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arxiv: 2605.16614 · v1 · pith:JIFTML4Inew · submitted 2026-05-15 · 🪐 quant-ph

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

Soumyojyoti Dutta , Tushar This is my paper

Pith reviewed 2026-05-20 18:00 UTC · model grok-4.3

classification 🪐 quant-ph
keywords magic secret sharingquantum resource theoryGHZ entanglementblind quantum computationnon-Clifford gatesthreshold schemesWigner distancedevice-independent protocols
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The pith

Quantum states carrying non-Clifford magic for universal computation can be threshold-shared so only authorized coalitions recover the exact magic content.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Magic Secret Sharing as a primitive that treats computational magic, the resource from non-stabilizer states, as the secret to be distributed. A pre-shared GHZ state plus one local phase gate creates an (n-1,n) threshold: every single party obtains the maximally mixed state with zero magic content, while the full coalition reconstructs the precise magic value needed for non-Clifford operations. This construction matters because it lets computational power itself be controlled cryptographically, supporting secure multi-server blind quantum computation without any party gaining advantage from local operations alone. The scheme further certifies delivery in a one-sided device-independent way and demonstrates the (2,3) case on an IBM processor with measured fidelities above 0.95.

Core claim

Magic Secret Sharing distributes the magic content C(phi) = (|sin(phi)| + |cos(phi)| - 1)/2 of a phase gate so that any individual party holds only the maximally mixed state I/2 with C = 0. The authorised coalition recovers this value exactly, enabling a logical T gate through gate teleportation. Among diagonal parametric gates, only phase gates meet the column-sum security condition. The protocol lifts to a one-sided device-independent setting by steering inequalities that certify magic delivery without trusting the coalition's devices.

What carries the argument

Pre-shared GHZ entanglement plus a local phase gate P(phi) that enforces the column-sum condition, distributing magic content so individuals receive zero while the coalition receives the full C(phi) value.

If this is right

  • Authorised coalitions can implement logical T gates via gate teleportation inside multi-server blind quantum computation.
  • The protocol extends to one-sided device-independent verification of magic delivery through steering inequalities.
  • Phase gates are the only diagonal parametric family that satisfy the exact security condition required for the threshold property.
  • Experimental runs on a 156-qubit processor confirm zero magic for individuals and state fidelities of 0.959-0.986 for the authorised party.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Threshold control over magic could be combined with other quantum resources to create layered access policies for distributed quantum algorithms.
  • The same GHZ-plus-phase construction might certify computational advantage in networks where parties must prove they possess non-Clifford power without revealing the full state.
  • Testing whether the scheme tolerates realistic noise levels on larger coalitions would show how far the zero-magic guarantee for individuals survives hardware imperfections.

Load-bearing premise

Any single party is left with exactly the maximally mixed state whose magic content is zero.

What would settle it

Measurement on any individual party's reduced state yielding a Wigner distance C greater than the reconstruction tolerance, or the coalition failing to recover the predicted numerical value of C(phi) for a chosen phase angle.

Figures

Figures reproduced from arXiv: 2605.16614 by Soumyojyoti Dutta, Tushar.

Figure 1
Figure 1. Figure 1: (2, 3) MSS circuit. Alice (q0) prepares a GHZ state (H + two CX gates), injects magic via P(φ), and measures X, broadcasting mA. Bob (q1) measures X and broadcasts mB. Charlie (q2) applies Z mB to obtain P(φ)|+⟩ with C = C(φ). Theorem 3 ((2, 3) threshold). Protocol 1 achieves: (i) Faithfulness: C(ρC) = C(φ). (ii) Security: ρB = I/2, so C(ρB) = 0. (iii) Key indistinguishability: ρB(φ1) = ρB(φ2) = I/2 ∀ vali… view at source ↗
Figure 2
Figure 2. Figure 2: Standard MSS on ibm marrakesh (4096 shots). (a) Measured C(ρC) (scatter, 1σ bootstrap error bars) against the theory curve C(φ) = (|sin φ| + | cos φ| − 1)/2 (line). All four φ values are faithful to within 0.034. (b) C(ρB) < 10−6 in every case — noise￾robust security from the I/2 invariance argument. BQC the relevant threat is unilateral computational universality, not information leakage; MSS is designed … view at source ↗
read the original 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) = 0.000, below LP reconstruction tolerance) holds in all runs, and state fidelity reaches 0.959-0.986 for the authorised party, with faithfulness confirmed for all four test values of phi including near-exact recovery (C = 0.154 vs theory 0.153) for phi = pi/8.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces Magic Secret Sharing (MSS), a quantum cryptographic primitive distributing computational magic (non-Clifford resource) via pre-shared GHZ states and a single local phase gate P(φ). It achieves an (n-1,n) threshold where any individual party receives the maximally mixed state I/2 with magic content C(I/2)=0, while the authorized coalition reconstructs exactly C(φ)=(|sin(φ)|+|cos(φ)|-1)/2. Phase gates are characterized as unique among diagonal parametric gates via an exact column-sum security condition. The protocol is lifted to a one-sided device-independent setting using a steering inequality, and the (2,3) instance is demonstrated experimentally on ibm_marrakesh with reported fidelities 0.959-0.986 and security (C=0.000) holding across runs.

Significance. If the central claims hold, the work meaningfully connects the resource theory of magic with threshold secret sharing and blind quantum computation, enabling distributed control of non-Clifford power. The experimental validation on superconducting hardware, the 1SDI certification, and the uniqueness characterization via column-sum condition are concrete strengths. The free parameter φ and the exact reconstruction for the authorized set add to the result's interest.

major comments (2)
  1. [Security derivation] § on security derivation (partial-trace step): the load-bearing claim that every single-party reduced state is exactly I/2 independent of φ must be shown explicitly. The partial trace over the GHZ state after one local P(φ) must be computed to confirm the resulting 2×2 matrix has vanishing off-diagonal elements and no residual φ dependence; any φ-dependent coherence would permit local magic extraction and collapse the (n-1,n) threshold.
  2. [Reconstruction claim] Reconstruction section: the assertion that the authorized coalition obtains exactly C(φ) requires the explicit sequence of operations or measurements performed by the coalition and the precise definition of how the Wigner distance is recovered from the reconstructed state.
minor comments (2)
  1. [Magic measure definition] Clarify the precise relation between the functional form C(φ) and the underlying Wigner-function definition of magic; the expression (|sin φ| + |cos φ| - 1)/2 should be derived from the resource-theoretic measure rather than introduced directly.
  2. [Experimental results] In the experimental section, specify the number of shots, any error-mitigation techniques, and how the reported C values (e.g., 0.154 vs. theory 0.153 for φ=π/8) are extracted from tomography or direct measurement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight important points for rigor, and we address each one below, indicating the revisions we will incorporate.

read point-by-point responses

  1. Referee: [Security derivation] § on security derivation (partial-trace step): the load-bearing claim that every single-party reduced state is exactly I/2 independent of φ must be shown explicitly. The partial trace over the GHZ state after one local P(φ) must be computed to confirm the resulting 2×2 matrix has vanishing off-diagonal elements and no residual φ dependence; any φ-dependent coherence would permit local magic extraction and collapse the (n-1,n) threshold.

    Authors: We agree that an explicit partial-trace calculation strengthens the security proof. In the revised manuscript we will add the following derivation in the security section. The initial GHZ state is |GHZ⟩ = (|0⟩⊗n + |1⟩⊗n)/√2. After applying P(φ) = diag(1, e^{iφ}) to the first qubit the state becomes (|0⟩|0⟩⊗(n−1) + e^{iφ}|1⟩|1⟩⊗(n−1))/√2. The reduced density operator for the first qubit is obtained by tracing out qubits 2 through n. The off-diagonal elements involve inner products ⟨0⊗(n−1)|1⊗(n−1)⟩ and ⟨1⊗(n−1)|0⊗(n−1)⟩, both of which vanish by orthogonality of the computational-basis states. Consequently the reduced state is exactly I/2 with no φ dependence, confirming that C(I/2) = 0 and that the (n−1,n) threshold is preserved. revision: yes

  2. Referee: [Reconstruction claim] Reconstruction section: the assertion that the authorized coalition obtains exactly C(φ) requires the explicit sequence of operations or measurements performed by the coalition and the precise definition of how the Wigner distance is recovered from the reconstructed state.

    Authors: We accept that the reconstruction procedure should be stated more explicitly. In the revised manuscript we will expand the reconstruction section to include the following steps: (i) the authorized parties perform a joint parity measurement on their GHZ shares (equivalent to a logical X measurement on the encoded qubit), (ii) they apply local phase corrections conditioned on the classical communication of the measurement outcome, thereby teleporting the action of P(φ) onto a target logical qubit, and (iii) the resulting logical state ρ is used to evaluate the magic monotone C(ρ) = (|sin φ| + |cos φ| − 1)/2 directly via its definition as the Wigner-distance measure introduced in the paper. This sequence recovers exactly the claimed magic content for the authorized set. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via explicit construction and partial-trace verification.

full rationale

The protocol constructs the (n-1,n) threshold explicitly from a shared GHZ state plus one local phase gate P(phi). Security follows from the direct partial-trace calculation that each single-party reduced density matrix equals I/2 independently of phi, which is a standard linear-algebra fact rather than a redefinition. The authorized coalition's reconstruction of the state (and its magic content) is likewise obtained by applying the inverse operations to the joint system; the explicit functional form C(phi) = (|sin(phi)| + |cos(phi)| - 1)/2 is the evaluated result of the chosen magic monotone on that reconstructed state, not an input that is then re-predicted. The column-sum uniqueness condition is derived from the I/2 requirement and then checked against the family of diagonal gates, constituting an independent characterization rather than a self-referential loop. No fitted parameters, self-citations, or ansatzes are load-bearing for the central claims.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The protocol rests on the resource theory of magic, the properties of GHZ states, and the characterization that phase gates alone satisfy the column-sum security condition among diagonal gates. The phase angle phi is a controllable parameter rather than a fitted constant.

free parameters (1)
  • phi
    Phase angle of the local gate P(phi) that sets the target magic content; chosen by the protocol designer.
axioms (1)
  • domain assumption Phase gates are the unique class of diagonal parametric gates satisfying the security condition via an exact column-sum condition.
    Invoked to establish that only phase gates meet the requirement that individual parties receive zero-magic states.
invented entities (1)
  • Magic Secret Sharing (MSS) primitive no independent evidence
    purpose: Threshold distribution of computational magic rather than classical or quantum data.
    New cryptographic object defined in the paper; no independent evidence outside this work is provided.

pith-pipeline@v0.9.0 · 5856 in / 1537 out tokens · 144625 ms · 2026-05-20T18:00:29.955672+00:00 · methodology

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Reference graph

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