[Paper Review] Classical Verification of Quantum Proofs

Zhengfeng Ji|arXiv (Cornell University)|May 27, 2015

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper presents a classical interactive protocol that verifies quantum proofs for the local Hamiltonian problem, demonstrating that approximating the non-local value of multi-player one-round games to inverse-polynomial precision is QMA-hard. The work establishes a deep connection between QMA-completeness, Hamiltonian complexity, and non-local games via entangled provers and partial rigidity of quantum stabilizer codes.

ABSTRACT

We present a classical interactive protocol that verifies the validity of a quantum witness state for the local Hamiltonian problem. It follows from this protocol that approximating the non-local value of a multi-player one-round game to inverse polynomial precision is QMA-hard. Our work makes an interesting connection between the theory of QMA-completeness and Hamiltonian complexity on one hand and the study of non-local games and Bell inequalities on the other.

Motivation & Objective

• To develop a classical interactive protocol that verifies quantum witness states for the local Hamiltonian problem.
• To establish a connection between QMA-completeness, Hamiltonian complexity, and non-local games with entangled provers.
• To show that approximating the non-local value of multi-player one-round games to inverse-polynomial precision is QMA-hard.
• To extend the theory of rigidity in quantum non-local games to general stabilizer codes with a designated special player.

Proposed method

• The protocol uses a multi-prover one-round game with entangled provers, where the verifier classically checks quantum proofs via non-local correlations.
• It employs a generalized stabilizer code game with a special player (t), using XZ-form generators to define questions and answer parities.
• The game is constructed using a 'π/4-trick' on the t-th qubit to generate four operators h₁⁽ᵗ⁾, h₂⁽ᵗ⁾, h₃⁽ᵗ⁾, h₄⁽ᵗ⁾ from X and Z generators of the stabilizer.
• Questions are encoded as labels in {*, 0, 1, 2, 3}, with * indicating no question, and answers are bits whose parity must match a predefined sign.
• The protocol relies on partial rigidity theorems, showing that near-optimal quantum strategies must be isomorphic to the ideal quantum strategy.
• The analysis uses state-dependent distance measures between reflections and trace inequalities to bound deviations from the ideal strategy.

Experimental results

Research questions

• RQ1 Can a classical verifier efficiently verify a quantum witness state for the local Hamiltonian problem using only classical communication and entangled provers?
• RQ2 Is approximating the non-local value of a multi-player one-round game to inverse-polynomial precision QMA-hard?
• RQ3 Can the rigidity of quantum non-local games be extended to general stabilizer codes with a designated special player?
• RQ4 What is the role of entanglement in enabling higher non-local values compared to classical strategies?

Key findings

• Approximating the non-local value of a multi-player one-round game to inverse-polynomial precision is QMA-hard.
• The special-player stabilizer game achieves partial rigidity, meaning any near-optimal quantum strategy must be isomorphic to the ideal quantum strategy.
• The protocol uses a classical verifier to verify quantum proofs via entangled non-local games, establishing a bridge between QMA and non-local game complexity.
• The analysis shows that the state-dependent distance between reflections and the ideal strategy is bounded by the square root of the expectation of (1 + Cₗ)², ensuring soundness.
• The construction generalizes the five-qubit code game to arbitrary stabilizer codes with distance ≥2 and non-fixed qubits.
• The method proves that the multi-linearity test is sound against entangled provers, supporting the containment of NEXP in MIP*.