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Phys-48 · physics · mathematics · Physics

Polynomial quantum synthesis of any unitary using a suitable Boolean oracle

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Is there one classically polynomial-time uniform family $A_n$ of quantum oracle circuits over H,T,T-dagger,CNOT, with polynomially many qubits, gates, calls and oracle input bits, such that for every n-qubit unitary U some Boolean oracle f makes its output channel within diamond distance 1/2 of conjugation by U? The oracle may depend on U but the circuit template may only depend on n.

Short English statement

Is there one classically polynomial-time uniform family $A_n$ of quantum oracle circuits over H,T,T-dagger,CNOT, with polynomially many qubits, gates, calls and oracle input bits, such that for every n-qubit unitary U some Boolean oracle f makes its output channel within diamond distance 1/2 of conjugation by U? The oracle may depend on U but the circuit template may only depend on n.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Phys-48-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Phys-48

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.

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

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