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Contents
  1. Unconditional Separation of Quantum and Classical Bounded-Error Polynomial Time (BQP != BPP)
  2. Definitions and mathematical statement
  3. Short English statement
  4. Research attempt
  5. Current frontier.
  6. Attack route.
  7. Attempt: a total factoring threshold language.
  8. Attempt: a quantitative oracle-to-input audit.
  9. Attempt: what a legitimate totalization needs.
  10. Stress test.
  11. Outcome.
  12. Sources

Unconditional Separation of Quantum and Classical Bounded-Error Polynomial Time (BQP != BPP)

Definitions and mathematical statement

Both classes decide total binary languages with error at most $1/3$ on every input and polynomial running time. The class $\mathsf{BPP}$ uses classical random bits; $\mathsf{BQP}$ uses uniform quantum computation over a fixed effective universal gate set. The target is \[ \exists L\subseteq\{0,1\}^*:\quad L\in\mathsf{BQP}\ \text{and}\ L\notin\mathsf{BPP}. \] The assertion is unconditional and unrelativized: a separation only in the presence of an oracle or under a conjectured hardness assumption does not meet it.

Formulation and concept references: [S1].

Short English statement

Can one prove, without computational assumptions, that quantum computers efficiently solve some decision problem beyond all efficient randomized classical computers?

Research attempt

Current frontier.
Research check: 27 September 2026. Shor supplies polynomial-time quantum factoring, not a classical lower bound. [E1]. The 2025 sampling/function paper in the original bibliography obtains conditional consequences involving collapse of the polynomial hierarchy. The September 2026 Fourier-hierarchy manuscript concerns oracle separations. Neither statement supplies the unconditional total language required here. [S2], [S3]. We distinguish these carefully before trying to convert a quantum algorithm into a separating language.
Attack route.
There are three distinct missing steps: prove classical hardness of a known quantum task; make an oracle lower bound survive an explicit description of the oracle; or turn a promise separation into a total-language separation. The last two steps are often concealed by a change of encoding. We pursue the first through an exact decision/search reduction, then test the other two against that reduction. All claims of hardness below remain hypotheses.
Attempt: a total factoring threshold language.
Fix a conventional, polynomial-time parsable binary encoding of pairs and put \[ L_{\rm fac}=\{\langle N,t\rangle:N\geq2,\ 2\leq t<N, \ \exists d\ (2\leq d\leq t,\ d\mid N)\}. \tag{19.1} \] Malformed encodings, primes, and out-of-range thresholds are no-instances. The language is total. A bounded-error quantum factorization algorithm, followed by multiplication checks on its output, decides it in quantum polynomial time. Complete factorization reveals the least prime factor; on a prime it reveals no proper factor. The usual amplification and polynomial-time truncation of the factoring algorithm give the required worst-case bounded-error time bound. This uses the established quantum algorithm, not an assumption about its classical complexity. [E1].

Here is the converse reduction with adaptive error accounted for. Suppose $L_{\rm fac}\in\mathsf{BPP}$, and let $N$ be composite of binary length $n$. The predicate in $19.1$, as a function of $t\in[2,N-1]$, changes from zero to one precisely at the least prime divisor $p$ of $N$. Binary search uses at most $n$ decision calls and returns $p$ if all answers are correct. Amplify every call to error at most $1/(12(n+1))$, using fresh random bits. Even though later queries depend on earlier answers, conditional on any past transcript the next query is a fixed input, so its error bound still holds. The union bound makes the entire search succeed with probability at least $11/12$. Division verifies the output; a failed verification can be reported as failure. The running time is polynomial in $n$, including amplification and arithmetic. Consequently \[ \begin{split} &\text{no randomized polynomial-time algorithm finds a proper factor}\\[-2pt] &\text{of every composite with success at least }2/3 \quad\Longrightarrow\quad L_{\rm fac}\in\mathsf{BQP}\setminus\mathsf{BPP}. \end{split} \tag{19.2} \] This is a complete conditional implication, not an unconditional separation. In particular, a negative answer to UnsolvedMath problem 8400004's deterministic factoring question would not supply the stronger randomized lower bound assumed here.

Attempt: a quantitative oracle-to-input audit.
Consider an oracle $f:\{0,1\}^m\to\{0,1\}^m$ in Simon's promise problem: $f(x)=f(y)$ exactly when $y=x$ or $y=x\oplus s$, for a nonzero hidden $s$. The quantum algorithm obtains random linear equations orthogonal to $s$; classical queries require a collision to expose it in the black-box model. The established query-complexity framework, including the distinction between oracle and ordinary machines, is essential here. [E3].

If the entire table of $f$ is the input, its length is $B=m2^m$. A classical algorithm can read and sort the pairs $(f(x),x)$ in time polynomial in $B$, recover a collision, and obtain $s$. Thus an $\exp(\Omega(m))$ query lower bound is not a superpolynomial lower bound in this encoding. A proof that charged time in $m$ while giving the table as an input would have changed the target's size parameter.

If instead a size-$\operatorname{poly}(m)$ circuit $C$ describes $f$, quantum queries can be implemented by reversible evaluation of $C$, with its workspace uncomputed. But a classical algorithm now sees $C$, not just an oracle. As a concrete stress test, take a rank-$m-1$ binary matrix $A$ and $f(x)=Ax$. Its kernel is $\{0,s\}$, so it satisfies Simon's promise. Given the coefficient list defining $C$, Gaussian elimination finds $s$ in polynomial time. The oracle difficulty therefore does not hold uniformly for small descriptions, even though their truth tables have the same two-to-one pattern. A hard explicit family, together with a lower bound against algorithms inspecting its descriptions, is still required.

Attempt: what a legitimate totalization needs.
Suppose a promise problem has a polynomial-time decidable promise set $D$ and a bounded-error quantum algorithm on $D$. Define a total language by rejecting inputs outside $D$. The preliminary membership test for $D$ then gives a BQP algorithm for the totalization, and any BPP decider for it solves the original promise problem. This elementary lemma is sound. It is useless without a cheap test for the entire promise.

For an arbitrary circuit, "acceptance probability lies outside $(1/3,2/3)$" is a semantic promise, not a syntactic property that may be assumed efficiently decidable. Repeating the circuit does not decide membership in that promise with a fixed gap near either endpoint. Likewise, testing Simon periodicity by evaluating every pair uses an exponential number of evaluations in a succinct encoding. Rejecting invalid descriptions is therefore an uncharged operation in the naive construction. The total threshold language $19.1$ avoids that particular problem, but does not avoid the classical-hardness problem.

Stress test.
The reduction handles all composite inputs and all adaptive transcripts; it is not an average-case factoring assertion. Its hypothesis is stronger than merely excluding deterministic factoring. Oracle results and sampling results have not been substituted for a lower bound for $19.1$. The linear Simon example refutes only a universal description-to-oracle transfer principle, not the existence of other hard descriptions. Also, $\mathsf P=\mathsf{PSPACE}$ would imply $\mathsf{BQP}=\mathsf{BPP}=\mathsf P$, by the standard polynomial-space simulation of quantum computation; the target would then be false. [E2]. That dependency is another reason an unconditional hardness assertion here cannot be supplied by a query bound alone.
Outcome.
Outcome: Conditional reduction. The notebook gives a total-language witness conditional on worst-case randomized factoring hardness, with a fully charged adaptive reduction. It also isolates concrete failures of table encoding, succinct-oracle transfer, and unchecked promise totalization. No randomized factoring lower bound has been established. The conditional implication and the special examples are not claimed to be historically new.

Sources