Optimal universal eight-copy concentration to a CCZ state
- Field
- Topics
Problem
What is the optimal success probability for a fixed catalyst-free stabilizer protocol that converts eight copies of an arbitrary unknown pure qubit into an exact \(|CCZ\rangle\) state?
Let \(\mathcal U_8\) be the fixed protocols built from stabilizer ancillas, Clifford unitaries, Pauli measurements, classical feedforward, and discarding, chosen independently of the input, whose accepted branches map \(\psi^{\otimes8}\) exactly to
Acceptance may be zero on stabilizer inputs, and no magic catalyst is allowed. For a pure-state Bloch vector \((x,y,z)\), define \(p_8^*(\psi)=\sup_{\Lambda\in\mathcal U_8}p_\Lambda(\psi)\) and \(m_3(\psi)=(1-x^6-y^6-z^6)/2\). The proposed optimum is
Does Eq. (2) hold for every pure qubit \(\psi\), with the target state fixed by Eq. (1)?
Source
The question is explicitly posed or retained as open in the cited primary literature [Rizzo26]. The statement is rewritten here to make its hypotheses and success criterion self-contained.
Progress
Rizzo and Leone establish
\begin{equation} \frac23m_3(\psi)\leq p_8^*(\psi)\leq\frac76m_3(\psi) \tag{3} \end{equation}and explicitly leave eight-copy optimality open [Rizzo26]. Equality with the lower bound is a candidate, not a published theorem or asserted consensus conjecture.
The displayed definitions, constraints, and target bounds are recorded in Eqs. (3).
Status: open in the August 2026 preprint; no subsequent resolution identified.
Comment
This finite-copy question offers a bounded input space, but “universal” is essential: optimizing a circuit for one known input is a different task. Its short history warrants more caution about community prominence than longer-established magic-state problems.
Here stabilizer protocols are built from stabilizer ancillas, Clifford unitaries, Pauli measurements, classical feedforward, and discarding; they are not arbitrary stabilizer-preserving channels [Zurel26][Fang26].