Non-local Quantum Computation (NLQC)

• Non-local quantum computation (NLQC) is the process of implementing joint quantum operations on separated systems using local operations, pre-shared entanglement, and a single round of communication.
• Canonical protocols such as teleport*-and-correct and port-based teleportation demonstrate NLQC’s methodology by managing errors via the diamond norm and optimizing entanglement use.
• NLQC’s insights into entanglement trade-offs, circuit structure sensitivity, and cryptographic applications underscore its significance for secure and scalable distributed quantum computing.

Non-local quantum computation (NLQC) is the task of implementing a joint quantum operation on spatially separated systems without physically bringing them together, using local quantum operations, pre-shared entanglement, and a single simultaneous round of communication. In the modern one-round formulation, correctness is typically measured in diamond norm, and the primary resource is the entanglement cost needed to replace a direct interaction (May, 4 May 2026). A closely related operational paradigm is instantaneous nonlocal quantum computation (INQC), in which spacelike separated parties perform local measurements and later broadcast outcomes so that local corrections reconstruct the target operation while preserving causality; within this setting, local operations and broadcast communication (LOBC) is the non-interactive analogue of LOCC (Gonzales et al., 2018).

1. Operational model and variants

A general one-round NLQC protocol can be written as a bipartite channel of the form

$$\mathcal{N}_{AB\to A'B'}=(\mathcal{W}^L_{K_a M_a\to A'}\otimes \mathcal{W}^R_{K_b M_b\to B'})\circ (\mathcal{V}^L_{AL\to K_a M_b}\otimes \mathcal{V}^R_{RB\to M_a K_b})\circ (I_{AB}\otimes \mathcal{S}_{\emptyset\to LR}),$$

where $\mathcal{S}_{\emptyset\to LR}$ prepares the shared resource $\Psi_{LR}$, $\mathcal{V}^L,\mathcal{V}^R$ generate the simultaneous messages, and $\mathcal{W}^L,\mathcal{W}^R$ complete the computation after the single exchange (Bluhm et al., 24 Jun 2026). In the equivalent formalism emphasized in the book-length treatment of the subject, a one-round protocol is an NLQC for a target channel $M$ when $\|N-M\|_\diamond\le \epsilon$, and the entanglement cost is the minimum number of ebits sufficient for such an $\epsilon$-correct implementation (May, 4 May 2026).

INQC is the classical-communication-only specialization in which the quantum part of the protocol is completed before communication, and only a single later exchange of classical outcomes is allowed. For a target unitary $U$, the INQC objective is

$$|\psi\rangle\otimes|\eta\rangle \to |\phi_m\rangle \overset{LU(m)}{=} U|\psi\rangle \qquad \forall m,$$

where the local correction $\mathcal{S}_{\emptyset\to LR}$0 depends only on the broadcast transcript $\mathcal{S}_{\emptyset\to LR}$1 (Gonzales et al., 2018). This model is causality-compatible because the pre-correction states cannot be used for superluminal signaling, and it is strictly less powerful than general LOCC because it forbids interactive adaptivity (Gonzales et al., 2018).

Several task families recur throughout the literature. General channel simulation and bipartite unitary implementation are the most direct formulations (May, 4 May 2026). Classically controlled tasks include $\mathcal{S}_{\emptyset\to LR}$2-route, where a quantum register is routed according to a Boolean function $\mathcal{S}_{\emptyset\to LR}$3, and $\mathcal{S}_{\emptyset\to LR}$4-measure, where the measurement basis depends on $\mathcal{S}_{\emptyset\to LR}$5 (Bluhm et al., 29 May 2025). Coherently controlled variants include $\mathcal{S}_{\emptyset\to LR}$6-PHASE and $\mathcal{S}_{\emptyset\to LR}$7-$\mathcal{S}_{\emptyset\to LR}$8, where the global unitary applied to target systems is controlled by a distributed classical predicate (Bluhm et al., 29 May 2025).

2. Canonical constructions and upper bounds

The most basic one-round construction is teleport$\mathcal{S}_{\emptyset\to LR}$9-and-correct: Bob teleport$\Psi_{LR}$0s his input to Alice, Alice applies the joint operation, and Pauli-frame information is resolved after the simultaneous classical exchange. Because Clifford operations map Pauli operators to Pauli operators, any $\Psi_{LR}$1-qubit Clifford unitary admits a one-round NLQC with $\Psi_{LR}$2 ebits (May, 4 May 2026).

For arbitrary channels, the canonical black-box construction is port-based teleportation (PBT). In the standard PBT-based upper bound, the teleportation channel satisfies

$\Psi_{LR}$3

and therefore any bipartite quantum channel on $\Psi_{LR}$4 qubits admits a one-round PBT-based NLQC with $\Psi_{LR}$5, yielding an exponential entanglement upper bound in $\Psi_{LR}$6 (May, 4 May 2026). Earlier PBT-based INQC constructions already reduced Vaidman-style doubly-exponential entanglement overhead to singly-exponential overhead (Beigi et al., 2011). Later work gave the first efficient quantum algorithm for the PBT pretty-good measurement, with local complexity polynomial in the number of ports and the port dimension, and used this to reduce a previously known triple-exponential gap in the entanglement–complexity relationship to a double-exponential gap (Fei et al., 2023).

A distinct line of work replaces black-box dependence on system size by dependence on circuit structure. For Clifford+$\Psi_{LR}$7 circuits of $\Psi_{LR}$8-depth $\Psi_{LR}$9, instantaneous non-local computation can be achieved with

$\mathcal{V}^L,\mathcal{V}^R$0

and for $\mathcal{V}^L,\mathcal{V}^R$1-count $\mathcal{V}^L,\mathcal{V}^R$2 there is a protocol using $\mathcal{V}^L,\mathcal{V}^R$3 EPR pairs (Speelman, 2015). When the circuit has small past light cones, the cost can be made to scale with the non-locality of the unitary rather than with $\mathcal{V}^L,\mathcal{V}^R$4 alone: the entanglement cost is $\mathcal{V}^L,\mathcal{V}^R$5, where $\mathcal{V}^L,\mathcal{V}^R$6 is the maximum volume of a past light cone in a circuit implementing the unitary (Dolev et al., 2022). A multipartite extension with restricted Clifford light cones gives

$\mathcal{V}^L,\mathcal{V}^R$7

for $\mathcal{V}^L,\mathcal{V}^R$8 parties, $\mathcal{V}^L,\mathcal{V}^R$9-depth $\mathcal{W}^L,\mathcal{W}^R$0, total qubit count $\mathcal{W}^L,\mathcal{W}^R$1, and backward-light-cone size $\mathcal{W}^L,\mathcal{W}^R$2 (Girish et al., 10 Jun 2026).

For two-qubit unitaries, substantially sharper bounds are known in the LOBC model. Any two-qubit unitary can be implemented under LOBC with success probability $\mathcal{W}^L,\mathcal{W}^R$3 using $\mathcal{W}^L,\mathcal{W}^R$4 ebits, and consequently with diamond-norm error $\mathcal{W}^L,\mathcal{W}^R$5 using at most

$\mathcal{W}^L,\mathcal{W}^R$6

ebits, an exponential improvement over the previously known $\mathcal{W}^L,\mathcal{W}^R$7 behavior for two-qubit protocols (Gonzales et al., 2018). The same paper also shows that any Hermitian controlled gate $\mathcal{W}^L,\mathcal{W}^R$8 with $\mathcal{W}^L,\mathcal{W}^R$9 can be implemented by LOBC using a single shared ebit (Gonzales et al., 2018).

Setting	Representative upper bound	Source
Arbitrary bipartite channel via PBT	$M$0, $M$1	(May, 4 May 2026)
Clifford unitary	$M$2 ebits	(May, 4 May 2026)
Clifford+$M$3, $M$4-depth $M$5	$M$6 ebits	(Speelman, 2015)
Small-light-cone circuit	$M$7 ebits	(Dolev et al., 2022)
Arbitrary two-qubit LOBC	$M$8 ebits	(Gonzales et al., 2018)

These constructions establish a recurrent theme: general-purpose NLQC remains expensive, but the resource can collapse from exponential-in-$M$9 to polynomial or quasi-polynomial when the target operation has special algebraic, Clifford, $\|N-M\|_\diamond\le \epsilon$0-depth, or light-cone structure.

3. Lower bounds and entanglement cost

Lower bounds in NLQC have historically been more difficult than upper bounds. A basic early result proved that a linear number of ebits is necessary for a certain instantaneous non-local measurement, and used mutually unbiased bases to show that constant-accuracy implementations require $\|N-M\|_\diamond\le \epsilon$1 entanglement (Beigi et al., 2011). The later literature develops several more systematic lower-bound paradigms.

For LOBC gate implementation, a foundational result concerns controlled unitaries on $\|N-M\|_\diamond\le \epsilon$2. For

$\|N-M\|_\diamond\le \epsilon$3

with all phases $\|N-M\|_\diamond\le \epsilon$4 distinct, any LOBC implementation requires at least $\|N-M\|_\diamond\le \epsilon$5 ebits; equivalently, for a pure resource $\|N-M\|_\diamond\le \epsilon$6,

$\|N-M\|_\diamond\le \epsilon$7

This yields an unbounded LOBC–LOCC separation, since interactive LOCC implements the same gate with two ebits by teleporting Alice’s qubit to Bob and back (Gonzales et al., 2018). That work emphasizes that the bound is on entanglement entropy, not merely on local dimension, and identifies it as the first such entropy-based lower bound for instantaneous nonlocal gate implementation (Gonzales et al., 2018).

A more general 2026 framework introduces two unitary-dependent lower-bound quantities: controllable correlation (CC) and controllable entanglement (CE). If a unitary $\|N-M\|_\diamond\le \epsilon$8 has $\|N-M\|_\diamond\le \epsilon$9-controllable correlation, then any protocol implementing $\epsilon$0 within diamond error $\epsilon$1 using resource $\epsilon$2 obeys

$\epsilon$3

where

$\epsilon$4

If $\epsilon$5 has $\epsilon$6-controllable entanglement, then for small enough $\epsilon$7,

$\epsilon$8

These bounds apply even when the single communication round is quantum rather than classical, are additive under parallel repetition in the stated regimes, and give a tight lower bound $\epsilon$9 for CNOT (Cleve et al., 30 Jan 2026).

Representative gate-level consequences are already nontrivial. For Haar-random two-qubit unitaries, the CC lower bound was reported as almost always nonzero over 100,000 samples, with mean $U$0; the paper also gives lower bounds for DCNOT, $U$1, iSWAP, $U$2, Sycamore, ECR, CS, and CT (Cleve et al., 30 Jan 2026). The book-length survey synthesizes these and related paradigms, including monogamy-of-entanglement bounds for BB84 measurement tasks and the observation that SWAP still lacks a precise one-round entanglement characterization (May, 4 May 2026).

For classically controlled one-round tasks, perfect-correctness lower bounds can be expressed through matrix rank. For $U$3-routing, the log Schmidt rank cost obeys

$U$4

and for perfect $U$5-BB84,

$U$6

where $U$7 is constrained to vanish exactly on one side of the Boolean support (Asadi et al., 2024). This yields linear-in-$U$8 lower bounds on the log Schmidt rank for natural functions such as Equality, Greater-Than, and Set-Disjointness (Asadi et al., 2024).

Setting	Representative lower bound	Source
Generic controlled gate on $U$9	$$|\psi\rangle\otimes|\eta\rangle \to |\phi_m\rangle \overset{LU(m)}{=} U|\psi\rangle \qquad \forall m,$$0 ebits	(Gonzales et al., 2018)
CNOT	$$|\psi\rangle\otimes|\eta\rangle \to |\phi_m\rangle \overset{LU(m)}{=} U|\psi\rangle \qquad \forall m,$$1, tight	(Cleve et al., 30 Jan 2026)
$$|\psi\rangle\otimes|\eta\rangle \to |\phi_m\rangle \overset{LU(m)}{=} U|\psi\rangle \qquad \forall m,$$2-routing, perfect correctness	$$|\psi\rangle\otimes|\eta\rangle \to |\phi_m\rangle \overset{LU(m)}{=} U|\psi\rangle \qquad \forall m,$$3-type lower bound	(Asadi et al., 2024)
Certain instantaneous non-local measurements	$$|\psi\rangle\otimes|\eta\rangle \to |\phi_m\rangle \overset{LU(m)}{=} U|\psi\rangle \qquad \forall m,$$4 ebits	(Beigi et al., 2011)

These lower bounds indicate that interaction can be traded for entanglement, but not freely: in several settings, removing interaction forces entanglement costs that are provably logarithmic, linear, or task-dependent through rank, mutual information, or entanglement generation.

4. Task families, reductions, and an emerging complexity theory

A major recent development is the treatment of NLQC task families through reductions. In this perspective, one studies the relative hardness of tasks rather than attempting an immediate complete characterization of entanglement cost. The central 2025 result is that the two principal classically controlled tasks, $$|\psi\rangle\otimes|\eta\rangle \to |\phi_m\rangle \overset{LU(m)}{=} U|\psi\rangle \qquad \forall m,$$5-route and $$|\psi\rangle\otimes|\eta\rangle \to |\phi_m\rangle \overset{LU(m)}{=} U|\psi\rangle \qquad \forall m,$$6-measure, are equivalent under $$|\psi\rangle\otimes|\eta\rangle \to |\phi_m\rangle \overset{LU(m)}{=} U|\psi\rangle \qquad \forall m,$$7-overhead reductions; more precisely, $$|\psi\rangle\otimes|\eta\rangle \to |\phi_m\rangle \overset{LU(m)}{=} U|\psi\rangle \qquad \forall m,$$8-route, $$|\psi\rangle\otimes|\eta\rangle \to |\phi_m\rangle \overset{LU(m)}{=} U|\psi\rangle \qquad \forall m,$$9-measure, and CDQS are all interconvertible with constant-factor overhead in entanglement cost (Bluhm et al., 29 May 2025). This transports upper bounds, lower bounds, amplification results, and cryptographic consequences across the three models.

An earlier line of work connected these routing and measurement tasks to information-theoretic cryptography. In particular, $\mathcal{S}_{\emptyset\to LR}$00-routing is equivalent, up to small overhead, to the quantum analogue of conditional disclosure of secrets, while coherent function evaluation induces efficient private simultaneous message protocols (Allerstorfer et al., 2023). One concrete consequence is a worst-case subexponential upper bound

$\mathcal{S}_{\emptyset\to LR}$01

for the entanglement cost of $\mathcal{S}_{\emptyset\to LR}$02-routing, which then transfers to equivalent classically controlled NLQC tasks (Allerstorfer et al., 2023). The 2025 reduction theory makes the same subexponential bound immediately available for $\mathcal{S}_{\emptyset\to LR}$03-measure for every Boolean $\mathcal{S}_{\emptyset\to LR}$04, and also imports efficient protocols for functions in $\mathcal{S}_{\emptyset\to LR}$05 (Bluhm et al., 29 May 2025).

Beyond the classically controlled regime, recent work establishes a web of reductions “beyond Clifford operations.” In the setting of large classical inputs $\mathcal{S}_{\emptyset\to LR}$06 and fixed-size quantum inputs, $\mathcal{S}_{\emptyset\to LR}$07-route reduces to controlled single-qubit measurements; all non-trivial single-qubit $\mathcal{S}_{\emptyset\to LR}$08-measure tasks are equivalent under $\mathcal{S}_{\emptyset\to LR}$09 oracle reductions; $\mathcal{S}_{\emptyset\to LR}$10-measure$\mathcal{S}_{\emptyset\to LR}$11 is equivalent to $\mathcal{S}_{\emptyset\to LR}$12-Bell; $\mathcal{S}_{\emptyset\to LR}$13-Bell yields controlled Clifford measurements and controlled Clifford unitaries; and $\mathcal{S}_{\emptyset\to LR}$14-measure$\mathcal{S}_{\emptyset\to LR}$15 reduces to controlled application of any unitary of the form

$\mathcal{S}_{\emptyset\to LR}$16

with $\mathcal{S}_{\emptyset\to LR}$17 Clifford and $\mathcal{S}_{\emptyset\to LR}$18 diagonal (Bluhm et al., 24 Jun 2026). The diagonal component is synthesized through phase gadgets and non-adaptive MBQC-style constructions, showing that many position-verification-motivated tasks have the same asymptotic entanglement cost (Bluhm et al., 24 Jun 2026).

The same reductionist viewpoint now extends into private simultaneous message passing. In the quantum PSM setting, new lower bounds use privacy rather than correctness alone: Nečiporuk’s measure lower-bounds the entanglement required for $\mathcal{S}_{\emptyset\to LR}$19-player quantum PSM with perfect correctness, and the rank of the communication matrix lower-bounds two-player quantum PSM with perfect privacy and imperfect correctness (Girish et al., 10 Jun 2026). On the upper-bound side, if $\mathcal{S}_{\emptyset\to LR}$20 is computed by a quantum circuit of size $\mathcal{S}_{\emptyset\to LR}$21, depth $\mathcal{S}_{\emptyset\to LR}$22, and $\mathcal{S}_{\emptyset\to LR}$23 players each hold $\mathcal{S}_{\emptyset\to LR}$24 bits, then

$\mathcal{S}_{\emptyset\to LR}$25

again reinforcing the role of circuit structure in one-round non-local resource costs (Girish et al., 10 Jun 2026).

5. Cryptography, communication complexity, and holography

NLQC is the standard cheating paradigm for quantum position verification (QPV). Already in 2011, PBT-based INQC showed that one-round position-verification schemes can be broken with entanglement exponential in the number of communicated qubits, improving over earlier doubly-exponential attacks (Beigi et al., 2011). Later work connected this to $\mathcal{S}_{\emptyset\to LR}$26-routing and related cryptographic primitives, giving the first subexponential upper bound $\mathcal{S}_{\emptyset\to LR}$27 on the worst-case cost of $\mathcal{S}_{\emptyset\to LR}$28-routing and the first efficient $\mathcal{S}_{\emptyset\to LR}$29-routing protocol for a function believed to lie outside $\mathcal{S}_{\emptyset\to LR}$30 (Allerstorfer et al., 2023). The reduction theory around $\mathcal{S}_{\emptyset\to LR}$31-measure and $\mathcal{S}_{\emptyset\to LR}$32-route implies that many practically implementable QPV schemes therefore have the same asymptotic entanglement cost and hence similar security levels (Bluhm et al., 29 May 2025, Bluhm et al., 24 Jun 2026).

NLQC also interfaces directly with communication and circuit complexity. The small-light-cone protocol makes the entanglement cost depend on the circuit’s maximum past-light-cone volume $\mathcal{S}_{\emptyset\to LR}$33, yielding polynomial entanglement for $\mathcal{S}_{\emptyset\to LR}$34 and quasi-polynomial entanglement for $\mathcal{S}_{\emptyset\to LR}$35 (Dolev et al., 2022). The low-$\mathcal{S}_{\emptyset\to LR}$36-depth and low-light-cone constructions subsequently feed into private simultaneous message upper bounds and into classical simulations of certain one-round entanglement-assisted communication models (Speelman, 2015, Girish et al., 10 Jun 2026). This suggests that NLQC is best viewed not as a single protocolic trick, but as a family of structure-sensitive simulations whose cost tracks non-Clifford depth, light-cone growth, or algebraic control complexity.

A distinct application domain is holography and quantum gravity. For finite-memory quantum systems on a circular lattice, a single simultaneous round of quantum communication plus pre-shared entanglement equal to the entropy across the $\mathcal{S}_{\emptyset\to LR}$37 bipartition suffices to implement a “pseudo-bulk” channel when the spread of the dynamics is at most $\mathcal{S}_{\emptyset\to LR}$38; in holographic CFT simulations, this reproduces the channel induced by local bulk dynamics (Dolev et al., 2022). Under plausible assumptions about bulk computation, this implies that any polynomially complex unitary can be realized by boundary NLQC with polynomial entanglement (Dolev et al., 2022). In two-sided black-hole geometries, early behind-the-horizon collisions correspond to one-round boundary NLQC using the thermofield double as the entanglement resource, and they imply a boundary mutual-information signature $\mathcal{S}_{\emptyset\to LR}$39 when the collision occurs before the relevant connected extremal surface (May et al., 2023). Efficient PBT is explicitly relevant here because bulk computations are realized as boundary NLQC, and the efficient PBT algorithm was motivated in part by this AdS/CFT connection (Fei et al., 2023).

6. Conceptual limits and open problems

Several sharp open problems remain. In LOBC, it is open whether every two-qubit unitary admits an exact deterministic protocol with finite entanglement; the known $\mathcal{S}_{\emptyset\to LR}$40 construction is deterministic only for special angle families, while the general protocol has failure probability $\mathcal{S}_{\emptyset\to LR}$41 (Gonzales et al., 2018). For generic controlled gates on $\mathcal{S}_{\emptyset\to LR}$42, the best known lower bound is $\mathcal{S}_{\emptyset\to LR}$43 ebits, while known upper bounds scale linearly in $\mathcal{S}_{\emptyset\to LR}$44, leaving an exponential gap (Gonzales et al., 2018). For general one-round NLQC, the exact one-round entanglement cost of SWAP remains open (May, 4 May 2026).

The lower-bound side is equally incomplete. CC and CE provide the first general lower-bound techniques that can be evaluated for arbitrary bipartite unitaries, but channels beyond unitaries, multipartite extensions, tighter additivity statements, and explicit dependence on Cartan/KAK parameters remain open (Cleve et al., 30 Jan 2026). For classically controlled tasks, rank lower bounds presently require perfect correctness, and extending them robustly to noisy implementations remains unresolved (Asadi et al., 2024). In private simultaneous quantum message passing, tightening privacy-based lower bounds, extending Nečiporuk-style methods beyond perfect correctness, and optimizing the dependence on circuit depth are explicit open directions (Girish et al., 10 Jun 2026).

A more conceptual challenge concerns operational meaning. One 2025 result argues that, in a black-box MIP*/LOCC setting, it is Turing-undecidable to decide from finite operational data whether a physical computation genuinely used nonlocal resources, or whether apparently separated components were merely subsystems of a monolithic quantum device (Fields et al., 24 Jan 2025). This does not invalidate the formal resource theory of NLQC, but it places a strict limit on device-independent certification of “non-local resource usage” as a physical fact (Fields et al., 24 Jan 2025).

Taken together, these results delineate a field with a highly nontrivial resource landscape. General-purpose one-round simulation is possible but often exponentially expensive; structured families admit substantially cheaper constructions; interaction can sometimes be exchanged for entanglement, but only within quantitatively sharp limits; and the resulting theory now spans quantum cryptography, communication complexity, many-body dynamics, and holography (May, 4 May 2026).