## Abstract

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 QMA_{1}-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 language | American English |
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Pages (from-to) | 1048-1059 |

Number of pages | 12 |

Journal | Quantum Information and Computation |

Volume | 15 |

Issue number | 11-12 |

DOIs | |

State | Published - 1 Sep 2015 |

Externally published | Yes |

## Keywords

- Complexity
- QMA
- Quantum satisfiability

## All Science Journal Classification (ASJC) codes

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