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Independent runs of a BQP algorithm with fresh ancilla qubits can be combined by a classical threshold rule. If completeness exceeds soundness by $\gamma$, a Hoeffding inequality bounds the error after $r$ runs by $e^{-r{\gamma }^2/2}$. Polynomial repetition handles inverse-polynomial gaps. This does not automatically preserve the restricted gate and measurement rules of a stoquastic circuit.

BQP consists of promise problems decided by a uniform family of polynomial-size quantum circuits. On input $x$, with polynomially many zero-initialized ancilla qubits, a designated output measurement accepts with probability at least $2/3$ on YES inputs and at most $1/3$ on NO inputs. Uniformity means a classical polynomial-time procedure produces the quantum circuit description from the input length, or equivalently produces the verifier quantum circuit with $x$ supplied as input.

For BQP error reduction, run independent copies with freshly initialized registers. If the completeness and soundness thresholds are any constants $a>b$, accept when the fraction of accepting trials exceeds $(a+b)/2$. A Hoeffding inequality bounds either error by $\exp [-r(a-b{)}^2/2]$, with $r$ repetitions. The threshold need not be a simple majority when both $a$ and $b$ lie on the same side of $1/2$.

Choosing $r$ sufficiently large gives the usual $2/3,1/3$ thresholds, or any other fixed separated thresholds. Even an inverse-polynomial gap can be amplified using polynomially many repetitions. Classical threshold evaluation can be incorporated into the uniform computation. Thus fixed separated acceptance probabilities define the same BQP class. This argument concerns unrestricted BQP quantum circuits and does not assume such amplification is available for restricted stoquastic circuits.

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.