Secret-shared teleportation of a two-qudit gate with a deferred recipient

Abstract

An unknown quantum operation, like an unknown quantum state, cannot be perfectly copied or stored for later use. A specific gate can be embedded in an entangled resource and teleported, but its destination is fixed when the resource is created. Here we show that the capability to apply a two-qudit entangling gate can instead be secret-shared among many parties: the shares are distributed in advance, and a later key release lets exactly one of them, chosen after distribution, exercise it. A provider encrypts half of a Bell pair held with a user into $n$ shares, one per candidate node. The selected node, given the key qudits, completes an exact non-local $\mathtt{CSUM}^{k}$ gate with the user's data, in every measurement branch and for any qudit dimension, with the power $k$ chosen by the user alone. Any coalition of fewer than $n$ candidate nodes, or the key register alone, holds no information, unselected nodes cannot tell whether a selection occurred, and the key cannot decrypt twice. For two candidates the registers form a $((3,5))$ threshold scheme. Encoder and decoder are explicit $O(n)$ circuits. The construction supports jointly authorized and anonymous gate application, advancing distributed quantum computing and quantum network capabilities.

Can the ability to perform a nonlocal entangling gate be distributed before its recipient is chosen, without requiring unselected nodes to participate? The paper composes qudit encrypted cloning with gate teleportation. A provider encodes half of a Bell pair shared with Alice into $n$ candidate shares and retains $n$ quantum-key qudits. Releasing the keys to one candidate lets that node recover Alice’s Bell partner and implement $\mathtt{CSUM}^{k}$ on their respective data qudits, with $k$ chosen locally by Alice. For every $d\ge2$ and $n\ge2$, the ideal protocol is exact on arbitrary reference-entangled inputs: each measurement branch has probability $1/d^2$ and yields the same corrected gate. Every proper coalition of candidate shares, and the key register alone, carries no information about the shared resource. Decoding removes the resource correlations needed for another redemption. The encoder and decoder use $4n$ and $3n$ two-qudit gates, respectively; gate completion exchanges two classical dits. Weyl expansions, quadratic Gauss sums and stabilizer counting establish recovery, phase-angle conditions and the access structure, with numerical cross-checks. For $n=2$, the five-register scheme is $((3,5))$ threshold when $\gcd(2r^2-1,d)=1$, where the encoding angle is $\pi r/d$. The guarantees assume trusted preparation, semi-honest candidates and authenticated quantum-key delivery and classical messages. Selection requires transmitting $n$ key qudits. Anonymity covers unselected nodes’ registers and transcript values, not communication metadata. The construction covers a Clifford-gate family; its depolarizing-noise analysis assumes ideal gates and measurements, not fault tolerance.