Minimum Clique-Optimised Quantum Algorithm

• Minimum Clique-Optimised Quantum Algorithm is a quantum algorithm that reformulates causal querying in multiloop Feynman diagrams as a graph-theoretic optimization over Boolean clause sets.
• The approach uses greedy clique-cover heuristics to compress the quantum oracle by grouping mutually exclusive clauses, significantly reducing ancilla requirements and circuit depth.
• Classical preprocessing transforms diagram edges into clause groups, enabling resource optimization and improved circuit metrics vital for practical NISQ implementations.

Searching arXiv for the cited MCA and related clique-optimization papers to ground the article. The Minimum Clique-optimised quantum Algorithm (MCA) is a gate-based quantum algorithm for querying causal structures in multiloop Feynman diagrams by reformulating the causal-consistency problem as a graph-theoretic optimization over Boolean clause sets. In the formulation introduced in “Graph theory-based automated quantum algorithm for efficient querying of acyclic and multiloop causal configurations” (Ochoa-Oregon et al., 6 Aug 2025), MCA exploits an analogy with the Minimum Clique Partition problem to reduce ancilla usage in the oracle: mutually exclusive clause families are grouped into cliques, and each clique can share a single ancillary qubit. The resulting workflow combines classical preprocessing on clause graphs with Grover-style oracle querying, and its reported evaluation is based on transpiled quantum circuit depth and circuit area (Ochoa-Oregon et al., 6 Aug 2025).

1. Problem setting and graph-theoretic formulation

In MCA, a multiloop Feynman vacuum diagram is associated with a reduced graph $G_{\rm red}=(V,E)$, where $V$ is the set of interaction vertices and $E$ is the set of undirected edges, each edge $e_i$ representing one or more propagators whose energy flows must be aligned (Ochoa-Oregon et al., 6 Aug 2025). A causal configuration is encoded by assigning a Boolean direction variable to each edge qubit, $e_i\in\{0,1\}$, and causality is identified with acyclicity in the directed graph obtained by orienting the edges.

The algorithm expresses acyclicity through a family of Boolean clauses $\{c_k\}$ representing directed cyclic substructures, or “eloops,” and requires that all such cyclic conditions be false. The condition is written as

$$\mathcal{A}\;=\;\bigwedge_{k=0}^{m-1}\neg\,c_k\;=\;1 \quad\Longleftrightarrow\quad \text{acyclic (causal) configuration}\,.$$

This supplies the logical target of the oracle: mark assignments of edge directions that satisfy the global acyclicity predicate (Ochoa-Oregon et al., 6 Aug 2025).

The minimum-clique component enters through the mutual-exclusion graph

$$G_{\rm ME}=(\mathcal C,\;A)\,,\qquad (c_i,c_j)\in A\;\Longleftrightarrow\; c_i\wedge c_j=0\quad\forall\text{ assignments,}$$

defined on the set of eloop clauses $\mathcal C=\{c_1,\dots,c_m\}$ (Ochoa-Oregon et al., 6 Aug 2025). A clique in $G_{\rm ME}$ is a subset of clauses that are pairwise mutually exclusive. Because clauses in such a clique can never be simultaneously true, they may be evaluated onto a single ancillary qubit in the quantum oracle. The Minimum Clique Partition problem on $V$0 therefore directly controls ancilla minimization, and the partition size equals the chromatic index of the complement graph in the formulation given in the paper (Ochoa-Oregon et al., 6 Aug 2025).

This graph-theoretic reduction places MCA within a broader pattern in quantum algorithm design: clique and clique-cover structure is used not only for combinatorial search but also for oracle compression and measurement reduction. A related example is the use of minimum clique cover for grouping qubit-wise commuting Pauli strings in VQE measurement optimization, where a clique in the qubit-wise-commutation graph corresponds to terms measurable in a single basis (Verteletskyi et al., 2019). This suggests that MCA’s core principle is not clique search per se, but resource optimization by exploiting graph structure in the algebra of constraints.

2. Classical preprocessing pipeline

The classical front end of MCA converts the diagram into clause structures suitable for a compressed oracle. The starting point is a partition of edges into sets $V$1 that share loop-momentum combinations. From these, Boolean sub-eloop tags are defined as

$V$2

Eloop clauses are then built via

$V$3

These constructions supply the clause family on which the mutual-exclusion analysis is performed (Ochoa-Oregon et al., 6 Aug 2025).

The preprocessing stage then builds the mutual-exclusion adjacency matrix, denoted MutualAuxMatrix, for the clause set. The paper states that MCA approximates the Minimum Clique Partition by a greedy “find largest clique → remove → repeat” routine called GraphConditionCombination, producing the clique family $V$4 (Ochoa-Oregon et al., 6 Aug 2025). The number $V$5 of such cliques is the number of ancilla qubits required in the oracle.

A second clause graph, MutualClausesMatrix, is constructed to group clauses by shared $V$6 or $V$7 terms. From this graph, the algorithm extracts another clique cover, denoted $V$8, and then optimizes clause ordering via a Bayesian hyperparameter optimizer, specified as Optuna/TPE, to obtain $V$9 (Ochoa-Oregon et al., 6 Aug 2025). This ordering step is part of the automation of the oracle-construction workflow rather than the underlying causal predicate itself.

The use of greedy clique-cover heuristics is consistent with the treatment of minimum clique cover in measurement optimization for VQE. In that setting, the exact Minimum Clique Cover problem is NP-hard, and practical implementations rely on polynomial heuristics such as Greedy, Largest-First, Smallest-Last, DSATUR, Recursive Largest-First, Dutton–Brigham, COSINE, or repeated clique removal (Verteletskyi et al., 2019). A plausible implication is that MCA’s preprocessing inherits the same trade-off: suboptimal clique covers do not invalidate the construction, but they increase the ancilla count and therefore enlarge oracle resources.

3. Quantum circuit architecture

After classical preprocessing, MCA constructs a Grover-style circuit with three registers. The edge register is $E$0, containing $E$1 qubits; the ancilla register is $E$2, containing one qubit per clique in $E$3; and there is a single output marker qubit $E$4 (Ochoa-Oregon et al., 6 Aug 2025). Initialization is specified as

$E$5

The oracle $E$6 iterates over the ancilla cliques $E$7. For each clique, the circuit applies $E$8 gates to edge qubits appearing in $E$9 clauses when needed, and then uses a multi-controlled Toffoli to map the conjunction of the relevant edge bits onto ancilla $e_i$0 (Ochoa-Oregon et al., 6 Aug 2025). After all ancillas have been populated, one additional multi-controlled Toffoli, controlled on all $e_i$1 and on the tagged edge $e_i$2, targets the output qubit. The ancillas are then uncomputed in reverse order.

The oracle action is written as

$e_i$3

with binary oracle function

$e_i$4

The explicit fixation of one edge, $e_i$5, is described in the paper as a symmetry-breaking device that halves the number of solutions (Ochoa-Oregon et al., 6 Aug 2025).

The diffusion stage acts on the edge register alone, using

$e_i$6

Thus a Grover iteration consists of the compressed oracle followed by a standard diffuser on the edge qubits (Ochoa-Oregon et al., 6 Aug 2025).

The rationale for ancilla compression is structural. Each clique in $e_i$7 contains eloop clauses that are pairwise mutually exclusive, so only one of them can ever fire; consequently, a single ancilla suffices for that entire clique (Ochoa-Oregon et al., 6 Aug 2025). This is closely analogous to clique-based compression in VQE measurement grouping, where each clique in the qubit-wise-commutation graph can be measured in one basis (Verteletskyi et al., 2019), although the operational meaning of the clique differs between the two settings.

4. Complexity measures and reported performance

The paper analyzes MCA using circuit depth and circuit area rather than asymptotic runtime alone. Let $e_i$8, $e_i$9, and $e_i\in\{0,1\}$0. The theoretical depth of one oracle call is given as

$e_i\in\{0,1\}$1

while the full Grover iteration depth is

$e_i\in\{0,1\}$2

The circuit area is

$e_i\in\{0,1\}$3

These expressions make explicit that the clique partition affects not only qubit count but also depth through the number and size of clause groups (Ochoa-Oregon et al., 6 Aug 2025).

The paper also contrasts MCA with a more direct MCX-based oracle construction. For the three-eloop, twelve-edge topology, MCA uses $e_i\in\{0,1\}$4 qubits versus 21 for MCX, with theoretical depth 23 versus 31 (Ochoa-Oregon et al., 6 Aug 2025). After Qiskit transpilation at optimization level 2 on IBM Brisbane, the reported values are approximately depth 25 and area 400 for MCA, compared with approximately depth 28 and area 588 for MCX. Similar gains, described as approximately 10% in depth and approximately 15% in area, are reported for four- and five-eloop tests (Ochoa-Oregon et al., 6 Aug 2025).

The paper further states that classical DAG enumeration costs $e_i\in\{0,1\}$5, whereas MCA’s Grover speedup requires $e_i\in\{0,1\}$6 iterations with $e_i\in\{0,1\}$7 and $e_i\in\{0,1\}$8 in the scenario analyzed (Ochoa-Oregon et al., 6 Aug 2025). This places MCA in the family of amplitude-amplified search procedures whose advantage depends on a structured oracle.

Related gate-based clique-search work provides useful context for how such oracle costs can dominate practical feasibility. In “Finding Small and Large $e_i\in\{0,1\}$9-Clique Instances on a Quantum Computer,” the authors compare checking-based and incremental-based oracle constructions, quantify gate counts, depths, and qubit requirements for triangle-finding circuits, and emphasize that constant factors and state-preparation overhead are decisive on NISQ hardware (Metwalli et al., 2020). This suggests that MCA’s clique-based ancilla compression is best understood as a circuit-engineering response to exactly that regime.

5. Relationship to other clique-based quantum methods

Although MCA in the strict sense refers to the causal-query algorithm of (Ochoa-Oregon et al., 6 Aug 2025), minimum-clique and clique-cover ideas recur in several quantum subfields.

In VQE measurement optimization, the Hamiltonian is written as

$\{c_k\}$0

and a graph is built whose vertices are Pauli strings and whose edges connect qubit-wise commuting pairs (Verteletskyi et al., 2019). A clique in this graph is a set of terms measurable in a single basis of one-qubit projective measurements, so the grouping problem becomes a Minimum Clique Cover problem. The paper reports that, on average, grouping qubit-wise commuting terms reduced the number of operators to measure three times compared to the total number of terms in the considered Hamiltonians (Verteletskyi et al., 2019). The conceptual affinity with MCA is direct: both reduce quantum-resource overhead by replacing a term-wise treatment with clique-wise aggregation on an auxiliary graph.

Clique-based ideas also appear in quantum algorithms for graph problems themselves. “Finding Small and Large $\{c_k\}$1-Clique Instances on a Quantum Computer” formulates $\{c_k\}$2-clique detection as Grover search over either the full subset space or a limited-weight Dicke-state subspace, with oracle constructions that check edge counts and node counts (Metwalli et al., 2020). The paper gives

$\{c_k\}$3

with $\{c_k\}$4 for full search and $\{c_k\}$5 for limited search, and analyzes amplitude amplification through

$\{c_k\}$6

The authors also discuss how one could adapt this framework toward a “Minimum-Clique-Optimised Quantum Algorithm” by looping over clique sizes and, for unknown solution counts, incorporating quantum counting (Metwalli et al., 2020). That proposal is distinct from the causal MCA of (Ochoa-Oregon et al., 6 Aug 2025), but it shows that “minimum clique optimization” has been used as a design motif in quantum search beyond the specific Feynman-diagram application.

A separate line of work analyzes clique optimization on quantum annealers. In “An Analysis of Quantum Annealing Algorithms for Solving the Maximum Clique Problem,” maximum clique is encoded as a QUBO on the complement graph, with

$\{c_k\}$7

and energy

$\{c_k\}$8

using $\{c_k\}$9 and $$\mathcal{A}\;=\;\bigwedge_{k=0}^{m-1}\neg\,c_k\;=\;1 \quad\Longleftrightarrow\quad \text{acyclic (causal) configuration}\,.$$0 (Gherardi et al., 2024). The same summary describes how this framework might be redefined for a minimum-clique objective by adding size-constraint slack variables. This is again conceptually related but methodologically different: it is an annealing-based optimization of clique size, whereas MCA (Ochoa-Oregon et al., 6 Aug 2025) is a gate-based oracle algorithm for causal querying.

6. Limitations, assumptions, and extensions

The MCA paper states several limitations explicitly. First, the Minimum Clique Partition problem is NP-hard, and the implementation therefore uses a greedy approximate clique cover; suboptimal covers increase the number $$\mathcal{A}\;=\;\bigwedge_{k=0}^{m-1}\neg\,c_k\;=\;1 \quad\Longleftrightarrow\quad \text{acyclic (causal) configuration}\,.$$1 of ancillas (Ochoa-Oregon et al., 6 Aug 2025). Second, the symmetry-breaking choice $$\mathcal{A}\;=\;\bigwedge_{k=0}^{m-1}\neg\,c_k\;=\;1 \quad\Longleftrightarrow\quad \text{acyclic (causal) configuration}\,.$$2 is fixed to halve the solutions, but more general symmetry-breaking may be required for atypical diagrams. Third, the depth of multi-controlled Toffoli gates grows with the number of controls, so advanced ancilla-assisted decompositions may reduce depth at the cost of more qubits. Finally, noise and hardware connectivity constrain the loop orders presently feasible on NISQ devices (Ochoa-Oregon et al., 6 Aug 2025).

These caveats are consistent with empirical lessons from related gate-based clique-search circuits. Simulations and IBMQ experiments for triangle finding reported that success probability on real hardware dropped to 4%–7%, and that noise models closely matched real runs, indicating that realistic device noise can strongly damp the marked-state amplitude (Metwalli et al., 2020). The same study connected feasibility to IBM’s Quantum Volume forecast, using

$$\mathcal{A}\;=\;\bigwedge_{k=0}^{m-1}\neg\,c_k\;=\;1 \quad\Longleftrightarrow\quad \text{acyclic (causal) configuration}\,.$$3

and argued that even the smallest high-tolerance circuit required substantially stronger hardware than was then available (Metwalli et al., 2020). This suggests that MCA’s practical advantage depends not only on fewer ancillas, but also on whether the compressed oracle materially reduces transpiled depth under the target backend’s coupling map and noise profile.

The paper outlines several extensions: improved clique-cover algorithms, including semidefinite relaxations and ILP, to reduce $$\mathcal{A}\;=\;\bigwedge_{k=0}^{m-1}\neg\,c_k\;=\;1 \quad\Longleftrightarrow\quad \text{acyclic (causal) configuration}\,.$$4 further; hybrid variational subroutines replacing part of the oracle; dynamic rearrangement of clauses depending on partial measurements, described as “adaptive Grover”; application to scattering-amplitude integrand sampling with weighted amplitude amplification; and integration with error mitigation and circuit knitting for larger multiloop topologies (Ochoa-Oregon et al., 6 Aug 2025). A plausible implication is that MCA should be viewed less as a fixed circuit template than as a graph-compilation strategy for causal oracles.

In that sense, MCA occupies a specific position in the broader landscape of quantum algorithms using clique structure. It is not a generic clique solver, nor merely a measurement-grouping heuristic, but an automated causal-query algorithm in which minimum clique partitioning serves as an oracle-optimization primitive. Its technical significance lies in showing that clique decomposition can act directly on the logical structure of a quantum oracle, reducing ancilla count and improving transpiled circuit metrics in a problem domain—multiloop causal configurations—where the underlying constraint graph is itself physically motivated (Ochoa-Oregon et al., 6 Aug 2025).