Multi-slot overhead of virtual channel conjugation

Problem

What is the optimal quasiprobability overhead of implementing the complex conjugate of an unknown quantum channel from $n$ queries? Let $\mathcal N:\mathcal L(A)\to\mathcal L(B)$ be an unknown channel with $d_A:=\dim A$ and $d_B:=\dim B$, and fix orthonormal bases of $A$ and $B$. The complex conjugate of $\mathcal N$ is the channel

\begin{equation} \mathcal N^{*}(X):=\overline{\mathcal N(\overline X)}, \tag{1} \end{equation}

where the bar is entrywise complex conjugation in the fixed bases, so the Choi operator of $\mathcal N^{*}$ is the entrywise conjugate of that of $\mathcal N$. An $n$-slot quantum comb is a physically realizable circuit with $n$ open slots, each receiving one use of the unknown channel, whose overall action is again a channel from $A$ to $B$; write $\mathrm{Comb}_n$ for the set of such combs. An $n$-slot virtual comb is a real linear combination $\widetilde{\mathcal C}=\sum_i c_i\mathcal C_i$ with $\mathcal C_i\in\mathrm{Comb}_n$, and its base norm

\begin{equation} \|\widetilde{\mathcal C}\|_{\mathrm{base}} :=\min\Bigl\{\sum_i|c_i|: \widetilde{\mathcal C}=\sum_ic_i\mathcal C_i,\ c_i\in\mathbb R,\ \mathcal C_i\in\mathrm{Comb}_n\Bigr\} \tag{2} \end{equation}

is the sampling overhead: estimating an expectation value of the output of $\widetilde{\mathcal C}$ to additive error $\varepsilon$ by Monte Carlo sampling of the $\mathcal C_i$ costs $O(\|\widetilde{\mathcal C}\|_{\mathrm{base}}^{2}\varepsilon^{-2})$ runs. Define the optimal $n$-query overhead of universal conjugation by

\begin{equation} g_n(d_A,d_B) :=\inf\Bigl\{\|\widetilde{\mathcal C}\|_{\mathrm{base}}: \widetilde{\mathcal C}\text{ is an $n$-slot virtual comb with } \widetilde{\mathcal C}(\mathcal N^{\otimes n})=\mathcal N^{*} \text{ for every channel }\mathcal N\Bigr\}. \tag{3} \end{equation}

Since the extra slots may be discarded, $g_n\leq g_1$. Determine $g_n(d_A,d_B)$ in Eq. (3) for $n\geq2$: is $g_n(d_A,d_B)<g_1(d_A,d_B)$ for some $n$, and what is $\inf_{n}g_n(d_A,d_B)$?

Source

The question is implicit in Zhu, Tang, Zhen, Li, Bai, and Wang, who determine $g_1$ exactly and name multi-slot virtual protocols as future work [ZTZ+26].

Progress

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

No completely positive supermap using any finite number of queries implements $\mathcal N^{*}$ of Eq. (1) for every channel $\mathcal N$, and the same obstruction rules out a universal physical implementation of the adjoint $\mathcal N^{\dagger}$; the transpose, by contrast, admits a probabilistic single-query implementation. Hence any universal conjugation must be virtual, with an overhead of the form Eq. (2) [ZTZ+26].
A one-slot virtual comb implements complex conjugation, and its base norm is optimal among one-slot protocols:

\begin{equation} g_1(d_A,d_B)=d_Ad_B-d_A+1. \tag{4} \end{equation}

The optimality proof in Eq. (4) uses semidefinite duality for a single slot and does not extend to correlated multi-slot strategies [ZTZ+26].
The state-preparation case $d_A=1$ admits a multi-slot improvement. For $d=d_B\geq2$ and $n\geq d-1$, Theorem 3, Eqs. (14)–(15), of Brzić, Grinko, Studziński, and Quintino supplies CPTP maps $C_\eta$ on $n$ copies such that $C_\eta(\rho^{\otimes n})=\eta\rho^T+(1-\eta)I/d$ for every density operator $\rho$, for both $a=n/[n+d(d-1)]$ and $b=-1/(d-1)$ [BGSQ26]. Consequently, the virtual map

\begin{equation} \widetilde C=\frac{1-b}{a-b}C_a+\frac{a-1}{a-b}C_b \quad\text{satisfies}\quad \widetilde C(\rho^{\otimes n})=\rho^T. \tag{5} \end{equation}

These maps are valid combs for state-preparation slots. Summing the absolute coefficients in Eq. (5) gives

\begin{equation} 1\leq g_n(1,d)\leq1+\frac{2(d-1)^2}{n+d-1}, \qquad \inf_{n\geq1}g_n(1,d)=1. \tag{6} \end{equation}

The lower bound in Eq. (6) follows from trace preservation, which forces the coefficients of every exact decomposition to sum to one. Its upper bound is strictly below $g_1(1,d)=d$ when $n>d-1$; in particular $g_2(1,2)\leq5/3<2$. The theorem determines the physical white-noise visibility range; this argument supplies an upper bound on the virtual overhead, not a proof of its exact finite-$n$ optimum.
Composing virtual conjugation with the probabilistic transpose gives black-box access to $\mathcal N^{\dagger}$ and, for a unital channel, estimates expectation values of the Petz recovery map to error $\varepsilon$ with failure probability at most $\delta$ from $O(d_A^{3}d_B^{3}\varepsilon^{-2}\log(1/\delta))$ samples of the channel, so the value of $g_n$ directly controls the cost of such applications [ZTZ+26].

Comment

State preparation is included in the question. In that subcase, Eq. (6) proves a strict multi-slot advantage and determines the infimum over the number of queries. The exact finite-$n$ optima for state preparation and the general-channel values with $d_A>1$ remain unresolved by these results. One-slot optimality alone does not settle correlated multi-slot protocols. The bound on the base norm measures variance per run; each $n$-slot run itself consumes $n$ channel queries.

References

[ZTZ+26]: C. Zhu, Z. Tang, G. Zhen, Y. Li, G. Bai, and X. Wang, “Simulation of Adjoints and Petz Recovery Maps for Unknown Quantum Channels,” arXiv preprint (2026).arXiv
[BGSQ26]: V. Brzić, D. Grinko, M. Studziński, and M. T. Quintino, “Optimal pure state cloning and transposition are complementary channels,” arXiv preprint, version 2 (2026).arXiv

Multi-slot overhead of virtual channel conjugation

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