Quantum Leaky Integrate-and-Fire Neurons

• Quantum Leaky Integrate-and-Fire (QLIF) models are quantum or quantum-inspired adaptations of classical neurons that integrate, leak, threshold, and reset to emulate spiking behavior.
• Gate-based QLIF neurons utilize single-qubit rotations and T1 decay to implement spiking with low circuit depth, making them suitable for NISQ hardware.
• Canonical quantization of memristive circuits and quantum-inspired latency formulations improve timing fidelity and computational efficiency in neuromorphic applications.

Quantum Leaky Integrate-and-Fire (QLIF) denotes a family of leaky integrate-and-fire neuron models that transpose core LIF mechanisms—integration, leak, thresholded firing, reset, and in some cases adaptation—into quantum or quantum-inspired form. Recent arXiv literature uses the term in at least three distinct senses: a gate-based single-qubit spiking neuron whose state is the excited-state population of a qubit, a canonically quantized memristive LIF circuit formulated as an open quantum system, and a quantum-inspired latency model in which action-potential onset is treated as a probabilistic event distributed in time rather than a deterministic threshold crossing (Brand et al., 2024, Brand et al., 26 Jun 2025, Johnson et al., 3 Oct 2025, Marchisio et al., 18 May 2026).

1. Scope, terminology, and classical antecedent

Across the recent literature, QLIF is anchored to the classical leaky integrate-and-fire neuron, typically written either as

$\tau \frac{dU(t)}{dt} = -U(t) + I_{\mathrm{in}}R$

or as

$$C_m \frac{dV(t)}{dt} = -\frac{V(t)}{R} + I_{\mathrm{in}}(t),$$

with thresholding and reset appended to produce spikes (Brand et al., 2024, Brand et al., 26 Jun 2025). In discretized form, one standard update is

$$U[t] = \beta U[t-1] + (1-\beta)I_{\mathrm{in}}[t], \qquad \beta = \exp(-1/\tau),$$

and in spiking-network form the reset term is included as

$$U[t] = \beta U[t-1] + WX[t] - S_{\mathrm{out}}[t-1]U_{\mathrm{thr}},$$

with output spike generation determined by threshold comparison (Brand et al., 2024).

A notable source of ambiguity is that the same acronym is now used for different model classes. One line of work treats QLIF as a compact gate-based neuron for NISQ hardware; another derives a quantum memristive LIF from circuit quantum electrodynamics; a third uses “QLIF” or “QI-LIF” for a stimulus-accelerated, probabilistic onset-timing model motivated by experimental neurophysiology (Brand et al., 2024, Brand et al., 26 Jun 2025, Johnson et al., 3 Oct 2025, Marchisio et al., 18 May 2026). This suggests that QLIF is presently better understood as a family label than as a uniquely standardized architecture.

Formulation	Core mechanism	Representative role
Gate-based QLIF	Single-qubit excited-state population updated by $R_X$ rotations and $T_1$ relaxation	QSNN, QSCNN, hybrid recurrent forecasting
Canonically quantized memristive QLIF	Memristive leak represented by a transmission-line bath with Hamiltonian and GKSL dynamics	Quantum neuromorphic circuit model
Quantum-inspired QI-LIF / QLIF	Stimulus-dependent acceleration plus Gaussian onset-time distribution	Action-potential latency prediction

2. Quantum-inspired latency formulations

In the action-potential timing formulation, the starting point is the classical LIF membrane equation

$C_m \frac{dV(t)}{dt} = I_{inj} - \frac{V(t)}{R_m},$

with solution

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

where $\tau_m = R_m C_m$ (Johnson et al., 3 Oct 2025). Classical onset timing is then the deterministic threshold-crossing time $t_{LIF}$, obtained by imposing $$C_m \frac{dV(t)}{dt} = -\frac{V(t)}{R} + I_{\mathrm{in}}(t),$$0. In the paper’s framing, this deterministic threshold mechanism is too rigid to reproduce the sharply decreasing, saturating latency dependence observed when stimulus strength increases.

The proposed QLIF or QI-LIF modification introduces a stimulus-dependent effective membrane time constant,

$$C_m \frac{dV(t)}{dt} = -\frac{V(t)}{R} + I_{\mathrm{in}}(t),$$1

where $$C_m \frac{dV(t)}{dt} = -\frac{V(t)}{R} + I_{\mathrm{in}}(t),$$2 is normalized stimulus intensity and $$C_m \frac{dV(t)}{dt} = -\frac{V(t)}{R} + I_{\mathrm{in}}(t),$$3 is an empirical coupling coefficient (Johnson et al., 3 Oct 2025). The corresponding potential is written as

$$C_m \frac{dV(t)}{dt} = -\frac{V(t)}{R} + I_{\mathrm{in}}(t),$$4

yielding the onset-time expression

$$C_m \frac{dV(t)}{dt} = -\frac{V(t)}{R} + I_{\mathrm{in}}(t),$$5

The explicit modeling assumption is that stronger stimuli accelerate membrane charging dynamics and thereby shorten action-potential latency.

Its most distinctive feature is the quantum-inspired treatment of onset as a probability distribution in time rather than a single deterministic event. The onset-time density is modeled by a Gaussian wave packet,

$$C_m \frac{dV(t)}{dt} = -\frac{V(t)}{R} + I_{\mathrm{in}}(t),$$6

where $$C_m \frac{dV(t)}{dt} = -\frac{V(t)}{R} + I_{\mathrm{in}}(t),$$7 is the most likely onset latency and $$C_m \frac{dV(t)}{dt} = -\frac{V(t)}{R} + I_{\mathrm{in}}(t),$$8 captures timing uncertainty or variability (Johnson et al., 3 Oct 2025). For a symmetric Gaussian,

$$C_m \frac{dV(t)}{dt} = -\frac{V(t)}{R} + I_{\mathrm{in}}(t),$$9

The same work also notes that multiple inputs can be represented by superposing multiple Gaussian packets weighted by synaptic efficacy. In this formulation, the “quantum-inspired” element is not a quantum circuit or quantum hardware realization, but a probabilistic representation of onset timing.

The biological motivation summarized in the paper is that increasing stimulus amplitude shortens AP latency, increasing pulse duration also shortens latency, and spike amplitude remains essentially constant across conditions, so timing rather than spike size encodes stimulus strength (Johnson et al., 3 Oct 2025). The model is therefore targeted at timing codes, especially in regimes where strong or rapidly changing inputs produce saturating latency curves.

3. Gate-based single-qubit QLIF neurons

The gate-based QLIF introduced as a quantum spiking neuron replaces the scalar membrane potential with the excited-state population $$U[t] = \beta U[t-1] + (1-\beta)I_{\mathrm{in}}[t], \qquad \beta = \exp(-1/\tau),$$0 of a single qubit (Brand et al., 2024). The qubit starts in $$U[t] = \beta U[t-1] + (1-\beta)I_{\mathrm{in}}[t], \qquad \beta = \exp(-1/\tau),$$1, an input spike is encoded with an $$U[t] = \beta U[t-1] + (1-\beta)I_{\mathrm{in}}[t], \qquad \beta = \exp(-1/\tau),$$2 rotation, and recurrence is implemented by reconstructing the prior population through

$$U[t] = \beta U[t-1] + (1-\beta)I_{\mathrm{in}}[t], \qquad \beta = \exp(-1/\tau),$$3

When an input spike is present, the updated population is

$$U[t] = \beta U[t-1] + (1-\beta)I_{\mathrm{in}}[t], \qquad \beta = \exp(-1/\tau),$$4

with binary input

$$U[t] = \beta U[t-1] + (1-\beta)I_{\mathrm{in}}[t], \qquad \beta = \exp(-1/\tau),$$5

Leak is implemented by allowing the qubit to idle and relax under $$U[t] = \beta U[t-1] + (1-\beta)I_{\mathrm{in}}[t], \qquad \beta = \exp(-1/\tau),$$6 decay. The explicit decay encoding is

$$U[t] = \beta U[t-1] + (1-\beta)I_{\mathrm{in}}[t], \qquad \beta = \exp(-1/\tau),$$7

and the full discrete-time update rule is

$$U[t] = \beta U[t-1] + (1-\beta)I_{\mathrm{in}}[t], \qquad \beta = \exp(-1/\tau),$$8

Thresholding is implemented by comparing the measured excited-state population to a firing threshold, after which the neuron emits a spike and resets to $$U[t] = \beta U[t-1] + (1-\beta)I_{\mathrm{in}}[t], \qquad \beta = \exp(-1/\tau),$$9 if the threshold is exceeded (Brand et al., 2024).

A central architectural claim of this line of work is compactness: the neuron uses a single qubit, a maximum of 2 rotation gates per evaluation, and no CNOT gates (Brand et al., 2024). The model is therefore explicitly positioned as low-depth and NISQ-compatible. The same neuron is then used as the basic unit in a Quantum Spiking Neural Network (QSNN) and a Quantum Spiking Convolutional Neural Network (QSCNN), with spike trains generated from image intensities, typically using a Poisson process.

QLIF-CAST adapts this gate-based lineage to continuous-valued time-series forecasting by encoding excitation as

$$U[t] = \beta U[t-1] + WX[t] - S_{\mathrm{out}}[t-1]U_{\mathrm{thr}},$$0

and applying, at each timestep, the depth-2 circuit

$$U[t] = \beta U[t-1] + WX[t] - S_{\mathrm{out}}[t-1]U_{\mathrm{thr}},$$1

with

$$U[t] = \beta U[t-1] + WX[t] - S_{\mathrm{out}}[t-1]U_{\mathrm{thr}},$$2

and post-update probability

$$U[t] = \beta U[t-1] + WX[t] - S_{\mathrm{out}}[t-1]U_{\mathrm{thr}},$$3

(Marchisio et al., 18 May 2026). The input angle is

$$U[t] = \beta U[t-1] + WX[t] - S_{\mathrm{out}}[t-1]U_{\mathrm{thr}},$$4

with spike decision

$$U[t] = \beta U[t-1] + WX[t] - S_{\mathrm{out}}[t-1]U_{\mathrm{thr}},$$5

If no input spike is present, the decay angle is

$$U[t] = \beta U[t-1] + WX[t] - S_{\mathrm{out}}[t-1]U_{\mathrm{thr}},$$6

where $$U[t] = \beta U[t-1] + WX[t] - S_{\mathrm{out}}[t-1]U_{\mathrm{thr}},$$7 is learnable and $$U[t] = \beta U[t-1] + WX[t] - S_{\mathrm{out}}[t-1]U_{\mathrm{thr}},$$8 is fixed (Marchisio et al., 18 May 2026). Training uses an arctangent surrogate,

$$U[t] = \beta U[t-1] + WX[t] - S_{\mathrm{out}}[t-1]U_{\mathrm{thr}},$$9

for backpropagation through the non-differentiable spike operation.

4. Canonically quantized memristive LIF circuits

A distinct QLIF formulation starts from a classical memristive LIF circuit in which the leak resistor is replaced by a memristor (Brand et al., 26 Jun 2025). The classical equation is

$R_X$0

with memristor state evolution

$R_X$1

The paper uses the Strukov TiO$R_X$2-type memristor model, with

$R_X$3

and, equivalently, the charge-based form

$R_X$4

Under periodic driving, the standard memristive signature is a pinched hysteresis loop in the $R_X$5-$R_X$6 plane.

The central quantization difficulty is that a memristor is dissipative and classically irreversible. The solution adopted is an open-quantum-system construction in which the leak is represented by a semi-infinite lossless transmission line built from coupled $R_X$7 oscillators, akin to a Caldeira–Leggett bath picture (Brand et al., 26 Jun 2025). The transmission line has characteristic impedance

$R_X$8

and in the adiabatic regime this is identified with the memristive resistance through

$R_X$9

The membrane node is coupled to the line through a weak coupling capacitor $T_1$0.

The formulation is developed through a circuit Lagrangian in node fluxes and then canonically quantized. The canonical momenta are

$T_1$1

and quantization promotes flux and charge to operators satisfying

$T_1$2

(Brand et al., 26 Jun 2025). In the weak-coupling limit $T_1$3, Hamilton’s equations recover

$T_1$4

which reduces to the classical memristive LIF equation under the identification $T_1$5. This weak-coupling, adiabatic reduction is a defining claim of the paper: the quantum model reproduces the classical memristive LIF limit in the appropriate regime.

For simulation, the membrane node is represented as a single-mode oscillator governed by a time-dependent Gorini–Kossakowski–Sudarshan–Lindblad master equation,

$T_1$6

with

$T_1$7

(Brand et al., 26 Jun 2025). The memristor state is updated self-consistently from the expectation value of the node voltage/current via

$T_1$8

In this construction, memory enters through the history dependence of the leak rate.

Spiking in this model is not generated as a fundamentally quantum jump operator. Instead, the voltage expectation value $T_1$9 is monitored; when threshold is crossed, the solver is interrupted and the state is reset according to

$C_m \frac{dV(t)}{dt} = I_{inj} - \frac{V(t)}{R_m},$0

followed by a refractory period during which both input drive and memristor update are paused (Brand et al., 26 Jun 2025). The paper explicitly notes that this remains a classical threshold-and-reset rule layered on top of quantum circuit dynamics.

5. Architectures, benchmarks, and reported results

The quantum-inspired latency model is evaluated on synthetic data and on tabulated stimulus-response values motivated by experimental findings rather than on a new wet-lab experiment (Johnson et al., 3 Oct 2025). In the stimulus-voltage benchmark, amplitudes range from 10 V to 50 V, and the stated experimental pattern is that voltage increases reduce AP delay by about 1.8 ms per 10 V step, pulse width increases from 50 to 200 $C_m \frac{dV(t)}{dt} = I_{inj} - \frac{V(t)}{R_m},$1 reduce latency from about 4.2 ms to 1.5 ms, and AP amplitude stays within about $C_m \frac{dV(t)}{dt} = I_{inj} - \frac{V(t)}{R_m},$2 mV. The reported relative error metric is

$C_m \frac{dV(t)}{dt} = I_{inj} - \frac{V(t)}{R_m},$3

At 10 V, experimental latency is $C_m \frac{dV(t)}{dt} = I_{inj} - \frac{V(t)}{R_m},$4 ms, SA-LIF predicts $C_m \frac{dV(t)}{dt} = I_{inj} - \frac{V(t)}{R_m},$5 ms with relative error $C_m \frac{dV(t)}{dt} = I_{inj} - \frac{V(t)}{R_m},$6, and QI predicts $C_m \frac{dV(t)}{dt} = I_{inj} - \frac{V(t)}{R_m},$7 ms with relative error $C_m \frac{dV(t)}{dt} = I_{inj} - \frac{V(t)}{R_m},$8. At 23.3 V, the errors are $C_m \frac{dV(t)}{dt} = I_{inj} - \frac{V(t)}{R_m},$9 for SA-LIF and $$V(t) = V_\infty \left(1 - e^{-t/\tau_m}\right), \qquad V_\infty = I_{inj}R_m,$$0 for QI; at 32.2 V they are $$V(t) = V_\infty \left(1 - e^{-t/\tau_m}\right), \qquad V_\infty = I_{inj}R_m,$$1 and $$V(t) = V_\infty \left(1 - e^{-t/\tau_m}\right), \qquad V_\infty = I_{inj}R_m,$$2; at 50.0 V they are $$V(t) = V_\infty \left(1 - e^{-t/\tau_m}\right), \qquad V_\infty = I_{inj}R_m,$$3 and $$V(t) = V_\infty \left(1 - e^{-t/\tau_m}\right), \qquad V_\infty = I_{inj}R_m,$$4 (Johnson et al., 3 Oct 2025). The same trend appears in the spike-count synthetic benchmark built from 100 synthetic data points with spike counts from 5 to 50 and a saturating exponential target curve $$V(t) = V_\infty \left(1 - e^{-t/\tau_m}\right), \qquad V_\infty = I_{inj}R_m,$$5. The paper repeatedly emphasizes that QI-LIF keeps relative error below about 30% for most of the range, whereas SA-LIF often exceeds 50%.

The original gate-based QLIF is evaluated through QSNN and QSCNN models on MNIST, Fashion-MNIST, and KMNIST, each with 60,000 training images and 10,000 test images (Brand et al., 2024). In the fully connected setting, the reported QSNN test accuracies are 88.25 on MNIST, 75.25 on Fashion-MNIST, and 60.36 on KMNIST, with training time 2m14s. The QNN baseline reports 89.65, 82.90, and 67.18, with training time 54m22s. In the convolutional setting, QSCNN reports 90.62, 70.19, and 66.02, with training time 5m40s, while the Quanvolutional baseline reports 92.46, 83.99, and 71.83, with training time 6h27m8s. The paper’s main quantitative claim is a favorable trade-off rather than accuracy dominance: QLIF-based models are about 24× faster than the noiseless quantum competitors, about 68× faster than the noiseless quantum convolutional competitors, and about 146× and 333× faster, respectively, in noisy settings (Brand et al., 2024).

QLIF-CAST embeds 48 QLIF neurons in Layer 2 of a seven-layer hybrid quantum-classical recurrent architecture consisting of TimeDistributed Dense, QLIF neuronal layer, BatchNorm with dropout, LSTM with 24 hidden units, two regression heads, and a linear output layer (Marchisio et al., 18 May 2026). On the Weather History dataset with 96,453 hourly observations and a chronological split of 10,000 train and 2,000 test, the parameter-matched comparison reports for QLIF-CAST: MSE 17,897, MAE 35.54, RMSE 133.8, train time 87 s, epochs 12; and for the classical LIF baseline: MSE 21,152, MAE 37.18, RMSE 145.4, train time 36 s, epochs 12. The reported improvements are 15.4% lower MSE, 4.4% lower MAE, and 8.0% lower RMSE. The per-variable breakdown states that QLIF-CAST is better on Temperature and Pressure, whereas the classical LIF model is better on Humidity and Wind Speed. In cross-domain comparisons, QLIF-CAST versus QLSTM on Bangkok air quality yields MAE 15.73 versus 11.24 and RMSE 20.62 versus 15.06, but uses 3.8× fewer training epochs; against LSTM-QNN on wind speed, QLIF-CAST yields RMSE 5.58 versus 3.92 and MAE 4.14 versus 2.87, but is 94% faster in training time (Marchisio et al., 18 May 2026).

A distinct type of empirical result appears in the canonically quantized memristive model. Numerical integration of the GKSL dynamics over 2,000 time steps under a sinusoidal drive yields a pinched hysteresis loop in the $$V(t) = V_\infty \left(1 - e^{-t/\tau_m}\right), \qquad V_\infty = I_{inj}R_m,$$6-$$V(t) = V_\infty \left(1 - e^{-t/\tau_m}\right), \qquad V_\infty = I_{inj}R_m,$$7 plane, with the quantum loop described as similar to the classical one but less smooth because of weak-coupling and adiabatic approximations together with quantum fluctuations (Brand et al., 26 Jun 2025). For the full QLIF neuron, the same simulations show integration of positive current toward threshold, inhibitory response to negative current, spike generation, reset, and refractory gaps, with asymmetric spike-train behavior due to the timing and amplitude of the sinusoidal input.

Paper	Benchmark	Reported result
(Johnson et al., 3 Oct 2025)	AP onset timing, 50.0 V	SA-LIF error $$V(t) = V_\infty \left(1 - e^{-t/\tau_m}\right), \qquad V_\infty = I_{inj}R_m,$$8, QI error $$V(t) = V_\infty \left(1 - e^{-t/\tau_m}\right), \qquad V_\infty = I_{inj}R_m,$$9
(Brand et al., 2024)	QSNN on MNIST	88.25 test accuracy, 2m14s
(Marchisio et al., 18 May 2026)	Weather forecasting	15.4% lower MSE and 4.4% lower MAE than classical LIF
(Brand et al., 26 Jun 2025)	Quantum memristor simulation	Pinched hysteresis loop over 2,000 time steps

6. Conceptual distinctions, limitations, and open directions

A recurring misconception is that “quantum” has the same meaning across all QLIF papers. It does not. In the latency-prediction paper, the construction is explicitly quantum-inspired rather than a true quantum hardware implementation; its quantum element is the Gaussian wave-packet representation of uncertain spike timing (Johnson et al., 3 Oct 2025). In the canonically quantized memristive formulation, the circuit dynamics are quantum, but the spike event remains a classical threshold-and-reset rule applied to the expectation value of voltage (Brand et al., 26 Jun 2025). In the gate-based formulations, the neuron is quantum in the sense of single-qubit state evolution, but the networks remain hybrid trainable models built around shallow circuits, classical weights, surrogate gradients, and explicit measurement (Brand et al., 2024, Marchisio et al., 18 May 2026).

Each lineage also states clear limitations. The quantized memristive model is derived in a Born-Markov, weak-coupling, adiabatic regime; if $\tau_m = R_m C_m$0 becomes comparable to the core frequency $\tau_m = R_m C_m$1, or if $\tau_m = R_m C_m$2 varies on timescales comparable to $\tau_m = R_m C_m$3, then the derivation no longer holds and a fully quantum, likely non-Markovian theory is required (Brand et al., 26 Jun 2025). QLIF-CAST is trained in PennyLane simulation rather than end-to-end on actual QPU hardware, and its Phase 2 comparisons use published QLSTM and LSTM-QNN numbers rather than retraining those baselines under identical conditions (Marchisio et al., 18 May 2026). The original QSNN and QSCNN results show substantial runtime advantages but do not surpass classical ANN, SNN, CNN, or SCNN accuracy on the reported benchmarks (Brand et al., 2024). The latency-model paper, meanwhile, uses synthetic and tabulated stimulus-response data rather than new physiological experiments (Johnson et al., 3 Oct 2025).

These distinctions matter for interpretation. One branch of QLIF is concerned with compact gate-based quantum neuromorphic primitives; another with first-principles circuit quantization and memory-bearing open quantum systems; another with probabilistic timing models for biological spike onset. This suggests that the principal unifying idea is not a single equation but a common attempt to revise the LIF neuron so that leak, integration, timing, and thresholding are represented in a way that is either quantum-native or explicitly quantum-inspired. The shared research significance, as stated across the cited papers, lies in quantum neuromorphic computing, quantum spiking neural networks, brain-inspired computing, spiking neural networks, and quantum machine learning analogues (Brand et al., 2024, Brand et al., 26 Jun 2025, Johnson et al., 3 Oct 2025, Marchisio et al., 18 May 2026).