Quantum-Inspired LIF (QI-LIF) Models

• Quantum-Inspired LIF is a family of models that reformulate classical leaky integrate-and-fire dynamics by integrating probabilistic spike timing and qubit-inspired state representations.
• These models span diverse approaches, including probabilistic temporal packet modeling, qubit population encoding, and canonically quantized memristive circuits, each tailored for specific neural computations.
• QI-LIF approaches preserve event-driven spike logic while enriching state dynamics, offering improved timing accuracy and efficiency in both biological modeling and engineered quantum neural architectures.

Searching arXiv for the cited QI-LIF and closely related LIF papers to ground the synthesis. Using arXiv search to verify directly relevant QI-LIF and QLIF papers. Quantum-Inspired Leaky Integrate-and-Fire (QI-LIF) denotes a heterogeneous research direction in which the classical leaky integrate-and-fire template is retained, but its state variable, leak mechanism, threshold semantics, or spike-timing interpretation is reformulated using quantum-inspired probability packets, single-qubit population dynamics, or explicitly quantized circuit models. Recent arXiv usage does not refer to a single canonical model: one line treats first-spike onset as a probabilistic temporal packet, another encodes the neuron state as a qubit excited-state population updated by $R_x$ rotations and $T_1$-style relaxation, and a third derives a memristive LIF neuron by canonical quantization in circuit QED (Johnson et al., 3 Oct 2025, Brand et al., 2024, Brand et al., 26 Jun 2025).

1. Scope, usage, and terminology

The term is used unevenly across adjacent literatures. In directly relevant work, QI-LIF names a probabilistic extension of classical onset timing or a qubit-based LIF analogue. In neighboring work, similar abbreviations refer to different objects altogether: “QIF” denotes Quadratic Integrate-and-Fire, not quantum-inspired LIF, and “QB-LIF” denotes Quantized Burst Leaky Integrate-and-Fire, not a quantum neuron (Wan et al., 10 Nov 2025, Bai et al., 28 Apr 2026).

A concise way to separate the main strands is the following.

Strand	Core idea	Representative paper
Probabilistic QI-LIF	First-spike time represented by a Gaussian temporal packet	(Johnson et al., 3 Oct 2025)
Qubit-based QLIF	Neuron state encoded as single-qubit excited-state population	(Brand et al., 2024)
Hybrid forecasting QLIF	QLIF cell embedded in a classical recurrent regressor	(Marchisio et al., 18 May 2026)
Quantized memristive LIF	Canonically quantized circuit with Lindblad leak and classical threshold/reset	(Brand et al., 26 Jun 2025)
Adjacent but non-quantum lines	Integer LIF, burst quantization, event-based sampling	(Vidybida, 2015, Bai et al., 28 Apr 2026, Moser et al., 2024)

A recurrent source of confusion is the use of “quantization.” In the event-based ECG paper associated with (Moser et al., 2024), the operative mechanism is send-on-delta (SOD) sampling: an event is triggered whenever “a signal’s amplitude changed more than a threshold value $\theta>0$,” producing a signed spike train. That paper does not formalize a leaky integrate-and-fire operator, does not analyze leak or reset equations, and does not introduce any quantum formalism; its relevance is indirect and limited to threshold-based spike encoding (Moser et al., 2024).

2. Classical LIF substrate and discrete-state reformulations

The classical substrate remains the standard LIF mechanism: a scalar subthreshold state integrates input, decays toward rest, emits a spike at threshold, and resets. One explicit reference formulation uses

$$V(t+s)=e^{-s/\tau}V(t), \qquad V\mapsto V+h, \qquad \text{fire if } V\ge V_0,\qquad V\mapsto 0,$$

with subthreshold state space $V(t)\in[0;V_0[$ (Vidybida, 2015). QI-LIF proposals depart from this template chiefly by changing the state representation rather than abandoning thresholded event generation altogether.

A particularly relevant adjacent line is the integer-coded simulation of LIF. There, the continuum of subthreshold voltages is replaced by discrete representative values indexed by an integer pair $\{n,i\}$, with

$$V_{n,i}=\alpha^n V_0\left(\alpha+\frac{i}{N}(1-\alpha)\right),$$

and leak becomes the exact symbolic update

$\{n,i\}\xrightarrow{dt}\{n+1,i\}.$

This construction preserves the ordinary spike logic while making state identity exact at the representation level. The paper reports that $\delta V \le 2.0\times 10^{-11}$ was sufficient to guarantee identical spike times between integer-coded and floating-point LIF for all tested parameter combinations over one hour of simulated time (Vidybida, 2015).

This discrete-state line is not quantum-inspired in any formal sense. It nevertheless establishes two themes that recur in QI-LIF work: first, the membrane variable can be replaced by a bounded symbolic or latent state without losing spike-train semantics; second, exact state comparability becomes a design objective when recurrent dynamics, recurrence detection, or architecture-level state graphs matter. Those themes reappear, in a different mathematical idiom, when the membrane variable is replaced by a probability packet or a qubit excitation probability.

3. Probabilistic temporal QI-LIF

One explicit use of the term “quantum-inspired leaky integrate-and-fire” concerns action potential onset timing rather than recurrent spike trains. In that formulation, the classical LIF baseline predicts a deterministic first-spike latency from

$$C_m \frac{dV(t)}{dt} = I_{inj} - \frac{V(t)}{R_m}, \qquad V(t) = V_\infty \left(1 - e^{-t/\tau_m}\right),\qquad V_\infty = I_{inj}R_m,$$

with threshold-crossing time

$T_1$0

The paper then introduces a stimulus-accelerated latency model,

$T_1$1

and places a Gaussian temporal packet around the deterministic onset estimate: $T_1$2 The model is therefore a probabilistic timing layer built on top of LIF or stimulus-accelerated LIF, not a replacement of the subthreshold dynamics by a quantum equation (Johnson et al., 3 Oct 2025).

The biological motivation is latency coding. The paper reports that increasing stimulus voltage from 10 to 50 V reduces AP delay by about 1.8 ms per 10 V step, that increasing pulse width from 50 to 200 $T_1$3 reduces AP latency from 4.2 ms to 1.5 ms, and that AP amplitude remains stable within about $T_1$4 mV (Johnson et al., 3 Oct 2025). On the reported synthetic benchmarks, the quantum-inspired model improves onset-time prediction most clearly in the moderate-to-high stimulus regime. For example, at 50.0 V the table gives experimental latency 1.38 ms, SA-LIF 0.50 ms, QI 1.01 ms, with 63.89% versus 26.95% error; at 18.9 V, by contrast, SA-LIF is slightly better, with 3.98% versus 6.10% error (Johnson et al., 3 Oct 2025).

This version of QI-LIF is “quantum-inspired” only in a narrow formal sense. The paper explicitly does not provide a Schrödinger equation, a Hamiltonian, a complex wavefunction $T_1$5, or a measurement postulate. The imported object is the Gaussian wave packet in time, interpreted as a normalized probability density over firing times (Johnson et al., 3 Oct 2025). A common misconception is therefore to read this model as a quantum-mechanical neuron; it is more accurately a probabilistic first-spike latency model with quantum-inspired language.

4. Qubit-population QLIF neurons and network architectures

A second, more mechanistic strand replaces membrane voltage by the single-qubit excited-state population. In the original quantum LIF construction, the neuron state is

$T_1$6

interpreted as the probability of finding the qubit in $T_1$7. The previous state is reconstructed through the “memory” angle

$T_1$8

so that $T_1$9 reproduces the same excited-state population. Input spikes apply an $\theta>0$0 rotation, while leak is implemented by $\theta>0$1 relaxation or by an effective reverse rotation

$\theta>0$2

The resulting scalar recurrence is

$\theta>0$3

Thresholding and reset are described conceptually as in LIF: if the post-update population exceeds threshold, an output spike is emitted and the qubit is reset to $\theta>0$4 (Brand et al., 2024).

This construction is notable for its circuit minimality. The paper emphasizes that each QLIF neuron uses 1 qubit per neuron, at most 2 rotation gates per time step, and no CNOT gates (Brand et al., 2024). In classical simulation and network use, the model was instantiated as QSNN and QSCNN architectures. Reported test accuracies include 88.25% for QSNN on MNIST, 75.25% on Fashion-MNIST, and 60.36% on KMNIST, while QSCNN attains 90.62%, 70.19%, and 66.02% on the same datasets, with the paper emphasizing speed relative to other quantum baselines rather than superiority to classical ANN/CNN baselines (Brand et al., 2024).

The forecasting model QLIF-CAST pushes the same qubit-population idea into continuous regression. Each neuron executes the depth-2 circuit

$\theta>0$5

with

$\theta>0$6

and updated excitation

$\theta>0$7

Leak is again expressed through

$\theta>0$8

with fixed $\theta>0$9, and spiking uses a probability threshold of 0.75 (Marchisio et al., 18 May 2026). In the main weather benchmark, where architecture and parameter count were matched to a classical LIF baseline, QLIF-CAST achieved 15.4% lower MSE and 4.4% lower MAE. In cross-paper comparisons it occupied a speed-error trade-off regime, converging in up to 94% less training time than an LSTM-QNN baseline on wind speed, and its core two-gate circuit showed 1.2% average deviation from simulation on IBM Marrakesh (Marchisio et al., 18 May 2026).

Taken together, these qubit-population models define a distinctive QI-LIF family: bounded state $$V(t+s)=e^{-s/\tau}V(t), \qquad V\mapsto V+h, \qquad \text{fire if } V\ge V_0,\qquad V\mapsto 0,$$0, additive composition in angle space, exponential leak in population space, and conventional threshold/reset semantics layered on top.

5. Canonically quantized memristive LIF and open-system formulations

A third strand starts from a classical memristive LIF circuit and quantizes it. The classical starting point is

$$V(t+s)=e^{-s/\tau}V(t), \qquad V\mapsto V+h, \qquad \text{fire if } V\ge V_0,\qquad V\mapsto 0,$$1

where the leak element is a memristor rather than a fixed resistor. In the Strukov model,

$$V(t+s)=e^{-s/\tau}V(t), \qquad V\mapsto V+h, \qquad \text{fire if } V\ge V_0,\qquad V\mapsto 0,$$2

so the leak becomes explicitly history-dependent through $$V(t+s)=e^{-s/\tau}V(t), \qquad V\mapsto V+h, \qquad \text{fire if } V\ge V_0,\qquad V\mapsto 0,$$3 (Brand et al., 26 Jun 2025).

Because a dissipative memristor cannot be inserted directly into a closed Hamiltonian system, the leak is replaced by a weakly coupled semi-infinite transmission line. In the continuum limit the total Hamiltonian is

$$V(t+s)=e^{-s/\tau}V(t), \qquad V\mapsto V+h, \qquad \text{fire if } V\ge V_0,\qquad V\mapsto 0,$$4

with canonical commutator

$$V(t+s)=e^{-s/\tau}V(t), \qquad V\mapsto V+h, \qquad \text{fire if } V\ge V_0,\qquad V\mapsto 0,$$5

In the weak-coupling, adiabatic regime the reduced dynamics recover the LIF-like equation

$$V(t+s)=e^{-s/\tau}V(t), \qquad V\mapsto V+h, \qquad \text{fire if } V\ge V_0,\qquad V\mapsto 0,$$6

The reduced open-system description is then written as

$$V(t+s)=e^{-s/\tau}V(t), \qquad V\mapsto V+h, \qquad \text{fire if } V\ge V_0,\qquad V\mapsto 0,$$7

Leak is therefore implemented as state-dependent Lindblad damping, while memory enters through the time-dependent memristance (Brand et al., 26 Jun 2025).

Spiking, however, is still imposed classically. Thresholding is monitored via expectation values of the voltage operator, a spike is declared when the expectation crosses threshold, reset is implemented numerically as

$$V(t+s)=e^{-s/\tau}V(t), \qquad V\mapsto V+h, \qquad \text{fire if } V\ge V_0,\qquad V\mapsto 0,$$8

and the refractory period is imposed by pausing the input drive and memristor update (Brand et al., 26 Jun 2025). The paper thus occupies an intermediate position: its dynamical substrate is genuinely quantum and Hamiltonian, but its spike-generation nonlinearity remains a classical threshold/reset rule. That hybrid structure is one of the clearest indications of how a practical QI-LIF abstraction may be assembled: quantum or quantum-inspired subthreshold evolution plus conventional event logic.

6. Foundations, neighboring lines, and unresolved issues

Two older reduction papers show that effective LIF variables need not be literal membrane voltages. In the stochastic Morris–Lecar model, subthreshold dynamics near the stable point are approximated by a two-dimensional Ornstein–Uhlenbeck modulation of constant circular motion, and the radial OU process serves as a pre-firing LIF-type variable (Ditlevsen et al., 2011). In stochastic FitzHugh–Nagumo, after proving existence of a global random pullback attractor, local linearization near the stable equilibrium yields a damped rotation whose radial process satisfies

$$V(t+s)=e^{-s/\tau}V(t), \qquad V\mapsto V+h, \qquad \text{fire if } V\ge V_0,\qquad V\mapsto 0,$$9

with firing represented through a state-dependent hazard

$V(t)\in[0;V_0[$0

These embeddings suggest that a LIF analogue can legitimately be built on amplitude variables in transformed phase space rather than raw voltage, a point directly relevant to QI-LIF formulations based on bounded excitation norms or phase-amplitude decompositions (Yamakou et al., 2018).

A second neighboring foundation is event-based signal processing. For bandlimited reconstruction from LIF encoding, the spike train induces generalized samples

$V(t)\in[0;V_0[$1

and projection onto convex sets converges to a weighted pseudo-inverse of the corresponding sampling operator (Thao et al., 2022). A separate hardware line uses a clocked phase-encoding LIF circuit with adaptive refractory control

$V(t)\in[0;V_0[$2

to generate one spike per sampling window for a downstream spiking Fourier transform (Lopez-Randulfe et al., 2023). These lines are classical, but they reinforce an important theme: spike timing can be treated as a structured representation of analog information rather than as a mere by-product of threshold crossing.

Several limitations remain consistent across the QI-LIF literature. First, terminology is unstable: “QIF” is Quadratic Integrate-and-Fire, not quantum-inspired LIF, and “QB-LIF” is Quantized Burst, not a quantum neuron (Wan et al., 10 Nov 2025, Bai et al., 28 Apr 2026). Second, directly relevant QI-LIF papers often leave important mechanisms underspecified. The probabilistic first-spike model does not provide explicit reset or refractory equations, nor a fully specified hazard or decision rule beyond the Gaussian density (Johnson et al., 3 Oct 2025). The circuit-level quantum LIF formulations retain classical threshold/reset logic rather than deriving autonomous quantum spiking (Brand et al., 2024, Brand et al., 26 Jun 2025). Third, adjacent threshold-encoding papers can easily be overread: the ECG SOD work associated with (Moser et al., 2024) is about event-based sampling and downstream detection, not formal LIF analysis and not quantum mechanics (Moser et al., 2024).

The most stable encyclopedia-level characterization is therefore cautious. QI-LIF is not a single standard neuron model, but a family of LIF extensions organized around three recurring ideas: bounded latent state representations, nonclassical or probabilistic spike-time semantics, and leak mechanisms reformulated through qubit-like relaxation or open-system dissipation. What unifies these models is not a shared quantum ontology, but a shared attempt to preserve the event-driven logic of LIF while replacing the classical membrane-voltage picture by richer state spaces and richer timing laws.