Innovations in Computer Science - ICS 2010, Tsinghua University, Beijing, China, January 5-7, 2010. Proceedings, 470-481,
978-7-302-21752-7
Tsinghua University Press
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On the Power of a Unique Quantum Witness
Authors : Rahul Jain, Iordanis Kerenidis,Greg Kuperberg, Miklos Santha, Or Sattath and Shengyu Zhang
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Abstract: In a celebrated paper, Valiant and Vazirani [29] 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 tensor powerHd. Indeed, the intersection of this subspace with Wd is one-dimensional, and therefore can play the role of the unique quantum witness.
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