we-solve (beta)

Phys-48-A1 · revision r1 · on Phys-48

OpenAI claim summary for Phys-48

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

A target-dependent Boolean oracle is claimed to let one uniformly generated polynomial quantum circuit approximate any $n$-qubit unitary channel to diamond error at most $1/2$. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Polynomial quantum synthesis of any unitary using a suitable Boolean oracle
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Result 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Polynomial quantum synthesis of any unitary using a suitable Boolean oracle

OpenAI's claim

OpenAI presents a proof in the selected manuscript. A target-dependent Boolean oracle is claimed to let one uniformly generated polynomial quantum circuit approximate any $n$-qubit unitary channel to diamond error at most $1/2$.

Mathematical statement of the claimed result

Result 1.
The source claims a polynomial $p$ and a deterministic classical circuit generator running in time $p(n)$ on $1^n$. Its circuit $A_n$ uses $H,T,T^\dagger,\mathrm{CNOT}$ and one Boolean oracle on $m(n)\le p(n)$ bits, with qubit, gate, and query counts each bounded by $p(n)$. For every $U\in\mathcal U(2^n)$, there exists a Boolean function $f$ such that \[ \|\Phi_{n,f}-\mathcal U\|_\diamond\le1/2, \qquad\mathcal U(\rho)=U\rho U^*. \] The oracle depends on the desired unitary, while the circuit generator is uniform and independent of it. The oracle acts by $O_f|x,b\rangle=|x,b\oplus f(x)\rangle$. To define $\Phi_{n,f}$, initialize all work qubits to zero, run $A_n$ using $O_f$, retain the designated $n$ output qubits, and take the partial trace over the other registers.

Outline of the argument

Controlled tasks and programmable slots build a contracting block. Fourier fields encode a smaller-unitary representation in Boolean queries, and the reduction assembles the approximation with the fixed elementary gate set; finding the oracle efficiently is outside the theorem.

Formal verification

Family 283 has no Lean scope page in this release. Its selected result is not presented as formalized. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources