2009/06/24 by Rahul Jain, Iordanis Kerenidis, Jain, Rahul +10
Computer Science · Physics and Astronomy · #Complexity and Algorithms in Graphs #FOS: Physical sciences #Machine Learning and Algorithms #Quantum Computing Algorithms and Architecture #Quantum Physics (quant-ph) #quant-ph
paper · pdf · doi:10.48550/arxiv.0906.4425
The results in version 2 are mostly unchanged from version 1 by Jain, Kerenidis, Santha and Zhang. Some of this work was done independently by Sattath and Kuperberg and hence version 2 includes all authors
openalex publication_date 2009/06/24 · arxiv created 2011/01/19 · arxiv updated 2011/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a celebrated paper, Valiant and Vazirani raised the question of whether the difficulty of NP-complete problems was due to the wide variation of the number of witnesses of their instances. They gave a strong negative answer by showing that distinguishing between instances having zero or one witnesses is as hard as recognizing NP, under randomized reductions. We consider the same question in the quantum setting and investigate the possibility of reducing quantum witnesses in the context of the complexity class QMA, the quantum analogue of NP. The natural way to quantify the number of quantum witnesses is the dimension of the witness subspace W in some appropriate Hilbert space H. We present an efficient deterministic procedure that reduces any problem where the dimension d of W is bounded by a polynomial to a problem with a unique quantum witness. The main idea of our reduction is to consider the Alternating subspace of the d-th tensor power of H. Indeed, the intersection of this subspace with the d-th tensor power of W is one-dimensional, and therefore can play the role of the unique quantum witness.