Distributed Quantum-Enhanced Optimization

• D-QEO is a framework that distributes quantum variational methods over multiple QPUs, balancing fidelity, communication costs, and resource allocation for complex optimization tasks.
• It integrates techniques like distributed VQE, coherent entanglement sharing, and hybrid quantum-classical workflows to address both separable and coupled problem structures.
• Empirical studies report significant improvements in circuit depth reduction, gate minimization, and overall system performance, underscoring D-QEO’s scalability potential.

Searching arXiv for the cited D-QEO-related papers and nearby work to ground the article in current literature. Distributed Quantum-Enhanced Optimization (D-QEO) denotes a family of methods in which optimization is carried out, approximated, or made more executable under distributed quantum-computing constraints. In the cited literature, the term spans at least four closely related usages: distributed variational optimization over multiple logical QPUs; coherent distributed search coordinated by shared entanglement; hybrid quantum-classical decomposition workflows for large optimization instances; and optimization of the distributed quantum execution stack itself, including circuit rewriting, qubit placement, and resource allocation (Soós et al., 22 Apr 2026, Hasanzadeh et al., 24 Aug 2025, Huang et al., 8 Mar 2026, Sünkel et al., 9 Sep 2025). Across these usages, the unifying concern is that multi-QPU execution introduces communication, partitioning, fidelity, routing, and topology costs that must be balanced against the optimization objective rather than treated as a purely downstream implementation detail.

1. Conceptual scope and terminology

In the cited work, D-QEO is not a single standardized algorithm. One line of work uses the label directly for a hybrid framework in which the QPU acts as a “topographical preconditioner,” locating promising basins of attraction and generating seed points for a classical GPU-accelerated solver (Soós et al., 22 Apr 2026). A second line studies distributed variational algorithms such as distributed VQE (DVQE), enhanced distributed VQE (EDVQE), distributed QAOA (DQAOA), and QESTO, where the optimization routine itself is distributed over multiple quantum subsystems or QPUs (Hasanzadeh et al., 24 Aug 2025, Lin et al., 26 Dec 2025, Xu et al., 12 Jun 2025, Matwiejew et al., 3 Jun 2026). A third line focuses on structure-aware coherent distributed search, using factor graphs, separators, teleportation, and shared entanglement to preserve Grover-like scaling while reducing per-processor qubit requirements (Huang et al., 8 Mar 2026). A fourth line treats D-QEO more broadly as optimization for distributed quantum systems, including circuit-level reduction of communication overhead, heterogeneous compilation, and stochastic resource allocation over future quantum networks (Sünkel et al., 9 Sep 2025, Zhou et al., 21 Aug 2025, Ngoenriang et al., 2022).

This breadth matters because it prevents a common misreading: D-QEO does not necessarily mean that a quantum computer directly solves a classical optimization problem end-to-end. In some formulations, the quantum stage preserves a monolithic variational family across multiple QPUs (Hasanzadeh et al., 24 Aug 2025). In others, it implements a communication primitive based on persistent Bell pairs (Matwiejew et al., 3 Jun 2026). In still others, it reduces the cost of distributed execution before compilation or execution begins (Sünkel et al., 9 Sep 2025). A plausible implication is that D-QEO is best understood as a systems-and-algorithms umbrella for optimization under distributed quantum constraints, rather than as a single variational template.

Approach	Distributed mechanism	Reported outcome
EA-based circuit optimization	Circuit rewriting for DQC communication reduction	More than 89% global-gate reduction; up to 19% hop reduction
DVQE	Distributed ansatz with TeleGate-style remote entanglement	Fidelity = 1; matching energy trajectories
QESTO	Persistent pre-shared Bell pairs; no non-local gates after initialization	Stronger convergence than equivalently partitioned QAOA at depth two or higher
EDVQE	Distributed VQE with classical-quantum perturbation	Up to 1000 vertices using only 10 qubits
DQAOA	Sub-QUBO decomposition with MPI and GPU simulation	Up to 10x speedup over CPU-based simulations
Factor-graph framework	Separator-based coherent search with shared entanglement	$O(\sqrt{N})$ query complexity up to processor- and separator-dependent factors

2. Distributed execution models

The distributed substrate differs substantially across the literature, but several architectural motifs recur. DVQE distributes a parameterized ansatz over multiple logical QPUs while preserving the variational family of the monolithic circuit. Each QPU reserves one communication qubit and uses the remaining qubits as compute qubits; inter-QPU entangling gates are emulated by a TeleGate-like protocol in which local compute qubits are entangled with communication qubits, the communication qubits are entangled across QPUs, and classical communication via a “cat-measure bus” conditions the remote operation (Hasanzadeh et al., 24 Aug 2025). This is explicitly state-preserving rather than approximate: the distributed circuit is designed to reproduce the monolithic state.

QESTO uses a more restrictive but lower-overhead communication model. The problem graph is partitioned across QPUs, each distributed edge is assigned one Bell pair $\ket{\Phi^+}=\frac{\ket{00}+\ket{11}}{\sqrt{2}}$, and after Bell-pair initialization the runtime circuit uses only local operations on problem registers and communication qubits (Matwiejew et al., 3 Jun 2026). The method is framed as LOSE—local operations with shared entanglement—rather than repeated gate teleportation. The reported resource claim is correspondingly sharp: one Bell pair per distributed edge, and no non-local gates after initialization.

The factor-graph framework adopts a coordinator-worker network model. A factor graph $G(F,V,E,X)$ is cut along a separator of boundary variables; the coordinator prepares a superposition over boundary assignments, teleports boundary information to workers, each worker solves a conditioned local maximization subproblem, and the coordinator performs a Grover-like threshold test using amplitude amplification and phase estimation (Huang et al., 8 Mar 2026). This architecture is then lifted recursively into a hierarchical divide-and-conquer tree with fully coherent and hybrid modes.

A different kind of distribution appears in DQAOA. There, the workload is decomposed into sub-QUBOs, distributed asynchronously with MPI, simulated on CPU/GPU resources on the Frontier supercomputer, and then reassembled by an iterative acceptance-based aggregation step (Xu et al., 12 Jun 2025). The distributed unit is not only the quantum circuit but the full hybrid workflow: decomposition, dispatch, simulation, and global update.

At the network-management level, DQC$^2$O models an operator choosing among reserved quantum computers, on-demand quantum computers, and oriented quantum links with static capacity (Ngoenriang et al., 2022). Teleportation, entangled qubits stored in quantum memories, Bell pair capacity, fidelity degradation, and uncertain availability are treated as first-class system variables. This suggests a broader systems interpretation of D-QEO in which optimization is performed not only over algorithm parameters but also over distributed quantum infrastructure.

3. Objective functions and optimization principles

A central distinction among D-QEO formulations is what exactly is being optimized. In distributed circuit optimization for DQC, the primary objective is communication reduction under a partition $p$ of qubits to QPUs. The number of global CX gates is

$$c(p, cx) = \sum_{(i, j) \in cx} 1 - \delta(p(i), p(j)),$$

with optimization target $\min_p c(p, cx)$. When network topology matters, the binary count is replaced by hop distance,

$$g(p, cx) = \sum_{(i, j) \in cx} \textnormal{dist}(p(i), p(j)).$$

State preservation is measured by

$F(|t\rangle, U) = |\langle t|U|0\rangle|^2,$

and the combined objective becomes

$$f(|t\rangle, p, U) = \alpha \cdot F(|t\rangle, U) - g(p, U_{cx}), \qquad \max_U f(|t\rangle, p, U),$$

with proxy variants for depth and CX count (Sünkel et al., 9 Sep 2025). This is explicitly multi-criteria: fidelity is increased while communication, depth, or CX count is decreased.

Distributed variational methods generally optimize an Ising or QUBO expectation value. DVQE starts from

$\ket{\Phi^+}=\frac{\ket{00}+\ket{11}}{\sqrt{2}}$0

uses $\ket{\Phi^+}=\frac{\ket{00}+\ket{11}}{\sqrt{2}}$1, $\ket{\Phi^+}=\frac{\ket{00}+\ket{11}}{\sqrt{2}}$2, and produces a Hamiltonian

$\ket{\Phi^+}=\frac{\ket{00}+\ket{11}}{\sqrt{2}}$3

The variational objective is

$\ket{\Phi^+}=\frac{\ket{00}+\ket{11}}{\sqrt{2}}$4

with ADAM updates driven by finite-difference gradients (Hasanzadeh et al., 24 Aug 2025). EDVQE uses the same diagonal-factorization logic for distributed energy evaluation, writing $\ket{\Phi^+}=\frac{\ket{00}+\ket{11}}{\sqrt{2}}$5 with $\ket{\Phi^+}=\frac{\ket{00}+\ket{11}}{\sqrt{2}}$6, $\ket{\Phi^+}=\frac{\ket{00}+\ket{11}}{\sqrt{2}}$7, and for tensor-product states $\ket{\Phi^+}=\frac{\ket{00}+\ket{11}}{\sqrt{2}}$8,

$\ket{\Phi^+}=\frac{\ket{00}+\ket{11}}{\sqrt{2}}$9

For MaxCut it uses

$G(F,V,E,X)$0

and augments the variational loop with classical neighborhood search and quantum perturbation by $G(F,V,E,X)$1 (Lin et al., 26 Dec 2025).

QESTO departs from standard QAOA-style remote cost evaluation. On a graph $G(F,V,E,X)$2, the objective is

$G(F,V,E,X)$3

with

$G(F,V,E,X)$4

Constraint labels are coherently encoded into Bell halves by configuration-controlled $G(F,V,E,X)$5 rotations, followed by a transport unitary and uncomputation. The distributed communication layer is therefore not an approximation of a remote CNOT but a compatibility-selective amplitude-transport mechanism (Matwiejew et al., 3 Jun 2026).

The factor-graph framework formulates optimization as

$G(F,V,E,X)$6

then reduces the global problem to a boundary search over separator assignments $G(F,V,E,X)$7 with

$G(F,V,E,X)$8

Its main theoretical claim is $G(F,V,E,X)$9 query complexity up to processor- and separator-dependent factors, with explicit success and approximation guarantees (Huang et al., 8 Mar 2026).

The topographical-preconditioning formulation is narrower in problem class but different in intent. For separable objectives

$^2$0

independent quantum registers perform landscape probing rather than exact global minimization. The variational objective is based on Conditional Value-at-Risk,

$^2$1

with $^2$2 and $^2$3, so the QPU biases sampling toward low-energy regions and outputs a bounded search region for classical refinement (Soós et al., 22 Apr 2026).

4. Algorithmic families and representative methods

The EA-based circuit optimizer encodes a candidate quantum circuit as a one-dimensional list of gate objects, seeds the initial population with the target circuit, and evolves that population with single-point and uniform crossover plus six mutation operators: gate flip, delete gate, swap gate, shuffle, add gate, and remove CX gate. The reported hyperparameters are population size $^2$4, generations $^2$5, crossover rate $^2$6, mutation rate $^2$7, child rate $^2$8, and replace rate $^2$9. The method is explicitly a pre-compilation or pre-processing step for DQC rather than a full quantum optimization algorithm (Sünkel et al., 9 Sep 2025).

DVQE is an end-to-end pipeline: input QUBO data, convert QUBO to an Ising Hamiltonian, choose a hardware topology or QPU allocation, build a monolithic or distributed ansatz, initialize parameters with either random or metaheuristic warm starts, train with ADAM, sample the final circuit, and return the best bitstring. The framework is packaged as raiselab, exposed by a single main entry point DVQE(...), and includes an internal greedy algorithm that assigns qubits iteratively to the QPU with the current least load (Hasanzadeh et al., 24 Aug 2025).

EDVQE extends distributed VQE by alternating between classical intensification and quantum diversification. It begins with DVQE, applies a local search called CNS-1 based on single-vertex flips, embeds the resulting bitstring into a biased quantum initial state using $p$0, and introduces a QP-2 perturbation built from $p$1 gates between vertices in opposite partitions. Warm-start EDVQE further initializes from the best solutions from 10 independent runs of Goemans-Williamson with $p$2 projections (Lin et al., 26 Dec 2025).

DQAOA distributes a large QUBO into sub-QUBOs, solves them in parallel using QAOA instances with $p$3 and COBYLA, and iteratively updates the global solution by accepting only energy-reducing sub-solution replacements. Its decomposition methods include random selection, impact factor directed (IFD), BFS, and PFS, and its orchestration layer uses MPI alongside Qiskit Aer, NWQ-Sim, and QFw on Frontier CPU/GPU resources (Xu et al., 12 Jun 2025).

The heterogeneous compilation framework based on clustering and annealing belongs to D-QEO only under a broad interpretation, but it is central to distributed execution quality. It performs pattern-aware circuit segmentation using interaction-density windows and Jaccard similarity, initial placement by time-aware clustering with exponential decay weights $p$4, capacity repair, and simulated annealing refinement with an objective combining inter-QPU communication cost, intra-QPU local routing cost, and movement cost between segments (Zhou et al., 21 Aug 2025).

Finally, the factor-graph framework and QESTO both treat distributed structure as algorithmic input rather than as a compilation afterthought. The former cuts the problem along natural separators and coordinates local maximization coherently (Huang et al., 8 Mar 2026); the latter embeds distributed-edge compatibility directly into persistent Bell pairs and uses those Bell pairs as active carriers of constraint information (Matwiejew et al., 3 Jun 2026). This suggests that a defining property of mature D-QEO methods is not merely decomposition, but decomposition that preserves or exploits problem structure.

5. Empirical findings

For distributed circuit optimization on Grover-style state-preparation circuits with 4, 5, 6, and 8 qubits, the EA reports depth reductions of 36%, 32%, and 27% for the 4-, 5-, and 6-qubit cases when optimizing for depth. In the same setting, global-gate reductions were 8%, over 26%, and 5%. When directly minimizing global gates, the reported improvements were 60.41%, 15.741%, and 33.836% for 4, 5, and 6 qubits, and the abstract highlights that required global gates were reduced by more than 89%. In network-aware experiments, communication cost measured by number of hops between QPUs was reduced by 14.33% in a 6-qubit, 3-node network and 19.575% in an 8-qubit, 4-node grid. Mean fidelities for the best circuits included 0.972087, 0.91407, and 0.915459 for the 4-qubit minimum-global-gates, minimum-CX, and minimum-depth settings, and 0.990059 and 0.984862 for the 6- and 8-qubit minimum-distance settings; the correct solution could still be extracted in all experiments (Sünkel et al., 9 Sep 2025).

DVQE reports exact state preservation in its validation cases: a 6-qubit depth-2 example with 3 QPUs and a 4-qubit depth-4 example with 4 QPUs both achieved fidelity = 1. In unit commitment QUBO scenarios, monolithic and distributed energy curves matched exactly. On brute-force QUBO examples up to 10 qubits, both monolithic and distributed DVQE recovered the same optimum bitstring, including $p$5 with cost $p$6 for a 3-qubit problem, $p$7 with cost $p$8 for a 4-qubit problem, $p$9 with cost $$c(p, cx) = \sum_{(i, j) \in cx} 1 - \delta(p(i), p(j)),$$0 for a 5-qubit problem, and all zeros with cost $$c(p, cx) = \sum_{(i, j) \in cx} 1 - \delta(p(i), p(j)),$$1 for 8- and 10-qubit problems. The paper also reports that random initialization often converges slowly or not fully, while BH, GWO, and ABC converge faster and more accurately; BH converges fastest, GWO yields the most precise final parameters, and ABC is a middle ground (Hasanzadeh et al., 24 Aug 2025).

QESTO’s empirical case is bounded weighted Wang tile-matching. On the $$c(p, cx) = \sum_{(i, j) \in cx} 1 - \delta(p(i), p(j)),$$2 ensemble, its mean normalized optimality gap was about $$c(p, cx) = \sum_{(i, j) \in cx} 1 - \delta(p(i), p(j)),$$3 and $$c(p, cx) = \sum_{(i, j) \in cx} 1 - \delta(p(i), p(j)),$$4 at depths $$c(p, cx) = \sum_{(i, j) \in cx} 1 - \delta(p(i), p(j)),$$5, while the equivalently partitioned subgraph-QAOA and monolithic QAOA baselines remained much worse; at $$c(p, cx) = \sum_{(i, j) \in cx} 1 - \delta(p(i), p(j)),$$6, the reported mean gaps were roughly $$c(p, cx) = \sum_{(i, j) \in cx} 1 - \delta(p(i), p(j)),$$7 and $$c(p, cx) = \sum_{(i, j) \in cx} 1 - \delta(p(i), p(j)),$$8. At the same depth, feasible probability reached about $$c(p, cx) = \sum_{(i, j) \in cx} 1 - \delta(p(i), p(j)),$$9, compared with around $\min_p c(p, cx)$0 and $\min_p c(p, cx)$1 for the baselines. On the $\min_p c(p, cx)$2 ensemble, the mean gap improved from $\min_p c(p, cx)$3 at $\min_p c(p, cx)$4 to $\min_p c(p, cx)$5 at $\min_p c(p, cx)$6, and feasible probability at $\min_p c(p, cx)$7 was about $\min_p c(p, cx)$8, versus $\min_p c(p, cx)$9 and $$g(p, cx) = \sum_{(i, j) \in cx} \textnormal{dist}(p(i), p(j)).$$0. For the $$g(p, cx) = \sum_{(i, j) \in cx} \textnormal{dist}(p(i), p(j)).$$1 ensemble at $$g(p, cx) = \sum_{(i, j) \in cx} \textnormal{dist}(p(i), p(j)).$$2, QESTO placed about $$g(p, cx) = \sum_{(i, j) \in cx} \textnormal{dist}(p(i), p(j)).$$3 probability on the globally optimal rank, compared with roughly $$g(p, cx) = \sum_{(i, j) \in cx} \textnormal{dist}(p(i), p(j)).$$4 and $$g(p, cx) = \sum_{(i, j) \in cx} \textnormal{dist}(p(i), p(j)).$$5 for the reference methods (Matwiejew et al., 3 Jun 2026).

EDVQE reports that weighted MaxCut instances with up to 1000 vertices can be solved using only 10 qubits, and that the method consistently outperforms Goemans-Williamson. On weighted cluster graphs it surpasses Goemans-Williamson with $$g(p, cx) = \sum_{(i, j) \in cx} \textnormal{dist}(p(i), p(j)).$$6 projections on nearly all sizes and, for graph sizes 400–1000, beats Goemans-Williamson with $$g(p, cx) = \sum_{(i, j) \in cx} \textnormal{dist}(p(i), p(j)).$$7 projections. In the haplotype-phasing application on the ABCA1 gene region, the instance size was 147,149 base pairs, 187 SNP sites, and 506 reads; the distributed architecture used 46 subsystems with only 11 qubits per subsystem. Reported outcomes included an average approximation ratio around 94.06%, a 40% success rate in matching Goemans-Williamson’s best cut value, and on a gold-standard benchmark 100% phasing completeness, 0 switch error, and 0 Hamming error (Lin et al., 26 Dec 2025).

DQAOA emphasizes systems performance. On Frontier it reports up to 10x speedup over CPU-based simulations. For QFw scaling examples, GHZ-30 runtime dropped from 352 s on 2 nodes, 4 procs/node to 30 s on 16 nodes, 8 procs/node, which the paper states is an 11.7× speedup or about a 91.5% reduction. QAOA-20 runtime improved from 47.5 s on 1 node, 2 procs/node to 24.8 s on 4 nodes, 8 procs/node, described as a 1.9× speedup or 47.8% reduction. On dense QUBOs from 300 to 1000 dimensions, IFD improved convergence and quality relative to random decomposition; on sparse Max-Cut instances from 100 to 1000 nodes, PFS performed best in approximate ratio, BFS also outperformed random, and IFD could perform worse because it might ignore pairwise structural interactions (Xu et al., 12 Jun 2025).

The factor-graph framework validates a different metric family: query and entanglement complexity. In single-level benchmarks, the coherent algorithm used fewer oracle queries than a classical-communication benchmark, with reported query reduction factors ranging from about $$g(p, cx) = \sum_{(i, j) \in cx} \textnormal{dist}(p(i), p(j)).$$8 to $$g(p, cx) = \sum_{(i, j) \in cx} \textnormal{dist}(p(i), p(j)).$$9. For topology, EPR usage increased with hop distance and the observed ordering was line $F(|t\rangle, U) = |\langle t|U|0\rangle|^2,$0 ring $F(|t\rangle, U) = |\langle t|U|0\rangle|^2,$1 tree $F(|t\rangle, U) = |\langle t|U|0\rangle|^2,$2 mesh $F(|t\rangle, U) = |\langle t|U|0\rangle|^2,$3 star. In a two-level hierarchy at $F(|t\rangle, U) = |\langle t|U|0\rangle|^2,$4, query multipliers relative to coherent were 4.27 for hybrid_root, $F(|t\rangle, U) = |\langle t|U|0\rangle|^2,$5 for hybrid_level1, and $F(|t\rangle, U) = |\langle t|U|0\rangle|^2,$6 for hybrid_all; EPR multipliers were 3.49 for hybrid_root, 0.236 for hybrid_level1, and zero for hybrid_all in the coherent-communication accounting model (Huang et al., 8 Mar 2026).

The topographical-preconditioning D-QEO reports dramatic search-space reduction on 10-dimensional Rastrigin and separable Ackley. For Rastrigin at $F(|t\rangle, U) = |\langle t|U|0\rangle|^2,$7, the original volume was $F(|t\rangle, U) = |\langle t|U|0\rangle|^2,$8, the preconditioned volume was $F(|t\rangle, U) = |\langle t|U|0\rangle|^2,$9, the reduction factor was $$f(|t\rangle, p, U) = \alpha \cdot F(|t\rangle, U) - g(p, U_{cx}), \qquad \max_U f(|t\rangle, p, U),$$0, the original minima count was $$f(|t\rangle, p, U) = \alpha \cdot F(|t\rangle, U) - g(p, U_{cx}), \qquad \max_U f(|t\rangle, p, U),$$1, and the preconditioned minima count was $$f(|t\rangle, p, U) = \alpha \cdot F(|t\rangle, U) - g(p, U_{cx}), \qquad \max_U f(|t\rangle, p, U),$$2. For Ackley at $$f(|t\rangle, p, U) = \alpha \cdot F(|t\rangle, U) - g(p, U_{cx}), \qquad \max_U f(|t\rangle, p, U),$$3, the original volume was $$f(|t\rangle, p, U) = \alpha \cdot F(|t\rangle, U) - g(p, U_{cx}), \qquad \max_U f(|t\rangle, p, U),$$4, the preconditioned volume was $$f(|t\rangle, p, U) = \alpha \cdot F(|t\rangle, U) - g(p, U_{cx}), \qquad \max_U f(|t\rangle, p, U),$$5, and the reduction factor was $$f(|t\rangle, p, U) = \alpha \cdot F(|t\rangle, U) - g(p, U_{cx}), \qquad \max_U f(|t\rangle, p, U),$$6. The method also reports that higher quantum evaluation budgets, $$f(|t\rangle, p, U) = \alpha \cdot F(|t\rangle, U) - g(p, U_{cx}), \qquad \max_U f(|t\rangle, p, U),$$7, $$f(|t\rangle, p, U) = \alpha \cdot F(|t\rangle, U) - g(p, U_{cx}), \qquad \max_U f(|t\rangle, p, U),$$8, and $$f(|t\rangle, p, U) = \alpha \cdot F(|t\rangle, U) - g(p, U_{cx}), \qquad \max_U f(|t\rangle, p, U),$$9, progressively tighten the quantum distribution and improve downstream classical convergence, while substantially reducing the number of classical BFGS iterations required after preconditioning (Soós et al., 22 Apr 2026).

6. Limitations, misconceptions, and research directions

Several recurring limitations bound the current state of D-QEO. Many of the strongest results are simulation-based rather than hardware demonstrations. DVQE validates fidelity preservation, convergence, and exact bitstring recovery in simulation and explicitly notes that TeleGate is treated as an idealized mechanism; real deployment will face communication latency, imperfect entanglement distribution, decoherence, synchronization overhead, and noise in remote gate emulation (Hasanzadeh et al., 24 Aug 2025). DQAOA’s principal acceleration claim is likewise based on GPU-accelerated quantum simulation on an HPC system, not on physical QPU execution (Xu et al., 12 Jun 2025). The topographical-preconditioning framework states that its results do not constitute a wall-clock speedup claim because the quantum circuits were classically simulated (Soós et al., 22 Apr 2026).

A second limitation is scope of applicability. The topographical-preconditioning D-QEO makes its scalable claim specifically for separable functions, where $\ket{\Phi^+}=\frac{\ket{00}+\ket{11}}{\sqrt{2}}$00 and no cross-register entanglement or tensor knitting is required; non-separable or coupled problems are explicitly not fully solved by that framework (Soós et al., 22 Apr 2026). QESTO is demonstrated on bounded weighted Wang tile-matching ensembles rather than on arbitrary graph optimization families (Matwiejew et al., 3 Jun 2026). The factor-graph framework requires a useful small separator and a network capable of shared entanglement and teleportation, with communication cost scaling linearly in network diameter (Huang et al., 8 Mar 2026).

A third issue is that approximate correctness is sometimes sufficient and sometimes not. The EA circuit optimizer preserves output states with high fidelity and retains extractability of the intended solution, but the optimized circuits are not necessarily strictly unitary-equivalent to the originals (Sünkel et al., 9 Sep 2025). This is appropriate for the reported state-preparation and Grover-type tasks, but it would be a stronger constraint for applications requiring exact unitary equivalence. A related misconception is to assume that “distributed” implies only more communication. QESTO and the factor-graph framework both argue the opposite: entanglement can be used as a reusable communication substrate or as a coherence-preserving coordination mechanism, reducing repeated remote-gate costs when structure is exploited (Matwiejew et al., 3 Jun 2026, Huang et al., 8 Mar 2026).

Terminological ambiguity is itself a substantive issue. The heterogeneous compilation framework explicitly states that it is an instance of D-QEO only under a broad interpretation, because its optimization engine is classical—clustering plus simulated annealing—even though the target is a distributed quantum system (Zhou et al., 21 Aug 2025). DQC$\ket{\Phi^+}=\frac{\ket{00}+\ket{11}}{\sqrt{2}}$01O similarly addresses collaborative optimization by adaptive quantum resource allocation rather than by a direct distributed quantum optimization algorithm (Ngoenriang et al., 2022). This suggests that the phrase “quantum-enhanced” can refer either to a quantum method that improves optimization quality or to optimization machinery that improves the deployability of distributed quantum computation.

The open problems identified across the cited work are largely infrastructural and structural. DQC$\ket{\Phi^+}=\frac{\ket{00}+\ket{11}}{\sqrt{2}}$02O emphasizes efficient routing for collaborative quantum computers, quantum repeaters, quantum routers, coexistence of heterogeneous distributed algorithms, adaptive quantum resource allocation, and distributed quantum machine learning (Ngoenriang et al., 2022). The compilation literature points to topology-aware and heterogeneity-aware cost modeling beyond mere remote-gate minimization (Zhou et al., 21 Aug 2025). The factor-graph framework introduces an explicit coherent-versus-hybrid trade-off, suggesting that near-term scalability may depend on selective insertion of measurements to cap depth even when this sacrifices query complexity (Huang et al., 8 Mar 2026). A plausible implication is that future D-QEO systems will be judged less by any single variational template than by how effectively they integrate decomposition, communication primitives, hardware topology, and hybrid optimization into one distributed control stack.