance is satisfiable. Deciding (k, f *(k))-QSAT is computationally easy: by definition the answer is “satisfiable”. But, by relaxing the conditions slightly, we show that (k, f *(k)+2)-QSAT is QMA1-hard, for k ≥ 15. This is a quantum analogue of a classical result by Kratochvíl et al. [1]. We use the term “an almost sudden jump” to stress that the complexity of (k, f *(k) + 1)-QSAT is open, where the jump in the classical complexity is known to be sudden. We present an implication of this finding to the quantum PCP conjecture, arguably one of the most important open problems in the field of Hamiltonian complexity. Our implication imposes some constraints on one possible way to refute the quantum PCP.

Original languageEnglish
Pages (from-to)1048-1059
Number of pages12
JournalQuantum Information and Computation
Issue number11-12
StatePublished - 1 Sep 2015
Externally publishedYes


  • Complexity
  • QMA
  • Quantum satisfiability

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Statistical and Nonlinear Physics
  • Nuclear and High Energy Physics
  • Mathematical Physics
  • General Physics and Astronomy
  • Computational Theory and Mathematics


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