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gpt 6 astra pro attempt on CS-8

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Unverified LLM-generated attempt. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Manuscript

Contents
  1. Unconditional Separation of BQP from the Polynomial Hierarchy
  2. Definitions and mathematical statement
  3. Short English statement
  4. Research attempt
  5. Current frontier.
  6. Attack route.
  7. Attempt.
  8. Stress test.
  9. Outcome.
  10. Sources

Unconditional Separation of BQP from the Polynomial Hierarchy

Definitions and mathematical statement

The polynomial hierarchy is $\mathsf{PH}=\bigcup_{k\ge0}\Sigma_k^{\mathsf P}$, where a language in $\Sigma_k^{\mathsf P}$ has a polynomial-time predicate preceded by $k$ alternating blocks of polynomially bounded existential and universal quantifiers, starting existentially. The target is \[ \exists L\in\mathsf{BQP}\quad L\notin\Sigma_k^{\mathsf P}\text{ for every fixed }k. \] This is a statement about ordinary uniform computation. Exhibiting an oracle relative to which the containment fails is a different statement.

Formulation and concept references: [S1].

Short English statement

Can efficient quantum computation be shown to solve a problem beyond every fixed level of the polynomial hierarchy?

Research attempt

Current frontier.
The established Raz--Tal result is an oracle separation $\mathsf{BQP}^O\not\subseteq\mathsf{PH}^O$, obtained from a distributional query problem; deleting the superscripts is not a valid inference. [E1]. Buzet--Chailloux's August 2026 version implements 2-Forrelation with restricted IQP resources and strengthens an oracle separation, not the present unrelativized statement. [S3], [E2]. Miloschewsky--Podder--Rudolph's September 2026 manuscript constructs an oracle where quantum and deterministic language classes agree while their promise versions do not. Its abstract makes particularly clear why an oracle promise must not silently become a total language. [S4], [E3]. The recent statements were checked; their full proofs were not independently audited. Aaronson's original formulation remains the target here. [S1].
Attack route.
A plausible route is to replace the random oracle functions in Forrelation by uniformly evaluable, succinct functions, then prove that every fixed level of the polynomial hierarchy fails on the resulting total decision problem. This requires both a total gap convention and a lower bound that survives access to the functions' descriptions. Query hardness by itself supplies neither. I test two natural explicit replacements, Fourier-sparse functions and quadratic phases, before attempting a more ambitious derandomization. The calculations below are deductions in this notebook, not new claims about the published oracle construction.
Attempt.
Set $N=2^m$, take $f,g:{\mathbb{F}}_2^m\to\{-1,1\}$, and use the normalization \begin{equation}\label{wsref-0-72-33-35-3a-70-68-69} \Phi(f,g)=N^{-3/2}\sum_{x,y\in{\mathbb{F}}_2^m} f(x)(-1)^{x\cdot y}g(y). \end{equation} This is the inner product of two normalized states after a normalized Hadamard transform, so $|\Phi|\le1$. With reversible evaluation circuits for $f,g$, a Hadamard test estimates this real amplitude to fixed additive accuracy with polynomially many gates in the description lengths. This explains the attraction of a constant-gap promise; it does not define bounded-error answers for inputs arbitrarily close to a threshold. The Forrelation setup and its query interpretation are discussed in [S1] and [E1].

Lemma 35.1 (a sparsity obstruction). Define $\widehat f(y)=N^{-1}\sum_x f(x)(-1)^{x\cdot y}$. If at most $k$ Fourier coefficients are nonzero, then \begin{equation}\label{wsref-0-72-33-35-3a-73-70-61-72-73-65} |\Phi(f,g)|\le\sqrt{k/N}. \end{equation} Proof. Orthogonality gives $\sum_y\widehat f(y)^2=1$, and $\Phi=N^{-1/2}\sum_y\widehat f(y)g(y)$. Cauchy--Schwarz on its support proves the assertion. Thus $\Phi\ge c>0$ requires $k\ge c^2N$. A family with polynomially many Fourier coefficients in $m$ cannot produce constant positive Forrelation in this normalization. This says nothing against succinct functions with exponentially large Fourier support.

Lemma 35.2 (quadratic phases admit exact classical evaluation). Suppose $f(x)=(-1)^{q(x)}$ and $g(y)=(-1)^{r(y)}$, with explicitly given multilinear quadratic polynomials over ${\mathbb{F}}_2$. Then $\Phi(f,g)$ can be computed exactly in time polynomial in $m$, including its sign.

Proof. The numerator of (r35:phi) is $S(Q)=\sum_{z\in{\mathbb{F}}_2^{2m}}(-1)^{Q(z)}$, where $Q(x,y)=q(x)+x\cdot y+r(y)$ is quadratic. Here is an elimination algorithm that does not enumerate the $2^{2m}$ points. If a cross term $uv$ occurs, write, with $z$ denoting the other variables, \[ Q(u,v,z)=uv+uA(z)+vB(z)+C(z), \qquad \deg A,\deg B\le1,\quad\deg C\le2. \] For fixed $z$, direct summation gives \begin{equation}\label{wsref-0-72-33-35-3a-65-6c-69-6d-69-6e-61-74-65} \sum_{u,v\in{\mathbb{F}}_2}(-1)^{uv+uA+vB+C}=2(-1)^{C+AB}. \end{equation} Replace $Q$ by $C+AB$, reducing $z_i^2=z_i$, and record a factor two. The degree remains at most two. If no cross term remains, an affine polynomial has sum zero when any linear coefficient is nonzero, and otherwise sum $(-1)^{Q(0)}2^{\#z}$. At most $m$ eliminations occur; coefficient bookkeeping uses polynomially many bit operations. The answer is an integer divided by $2^{3m/2}$, representable exactly in ${\mathbb{Q}}(\sqrt2)$.

There is a useful independent magnitude check. Let $B(u,v)=Q(u+v)+Q(u)+Q(v)+Q(0)$ have rank $\rho$ on ${\mathbb{F}}_2^d$, $d=2m$. Changing variables from a pair of summation points to their difference gives \[ S(Q)^2=2^d\sum_{h\in\ker B}(-1)^{Q(h)+Q(0)}. \] The exponent restricts to a linear functional on $\ker B$. If it is nonzero the sum vanishes; otherwise $|S(Q)|=2^{d-\rho/2}$. The elimination algorithm supplies the sign that this square identity does not determine. $\square$

The obstruction is not simply the absence of strongly correlated examples. For even $m$, put $q(x)=x_1x_2+\cdots+x_{m-1}x_m$. Summing separately over pairs shows $\widehat f(y)=N^{-1/2}f(y)$. Hence \[ \Phi(f,f)=1,\qquad \Phi\left(f,\,y\mapsto f(y)(-1)^{y_1}\right)=0. \] These have maximally dense Fourier support and a large decision gap, but Lemma 35.2 evaluates both efficiently. Thus merely combining an explicit bent phase with an exact promise does not establish the desired hardness.

Where succinctness and totality intervene. Supplying full tables has input length $2N$. The fast Walsh transform computes (r35:phi) in $O(N\log N)$ arithmetic operations on integers with polynomially bounded bit lengths, so the query lower bound does not become a superpolynomial bound in that encoding. Supplying small circuits instead avoids the table-size cost, but allows classical algorithms to exploit the circuit syntax, as in Lemma 35.2.

For arbitrary circuit descriptions, the test $\Phi\ge a$ versus $\Phi\le b$ with fixed $a>b$ is a promise problem. Defining the remaining inputs to be no-instances does not make the quantum tester a total bounded-error algorithm: amplitudes can lie arbitrarily close to the new yes/no boundary. A successful construction must specify a total language and show its algorithm has error at most $1/3$ on every input, or provide a polynomial-time recognizable encoding whose valid descriptions have a proved gap and whose invalid descriptions receive deterministic answers. The oracle/promise distinction emphasized in [E3] is therefore a live issue here, not a cosmetic qualification.

The strongest plausible continuation is an explicit nonlinear family for which the gap is certified from syntax but no polynomial-hierarchy algorithm decides the associated label. To establish the target it would have to produce one language outside every fixed level, not a sequence $L_k$ with no uniform method for assembling them. No such family or lower-bound argument has been found in this attempt.

Stress test.
The elimination identity is checked in the accompanying script for all quadratic polynomials in four binary variables, against exhaustive summation. The dense examples are checked at even $m=2,4,6$. These are exact integer checks, not tests of a complexity-class separation. The exponent-sum algorithm is restricted to degree two; claiming its extension to arbitrary polynomial-size Boolean circuits would discard the essential invariant that $C+AB$ is still quadratic.

An oracle may contain information with no efficient uniform description. Replacing it by a random seed without bounding both seed length and evaluation cost is not de-oracularization. Fixing a successful seed nonconstructively may introduce advice. Neither of those steps is used here. The desired conclusion would in particular exclude $\mathsf{BQP}\subseteq\mathsf{NP}$, since $\mathsf{NP}\subseteq\mathsf{PH}$. No such unrelativized exclusion follows from either of the lemmas.

\Needspace{8\baselineskip}

Outcome.
Outcome: Exploratory approach with unresolved gap.

The attempt proves two explicit obstructions to simple replacements of the Forrelation oracle: Fourier sparsity rules out a constant signal, whereas quadratic phases allow a constant signal but admit exact polynomial-time classical evaluation. The remaining tasks are a uniform total-gap construction and a lower bound against all fixed hierarchy levels that uses more than oracle indistinguishability. Neither is supplied. The auxiliary calculations are not asserted to be historically new, and no complete separation candidate is claimed.

Sources

  • S1 Scott Aaronson, BQP and the Polynomial Hierarchy, arXiv:0910.4698, Abstract $2009$. https://arxiv.org/abs/0910.4698. Catalogue formulation source.
  • S2 BQP and the Polynomial Hierarchy. https://www.scottaaronson.com/papers/stoc114-aaronson.pdf. Catalogue research/status reference; not a proof endorsement.
  • S3 IQP circuits for 2-Forrelation. https://arxiv.org/abs/2604.15248. Catalogue research/status reference; not a proof endorsement.
  • S4 Promises should be taken seriously: On relativization with promise problems. https://arxiv.org/abs/2609.07945. Catalogue research/status reference; not a proof endorsement.
  • E1 Ran Raz and Avishay Tal, Oracle Separation of BQP and PH, ECCC TR18-107 $2018$; Journal of the ACM 69$4$ $2022$, article 30. https://eccc.weizmann.ac.il/report/2018/107/. https://doi.org/10.1145/3530258. Oracle scope and distributional statement checked 27 September 2026.
  • E2 Quentin Buzet and André Chailloux, IQP circuits for 2-Forrelation, arXiv:2604.15248v2, 24 August 2026, abstract and resource statement. https://arxiv.org/abs/2604.15248v2. Version-specific supplement to [S3]; not an unrelativized separation. Checked 27 September 2026.
  • E3 David Miloschewsky, Supartha Podder and Dorian Rudolph, Promises should be taken seriously: On relativization with promise problems, arXiv:2609.07945v1, 7 September 2026, abstract and language/promise separation statement. https://arxiv.org/abs/2609.07945v1. Recent manuscript, statement-level consultation 27 September 2026; not a complete proof audit.