Papers

Device-independent quantum secret sharing (DI-QSS) provides security against untrusted quantum devices. While device-independent quantum key distribution (DI-QKD) using Mermin-Peres magic square game [Zhen et al., Phys. Rev. Lett, 2023] has been proposed, we present the first DI-QSS protocol based on a pseudo-telepathy parity game without requiring dedicated rounds and specific basis configuration, unlike CHSH-based DI-QSS schemes [Zhang et al., Phys. Rev. A, 2024]. Our scheme simultaneously certifies device-independence and key generation using a single test, achieving optimal performance for a seven-qubit GHZ state configuration. Security against collective attacks is analyzed, and a positive key rate is obtained under white noise and photon loss. Moreover, we show a bitwise advantage over the previous protocol for producing the same raw key length.

Three-body interactions are fundamental for realizing novel quantum phenomena beyond pairwise physics, yet their implementation – particularly among distinct quantum systems – remains challenging. Here, we propose a hybrid quantum architecture comprising a magnonic mode (in a YIG sphere), an Andreev spin qubit (ASQ), and a superconducting qubit (SCQ), to realize a strong three-body interaction at the single-quantum level. Leveraging the spin-dependent supercurrent and circuit-integration flexibility of the ASQ, it is possible to engineer a strong tripartite coupling that jointly excites both qubits upon magnon annihilation (or excites magnons and SCQs upon ASQ deexcitation). Through analytical and numerical studies, we demonstrate that this interaction induces synchronized collapse and revival in qubit populations when the magnon is initially prepared in a coherent state. Notably, during the collapse region – where populations remain static – the entanglement structure undergoes a dramatic and continuous reorganization. We show that the genuine tripartite entanglement is redistributed into bipartite entanglement between the two qubits, and vice versa, with the total entanglement conserved. These phenomena, unattainable via two-body couplings, underscore the potential of three-body interactions for exploring intrinsically new quantum effects and advancing hybrid quantum information platforms.

This paper introduces a quantum-classical hybrid key distribution protocol applicable to any stabilizer-based quantum error-correcting code. Named the Pattern-Based Stabilizer Tripwire, the framework embeds logical data directly within the commutation relationships of physical Pauli operators while masking the underlying qubit layout using a dynamic, secret automorphism group of physical permutations. This coordinate-frame alignment ensures a 100% sifting efficiency during normal operation without discarding transmission slots. However, any active interception by an eavesdropper introduces a destructive structural logical error. Because the adversary cannot align with the hidden spatial frame transformation, the receiver's look-up decoder is blinded by a mismatched syndrome. This architectural misalignment forces an uncorrectable 50% Quantum Bit Error Rate upon interception, converting subtle anomalies into a deterministic cryptographic tripwire. To secure private key generation, the algorithm utilizes a piece-wise conditional XOR framework to dynamically modulate data commutation relationships via public classical error syndromes. By executing an algebraic XOR operation that selectively fuses or withholds local channel noise syndromes and targeted, commutation-changing operator sub-vectors, the public classical channel functions as an information-theoretic blinding mask. Because these public syndromes consist of discrete, commuting error correction patches rather than non-commuting geometric basis frames ([M, S_total] = 0), the protocol enforces a strict classical information bottleneck on the adversary, completely neutralizing the threat of retroactive quantum memory or collective attacks. When combined with a chronological 50-50 channel split, privacy amplification efficiently compresses the fragmented data, achieving an absolute collapse of the adversary's mutual information to zero.

We study the possibility of detection of “spin-boson” entanglement by qubit only measurements. Such entanglement is impossible to detect by previously proposed schemes that involve a fixed system-environment interaction, because of inherent symmetries within the coupling and the initial state of the environment. We take advantage of the possibility of tuning of qubit-environment coupling, that is available in some qubit realizations. As an example we study a superconducting transmon qubit interacting with a microwave cavity, which is one of such systems and is, furthermore, essential in the context of quantum information processing. We propose suitable Hamiltonian parameters for the preparation and measurement phases of the detection scheme that allow for an experimental test, and verify that the reported signal is nonnegligibly large still at finite temperatures.

Distributed, large-scale quantum computing will need architectures that combine matter-based qubits with photonic links, but today's software stacks target either gate-based chips or linear-optical devices in isolation. We introduce Optyx, an open-source Python framework offering a unified language to program, simulate, and prototype hybrid, networked systems: users create experiments that mix qubit registers, discrete-variable photonic modes, lossy channels, heralded measurements, and real-time feedback; Optyx compiles them via ZX/ZW calculus into optimised tensor-network forms, and executes with state-of-the-art contraction schedulers based on Quimb and Cotengra. Benchmarking on exact multi-photon circuit simulations shows that, versus permanent-based methods, tensor network contraction can deliver speedups of orders of magnitude for low-depth circuits and entangled photon sources, and natively supports loss and distinguishability – establishing it as both a high-performance simulator and a rapid-prototyping environment for next-generation photonic-network experiments.

We present a new model of quantum computation rooted in the representation theory of the mass less sector of unitary irreducible representations of the extended Poincare group developed in [1].

In this study, we propose the Quantum Gradient Flow Algorithm (QGFA), a novel quantum algorithm for solving symmetric positive definite (SPD) linear systems based on the variational formulation and time-evolution dynamics. Conventional quantum linear solvers, such as the quantum matrix inverse algorithm (QMIA), focus on approximating the matrix inverse through quantum signal processing (QSP). However, QMIA suffers from a crucial drawback: its computational efficiency deteriorates as the condition number increases. In contrast, classical SPD linear solvers, such as the steepest descent and conjugate gradient methods, are known for their fast convergence, which stems from the variational optimization principle of SPD systems. Inspired by this, we develop QGFA, which obtains the solution vector through the gradient-flow process of the corresponding quadratic energy functional. To validate the proposed method, we apply QGFA to the displacement-based finite element method (FEM) for two-dimensional linear elastic problems under plane stress conditions. The algorithm demonstrates accurate convergence toward classical FEM solutions even with a moderate number of QSP phase factors. Compared with QMIA, QGFA achieves lower relative errors and faster convergence when initialized with suitable initial states, demonstrating its potential as an efficient preconditioned quantum linear solver. The proposed framework provides a physically interpretable connection between classical iterative solvers and quantum computational paradigms. These findings suggest that QGFA can serve as a foundation for future developments in Quantum Computer-Aided Engineering (Quantum CAE), including nonlinear and multiphysics simulations.

Conservation laws are of paramount importance in our understanding of classical and quantum dynamics. Here, we present a general framework for constructing exact quantum integrals of motion with the desired locality and quantum numbers, which will be illustrated for the case of many-body-localization (MBL). The latter has been understood theoretically in terms of the existence of an extensive number of local integrals of motion (LIOMs). Using our approach, we show that one can express the task of finding LIOMs as an optimization problem. For some specifications, this problem surprisingly connects to the question of finding classical ground states of spin-glass models. Our work unifies previous results obtained in the MBL context and reveals intriguing connections between many-body localization, spin-glass physics, and constrained optimization problems.

Quantum circuit design is a key bottleneck for practical quantum machine learning on complex, real-world data. We present an automated framework that discovers and refines variational quantum circuits (VQCs) using graph-based Bayesian optimization with a graph neural network (GNN) surrogate. Circuits are represented as graphs and mutated and selected via an expected improvement acquisition function informed by surrogate uncertainty with Monte Carlo dropout. Candidate circuits are evaluated with a hybrid quantum-classical variational classifier on the next generation firewall telemetry and network internet of things (NF-ToN-IoT-V2) cybersecurity dataset, after feature selection and scaling for quantum embedding. We benchmark our pipeline against an MLP-based surrogate, random search, and greedy GNN selection. The GNN-guided optimizer consistently finds circuits with lower complexity and competitive or superior classification accuracy compared to all baselines. Robustness is assessed via a noise study across standard quantum noise channels, including amplitude damping, phase damping, thermal relaxation, depolarizing, and readout bit flip noise. The implementation is fully reproducible, with time benchmarking and export of best found circuits, providing a scalable and interpretable route to automated quantum circuit discovery.

Exceptional points are singularities in the spectrum of non-Hermitian systems in which several eigenvectors are linearly dependent and their eigenvalues are equal to each other. Usually it is assumed that the order of the exceptional point is limited by the number of degrees of freedom of a non-Hermitian system. In this letter, we refute this common opinion and show that non-Markovian effects can lead to dynamics characteristic of systems with exceptional points of higher orders than the number of degrees of freedom in the system. This takes place when the energy returns from reservoir to the system such that the dynamics of the system are divided into intervals in which it describes by the product of the exponential and a polynomial function of ever-increasing order. We demonstrate that by choosing the observation time, it is possible to observe exceptional points of arbitrary high orders.

Uncertainty relations are fundamental to quantum mechanics, encoding limits on the simultaneous measurement of conjugate observables. Violations of joint uncertainty bounds can certify entanglement – a resource critical for quantum information protocols and increasingly relevant in strong-field physics. Here, we investigate the pairwise time-delay and frequency-bandwidth uncertainties for arbitrary multimode quantum states of light, deriving a general lower bound for their joint product. We find that the nonclassical correction scales inversely with the average photon number, a behavior rooted in the so-called “monogamy of entanglement”. These results clarify the intensity scaling of quantum advantages in nonclassical light states and highlight the interplay between entanglement and photon statistics.

Witnessing and tracking topological phase transitions induced by interactions with the environment is a crucial challenge. Among the various experimental approaches to detect topological properties, the Mean Chiral Displacement (MCD) has emerged as a powerful bulk probe in one-dimensional chiral systems, allowing the extraction of the topological invariant from single-particle dynamics. Here we study the dynamics of a single particle in a one-dimensional Su-Schrieffer-Heeger chain coupled to multiple cavity modes via inter-cell hopping terms, focusing on the out-of-equilibrium behavior of the MCD. We show that, whenever the frequency is larger than the static hopping amplitudes, the coupling induces a discontinuous jump in the MCD, already at small times, signaling that such a coupling also leaves a signature in the survival edge probability when the dynamics are initialized at one of the two edges. For frequencies comparable to the static hopping amplitudes, topological order competes with dissipative effects, which makes the MCD behave smoothly, retaining information about the driven-dissipative topology.

Accurate knowledge of the uneven free spectral range of an optical microresonator, which provides direct insight into group velocity dispersion, is essential for understanding and controlling Kerr frequency comb dynamics. In this work, we present a simple and highly precise method formeasuring the free spectral range over a 5 THz bandwidth in silicon nitride microresonators, leveraging a wavemeter with 0.4 MHz resolution. Our fully fibered plug-and-play experimental setup enables the accurate extraction of resonance frequencies. By carefully analyzing the spectral position of each resonance, we measure both second- and third-order free spectral range expansion coefficients. This approach offers a robust and accessible tool for dispersion characterization in integrated photonic circuits, paving the way for next-generation of Kerr comb sources and quantum photonic technologies.

Quantum neural networks (QNNs) and parameterized quantum circuits (PQCs) are key building blocks for near-term quantum machine learning. However, their scalability is constrained by excessive parameters, barren plateaus, and hardware limitations. We propose LiePrune, the first mathematically grounded one-shot structured pruning framework for QNNs that leverages Lie group structure and quantum geometric information. Each gate is jointly represented in a Lie group–Lie algebra dual space and a quantum geometric feature space, enabling principled redundancy detection and aggressive compression. Experiments on quantum classification (MNIST, FashionMNIST), quantum generative modeling (Bars-and-Stripes), and quantum chemistry (LiH VQE) show that LiePrune achieves over $10\times$ compression with negligible or even improved task performance, while providing provable guarantees on redundancy detection, functional approximation, and computational complexity.

When light propagates through complex media, its output spatial distribution is highly sensitive to its wavelength. This fundamentally limits the bandwidth of applications ranging from imaging to communication. Here, we demonstrate analytically and numerically that the spatial correlations of hyper-entangled photon pairs, simultaneously entangled spatially and spectrally, remain stable across a broad bandwidth: The chromatic modal dispersion experienced by one photon is canceled to first order by its spectrally anti-correlated twin, defining a "two-photon bandwidth" that can far exceed its classical counterpart. We illustrate this modal dispersion cancellation in multimode fibers, thin diffusers and blazed gratings, and demonstrate its utility for broadband wavefront shaping of quantum states. These findings advance our fundamental understanding of quantum light in complex media with applications in quantum imaging, communication, and sensing.

Quantum frequency converters are key enabling technologies in photonic quantum information science to bridge the gap between quantum emitters and telecom photons. Here, we report a coherent frequency converter scheme combining a fiber-coupled nonlinear optical Lithium Niobate waveguide with a fiber-pigtailed single-photon source based on semiconductor quantum dots. Single and indistinguishable photons are converted from 925.7 nm to the telecommunication C-band, with a 48.4% end-to-end efficiency and full preservation of single-photon purity and indistinguishability. The integration of the two fiber-based modules achieving top-level performance represents an important step toward the practical interconnection of future quantum information processing systems operating at different wavelengths.

Analyzing routes of transport for open quantum systems with non-equilibrium initial conditions is extremely challenging. The state-to-state approach [A. Bose, and P.L. Walters, J. Chem. Theory Comput. 2023, 19, 15, 4828-4836] has proven to be a useful method for understanding transport mechanisms in quantum systems interacting with dissipative thermal baths, and has been recently extended to non-Hermitian systems to account for empirical loss. These non-Hermitian descriptions are, however, not capable of describing empirical processes of more general nature, including but not limited to a variety of pumping processes. We extend the state-to-state analysis to account for Lindbladian descriptions of generic dissipative, pumping and decohering processes acting on a system which is exchanging energy with a thermal bath. This Lindblad state-to-state method can elucidate routes of transport in systems coupled to a bath and additionally acted upon by Lindblad jump operators. The method is demonstrated using examples of excitonic aggregates subject to incoherent pumping and draining processes. Using this new state-to-state formalism, we demonstrate the establishment of steady-state excitonic currents across molecular aggregates, yielding a different first-principles approach to quantifying the same.

The Mpemba effect (MpE), where a far-from-equilibrium state of a system relaxes faster compared to a state closer to it, is a well-known counterintuitive phenomenon in classical and quantum systems. Various system-specific theories have been proposed to explain this anomalous behavior in driven systems, though the fundamental mechanism of MpE in undriven systems, where MpE was first observed, remains unresolved. This paper provides a generic model of MpE for a quantum spin-$1/2$ system following Markovian relaxation dynamics, regardless of system structure or environment. The key lies in the overlap of initial states with the fast relaxation mode; here, the constituents create a fast decay mode via interaction through the shared environment to show MpE, indicating that MpE happens due to the collective behavior of the system. I also show that a system with anisotropic relaxation naturally exhibits MpE, even without a shared environment among the particles.

We theoretically propose an experimentally viable scheme to explore the transfer of nonclassical correlations from a dipolar Bose-Einstein condensate (BEC) to a pair of impurities immersed in it. Operating at ultra-low temperature, density fluctuations of the dipolar BEC emulate a vacuum field with Lorentz-violating dispersion, while the two impurities function as Unruh-DeWitt detectors for the BEC quasiparticles. We study the harvesting of entanglement from the quantum vacuum of this analogue Lorentz-violating quantum field by spatially separated Unruh-DeWitt detectors. Our analysis reveals key parameter dependencies that optimize the harvesting of entanglement. In particular, unlike the Lorentz-invariant case, smoother detector switchings does not enhance the entanglement harvesting efficiency from the Lorentz-violating quantum field vacuum. Moreover, the strength of the Lorentz-invariant violation can shift the optimal energy structure of the detectors for harvesting entanglement from the Lorentz-violating quantum field vacuum-a clear deviation from the Lorentz-invariant scenario. As a fundamental quantum mechanical setup, our quantum fluid platform provides an experimentally realizable testbed for examining the entanglement harvesting protocol from an effective Lorentz-violating quantum field vacuum using a pair of impurity probers, which may also has potential implications for exploring the Lorentz-invariant violation in quantum field theory.

We study a unitary deformation of the relation between a positive self-adjoint reference Hamiltonian $H$ and the generator of phase evolution, \begin{equation*} F_\beta(H) = H + \beta H \log(H/E_*). \end{equation*} Since $F_\beta(H)$ is a real function of $H$, it has the same eigenvectors, and it preserves unitarity, purity and the Born rule. It does not cause decoherence in the sense of a loss of purity. Its effect is a nonlinear distortion of the spectrum. Changing the reference scale $E_*$ adds a term linear in $H$, which cannot be distinguished from a rescaling of the spectrum. As a result, a single transition frequency does not determine $\beta$, and neither does the static dephasing of an ensemble; a finite-width Ramsey model makes this explicit. The part of the deformation that does not depend on $E_*$ is contained in second and higher spectral differences, which sample the curvature of the spectral map. For a nominally harmonic ladder $E_n=\omega(n+\delta)$, the adjacent-transition anharmonicity is $A_n/\omega=\beta c_n(\delta)$, with $c_n\sim(n+\delta)^{-1}$. This decreasing template can be distinguished in a joint fit from ideal Kerr and specified polynomial nonlinearities using sufficiently many resolved levels, subject to platform-specific modeling of other effects. We describe a spectroscopy protocol, estimate the precision of a joint fit, and show a synthetic example. The result depends on the choice of energy origin, which second differences do not remove, and it applies only to a single system during free evolution, because the deformation is not additive over subsystems.

The nitrogen-vacancy (NV) center can serve as a magnetic sensor for electron paramagnetic resonance (EPR) measurements. Benefiting from its atomic size, the diamond chip can integrate a tremendous amount of NV centers to improve the magnetic-field sensitivity. However, EPR spectroscopy using NV ensembles is less efficient due to inhomogeneities in both sensors and targets. Spectral line broadening induced by ensemble averaging is even detrimental to spectroscopy. Here we show a kind of cross-relaxation EPR spectroscopy at zero field, where the sensor is tuned by an amplitude-modulated control field to match the target. The modulation makes detection robust to the sensor's inhomogeneity, while zero-field EPR is naturally robust to the target's inhomogeneity. We demonstrate an efficient EPR measurement on an ensemble of roughly 30000 NV centers. Our method shows the ability to not only acquire unambiguous EPR spectra of free radicals, but also monitor their spectroscopic dynamics in real time.

We present a Mathematica codebase for computing $\ell$-multilinear ranks ($\ell$-multiranks) of multiqudit quantum states using tensor-flattening techniques. By calculating the ranks of all bipartition-induced matricizations, the method provides an efficient criterion for detecting Genuine Multipartite Entangled (GME) states in systems with local dimension $d$. The code automatically generates all required tensor reshapes and outputs the full $\ell$-multirank profile, offering a practical tool for characterizing entanglement in high-dimensional multiqudit systems.

This text is written in remembrance of Vladimir (Volodia) Sidorenko, honoring his scientific work, his kindness, and the warm and humorous spirit he brought into the community.

Stabilizer-based simulation of quantum error-correcting codes typically relies on the Pauli-twirling approximation (PTA) to render non-Clifford noise classically tractable, but PTA can distort the behavior of physically relevant channels such as thermal relaxation. Physically accurate noise simulation is needed to train decoders and understand the noise suppression capabilities of quantum error correction codes. In this work, we develop an exact and stabilizer-compatible model of qubit thermal relaxation noise and show that the combined amplitude damping and dephasing channel admits a fully positive probability decomposition into Clifford operations and reset whenever $T_2 \leqslant T_1$. For $T_2 > T_1$, the resulting decomposition is negative, but allows a smaller sampling overhead versus independent channels. We further introduce an approximated error channel with reset that removes the negativity of the decomposition while achieving higher channel fidelity to the true thermal relaxation than PTA, and extend our construction to finite temperature relaxation. We apply the exact combined model to investigate large surface codes and bivariate bicycle codes on superconducting platforms with realistic thermal relaxation error. The differing logical performances across code states further indicate that noise-model-informed decoders will be essential for accurately capturing thermal-noise structure in future fault-tolerant architectures.

In this work, we investigate the interaction between a uniformly accelerated single qubit and a fermionic spinor field. Here we consider both the massless and the massive fermionic spinor fields. The qubit-field interaction occurs over a finite time and was evolved via perturbation theory. This approach yields the transition probability rates, from which we subsequently evaluate the quantum coherence of an Unruh-DeWitt (UDW) detector initially prepared in a qubit state. Our findings reveal that the UDW detector responds more when coupled with the fermionic field, and consequently, quantum coherence (for the fermionic case) degrades much more rapidly when compared to the case of the qubit linearly coupled with the scalar field. Moreover, the analysis suggests that particle mass plays a protective role against Unruh-induced decoherence as the rest mass energy becomes comparable to the detector's energy-level spacing, the detector's excitation probability and response decreases, which leads to the mitigation of quantum coherence degradation in accelerated quantum systems.

Spin squeezing is a powerful resource for quantum metrology, and recent hardware platforms based on interacting qubits provide multiple possible architectures to generate and reverse squeezing during a sensing protocol. In this work, we compare the sensing performance of three such architectures: quantum reservoir computers (QRCs), quantum perceptrons, and multi-layer quantum neural networks (QNNs), when used as squeezing-based field sensors. For all models, we consider a standard metrological sequence consisting of coherent-spin preparation, one-axis-twisting dynamics, field encoding via a weak rotation, time-reversal, and collective readout. We show that a single quantum perceptron generates the same optimal sensitivity as a QRC, but in the perturbative regime it benefits from accelerated squeezing due to steering by the output qubit. Stacking perceptrons into a QNN further amplifies this effect: in a 2-layer QNN with N_in input and N_out output qubits, the optimal squeezing time is reduced by a factor of N_out, while the achievable phase sensitivity remains Heisenberg-limited, Delta phi ~ 1/(N_in + N_out). Moreover, if the layers are used sequentially, first using the outputs to squeeze the inputs and then the inputs to squeeze the outputs, the two contributions to the response add constructively. This yields a sqrt(2) enhancement in sensitivity over a QRC when N_in = N_out and requires shorter total squeezing time. Generalizing to L layers, we show that the metrological gain scales as sqrt(L) while the required squeezing time decreases as 1/N_l, where N_l is the number of qubits per layer. Our results demonstrate that the structure of quantum neural networks can be exploited not only for computation, but also to engineer faster and more sensitive squeezing-based quantum sensors.

In two previous papers the author described “Islands of Instability" that may appear in wavefunction models with nonlinear evolution (of a type proposed originally in the context of the Measurement Problem). Such “IsoI" represent a new scenario for Hamiltonian systems implying so-called “chaos". Criteria was derived for, and shown to be fulfilled in, some finite-dimensional (multi-qubit) models, and generalized in the second paper to continuum models. But the only example produced of the latter was a model whose linear Schrodinger equation was exactly-solvable. As exact solutions of many-body problems are rare, here I show that the instability criteria can be verified by plugging test-functions into certain computable expressions, bypassing the solvability blockade. The method can accommodate realistic inter-molecular potentials and so may be relevant to instabilities in fluids and gasses.

We introduce an “entangled clock” in which time is defined operationally by discrete measurement registrations on a singlet state. Locally, each party's tick rate is fixed by the unbiased marginals. The nontrivial resource is the relational (coincidence-tick) stream: because the singlet's information budget is entirely exhausted by joint properties, the only definite temporal structure resides in the correlations between the two parties. Operationally, after exchanging time tags and outcomes, Alice and Bob identify synchronized events (that is, the $++$ channel) and thereby obtain a joint tick record. Comparing the $++$ coincidence rate R(\theta)=P_{++}(\vec a,\vec b) to Peres' isotropic bomb-fragment local-hidden-variable model (yielding R_{\mathrm{cl}}(\theta)=\theta/(2\pi)), we find that for obtuse analyzer separations the quantum prediction exceeds this natural classical benchmark, with a maximal relative excess of about 13.6% near \theta\approx 140.5^\circ. We emphasize that this “faster ticking” refers to the rate of identified coincidence ticks under a specific operational convention, not to an improved local clock rate, precision, or stability. Finally, by using multiple settings and a Bell test, we outline “Certified Private Time”: a device-independent certification of unpredictability/privacy of the relational time-stamp record against adversaries lacking foreknowledge of the settings, analogous to certified randomness generation.

When an ensemble of quantum emitters interacts with a common radiation field, their emission becomes collective, giving rise to superradiant and subradiant states, characterized by broadened and narrowed linewidths. In this work, we propose to harness subradiant states for quantum metrology; such states naturally arise in subwavelength-spaced atomic arrays in free space and in small ensembles of emitters coupled to one-dimensional waveguides. We demonstrate that their collective optical response yields sharp, narrow features in the transmittance spectrum, which can be used to enhance sensitivity to external perturbations. This improved sensitivity can be applied to atomic clock operation, spatially resolved imaging of emitter positions, and enables precise detection of both global and spatially varying detunings (such as those induced by electromagnetic fields or gravitational gradients).

Quantum point contacts (QPC) are a key instrument in investigating the physics of edge excitations in the quantum Hall effect. However, at not-so-high bias voltage values, the predictions of the conventional point QPC model often deviate from the experimental data both in the integer and (more prominently) in the fractional quantum Hall regime. One of the possible explanations for such behaviors is the dependence of the tunneling between the edges on energy, an effect not present in the conventional model. Here we introduce two models that take QPC spatial extension into account: wide-QPC model that accounts for the distance along which the edges are in contact; long-QPC model accounts for the fact that the tunneling amplitude originates from a finite bulk gap and a finite distance between the two edges. We investigate the predictions of these two models in the integer quantum Hall regime for the energy dependence of the tunneling amplitude. We find that these two models predict opposite dependences: the amplitude decreasing or increasing away from the Fermi level. We thus elucidate the effect of the QPC geometry on the energy dependence of the tunneling amplitude and investigate its implications for transport observables.