Equivalence of non-local computation tasks beyond Clifford operations

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.

• The paper establishes that non-trivial single-qubit f-controlled measurement tasks are equivalent via O(1) oracle reductions to f-routing, unifying their entanglement requirements.
• Detailed circuit analyses, gate teleportation, and MBQC techniques are applied to reduce controlled Clifford, non-Clifford, and diagonal unitary protocols to a common baseline.
• The results imply that enhancing quantum control with fixed-size quantum inputs does not increase QPV security, as all protocols exhibit similar subexponential entanglement vulnerabilities.

Equivalence Structures in Non-local Quantum Computation Beyond Cliffords

Overview and Motivation

The paper "Equivalence of non-local computation tasks beyond Clifford operations" (2606.26354) presents a systematic study of reductions among families of non-local quantum computation (NLQC) tasks, particularly targeting scenarios relevant for quantum position verification (QPV) where large classical inputs and fixed-size quantum inputs are most practical. The approach is analogous to computational complexity theory, mapping NLQC tasks under reductions that indicate their relative hardness and resource demands, specifically in terms of shared entanglement.

The authors establish a web of formal equivalences and reductions among NLQC paradigms, encompassing classical control, single-qubit measurements (including non-Clifford bases), and even NLQC protocols implementing unitaries beyond Clifford group circuits. Notably, they demonstrate that many tasks, apparently more complex or secure than classical-controlled SWAP ($f$-routing), are in fact equivalent or reducible in oracle terms to it, meaning attainable with similar asymptotic entanglement costs and thus similar security against adversaries in QPV contexts.

Formal Definitions and Reduction Constructs

NLQC tasks are defined as bipartite channels on distributed systems, required to be performed under the constraint of a single simultaneous round of quantum communication and pre-shared entanglement. For QPV, the archetype task is $f$-routing, where classical inputs $x$ (Alice), $y$ (Bob), and a quantum system $Q$ are manipulated—the goal being to transfer $Q$ to Bob or retain it with Alice per the value of a Boolean function $f(x, y)$.

The paper employs two notions of reduction:

• Resource state reduction: Implementation of task $G$ using a small number of copies of any NLQC resource state allowing task $F$ to be performed, plus a small auxiliary system.
• Oracle reduction: Implementation of task $G$ via black-box accesses to NLQC protocols for $f$0 (possibly in parallel), plus additional small resources.

The latter is stronger; all reductions in the paper belong to this class, implying both equivalence in entanglement complexity and in abstract protocol simulation.

Principal Equivalence Results: Single-Qubit and Clifford NLQC

A central technical achievement is the proof that all non-trivial single-qubit $f$1-controlled measurement NLQCs are equivalent under $f$2 oracle reductions to $f$3-measure$f$4 (computational vs. Hadamard)—and thereby to $f$5-routing. This encompasses tasks defined by arbitrary projective single-qubit measurements and extends to arbitrary pairs of bases, not limited to Clifford-related ones.

Detailed circuit analysis and decomposition demonstrates that for any $f$6-measure$f$7 task (for single-qubit unitaries $f$8), one can construct a reduction involving $f$9 calls to $x$0-measure$x$1, utilizing primitive operations (rotation splitting, angle addition, basis transformations) and repeated composition, all with bounded resource overheads. Conversely, $x$2-measure$x$3 can be simulated with any such $x$4-measure$x$5 protocol, given modest repetition and aggregation. The reduction overhead is independent of the classical input size $x$6.

This result implies that all plausible single-qubit QPV schemes leveraging $x$7-controlled measurements have identical asymptotic security properties (subexponential attacks), establishing a strong negative claim on the prospect of improving security via basis or measurement generalization in this regime.

The analysis further extends to multi-qubit tasks:

• Controlled Clifford measurements ($x$8-measure in bases related by Clifford operators) and controlled Clifford unitaries ($x$9-unitary between Clifford circuits) are reduced to $y$0-measure$y$1 via gate teleportation and measurement-based strategies.
• Controlled application of unitaries of the form $y$2, with $y$3 diagonal and $y$4 Cliffords (potentially high $y$5-depth), is shown, using measurement-based circuits and non-adaptive graph-state MBQC, to be oracle-reducible in $y$6 copies to $y$7-measure$y$8.

The technical machinery includes decomposition of diagonal unitaries into products of multi-qubit $y$9 rotations, controlled injection protocols, and utilization of Clifford commutation identities. The reductions are explicit, constructive, and utilize techniques from circuit identities, gate teleportation, and port-based teleportation.

Implications for Quantum Position Verification

A practical upshot is that feasible QPV schemes based on controlled single-qubit measurements, controlled Clifford gates, or even the broader class $Q$0 diagonal unitaries, do not escape the entanglement scaling (the best attacks require subexponential, not exponential, entanglement) known for classical-controlled SWAP ($Q$1-routing) [allerstorfer2024relating]. Upper bounds from garden-hose model, span program analysis, and circuit decompositions remain applicable. Thus, attempts to improve QPV protocol security via increased quantum control, provided the classical control structure is maintained and quantum inputs are $Q$2 size, are infeasible with present reductions.

This conclusion is corroborated by reductions propagating both upper bounds and known subexponential attacks. Theoretical lower bounds remain elusive even for $Q$3-routing, and no super-linear lower bounds are expected given ties to formula size and span program complexity.

Theoretical and Constructive Advances

The reduction framework provides a structured lens through which families of NLQC tasks can be classified according to their entanglement hardness. The introduction of explicit reduction protocols, using gate teleportation and MBQC-like gadgets, is constructive and generalizable, bridging areas of cryptography (CDS), communication complexity, and measurement-based computation. The formal equivalence strengthens previously informal connections between $Q$4-routing, $Q$5-measure, and other cryptographically-motivated primitives [bluhm2025complexity, allerstorfer2024relating].

The main conjecture left open is the status of NLQC tasks controlled by more general non-Clifford unitaries—notably whether layering diagonal gates can push security higher, or whether residual reductions apply.

Speculation on Future Developments

Given the equivalence established, future investigations may target:

• NLQC tasks involving more general input-output configurations or adaptive measurement structures, potentially evading reduction to $Q$6-route.
• Composite protocols layering diagonal unitaries or employing post-selection, perhaps with increased entanglement complexity.
• Robust lower bounds enhancement via novel complexity measures outside span programs or formula size.
• Applications in quantum cryptography, quantum gravity, and communication complexity leveraging reduced NLQC hardness, possibly for protocol design or attack construction.

Extensions to multi-party NLQC, multi-round communication, and hybrid classical-quantum control could also change the reduction landscape, necessitating new complexity-theoretic machinery.

Conclusion

This paper rigorously demonstrates that a wide array of NLQC tasks, including controlled single-qubit measurements (in arbitrary bases), controlled Clifford operations, and a substantial class of classically-controlled non-Clifford gates, are equivalent in terms of NLQC resource demands and oracle reducibility. The reductions leverage explicit circuit constructions and MBQC-inspired strategies, and firmly establish that increasing quantum control or measurement complexity does not yield enhanced security in quantum position verification within the considered paradigm. The results provide both theoretical clarity and practical guidance for protocol design, foregrounding the structural landscape of NLQC hardness beyond Clifford operations (2606.26354).