 Solution

The class $C$ uses polynomial-register real-arithmetic computation: its storage restriction counts real registers, without limiting their precision or the running time. Use the usual decision-model interpretation that computed real values may be compared with fixed thresholds, and that the fixed universal gate set's real constants are available. Without a way to test values, arithmetic instructions alone would not specify the intended decision model.

Let a BQP quantum circuit use $q=\mathrm{poly}(n)$ qubits and $T=\mathrm{poly}(n)$ gates. A matrix element can be evaluated by depth-first quantum circuit path summation:Enumerate the intermediate $q$-bit strings recursively. Store the current strings and loop positions, a partial product, and the partial sum at each depth. There are at most $T$ active levels, using $O(qT)$ registers or bits of loop information, not an exponentially long state vector. Each fixed-locality gate matrix element is computed from its few affected bits. Represent a complex number by two real registers; complex multiplication and addition require only real addition, subtraction and multiplication.

Recompute this amplitude separately for every final string $z$ whose output bit is one, accumulating the Born rule probabilityThe outer enumeration needs only another $O(q)$ bits and a real accumulator. Arbitrarily large running time is allowed, so repeated recomputation is harmless. Compare $p_1$ with $1/2$ to distinguish the promised BQP cases. This proves $\mathrm{BQP}\subseteq C$ using polynomial storage.

The printed probability hint omits the squares of the amplitude moduli. Summing their absolute values alone is not the Born rule and can exceed one. The corrected sum above is necessary for the containment proof. The argument is about the stated real-register model, rather than a claim that exponential-precision numbers are free in an ordinary bit-cost model.