⚛️ quantum physics

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

This paper presents a protocol for secret-sharing the capability to apply a two-qudit entangling gate among multiple parties, enabling a single recipient to be selected after distribution to perform an exact non-local operation while ensuring security against unauthorized coalitions and preserving recipient anonymity.

Original authors: Tanay Roy

Published 2026-11-02

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tanay Roy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of quantum physics, information behaves in ways that defy our everyday intuition. Unlike a photograph or a digital file, a piece of quantum information cannot be copied or stored for later use without being destroyed or altered. This rule, known as the no-cloning theorem, extends to the actions we perform on that information as well. If a scientist has a specific quantum operation—a kind of mathematical instruction that changes one quantum state into another—they cannot simply save a copy of that instruction to run again later. They can, however, embed a specific operation into a shared resource of entangled particles, a phenomenon where two particles remain linked across vast distances. By measuring one part of this pair and sending a classical signal, the operation can be "teleported" to a distant location. This is the foundation of a future quantum internet, allowing computers in different places to work together without physically moving their delicate data. Yet, a fundamental limitation has always existed: when this entangled resource is created, the destination of the operation is fixed. Whoever holds the necessary half of the entangled pair holds the power to perform the gate, and that power cannot be hidden or shared among a group until the moment of use.

A researcher at Fermi National Accelerator Laboratory has now demonstrated a way to break this limitation. They have devised a method to split the ability to perform a specific quantum gate among a group of people, such that the final user is not chosen until after the resources have already been distributed. Imagine a vault containing a single key that can unlock a door. In the traditional model, the key is handed to a specific person, and that person is the only one who can open the door. In this new scheme, the key is shattered into pieces and distributed to a group of candidates. None of the pieces work on their own, and no one knows who will eventually get to use the door. Later, a trusted provider releases a small set of instructions to just one person in that group. Only that chosen individual, holding both their piece of the shattered key and the new instructions, can reassemble the power to open the door. The others remain unaware that a choice was ever made, and the door remains locked for everyone else.

The researcher achieved this by applying a technique called encrypted cloning to the entangled resource itself, rather than to the data being processed. In their protocol, a provider creates a special entangled pair of particles and keeps one half while giving the other half to a user who holds the data. The provider then takes their half and splits it into many "shares," sending one share to each candidate node in a network. These shares are scrambled in such a way that they look like random noise; holding a single share, or even a group of shares, reveals absolutely nothing about the original entanglement or the data it is linked to. The provider retains a separate set of "key" particles. At a later time, when a decision is made to use the gate, the provider sends the key particles to exactly one of the candidates. This recipient can then combine their share with the keys to perfectly reconstruct the original entangled link. Once this link is restored, the recipient and the user can perform a non-local quantum gate, an operation that links their two separate pieces of data together, without ever having met or exchanged the data directly.

The study proves that this process works with perfect precision for any size of quantum system, not just the simplest two-level systems. The researcher showed that the reconstruction is exact, meaning the gate is applied with perfect fidelity, and that the privacy is absolute. If a group of candidates tries to pool their shares without the key, they learn nothing. Even if they possess the key but not the shares, they cannot do anything. Crucially, the unselected candidates cannot tell whether a selection has occurred; their view of the world remains completely unchanged, preserving the anonymity of the chosen user. Furthermore, the key can be used only once. Once a recipient has reconstructed the entanglement to perform the gate, the key is effectively spent. If a second person tries to use the same key, they will fail to reconstruct the link, obtaining only a random guess rather than the correct operation.

This work introduces a new level of flexibility to distributed quantum computing. It allows for a form of joint authorization where a gate can be distributed to many potential users, but only one can exercise it, and that choice can be made after the distribution is complete. The researcher also demonstrated that the power of the gate itself can be chosen by the user at the very last moment, without revealing that choice to the provider or the other candidates. The mathematical construction relies on specific properties of high-dimensional quantum systems, which the researcher utilized to ensure the protocol works for any dimension of the quantum particles involved. The circuits required to encode and decode this information are efficient, scaling linearly with the number of candidates, making the approach practical for current and near-future quantum hardware. By enabling anonymous, deferred, and jointly authorized quantum operations, this method advances the capabilities of quantum networks, offering a secure way to manage access to quantum resources in a future where quantum computers are connected across the globe.

Technical Summary: Secret-Shared Teleportation of a Two-Qudit Gate with a Deferred Recipient

Problem Statement
Quantum information cannot be copied, and unknown quantum operations cannot be stored for later execution. While specific gates can be embedded in entangled resources and teleported, traditional gate teleportation protocols bind the resource to a fixed destination at the moment of creation. This limitation prevents the "deferral" of the recipient: a provider cannot distribute a gate capability to multiple candidates and later select exactly one to exercise it without revealing the choice or requiring all candidates to participate. Existing schemes for sharing quantum operations (e.g., controlled remote implementation) typically fix the recipient in advance or require all controllers to measure and announce outcomes, allowing any single controller to veto the operation. The paper addresses the challenge of secret-sharing the capability to apply a non-local entangling gate such that:

The recipient is chosen only after the resource is distributed.
Unselected parties learn nothing about the data or the selection event.
The gate is applied exactly, in every measurement branch, for any qudit dimension $d$ .

Methodology
The protocol utilizes encrypted cloning applied not to data, but to one half of a gate-teleportation resource. It operates in three stages involving a provider, a user (Alice), and $n$ candidate nodes:

Resource Distribution (Stage 1):
- A provider prepares a maximally entangled pair $|\Phi\rangle_{B_A B_C}$ , giving $B_A$ to Alice and retaining $B_C$ .
- The provider prepares $n$ Bell pairs $|\Phi\rangle_{S_j K_j}$ .
- An encoder circuit $U_{enc}$ is applied to $B_C$ and the signal qudits $S_1, \dots, S_n$ . This circuit uses a ladder of controlled-sum (CSUM) gates, a quadratic phase gate $P_\theta$ , and Fourier transforms.
- The signal qudits $S_j$ are sent to $n$ candidate sites, while the key qudits $K_1, \dots, K_n$ are retained by the provider.
- Crucially, the phase angle $\theta$ must be a "flat angle" $\theta^*(d)$ (specifically $2\pi/d$ for odd $d$ and $\pi/d$ for even $d$ ) to ensure exact recovery.
Key Redemption (Stage 2):
- At a later time, the provider selects a specific node (Bob) and releases the key register $(K_1, \dots, K_n)$ to him via an authenticated quantum channel.
- Bob applies a decoder circuit $U_{dec}$ to his signal qudit $S_B$ and the key register.
- This decoder, consisting of $O(n)$ gates, recovers the original entanglement between $B_A$ and $B_C$ , effectively teleporting the entanglement to the pair $(B_A, S_B)$ with unit fidelity. All other shares remain maximally mixed and uncorrelated with the data.
Gate Completion (Stage 3):
- Alice and Bob perform local operations and classical communication to complete a non-local gate.
- Alice applies $CSUM^{k_1}(D_A \to B_A)$ and measures $B_A$ . Bob applies $CSUB^{k_2}(S_B \to D_B)$ and measures $S_B$ in the Fourier basis.
- After exchanging measurement outcomes, they apply Weyl corrections. The protocol implements an exact $CSUM^k$ gate where $k \equiv k_1 k_2 \pmod d$ . The user (Alice) can choose the power $k$ locally, and this choice remains hidden from the provider and candidates.

Key Contributions and Results

Threshold Secret Sharing of a Gate: The protocol establishes a quantum secret-sharing scheme where the "secret" is the entanglement resource required to teleport a gate. For $n=2$ candidates, the five distributed registers (the original pair plus two signal-key pairs) form a ((3, 5)) threshold scheme: any three registers can recover the secret, while any two hold no information.
Exactness and Universality: The protocol works for any qudit dimension $d \ge 2$ and any number of candidates $n \ge 2$ . It guarantees exact gate application in every measurement branch (no post-selection) and works for inputs entangled with arbitrary external systems.
Anonymity and Privacy:
- Unselected Nodes: Any coalition of fewer than $n$ candidates, or the key register alone, holds a state that is maximally mixed and uncorrelated with the user's data. Unselected nodes cannot distinguish whether a selection occurred or which node was selected.
- Single-Use: The key register is single-use. Once Bob decodes, the remaining shares are re-paired with their keys and lose all correlation with Alice's data. A second attempt to decode with the spent key yields only random guess fidelity ( $1/d^2$ ).
Deferred Recipient and Anonymous Power: The recipient is chosen after distribution. Furthermore, the power $k$ of the applied gate is chosen by the user after distribution and is not revealed in the transmitted registers or public messages, providing a form of operation hiding.
Explicit Circuits: The paper provides explicit $O(n)$ gate circuits for both the encoder and decoder, independent of the Hilbert space dimension $d^2(n+1)$ .
Noise Resilience: The process fidelity under depolarizing noise is analyzed. The fidelity scales linearly with the noise probability $p$ on the redeemed share and the key register, dropping from 1 to $1/d^2$ at $p=1$ . Noise on unselected shares has no effect on the gate.

Significance and Claims
The paper claims to advance distributed quantum computing and quantum network capabilities by enabling joint authorization, anonymity, and early distribution of gate capabilities.

Novelty: The construction is novel even for qubits ( $d=2$ ). For $d > 2$ , it relies on specific features of the qudit Clifford group arithmetic, particularly the necessity of "flat angles" for the quadratic-phase encoder and sign bookkeeping in the teleportation stage to support arbitrary powers of the gate.
Relation to Existing Work: The protocol is framed as "encrypted cloning of a gate" rather than a state. It extends the concept of teleporting unitary operations by allowing the destination and timing to be decided post-creation.
Practicality: The resource requirements are modest ( $2n+4$ qudits, $O(n)$ gates), making the scheme feasible for current qudit processors and photonic platforms.
Limitations: The protocol assumes semi-honest adversaries (who follow the protocol but may collude) and ideal operations for the encoder/decoder and measurements. It does not cover active attacks on the key or classical channels, nor does it currently extend to non-Clifford gates or fault-tolerant encodings of the full register.

The work provides a closed-form derivation of the access structure and proves that the scheme functions as a threshold secret-sharing scheme for a non-local two-qudit gate, offering a new primitive for secure, distributed quantum operations.

Technical Summary: Secret-Shared Teleportation of a Two-Qudit Gate with a Deferred Recipient

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