[Paper Review] Quantum Versus Classical Proofs and Advice

Scott Aaronson, Greg Kuperberg|arXiv (Cornell University)|Apr 10, 2006

Quantum Computing Algorithms and Architecture17 references4 citations

TL;DR

This paper establishes a quantum oracle separation between QMA and QCMA, demonstrating that quantum proofs offer an advantage over classical proofs in certain query complexity settings. It proves a lower bound of Ω(√(2ⁿ/(m+1))) queries for finding a marked n-qubit state even with classical description and a quantum oracle, and presents a QCMA protocol nearly matching this bound, while showing classical proofs suffice for the group non-membership problem previously thought to require quantum proofs.

ABSTRACT

This paper studies whether quantum proofs are more powerful than classical proofs, or in complexity terms, whether QMA=QCMA. We prove three results about this question. First, we give a "quantum oracle separation" between QMA and QCMA. More concretely, we show that any quantum algorithm needs $Ω(\sqrt{2^n/(m+1)})$ queries to find an $n$-qubit "marked state" $\lvertψ angle$, even if given an $m$-bit classical description of $\lvertψ angle$ together with a quantum black box that recognizes $\lvertψ angle$. Second, we give an explicit QCMA protocol that nearly achieves this lower bound. Third, we show that, in the one previously-known case where quantum proofs seemed to provide an exponential advantage, classical proofs are basically just as powerful. In particular, Watrous gave a QMA protocol for verifying non-membership in finite groups. Under plausible group-theoretic assumptions, we give a QCMA protocol for the same problem. Even with no assumptions, our protocol makes only polynomially many queries to the group oracle. We end with some conjectures about quantum versus classical oracles, and about the possibility of a classical oracle separation between QMA and QCMA.

Motivation & Objective

• To investigate whether quantum proofs (QMA) are more powerful than classical proofs (QCMA) in quantum complexity theory.
• To resolve the open question of whether QMA = QCMA by constructing a relativized world where they differ.
• To analyze the power of quantum versus classical advice in BQP/qpoly and BQP/poly classes.
• To examine whether quantum proofs provide exponential advantages in known problems, such as group non-membership.
• To explore the feasibility of classical oracle separations between QMA and QCMA and the role of quantum oracles in complexity separations.

Proposed method

• Introduces the concept of a quantum oracle as a sequence of unitary operations accessible in superposition, generalizing classical oracles.
• Proves a query lower bound of Ω(√(2ⁿ/(m+1))) for finding a marked n-qubit state, even with an m-bit classical description and a quantum oracle recognizing the state.
• Constructs a QCMA protocol that nearly achieves this lower bound, demonstrating near-optimality of classical proofs in this setting.
• Applies a novel algebraic technique involving unitary matrices with polynomial entries to bound the number of distinct values a unitary matrix can take over Boolean inputs.
• Uses the structure of Clifford algebras and anticommutation relations to derive constraints on matrix differences, leading to a bound on the number of non-zero perturbations.
• Demonstrates that for any unitary matrix U(X) with entries that are linear polynomials in k variables, U(X) can take at most 4^N distinct values over {0,1}^k.

Experimental results

Research questions

• RQ1 Is there a quantum oracle relative to which QMA and QCMA are separated?
• RQ2 Can classical proofs (QCMA) achieve the same query complexity as quantum proofs (QMA) in the marked state problem?
• RQ3 Does the group non-membership problem, previously solved via QMA, admit a QCMA protocol under plausible group-theoretic assumptions?
• RQ4 Can the techniques used to bound unitary matrix variation over Boolean inputs be extended to other complexity-theoretic separations?
• RQ5 Is there a classical oracle separation between QMA and QCMA, or is the quantum oracle approach necessary?

Key findings

• There exists a quantum oracle U such that QMA^U ≠ QCMA^U, establishing a quantum oracle separation between the two classes.
• Any quantum algorithm requires at least Ω(√(2ⁿ/(m+1))) queries to find a marked n-qubit state, even with an m-bit classical description and a quantum oracle.
• A QCMA protocol is constructed that nearly matches this lower bound, showing classical proofs are nearly as powerful as quantum ones in this context.
• For the group non-membership problem, a QCMA protocol is given under plausible group-theoretic assumptions, and even without assumptions, it makes only polynomially many queries to the group oracle.
• The number of distinct values a unitary matrix U(X) with entries that are degree-1 complex polynomials in k variables can take over {0,1}^k is at most 4^N, where N is the matrix size.
• The analysis shows that at most 2N of the k-bit perturbations can be non-zero, limiting the effective dependence of U(X) on the input variables.