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arXiv:2605.16614

Magic Secret Sharing: Threshold Control of Quantum Computational Power via GHZ Entanglement

Soumyojyoti Dutta, Tushar

quant-phSubmitted May 15, 2026• Announced May 19, 2026• 0 citations• Revised Oct 6, 2026 (v5)
v1 · May 15, 2026v2 · Aug 3, 2026v3 · Sep 21, 2026v4 · Sep 23, 2026v5 · Oct 6, 2026
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Abstract

We introduce Magic Secret Sharing (MSS), a quantum cryptographic primitive in which the secret is the computational capability of a quantum state rather than its classical description. In the resource theory of magic, non-stabilizer states fuel universal quantum computation via non-Clifford gates; MSS distributes this resource with an (n-1,n) threshold structure using a pre-shared GHZ state and a single local phase gate P(phi) = diag(1, exp(i*phi)). Any individual party holds the maximally mixed state I/2, with Wigner distance C(I/2) = 0, so no local operation can yield non-Clifford computational advantage regardless of what operations are applied or what noise acts on the device. The authorised coalition reconstructs magic content C(phi) = (|sin(phi)| + |cos(phi)| - 1)/2 exactly, enabling a logical T gate via gate teleportation in multi-server blind quantum computation (BQC). Among diagonal parametric gates, phase gates are the unique class satisfying the security condition, characterised via an exact column-sum condition. The protocol is elevated to a one-sided device-independent (1SDI) setting via a steering inequality: the assemblage produced on the recipient's side certifies magic delivery without trusting the coalition's devices. We demonstrate the (2,3) instance on ibm_marrakesh (156-qubit IBM Heron): security (C(rho_Bob) < 10^-6, the linear-programming solver tolerance) holds in every run and is independently reproduced on a second qubit assignment, and state fidelity reaches 0.959-0.973 for the authorised party, with faithfulness confirmed for all four test values of phi to within 0.025 in magic content, a residual that depolarising noise alone accounts for.

Comments
8 pages, 2 figures, 1 table. corrected threshold structure to (n,n); the one-sided DI functional is reinterpreted as a magic witness rather than a steering test; T-axis fidelities (0.937-0.959 vs 0.8585 threshold) reported without the >=8 sigma claim; coalition security stated as two-tier. Under review at Phys. Rev. A

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We introduce Magic Secret Sharing (MSS), a quantum cryptographic primitive in which the secret is the computational capability of a quantum state rather than its classical description. In the resource theory of magic, non-stabilizer states fuel universal quantum computation via non-Clifford gates; MSS distributes this resource with an (n-1,n) threshold structure using a pre-shared GHZ state and a single local phase gate P(phi) = diag(1, exp(i*phi)). Any individual party holds the maximally mixed state I/2, with Wigner distance C(I/2) = 0, so no local operation can yield non-Clifford computational advantage regardless of what operations are applied or what noise acts on the device. The authorised coalition reconstructs magic content C(phi) = (|sin(phi)| + |cos(phi)| - 1)/2 exactly, enabling a logical T gate via gate teleportation in multi-server blind quantum computation (BQC). Among diagonal parametric gates, phase gates are the unique class satisfying the security condition, characterised via an exact column-sum condition. The protocol is elevated to a one-sided device-independent (1SDI) setting via a steering inequality: the assemblage produced on the recipient's side certifies magic delivery without trusting the coalition's devices. We demonstrate the (2,3) instance on ibm_marrakesh (156-qubit IBM Heron): security (C(rho_Bob) < 10^-6, the linear-programming solver tolerance) holds in every run and is independently reproduced on a second qubit assignment, and state fidelity reaches 0.959-0.973 for the authorised party, with faithfulness confirmed for all four test values of phi to within 0.025 in magic content, a residual that depolarising noise alone accounts for.
Nonstabilizerness, or magic, is an archetypal \emph{quantum} resource that is necessary for quantum computational advantage. Here we uncover a phenomenon seemingly at odds with the quantum nature of magic: entirely nonlocal magic (ENM)---magic present only in correlations and absent from each party's marginal---can live without entanglement. We systematically study this separation and show it is universal and operationally reversible: every magical state or channel can be encoded into and recovered from a separable ENM realization using only local stabilizer processing and classical communication. We leverage this mechanism to devise an activation key protocol in which a classical key controls access to non-Clifford operations. We further formulate magic secret sharing, in which computational power inaccessible to any party alone becomes accessible through cooperation. On a superconducting quantum processor, we experimentally demonstrate activation key and network computing primitives, together with separable ENM state preparation and extraction protocols. Together, our results establish that magic can be classically activated, localized, and secret-shared without entanglement, providing new resource-control primitives for distributed quantum computation.
We initiate the study of quantum leakage resilience of unmodified Shamir secret sharing over prime fields. A well-studied leakage model for Shamir's secret sharing classically is single-bit local leakage from each share. We consider its quantum analogue where, for each party, a local leakage channel takes as input the party's share and outputs a leaked qubit. Without preshared entanglement, we show that the distinguishing advantage is 2−Ω(n)2^{-Ω(n)}2−Ω(n) when the threshold rate t/n=τt/n=τt/n=τ exceeds τ⋆≈0.73339τ_\star\approx0.73339τ⋆​≈0.73339 by a fixed positive margin. More generally, we allow disjoint entangled blocks of any fixed maximum size where there is no entanglement between different blocks or with the adversary, and each block emits at most a fixed number of qubits. Security holds when the threshold rate is high enough (sufficiently close to one). We then allow a specified set of devices to share entanglement with the adversary. We show that security holds even when a linear number of devices (αnαnαn for small α>0α>0α>0) share entanglement with each other and with the adversary for a large enough threshold rate. As a complementary negative result, we also show that even classical single-bit leakage makes Shamir scheme insecure if we allow arbitrarily large entanglement between the leakage devices. A GHZ state shared by exactly ttt leakage devices makes even classical one-bit leakage insecure, without any entanglement with the adversary. In this attack, each participating device emits only one classical bit, and their joint parity distinguishes any chosen pair of secrets with a constant advantage. Thus, for fixed threshold rates above τ⋆τ_\starτ⋆​, the maximum number of devices that may share arbitrary entanglement with one another and with the adversary while preserving security is linear in nnn up to constant factors, although the optimal support fraction remains open.
Quantum secret sharing is a cryptographic scheme that enables the secure storage and reconstruction of quantum information. While the theory of secret sharing is mature in its development, relatively few studies have explored the performance of quantum secret sharing on actual devices. In this work, we provide a pedagogical description of encoding and decoding circuits for different secret sharing codes, and we test their performance on IBM’s 127-qubit Brisbane system. We evaluate the quality of the implementation by performing a SWAP test between the decoded state and the ideal one, as well as by estimating how well the code preserves entanglement with a reference system. The results indicate that a ((3,5)) threshold secret sharing scheme and a non-threshold 7-qubit scheme perform similarly based on the SWAP test and entanglement fidelity, with both attaining a roughly 70–75% pass rate on the SWAP test for the reconstructed secret. We also investigate one implementation of a ((2,3)) qutrit threshold scheme and find that it performs the worst of all, which is expected due to the additional number of multi-qubit gate operations needed to encode and decode qutrits. A comparison is also made between schemes using mid-circuit measurement versus delayed-circuit measurement.
Quantum gate teleportation is a key technique in fault-tolerant quantum computation that uses resource states to implement logical gates. Here, we develop a theory of magic gate teleportation (MGT), a form of quantum gate teleportation that uses non-stabilizer resource states to implement non-Clifford unitary gates on arbitrary input states without revealing information about those states. This setting is motivated by the canonical approach to implementing logical non-Clifford gates in fault-tolerant quantum computation. We uncover a hidden structure within MGT---after backpropagating the Pauli measurements, MGT protocols can be viewed as encoding the input state into a stabilizer code heralded by the measurement outcomes, followed by a logical non-Clifford gate. Using this structure, we construct MGT protocols for any resource state obtained by applying commuting Pauli rotations to a stabilizer state, and provide an efficient algorithm for synthesizing their circuit implementations. Conversely, we prove that directly useful single-qubit resource states are necessarily Clifford-equivalent to diagonal states, and establish the same constraint for multi-qubit resource states used in a broad class of MGT protocols. In particular, the output state distilled from the [ ⁣[5,1,3] ⁣][\![5, 1, 3]\!][[5,1,3]] protocol is not directly useful for any MGT protocols, though it can be probabilistically converted into directly useful states. Finally, we identify conditions under which the feedforward operators can be implemented by Pauli operators, shedding light on the paradigm of algorithmic fault tolerance and simplifying the feedforward operations needed for quantum computing.
We give an example of a wide class of problems for which quantum-information protocols based on multisystem entanglement can be mapped into much simpler ones involving one system. Secret sharing is a cryptographic primitive which plays a central role in various secure multiparty computation tasks and management of keys in cryptography. In secret sharing protocols, a classical message is divided into shares given to recipient parties in such a way that some number of parties need to collaborate in order to reconstruct the message. Quantum protocols for the task commonly rely on multipartite GHZ entanglement. We present a multiparty secret sharing protocol which requires only sequential communication of a single quantum ddd-level system (for any prime ddd). It has huge advantages in scalability and can be realized with state-of-the-art technology.
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 nnn shares, one per candidate node. The selected node, given the key qudits, completes an exact non-local CSUMk\mathtt{CSUM}^{k}CSUMk gate with the user's data, in every measurement branch and for any qudit dimension, with the power kkk chosen by the user alone. Any coalition of fewer than nnn 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))((3,5))((3,5)) threshold scheme. Encoder and decoder are explicit O(n)O(n)O(n) circuits. The construction supports jointly authorized and anonymous gate application, advancing distributed quantum computing and quantum network capabilities.