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arxiv: 2606.26354 · v1 · pith:IBGDP3A5new · submitted 2026-06-24 · 🪐 quant-ph

Equivalence of non-local computation tasks beyond Clifford operations

Andreas Bluhm , Simon H\"ofer , Alex May , Florian Speelman , Philip Verduyn Lunel This is my paper

Pith reviewed 2026-06-26 01:22 UTC · model grok-4.3

classification 🪐 quant-ph
keywords non-local quantum computationreductionsposition verificationClifford operationsgate teleportationmeasurement-based quantum computationdiagonal unitariessimultaneous communication
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The pith

Redirecting a quantum system via classical control implies protocols for controlled arbitrary-basis measurements and controlled unitaries built from Cliffords plus diagonals.

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

The paper maps relationships among non-local quantum computation tasks when classical inputs are large and quantum inputs are fixed-size. It proves that a protocol for the basic redirection task can be turned into protocols for controlled single-qubit measurements in any basis, controlled Clifford gates, and controlled gates of the form C1 D C0. These reductions are obtained by adapting gate-teleportation and measurement-based quantum computation techniques to the simultaneous-communication model. The equivalences imply that many position-verification schemes based on these tasks have identical asymptotic entanglement requirements.

Core claim

Within the large-classical-input, fixed-quantum-input regime, an implementation of the simplest redirection task based on classical control yields implementations of controlled single-qubit measurements in arbitrary bases, of any controlled Clifford unitary, and of any controlled unitary of the form U = C1 D C0 where D is an arbitrary diagonal unitary and C0, C1 are Clifford circuits; the constructions use gate teleportation and measurement-based quantum computation ideas and incur no extra entanglement or communication cost in the non-local simultaneous setting.

What carries the argument

Reductions that convert a redirection protocol into the listed controlled operations by embedding gate-teleportation and MBQC circuits inside the non-local simultaneous-communication model.

If this is right

  • Position-verification schemes built from these tasks share the same asymptotic entanglement cost.
  • Relative hardness ordering among the tasks is preserved under the reductions.
  • Security levels of the corresponding position-verification protocols are equivalent up to the shared scaling.
  • Gate-teleportation and MBQC strategies become available as tools inside non-local computation.

Where Pith is reading between the lines

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

  • Security analyses for position verification can be reduced to studying only the simplest redirection task.
  • The same reduction pattern may apply to other NLQC families once the fixed-size quantum input restriction is relaxed.
  • Connections between MBQC and simultaneous-communication models could be explored in settings with multiple rounds.

Load-bearing premise

The gate-teleportation and MBQC constructions carry over to the non-local simultaneous-communication setting without extra entanglement or communication overhead.

What would settle it

An explicit non-local protocol for a controlled non-Clifford diagonal unitary whose entanglement cost is strictly lower than that of the basic redirection task, in the large-classical-input regime, would falsify the claimed implication.

Figures

Figures reproduced from arXiv: 2606.26354 by Alex May, Andreas Bluhm, Florian Speelman, Philip Verduyn Lunel, Simon H\"ofer.

Figure 1
Figure 1. Figure 1: (a) Local implementation of a channel N. (b) Non-local quantum computation. V L , V R, WL , and WR are channels and χ is a resource state. Figure reproduced from [3]. 3 Some 2 → 2 tasks 11 4 Single qubit f-measure protocols 13 4.1 Angle characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Addition of measurement angles . . . . . . . . . . . . . . . . . . . . . . . 15 4.3 Splittin… view at source ↗
Figure 2
Figure 2. Figure 2: Summary of the 2 → 2 tasks we study and the implications we prove. All implications are oracle implications. reductions and implications in section 2.3. As a partial converse to this result, [3] showed that access to a few copies of purifi￾cations of the f-measure resource state suffices to implement f-route. This does not quite satisfy the formal notion of a reduction, as we define it here. One consequenc… view at source ↗
Figure 3
Figure 3. Figure 3: An oracle reduction from an NLQC G to F. The protocol for Gn uses α instances of Fn, plus a β qubit resource system Φ. Remark 15 Consider an f-controlled 2 → 2 task with quantum input Q on the left and quantum input Q′ on the right. Consider a new task with the same inputs and outputs, but which has quantum inputs A on the left and B on the right, where HA ⊗ HB = HQ ⊗ HQ′ . Then there is always an (α, β, δ… view at source ↗
Figure 4
Figure 4. Figure 4: Circuit that implements f-measure(I, RX(θ)) with local pre-processing and oracle use of f￾measure(I, RX(2θ)) and two copies of f-measure(I, H). 4.3 Splitting the measurement angle Another way to build single qubit f-measure protocols is by reducing the rotation angle. This can be done with the help of additional oracles to f-measure(I, H) protocols. Lemma 25 For any θ, the f-measure(I, RX(θ)) protocol can … view at source ↗
Figure 5
Figure 5. Figure 5: (a) Protocol to implement f-Bell using f-measure(I, H). The blue box represents a use of an f￾measure(I, H) oracle. (b) Protocol to implement f-measure(I, H) using f-Bell. The blue box represents a use of an f-Bell oracle. on δ, θ1, θ2, since the required number of oracle calls depends on these parameters, and therefore by remark 14 the additional error incurred also depends on these parameters. Note that … view at source ↗
Figure 6
Figure 6. Figure 6: Protocol to implement f-measure(C0, C1) with C0, C1 Clifford using f-Bell as an oracle. For Cliffords on nQ qubits, nQ copies of f-Bell should be used. The curved wire denotes a maximally entangled state. Now Alice and Bob measure in the Bell basis, and output the first index i in their result |Ψij ⟩. We see that they get outcome 0 with probability |β0| 2 = | ⟨+⟩ ψ| 2 , and outcome 1 with probability |β1| … view at source ↗
Figure 7
Figure 7. Figure 7: Circuit for implementing f-unitary(C0, C1) using an f-measure oracle. The f-measure oracle, labelled B f(x,y) , measures either the first and second wires in the Bell basis (pairwise, for each of the nQ qubits in each wire), or the first and third wires in the Bell basis. This teleports the input into the selected unitary. c ⊕ b ′ |ψ⟩ C † 1 Thus if Alice and Bob compute b ′ (from b, a and given the choices… view at source ↗
Figure 8
Figure 8. Figure 8: Circuit identity we use to apply multi-qubit [PITH_FULL_IMAGE:figures/full_fig_p031_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Circuit showing the reduction from f-unitary(I, C1DC0) to oracle implementations of f￾measure(I, RX(θ)). We defined CZA[Si]Bk = ⊗j∈SiCZAjBk . We show a case where D is the product of two multi-qubit Z rotations for simplicity; the generalization to many rotations is straightforward. The full multi-qubit Z rotation gadget is shown in figure 8. The top wire denotes n qubits, to which we want to apply a class… view at source ↗
Figure 10
Figure 10. Figure 10: Properties of the reductions between non-local strategies of different (spatial) input distributions. [PITH_FULL_IMAGE:figures/full_fig_p037_10.png] view at source ↗
read the original abstract

Non-local quantum computation (NLQC) studies how two collaborating players can implement channels on distributed systems using a single simultaneous round of quantum communication and shared entanglement. NLQC has applications in diverse areas, ranging from quantum position-verification to quantum gravity. Recently, it has been realized that the relationships among families of NLQC tasks are highly structured: many seemingly distinct tasks are related by reductions, wherein implementations of one task can be used to efficiently implement a second task. This is analogous to the notion of reduction in complexity theory, and reveals the relative hardness of NLQC tasks. In this work we continue the study of reductions among NLQC tasks. We focus on NLQC examples of the greatest interest in quantum position-verification; in particular examples involving large classical inputs and fixed-size quantum inputs, since these constitute the most feasible protocols for position-verification schemes. Within this setting, we find many new relationships among NLQC tasks. For instance, protocols for the simplest example of redirecting a quantum system based on a classical control imply protocols for controlled single qubit measurements in arbitrary bases, the controlled application of any Clifford unitary, and even the controlled application of any unitary of the form $U=C_1DC_0$ with $D$ an arbitrary diagonal unitary and $C_0, C_1$ Clifford circuits. This implies that many feasible position-verification schemes have the same asymptotic scaling for their entanglement cost, and hence a similar level of security. Our techniques rely on ideas from gate teleportation and measurement based quantum computation, among other areas, bringing several new strategies into NLQC which may be of independent interest.

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 paper claims that within the NLQC model restricted to large classical inputs and fixed-size quantum inputs, a basic redirection protocol (quantum system redirected by classical control) implies protocols for controlled single-qubit measurements in arbitrary bases, controlled Clifford unitaries, and controlled unitaries of the form U = C1 D C0 (D arbitrary diagonal, C0/C1 Clifford). These reductions are constructed via adaptations of gate teleportation and MBQC ideas, implying that the corresponding position-verification schemes share the same asymptotic entanglement cost and security scaling.

Significance. If the reductions are shown to incur no extra entanglement or communication overhead, the result is significant: it establishes a web of equivalences among NLQC tasks of direct interest to position verification, unifying their resource costs. The explicit use of gate-teleportation and MBQC constructions is a strength that the paper correctly highlights as bringing new strategies to the NLQC literature; these could be of independent interest beyond the equivalences themselves.

major comments (2)
  1. [§3.2] §3.2 (MBQC adaptation for controlled Cliffords): the reduction from redirection to controlled Clifford application must explicitly verify that the standard MBQC pattern (which typically involves adaptive measurements) is realized with exactly one simultaneous round of quantum communication and no additional shared entanglement; if the adaptation introduces sequential steps or extra resources, the claimed implication for U = C1 D C0 fails.
  2. [§4] §4 (entanglement-cost equivalence): the statement that the new protocols have identical asymptotic scaling rests on the overhead being zero in the reductions; without a concrete accounting (e.g., explicit resource counts before/after each reduction), the security-scaling conclusion for position-verification schemes is not yet load-bearing.
minor comments (2)
  1. Notation for the decomposition U = C1 D C0 is introduced without an explicit small example (e.g., a single-qubit case); adding one would improve readability.
  2. [Abstract] The abstract states the reductions exist but the main text should cross-reference the specific equations or figures that contain the explicit constructions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the reductions and their implications for position verification. We address each major comment below and will revise the manuscript accordingly to strengthen the explicit verification of resource overheads.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (MBQC adaptation for controlled Cliffords): the reduction from redirection to controlled Clifford application must explicitly verify that the standard MBQC pattern (which typically involves adaptive measurements) is realized with exactly one simultaneous round of quantum communication and no additional shared entanglement; if the adaptation introduces sequential steps or extra resources, the claimed implication for U = C1 D C0 fails.

    Authors: We thank the referee for this observation. Our §3.2 construction adapts the MBQC pattern by replacing adaptive measurements with a single simultaneous round of quantum communication, where the redirection protocol supplies the classical control bits non-adaptively and gate teleportation handles the Clifford corrections without introducing sequential rounds or extra entanglement. The resulting protocol for controlled Cliffords (and hence for U = C1 D C0) therefore inherits the single-round property of the original redirection task. To address the referee's concern directly, we will add an explicit verification paragraph (with a small diagram) confirming that no additional shared entanglement or communication rounds are introduced beyond those already present in the redirection protocol. revision: yes

  2. Referee: [§4] §4 (entanglement-cost equivalence): the statement that the new protocols have identical asymptotic scaling rests on the overhead being zero in the reductions; without a concrete accounting (e.g., explicit resource counts before/after each reduction), the security-scaling conclusion for position-verification schemes is not yet load-bearing.

    Authors: We agree that an explicit resource accounting will make the asymptotic equivalence claim fully rigorous and load-bearing for the position-verification applications. In the revised manuscript we will insert a dedicated subsection (or table) that tabulates the entanglement and classical-communication resources for each task before and after the reductions. This accounting shows that every reduction incurs at most a constant-factor overhead independent of the input size, so the asymptotic entanglement scaling (and therefore the security scaling) remains identical across the family of tasks. We will also state the precise constant factors so that readers can verify the zero-asymptotic-overhead claim directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; reductions use standard independent constructions

full rationale

The paper derives relationships among NLQC tasks by adapting gate teleportation and MBQC to the non-local simultaneous-communication model. These are standard quantum-information techniques with no indication of self-definition, fitted inputs renamed as predictions, or load-bearing self-citations whose validity depends on the present work. The central claims rest on explicit constructions that map one protocol to another without reducing to the input by construction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claim rests on applicability of gate teleportation and measurement-based quantum computation constructions to the NLQC model with simultaneous communication; no free parameters or invented entities are visible from the abstract.

axioms (2)
  • domain assumption Gate teleportation and MBQC techniques extend directly to non-local simultaneous-round computation without additional resource costs.
    The abstract states that the new reductions rely on ideas from gate teleportation and measurement-based quantum computation.
  • standard math Standard quantum mechanics and the definition of NLQC tasks with classical inputs and fixed-size quantum inputs.
    Background framework assumed for all NLQC work.

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discussion (0)

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