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QMA is the class of promise problems for which a uniform polynomial-size quantum circuit checks a polynomial-size quantum witness. A YES instance has some quantum witness accepted with probability at least $2/3$; on a NO instance every quantum witness is accepted with probability at most $1/3$. The verifier initializes its own ancilla qubits to zero. Mixed quantum witnesses do not improve the maximum because acceptance is linear in their density operator.

For a verifier $U_x$ and quantum witness embedding $J|w⟩=|w⟩|0\cdots 0⟩$, define the witness acceptance operatorThe Rayleigh quotient gives ${\max }_{\Vert w\Vert =1}⟨w|F|w⟩={\lambda }_{\max }(F)$. Every entry of $F$ can be recomputed by the polynomial-storage path summation of part (b); no full quantum witness matrix needs to be stored.

Let the quantum witness have $m$ qubits, so its dimension is $D=2^m$. For this positive operator, trace-power witness optimization usesThis is the useful exponentiated form of the supplied logarithmic inequality. Positivity is required; such a logarithmic statement is not true for an arbitrary operator. No logarithms or roots need to be calculated.

Choose $d=2m+2$. On a NO instance,since $2^m(2/3{)}^{2m+2}=(4/9)(8/9{)}^m<1$. On a YES instance, $\mathrm{Tr}(F^d)\ge (2/3{)}^d>(1/2{)}^d$. Thus comparison of $2^d\mathrm{Tr}(F^d)$ with one distinguishes the cases using only multiplication and a final comparison.

Evaluate the trace byThere are $d=O(m)$ active quantum witness indices, each requiring $m$ bits. Accumulate one product at a time and recompute its matrix entries as required. Together with the verifier path workspace, this uses polynomially many real registers. The potentially enormous time and the numerical precision are unrestricted in $C$. Therefore $\mathrm{QMA}\subseteq C$, without solving an exponentially large eigenvalue problem by storing its matrix.

A quantum witness is a polynomial-size quantum state supplied to a verifier as evidence for a YES instance. The verifier fixes its own ancilla qubits independently. Pure quantum witnesses suffice when maximizing a linear acceptance functional on density operators. Quantum witnesses are not generally restricted to basis vectors or product states.

StoqMA is a restricted quantum-verifier class using stoquastic circuits and polynomial-size quantum witnesses. A YES instance has a quantum witness accepted with probability at least $a$; on a NO instance every quantum witness has acceptance at most $b$, with an inverse-polynomial gap $a-b$. An optimal quantum witness can be chosen with nonnegative amplitudes, since the witness acceptance operator is entrywise nonnegative. The quantum witness is generally a superposition, rather than a computational basis vector. The stoquastic acceptance floor explains the lower limit $b\ge 1/2$ for a nontrivial soundness promise.

If the output of a stoquastic circuit is $|0⟩|r_0⟩+|1⟩|r_1⟩$ with nonnegative amplitudes, its plus-outcome probability is $1/2+\mathrm{Re}⟨r_0|r_1⟩\ge 1/2$. Basis quantum witnesses with zero and plus ancilla qubits always have this property. A soundness threshold below one half cannot hold for every such input.

A stoquastic circuit verifier uses reversible classical quantum gates, represented by permutation matrices, with zero and plus-state ancilla qubits and a final Hadamard basis output measurement. A basis input remains an entrywise nonnegative state; a general quantum witness need not be a basis input. The resulting acceptance structure is used in StoqMA.

For a positive witness acceptance operator on dimension $D$, large trace powers approximate its largest eigenvalue. With an $m$-qubit QMA quantum witness and $d=2m+2$, the usual NO instances give $\mathrm{Tr}(F^d)<2^{-d}$ and YES instances give a value above $2^{-d}$. A closed product-index expansion computes the trace by recomputation with polynomial storage, avoiding logarithms and roots in a real-arithmetic implementation.

For quantum witness embedding $J$, verifier unitary $U$ and accepting projector $\Pi$, the witness acceptance operator is the positive contraction $F=J^{†}{U}^{†}\Pi UJ$. The maximum acceptance over normalized quantum witnesses is ${\lambda }_{\max }(F)$ by the Rayleigh quotient. Restricting to basis quantum witnesses only tests its diagonal entries and need not find that maximum.