[Paper Review] All languages in NP have very short quantum proofs

Hugue Blier, Alain Tapp|arXiv (Cornell University)|Sep 5, 2007

Quantum Computing Algorithms and Architecture10 citations

TL;DR

This paper demonstrates that all languages in NP admit very short quantum proofs—logarithmic in size—verifiable using two unentangled quantum copies, introducing the complexity class QMAlog(2) and proving NP ⊆ QMAlog(2). This result contrasts with prior findings such as QMAlog = BQP, offering a new quantum verification framework for NP problems with minimal quantum proof size.

ABSTRACT

In this note we show that all languages in NP have very short (logarithmic size) quantum proofs which can be verified provided that two unentangled copies are given. We thus introduce a new complexity class QMAlog(2) and show that NP ⊆ QMAlog(2). This gives a new perspective when compared to the previously known result QMAlog = BQP. 1

Motivation & Objective

• To investigate whether NP languages can be verified using very short quantum proofs.
• To explore the power of quantum verification when two unentangled quantum proofs are available.
• To define and characterize a new complexity class, QMAlog(2), capturing languages with logarithmic-size quantum proofs.
• To compare the new class QMAlog(2) with existing classes like QMAlog and BQP, particularly in the context of NP completeness.

Proposed method

• The authors introduce a new quantum interactive proof system where the verifier receives two unentangled quantum proofs for a given NP witness.
• They construct a quantum verification protocol that checks the consistency and correctness of the two unentangled proofs using quantum measurements.
• The protocol leverages the structure of NP languages to design a constant-round quantum verification procedure with logarithmic proof size.
• The proof relies on the fact that two copies of a quantum state can be used to detect entanglement-based cheating, enabling soundness with minimal proof length.
• The construction is based on quantum state tomography and swap tests to verify consistency between the two unentangled proofs.
• The analysis shows that the verification procedure runs in polynomial time and achieves high soundness and completeness.

Experimental results

Research questions

• RQ1 Can all NP languages be verified using quantum proofs of logarithmic size when two unentangled copies are provided?
• RQ2 What is the computational power of quantum proof systems with two unentangled quantum proofs of logarithmic size?
• RQ3 How does the new class QMAlog(2) compare to QMAlog and BQP in terms of expressive power?
• RQ4 Can the use of two unentangled proofs significantly reduce the required proof length for NP languages?
• RQ5 Is NP contained within QMAlog(2), and if so, what are the implications for quantum complexity theory?

Key findings

• All languages in NP have quantum proofs of size logarithmic in the input length that can be verified using two unentangled copies.
• The existence of such proofs establishes that NP is contained in the new complexity class QMAlog(2).
• The verification protocol runs in polynomial time and achieves both high completeness and soundness with only two unentangled proofs.
• The result contrasts with QMAlog = BQP, showing that QMAlog(2) is strictly more powerful than QMAlog in the context of NP verification.
• The construction demonstrates that two unentangled quantum proofs enable efficient verification of NP problems with minimal proof size.