Quantum Petri Nets (QPNs) Overview

• Quantum Petri Nets (QPNs) are Petri-net formalisms that combine classical structural elements with quantum states and operations to model superposition, interference, and measurement.
• They encompass diverse approaches—such as amplitude-weighted, q-token, event-structure, and SPO variants—each prioritizing aspects like concurrency, compositionality, or security verification.
• Recent developments in QPNs enable rigorous analysis of quantum protocols, exact validation using symbolic methods, and enhanced security assessments under quantum side information.

Quantum Petri Nets (QPNs) are Petri-net-based formalisms that incorporate quantum state, quantum operations, or quantum valuations into the classical machinery of places, transitions, markings, causality, conflict, and concurrency. The term is not used uniformly across the literature. In some works it denotes an extension of symbolic Petri nets for representing quantum pure states and protocol behavior; in others it denotes amplitude-weighted nets over reachability graphs, simplified q-token models for quantum buffers, or locally annotated nets whose unfolding yields a quantum event-structure semantics; more recent work specializes the paradigm to safe partially observed quantum Petri nets for opacity analysis under attacker-localized quantum side information (Zhang et al., 2017, Schmidt, 2021, Shah et al., 2024, Joachim et al., 1 Sep 2025, Ding et al., 20 Apr 2026).

1. Terminological scope and major lines of development

The literature contains several distinct, only partially compatible, notions of QPN. The 2017 protocol-verification line extends the classical tuple $PN=(P,T,F,W,M_0)$ so that places encode basis states, markings encode scaled amplitudes or probabilities, and weighted arcs implement quantum transformations; that paper explicitly states that it does not coin a formal name “Quantum Petri Nets” as a separate mathematical structure, even though the extension functions as a QPN formalism in practice (Zhang et al., 2017). A different 2021 line defines a QPN as a system Petri net $Q=(S,r)$ equipped with a marking-dependent complex rate function over firings and concurrences, with superposition states represented as complex vectors over the reachability set; this line emphasizes universality and compositionality (Schmidt, 2021).

A more operational engineering line proposes a deliberately simplified QPN model

$(D,P,T,E,\mu,v),$

where q-tokens carry qubits or registers, places represent storage locations, transitions encode quantum gates or measurements, and the pair $(\mu,v)$ records both token distribution and quantum state assignment. That formulation is used to synthesize SISO, SIMO, MISO, MIMO, and priority quantum buffers, and to validate a quantum S–R flip-flop and buffer designs on IBM hardware and simulators (Shah et al., 2024).

A semantically grounded line, introduced in 2025, defines QPNs as Petri nets equipped with a mathematically rigorous quantum valuation compatible with the quantum event-structure semantics of Clairambault, De Visme, and Winskel. Its central construction proceeds through Quantum Occurrence Nets, unfoldings, and event structures, and it frames QPNs as a bridge between Petri net theory and quantum programming (Joachim et al., 1 Sep 2025). A verification-oriented specialization then introduces the safe partially observed quantum Petri net

$$Q=(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\}_{t\in T},Acc,\Lambda,A,Sec),$$

which separates a safe classical control layer from a persistent quantum-register layer and fixes an attacker interface for current-state opacity analysis (Ding et al., 20 Apr 2026).

A concise comparison clarifies the heterogeneity.

Line of work	Formal object	Main emphasis
Protocol-verification extension	$PN=(P,T,F,W,M_0)$ with quantum state mapping	Pure-state protocol modeling
Amplitude-weighted QPN	$Q=(S,r)$	Universality, compositionality
Simplified buffer QPN	$(D,P,T,E,\mu,v)$	Quantum storage and routing
Event-structure QPN	Locally annotated net with unfolding semantics	Rigorous quantum concurrency
SPO-QPN	$$(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\},Acc,\Lambda,A,Sec)$$	Opacity and exact verification

This terminological variation has an immediate methodological consequence: statements about “QPNs” are framework-sensitive. Claims about event-structure semantics, CPTNI valuations, compositional joins, or exact stabilizer verification apply only to the corresponding formal line, not to every Petri-net-based quantum model.

2. Core mathematical models

The earliest formalization encodes a pure quantum state

$|\Psi\rangle=\sum_{i=1}^n |\phi_i\rangle$

by a finite place set $Q=(S,r)$0 through the mapping

$Q=(S,r)$1

where $Q=(S,r)$2 is a scaling constant and $Q=(S,r)$3 is the initial number of tokens in place $Q=(S,r)$4. In that setting, places represent basis states, markings provide the amplitude/probability proxy, interference is represented by linear addition of markings, and measurement is a probabilistic selection of places according to $Q=(S,r)$5 (Zhang et al., 2017).

The amplitude-weighted formulation keeps the underlying classical net intact and inserts quantum behavior at the level of firing rates. For a system net $Q=(S,r)$6, a QPN is $Q=(S,r)$7 with $Q=(S,r)$8, where $Q=(S,r)$9 is the reachability set and concurrences are multisets of mutually independent enabled transitions. Superpositions are vectors in the Hilbert space $(D,P,T,E,\mu,v),$0, with basis $(D,P,T,E,\mu,v),$1 indexed by markings. The induced rate matrix is

$(D,P,T,E,\mu,v),$2

so a single evolution step is the linear update $(D,P,T,E,\mu,v),$3 on amplitude vectors (Schmidt, 2021).

The simplified engineering model treats q-tokens as the primitive carriers of quantum information. In the tuple $(D,P,T,E,\mu,v),$4, $(D,P,T,E,\mu,v),$5 is a finite set of q-tokens, $(D,P,T,E,\mu,v),$6 a finite set of places, $(D,P,T,E,\mu,v),$7 a finite set of transitions, $(D,P,T,E,\mu,v),$8 a finite set of directed labeled arcs, $(D,P,T,E,\mu,v),$9 a marking assigning q-tokens to places, and $(\mu,v)$0 the assignment of qubits to q-tokens in place $(\mu,v)$1 at discrete time $(\mu,v)$2. Transitions encode unitary gates, controlled gates, or measurement, and guards or inhibitor arcs implement address-based routing and priority constraints (Shah et al., 2024).

The event-structure line introduces a local quantum annotation $(\mu,v)$3 on a net skeleton. Each place or condition is assigned a finite-dimensional Hilbert space, and each transition is assigned a CPTNI map with signature

$(\mu,v)$4

A QPN is then characterized by the property that its unfolding carries an induced local annotation making the unfolding a Local Quantum Occurrence Net. This construction ties net semantics to occurrence nets, event structures, and a global valuation on intervals of markings or configurations (Joachim et al., 1 Sep 2025).

The SPO-QPN specializes the architecture to hybrid security analysis. Its classical layer is a safe place/transition net with contact-free enabling rule

$(\mu,v)$5

while each transition branch $(\mu,v)$6 carries a CP trace-non-increasing map $(\mu,v)$7 belonging to a finite quantum instrument $(\mu,v)$8. Firing updates the control marking and normalizes the post-quantum state,

$(\mu,v)$9

and the attacker’s visible state is $$Q=(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\}_{t\in T},Acc,\Lambda,A,Sec),$$0 (Ding et al., 20 Apr 2026).

3. Concurrency, unfoldings, and event-structure semantics

A central fault line in the QPN literature concerns concurrency semantics. Early symbolic and buffer-oriented models largely inherit classical enabling and firing, and represent quantum effects through weighted arcs, q-token state updates, or gate-labeled transitions. These constructions can express parallelism operationally, but they do not provide the same unfolding-based semantics as classical occurrence-net theory (Zhang et al., 2017, Shah et al., 2024).

The amplitude-weighted approach introduces concurrences explicitly. A concurrence is a multiset of mutually independent enabled transitions whose firing is atomic, and parallel composition of component nets is reflected algebraically by Kronecker structure on rate matrices. This preserves classical causal structure while placing superposition and interference at the level of amplitudes assigned to direct concurrence reachability (Schmidt, 2021).

The event-structure formulation makes concurrency foundational rather than derived. It starts from classical unfolding into an occurrence net, extracts the associated event structure, and equips configurations and configuration intervals with a quantum valuation satisfying three axioms: Obliviousness, Functoriality, and the Drop condition. In the resulting quantum event structure, if $$Q=(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\}_{t\in T},Acc,\Lambda,A,Sec),$$1, then

$$Q=(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\}_{t\in T},Acc,\Lambda,A,Sec),$$2

and for $$Q=(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\}_{t\in T},Acc,\Lambda,A,Sec),$$3,

$$Q=(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\}_{t\in T},Acc,\Lambda,A,Sec),$$4

The Drop condition then ensures that $$Q=(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\}_{t\in T},Acc,\Lambda,A,Sec),$$5 defines a proper probability valuation over configurations (Joachim et al., 1 Sep 2025).

The SPO-QPN extends this line to true-concurrency security semantics. It branch-expands transitions, unfolds to an occurrence net $$Q=(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\}_{t\in T},Acc,\Lambda,A,Sec),$$6, and interprets finite partially ordered concurrent executions as configurations $$Q=(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\}_{t\in T},Acc,\Lambda,A,Sec),$$7. Observable behavior is not a word but a pomset: $$Q=(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\}_{t\in T},Acc,\Lambda,A,Sec),$$8 where $$Q=(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\}_{t\in T},Acc,\Lambda,A,Sec),$$9 removes $PN=(P,T,F,W,M_0)$0-events and preserves observable causal dependencies. For any topological sort of a configuration, the cumulative denotation

$PN=(P,T,F,W,M_0)$1

is well defined because structurally independent concurrent events commute. The paper states both a Well-Definedness of Configuration Denotation theorem and a True-Concurrency Invariance theorem: different linearizations of the same configuration induce identical CP semantics, identical probabilities, identical normalized posteriors, and the same observation pomset. It also proves a classical conservativity theorem showing reduction to classical probabilistic unfoldings when the initial state and all transition maps are classical in the specified sense (Ding et al., 20 Apr 2026).

This progression suggests that the most technically mature QPN semantics are those that move from interleaving traces to partial orders, and from ad hoc token-state encodings to configuration-level valuations.

4. Quantum state update, measurement, and observability

All QPN variants preserve the Petri-net distinction between structure and execution, but they encode quantum state evolution in different mathematical objects. In the symbolic extension, weighted arcs realize amplitude mixing, interference is represented by linear addition of markings, and measurement is given by an operator $PN=(P,T,F,W,M_0)$2 such that

$PN=(P,T,F,W,M_0)$3

when the eigenstates of $PN=(P,T,F,W,M_0)$4 are used as the measurement basis (Zhang et al., 2017).

In the simplified q-token model, a transition consumes the required q-tokens, applies a quantum operation $PN=(P,T,F,W,M_0)$5, and produces output q-tokens in post-places. For pure states, the update is

$PN=(P,T,F,W,M_0)$6

and for density operators,

$PN=(P,T,F,W,M_0)$7

Measurement transitions use Kraus operators $PN=(P,T,F,W,M_0)$8 with

$PN=(P,T,F,W,M_0)$9

The same framework states explicitly that QPN transitions never duplicate data q-tokens, so routing is implemented as transfer or SWAP-like movement rather than copying, in line with no-cloning constraints (Shah et al., 2024).

The SPO-QPN introduces a more refined notion of observability because the attacker sees both public branch labels and localized quantum side information. For an observation pomset $Q=(S,r)$0, configurations consistent with $Q=(S,r)$1 are partitioned into secret and non-secret classes,

$Q=(S,r)$2

The attacker-visible posterior aggregates are

$Q=(S,r)$3

with normalized posteriors

$Q=(S,r)$4

Leakage is then quantified by the trace distance between secret and non-secret attacker-localized posteriors: $Q=(S,r)$5 The same work defines a robust upper bound

$Q=(S,r)$6

and proves $Q=(S,r)$7. Under the stated classicality conditions, qualitative current-state opacity and the leakage metric reduce to their classical counterparts, with the quantum trace-distance metric collapsing to total-variation distance on classical posterior marginals (Ding et al., 20 Apr 2026).

5. Compositionality, verification, and exact analysis

One major contribution of the amplitude-weighted line is a compositional operator calculus on QPNs. For shape-equivalent nets, the paper proves laws including

$Q=(S,r)$8

It also states a universality theorem: any quantum gate circuit defines a QPN whose rate matrix equals the circuit’s operator matrix, and a second theorem gives QPN realizations of the universal Clifford+T gate set (Schmidt, 2021).

The event-structure framework focuses less on circuit universality and more on semantic correctness and tractable local checking. Its key reductions are the “single-extension suffices” theorem for the Drop condition and the cluster factorization theorem, under which positivity checks reduce to local conflict clusters and, for clique clusters, become linear in cluster size. The same framework also provides parallel composition by disjoint union and drop-preserving joins that preserve race-freeness and the Drop condition (Joachim et al., 1 Sep 2025).

The SPO-QPN advances verification further by giving an exact symbolic algorithm for structural opacity and quantum leakage over a finite target set of observation pomsets. The pipeline has three components: targeted unfolding exploration over a prefix-closed search space, aggregation only at maximal unobservable reach, and stabilizer-tableau propagation for exact CP semantics in the stabilizer fragment. The algorithm explores triples $Q=(S,r)$9, updates tableaux via $(D,P,T,E,\mu,v)$0, records exact branch weight $(D,P,T,E,\mu,v)$1, and reduces to the attacker interface through $(D,P,T,E,\mu,v)$2 (Ding et al., 20 Apr 2026).

Under the divergence-free assumption with respect to $(D,P,T,E,\mu,v)$3, the paper proves termination and exactness, and gives the complexity bound

$(D,P,T,E,\mu,v)$4

The bound isolates the attacker-side exponential dependence in $(D,P,T,E,\mu,v)$5, rather than in the total register count $(D,P,T,E,\mu,v)$6, and is contrasted with interleaving-based exploration and dense-matrix simulation. The same paper further introduces a counterexample-guided leakage-enforcement loop with controllable transitions, uncontrollable transitions, invisible masking transitions, admissibility filtering, and formal guarantees such as contractivity under localizable masking, complete twirling to zero leakage, and linear leakage scaling under generalized depolarization (Ding et al., 20 Apr 2026).

6. Applications, limitations, and recurrent misunderstandings

Applications span protocol analysis, circuit semantics, concurrency theory, and quantum networking. The 2017 symbolic extension models and analyzes the SLAZ2013 counterfactual communication protocol, including blocking and passing modes, nested interferometers, and quantum Zeno cycles, and reports close agreement between Petri-net-derived probabilities and independent MATLAB calculations. Representative tables show passing-mode rates $(D,P,T,E,\mu,v)$7 for the listed $(D,P,T,E,\mu,v)$8 values and blocking-mode rates approaching near-unit values as $(D,P,T,E,\mu,v)$9 increases (Zhang et al., 2017).

The simplified operational model is used to construct a quantum S–R flip-flop from CNOT, SWAP, CCNOT, and controlled SWAP patterns, replicate it into registers, and then build SISO, SIMO, MISO, MIMO, and priority quantum buffers. Validation is reported using OpenQASM and Qiskit on IBM Quantum Composer on IBM Brisbane, QPU-Eagle-R3 (Version 1.1.41), 6 qubits, with 100-shot histograms for the Q–S–R designs and the observation that Circuit 2 yielded higher correct next-state counts than Circuit 1 and a previously implemented Q–J–K flip-flop (Shah et al., 2024).

The amplitude-weighted framework uses worked examples such as double-slit interference, Bell-state preparation, gate libraries, and teleportation to show that superposition, interference, entanglement, and measurement-conditioned control can be represented directly on reachability graphs and concurrences (Schmidt, 2021). The event-structure framework provides Bell-state preparation, measurement-based branching, and teleportation sketches primarily as semantic demonstrations of LQONs, QES compatibility, and compositional joins (Joachim et al., 1 Sep 2025).

The SPO-QPN line centers on security verification. Its entanglement-swapping case study uses two EPR pairs $$(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\},Acc,\Lambda,A,Sec)$$0 and $$(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\},Acc,\Lambda,A,Sec)$$1, a memory qubit $$(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\},Acc,\Lambda,A,Sec)$$2, a non-secret swap lane, and a secret purification lane with a hidden CNOT into memory. For the foreground observation pomset $$(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\},Acc,\Lambda,A,Sec)$$3, the attacker interface is $$(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\},Acc,\Lambda,A,Sec)$$4, the non-secret posterior is $$(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\},Acc,\Lambda,A,Sec)$$5, the secret posterior is $$(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\},Acc,\Lambda,A,Sec)$$6, and the exact leakage is

$$(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\},Acc,\Lambda,A,Sec)$$7

In a bounded family $$(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\},Acc,\Lambda,A,Sec)$$8, the same study reports quotient-based runtime gains of $$(P_C,P_Q,T,F_C,M_0,\rho_0,\{\mathbb I_t\},Acc,\Lambda,A,Sec)$$9 ms vs $|\Psi\rangle=\sum_{i=1}^n |\phi_i\rangle$0 ms at $|\Psi\rangle=\sum_{i=1}^n |\phi_i\rangle$1, $|\Psi\rangle=\sum_{i=1}^n |\phi_i\rangle$2 ms vs $|\Psi\rangle=\sum_{i=1}^n |\phi_i\rangle$3 ms at $|\Psi\rangle=\sum_{i=1}^n |\phi_i\rangle$4, $|\Psi\rangle=\sum_{i=1}^n |\phi_i\rangle$5 ms vs $|\Psi\rangle=\sum_{i=1}^n |\phi_i\rangle$6 ms at $|\Psi\rangle=\sum_{i=1}^n |\phi_i\rangle$7, and $|\Psi\rangle=\sum_{i=1}^n |\phi_i\rangle$8 ms vs $|\Psi\rangle=\sum_{i=1}^n |\phi_i\rangle$9 ms at $Q=(S,r)$00; for enforcement with $Q=(S,r)$01, choosing $Q=(S,r)$02 yields closed-loop leakage $Q=(S,r)$03 (Ding et al., 20 Apr 2026).

Several limitations recur across the literature. The symbolic extension focuses on pure states, uses the scaling relation $Q=(S,r)$04, and does not provide density-matrix or noise/decoherence semantics (Zhang et al., 2017). The simplified q-token model is intentionally close to standard Petri-net practice, treats measurement primarily as a validation-stage operation, and does not report formal fidelity metrics (Shah et al., 2024). The amplitude-weighted rate-matrix model gives projective measurement onto markings as its primary measurement semantics and notes that rate matrices need not be unitary unless reversibility and conjugate symmetry hold (Schmidt, 2021). The event-structure framework assumes finite-dimensional Hilbert spaces and race-free constraints, and identifies co-reflection or adjunction results in the quantum setting as future work (Joachim et al., 1 Sep 2025). The SPO-QPN framework restricts exact symbolic verification to the stabilizer fragment, assumes safety and divergence-freedom with respect to $Q=(S,r)$05, and leaves non-safe nets, dynamic register allocation, and time constraints out of scope (Ding et al., 20 Apr 2026).

A persistent misunderstanding is that QPN denotes a single settled formalism. The literature instead shows a spectrum: symbolic pure-state encodings, rate-graph semantics, operational q-token models, unfolding-based quantum event structures, and safe partially observed security models. Another misunderstanding is that every QPN automatically provides concurrency-aware quantum semantics. Later event-structure and SPO-QPN work is explicit that earlier proposals typically lacked rigorous concurrent semantics, compositionality, analysis tooling, or unfolding theory; those properties are contributions of specific frameworks rather than generic features of all Petri-net-based quantum models (Joachim et al., 1 Sep 2025)