One-way state generation from private Hamiltonian phase states

Problem

Do private-architecture Hamiltonian phase states yield a one-way state generator for explicit polynomial parameters?

For uniformly random \(A\in\mathbb F_2^{m\times n}\) and independent uniform phases \(\theta_i\in\{2\pi j/q:0\leq j<q\}\), define

\begin{equation} |\phi_{A,\theta}\rangle =\exp\!\left(i\sum_{i=1}^m\theta_iZ^{A_i}\right)|+\rangle^{\otimes n}, \qquad Z^{A_i}=\bigotimes_{j=1}^n Z^{A_{ij}}. \tag{1} \end{equation}

Do explicit polynomially bounded \(m(n)\) and growing \(q(n)\) exist such that, for every polynomial \(t\) and every quantum polynomial-time inverter \(\mathcal I\),

\begin{equation} \mathbb E\!\left[ \left|\langle\phi_{A',\theta'}|\phi_{A,\theta}\rangle\right|^2 \right]\leq\operatorname{negl}(n), \qquad (A',\theta')\leftarrow\mathcal I(1^n,|\phi_{A,\theta}\rangle^{\otimes t(n)})? \tag{2} \end{equation}

Here \(\operatorname{negl}(n)<n^{-a}\) eventually for every constant \(a>0\), and the architecture \(A\) in Eq. (1) is hidden. In Eq. (2), the inverter must output a valid description in the same parameter space; invalid outputs fail verification.

Source

This precise formulation is editor wording based on the unresolved direction and limitations documented in the cited primary literature [Bostanci25]; it is not presented as a verbatim conjecture of those authors.

Progress

Reports do not certify correctness or automatically change the problem's status. Progress policy.

Status: an unresolved quantum cryptographic hardness assumption. The originating work gives restricted worst-to-average-case reductions and bounded-copy design evidence, but not a proof of general search hardness. It also explains that polynomially many copies suffice information-theoretically: the conjectured obstacle is computation, not an absence of information. [Bostanci25], Sections 5.1–5.3
Later work develops measurement-assisted shallow preparation of these states and analyzes statistical properties. These are advances in realizing the ensemble, not proofs that polynomial-time quantum inversion is impossible. [Cao26]
The cited work leaves both general inversion and security unresolved. The original paper separately proposes decision HPS, concerning indistinguishability from Haar states; that different security task is not identified here with search HPS. [Bostanci25], Sections 4.2
Reported progress: PR #82.

Comment

This is the closest match to a quantum-information version of a one-way function: a short classical description prepares a state, but copies allegedly do not permit efficient reverse engineering.

The fidelity verifier matters. Recovering a different description of almost the same state is a successful attack; merely showing that the original labels are nonunique is not evidence of security. Conversely, tomography with an exponentially expensive reconstruction stage does not refute a polynomial-time hardness claim.

Scope caution: the security parameter regime is part of the research problem. The statement does not endorse an arbitrary choice such as \(m=n\), nor a version in which the architecture is public. The cited results establish partial evidence rather than security for an explicit general parameter regime.

References

[Bostanci25]: John Bostanci, Jonas Haferkamp, Dominik Hangleiter, and Alexander Poremba, Efficient Quantum Pseudorandomness from Hamiltonian Phase States. TQC 2025, LIPIcs 350, article 9. Pinpoints refer to the arXiv full text: §4.1, §4.2, and §§5.1–5.3.link link
[Cao26]: Chenfeng Cao and Jens Eisert, Measurement-Driven Quantum Advantages in Shallow Circuits. Physical Review Letters 136, 080601 (2026). See “Measurement-prepared Hamiltonian phase states” and the discussion.link

One-way state generation from private Hamiltonian phase states

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