Optimal asymptotic distillation yield of dephased T states

Problem

What is the optimal catalyst-free asymptotic \(T\)-state yield of a dephased single-qubit \(T\) state?

Let

\begin{equation} |T\rangle=\frac{|0\rangle+e^{i\pi/4}|1\rangle}{\sqrt2}, \qquad \tau=|T\rangle\langle T|, \qquad \rho_p=(1-p)\tau+pZ\tau Z. \tag{1} \end{equation}

For \(0\leq p\leq1\), define the catalyst-free stabilizer-operation yield

\begin{equation} D_T(\rho_p)=\sup\left\{R: \exists\ \Lambda_n,\ \frac12\left\|\Lambda_n(\rho_p^{\otimes n}) -\tau^{\otimes\lfloor Rn\rfloor}\right\|_1\longrightarrow0\right\}, \tag{2} \end{equation}

where \(\Lambda_n\) uses stabilizer ancillas, Clifford unitaries, Pauli measurements, classical feedforward, and discarding, but no magic catalyst. Determine the quantity in Eq. (2), with matching achievable and converse bounds, already for a fixed noise level such as \(p=0.01\) in Eq. (1).

Source

This precise formulation is editor wording based on the unresolved direction and limitations documented in the cited primary literature [Wills25][Rubboli24]; it is not presented as a verbatim conjecture of those authors.

Progress

Constant-overhead existence is solved. Wills, Hsieh, and Yamasaki constructed protocols with positive asymptotic yield for sufficiently good inputs; the paper appeared online in Nature Physics on September 16, 2025. The open issue here is the largest yield for a specified input, not whether overhead exponent zero is possible [Wills25]. A fixed preliminary purification stage extends such constructions to inputs within an established distillation basin.
A computable converse is available. Let

\begin{equation} q=\cos^2(\pi/8)=\frac{1+1/\sqrt2}{2}. \tag{3} \end{equation}

The displayed definitions, constraints, and target bounds are recorded in Eqs. (3).
The stabilizer boundary on this axis occurs at \(p_*=1-q\). Additivity and conversion bounds for relative entropy of magic give, for \(0\leq p<p_*\),

\begin{equation} D_T(\rho_p)\leq \frac{(1-p)\log_2\frac{1-p}{q} +p\log_2\frac{p}{1-q}} {-\log_2q}. \tag{4} \end{equation}

The displayed definitions, constraints, and target bounds are recorded in Eqs. (4).
This specializes the results of Rubboli, Takagi, and Tomamichel to the commuting optimizer \(q\tau+(1-q)Z\tau Z\) [Rubboli24] (Sections 4, 6 and 8). Direct substitution yields

\begin{equation} D_T(\rho_{0.01})\leq0.757659\ldots. \tag{5} \end{equation}

The displayed definitions, constraints, and target bounds are recorded in Eqs. (5).
This number is a converse bound, not an achieved yield or a conjectured equality.
May 28, 2026. Ehara and Takagi construct protocols with output counts arbitrarily close to linear in a sublinear sense, including \(n^{1-\eta}\) for every fixed \(\eta>0\). Their result concerns the relation between rate scaling and overhead exponents; it does not identify the optimal constant multiplying \(n\) [Ehara26].
Catalysis is a different task. Fang and Liu’s work, updated September 2, 2026, can reduce one-shot input count to one using a catalyst and reduced success probability. It neither supplies the missing catalyst-free optimal yield nor justifies ignoring failed attempts in rate accounting [Fang26] (Theorem 7).
August 10, 2026. New binary-extension-field code constructions target practical multi-qubit magic-state protocols; they do not establish a matching optimum for the quantity above [Gong26].
Status: open quantitative optimization, despite the solved scaling-exponent problem.

Comment

This problem addresses an operationally meaningful constant that overhead exponents conceal. Two protocols can both have constant overhead yet consume very different numbers of noisy states for a given output volume.

A manageable starting point is to tighten either side for one fixed rational \(p\), with every auxiliary magic state and failed attempt included in the accounting. An apparent formula derived under all stabilizer-preserving maps, nonzero-probability postselection, or borrowed magic should not be substituted for the catalyst-free, vanishing-global-error definition.

The operation class is stabilizer operations rather than all stabilizer-preserving channels; this distinction matters for conversion bounds [Zurel26].

References

[Zurel26]: M. Zurel, S. Jana, and N. de Silva, High-threshold magic state distillation with quantum quadratic residue codes, March 19, 2026. Introduction explicitly states the open conjectures; Sections 4.2–4.3 give the new constructions.link
[Fang26]: K. Fang and Z.-W. Liu, One-shot distillation with constant overhead using catalysts, Nature Communications 17, 6010 (2026); September 2, 2026. See Definitions 1–2 and Theorem 7.link
[Wills25]: A. Wills, M.-H. Hsieh, and H. Yamasaki, Constant-Overhead Magic State Distillation, Nature Physics 21, 1842–1846 (2025); The extended arXiv version explicitly asks for optimal overhead and asymptotically optimal protocols in Open Question 4.link
[Rubboli24]: R. Rubboli, R. Takagi, and M. Tomamichel, Mixed-state additivity properties of magic monotones based on quantum relative entropies for single-qubit states and beyond, Quantum 8, 1492 (2024);link
[Ehara26]: K. Ehara and R. Takagi, Asymptotic magic state distillation with almost linear rate, May 28, 2026. See Equation (1), Theorem 1, and the success-probability analysis.link
[Gong26]: A. Gong, C. A. Pattison, P. Rall, and A. Wills, Magic State Distillation via Codes over Binary Extension Fields, August 10, 2026.arXiv

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