QCMA ⊆ QMA

QCMA is contained in QMA.

Statement

Migrated verbatim from quantum-classical-merlin-arthur § Quantum-Classical Merlin-Arthur:

QCMA sits between MA (classical verifier) and QMA (quantum witness allowed) in the hierarchy of proof systems.

Migrated verbatim from quantum-classical-merlin-arthur § Known relationships:

• $\mathbf{MA}\subseteq \mathbf{QCMA}\subseteq \mathbf{QMA}$: any MA protocol is a QCMA protocol (ignore the quantum capabilities of the verifier); any QCMA protocol is a QMA protocol (quantum states can encode classical strings).

Migrated verbatim from quantum-merlin-arthur § Known relationships:

• $\mathbf{NP}\subseteq \mathbf{MA}\subseteq \mathbf{QCMA}\subseteq \mathbf{QMA}$: classical proofs can be checked classically, classically by a quantum machine, or quantumly by a quantum machine.

Notes

source: folklore: the claim carried no citation on the page it was migrated from, and none was invented.

class: unstated: no citing page says which notion of reduction is meant. Recording a class the wiki does not state would add a claim.

class: free because a containment between complexity classes is proved by any argument at all; the reduction-class axis does not discriminate here, and unstated would wrongly suggest the information is missing.

Recorded during migration and not fixed — these are claims about the source text, not changes to it:

• ‘sits between’ is informal; the inclusions are never written as such.
• Uncited.
• Exact duplicate of line 24 sub-edge 1 on the same page.
• Uncited and unlabelled folklore.
• Duplicates line 18 of the same page and quantum-merlin-arthur.md:28 sub-edge 2.
• Duplicates quantum-classical-merlin-arthur.md:24 sub-edge 1 and :18 sub-edge 1.
• The bullet’s gloss offers three verification modes for four classes, so the mapping of justification to inclusion is not one-to-one.