Papers

Recently, a technique known as quantum symmetry test has gained increasing attention for detecting bipartite entanglement in pure quantum states. In this work we show that, beyond qualitative detection, a family of well-defined measures of bipartite and multipartite entanglement can be obtained with symmetry tests. We propose and benchmark several efficient methods to estimate these measures, and derive near-optimal sampling strategies for each. Despite the nonlinearity of the methods, we demonstrate that the sampling error scales no worse than $O(N_{\mathrm{tot}}^{-1/2})$ with the total number of copies $N_{\mathrm{tot}}$, which suggests experimental feasibility. By exploiting symmetries we compute our measures for large number of copies, and derive the asymptotic decay exponents for relevant states in many-body systems. Using these results we identify tradeoffs between estimation complexity and sensitivity of the presented entanglement measures, oriented to practical implementations.

Nonlocality shapes quantum correlations, revealed through the violation of Bell inequalities. The intersection of all valid Bell inequalities is the so-called local polytope. In multipartite systems, characterizing the local polytope quickly becomes an intractable task as the system size increases. Optimizing Bell inequalities to maximize the ratio between their quantum value and classical bound is key to understanding multipartite nonlocality. We propose a gradient-based method for this optimization. Numerical results indicate that local maxima of this ratio typically correspond to facet Bell inequalities of the local polytope. This enables an iterative search for tight and robust Bell inequalities. Applied to permutation-invariant scenarios, the method provides tight Bell inequalities with large quantum violations and facilitates experimental certification of Bell correlations without full knowledge of the local polytope. Moreover, analytical results of the maximum ratio are derived in the thermodynamic limit.

The measurement of the anomalous electron magnetic moment $g-2$ through quantum transitions of a single trapped electron is the most stringent test of quantum field theory. These experiments are now so precise that they must account for the effects of the cavity containing the electron. Classical calculations of this "cavity shift" must subtract the electron's divergent self-field, and thus require knowledge of the exact Green's function for the cavity's electromagnetic field. We perform the first fully quantum calculation of the cavity shift in a closed cavity, which instead involves subtracting linearly divergent cavity mode sums and integrals. Using contour integration methods, we find perfect agreement with existing classical results for both spherical and cylindrical cavities, justifying their current use. Moreover, our mode-based results can be naturally generalized to account for systematic effects, necessary to push future measurements to the next order of magnitude in precision.

This essay offers an epistemological reinterpretation of the foundational divide between matrix mechanics and wave mechanics. Though formally equivalent, the two theories embody distinct modes of knowing: procedural construction and recognitional verification. These epistemic architectures anticipate, in philosophical form, the logical asymmetry expressed by the P versus NP problem in computational complexity. Here, the contrast between efficient generation and efficient recognition is treated not as a mathematical taxonomy but as a framework for understanding how knowledge is produced and validated across physics, computation, and cognition. The essay reconstructs the mathematical history of quantum mechanics through the original derivations of Werner Heisenberg, Max Born, Pascual Jordan, Paul Dirac, Erwin Schrödinger, Paul Ehrenfest, and Wolfgang Pauli, culminating in John von Neumann's unification of both approaches within the formalism of Hilbert space. By juxtaposing Heisenberg's algorithmic formalism with Schrödinger's representational one, it argues that their divergence reveals a structural feature of scientific reasoning itself-the enduring tension between what can be procedurally constructed and what can only be recognized.

With steady progress in the development of quantum networks, the question on how to best provide end-to-end characterization of such networks (Quantum Network Tomography) is quickly becoming more pressing. Initial results demonstrated how we can utilize multipartite entanglement distribution to determine error probabilities of single-Pauli channels and depolarizing channels. In this work, we show how the analysis of quantum capacity regions can be used as a powerful new tool in quantum network tomography. As a first application of the proposed method, we demonstrate how we can characterize the loss on quantum channels in the network directly from quantum capacity region diagrams, even in the presence of bit-flip errors. Our results indicate that quantum capacity regions are not only valuable for network design, resource allocation, and protocol benchmarking, but also show promise for applications in quantum network tomography, particularly in loss tomography.

The recently proposed QAOA-GPT framework demonstrated that generative pre-trained transformers can learn mappings between problem graphs and optimized quantum circuits for the Quantum Approximate Optimization Algorithm (QAOA). In this work, we extend QAOA-GPT to Higher-Order Unconstrained Binary Optimization (HUBO) problems, focusing on spin-glass Hamiltonians that include cubic interaction terms. Using FEATHER graph embeddings to encode topological information, we train the model on graph-circuit pairs generated via ADAPT-QAOA and evaluate its performance on 8- and 16-qubit instances embedded on heavy-hex lattices. The generative model produces adaptive QAOA-like circuits and corresponding variational parameters in a single forward pass, bypassing the iterative classical optimization loop. The generated circuits achieve average approximation ratios exceeding 0.95, closely matching classically optimized ADAPT-QAOA results, while maintaining consistent parameter distributions across circuit depths. These results demonstrate that QAOA-GPT generalizes effectively to higher-order cost Hamiltonians and complex energy landscapes, establishing generative modeling as a scalable pathway toward autonomous variational circuit design and quantum algorithm discovery in the NISQ era.

The decoherence phenomenon arising from an environmental monitoring of the state of a quantum system, as opposed to monitoring of a preferred observable, is worked out in detail using two equivalent formulations, namely, repeated applications of universal tomographic measurements using positive operator-valued measures, and its continuous time unravelling from the Lindblad equation. The effect of decoherence is analysed by studying the evolution of Stratonovich-Weyl quasiprobability distributions on the state-space of the system. It is shown that decoherence makes an arbitrary-given quasiprobability distribution manifestly positive, thus modelling the emergence of classicality in some sense. The decoherence timescale, the minimum time that quasiprobability distributions of every initial state of the system become nonnegative, is shown to decrease in Hilbert-space dimension, and hence larger quantum systems decohere faster.

In Nature, 646, 813 (2025), Aziz and Howl claim that classical (unquantised) gravity produces entanglement. We show that their model does not produce entanglement. Even if the model produced entanglement, it would be mediated by the quantised matter interaction, and not gravity. Hence entanglement mediated by gravity remains an unambiguous witness of gravity's quantum features.

Efficient verification of quantum computational resources is crucial as experiments advance toward fault-tolerance. Universal quantum computation can be achieved by consuming resource states through simple Pauli measurements, yet a significant gap remains between states that are easy to certify and those required for universality. We focus on Clifford-enhanced Product States, a class of resource states obtained by applying Clifford circuits to a product of single-qubit, potentially magic, states. While essential for universal computation, the certification of such states has previously relied on query oracles that are \#P-hard to implement, leaving their efficient, oracle-free verification an open challenge. In this work, we demonstrate that such classically intractable resource states can be efficiently verified using only Pauli measurements. Our protocol achieves sample- and time-efficiency in both i.i.d.\ and adversarial settings. This work fills a gap in Pauli-based certification, providing a new practical pathway to verify resource states that drive universal Pauli-based quantum computation.

A quantum walk in an ordered medium exhibits ballistic propagation. A related process is the driven quantum walk, in which the number of walkers varies along the propagation. In this work, we show that a driven quantum walk of biphoton states in an array of nonlinear waveguides does not propagate ballistically, but instead presents two transport regimes: superballistic and superdiffusive. We also study the effects of off-diagonal and diagonal disorder. In addition to the observation of Anderson localization, both types of disorder lead to the disappearance of the superballistic regime.

We show that harnessing daemonic advantage is possible while charging a quantum battery by first time-evolving the battery collectively with an auxiliary charger, followed by an energy extraction via tracing out the charger. We define the difference between the minimum daemonic ergotropy and the maximum ergotropy of the battery as the daemonic gap at the time where the ergotropy of the battery is maximum. Considering a harmonic mode as the battery and a qubit as the auxiliary charger interacting via Jaynes-Cummings interaction, we show that the daemonic gap can be closed for specific initial passive states of the battery, including the ground state, truncated mixtures of low-lying states, and canonical thermal states. We further define the daemonic band as the difference between the maximum and the minimum daemonic ergotropy, and show that starting from the ground state of the single-mode battery, the daemonic band collapses. We also show that achieving a complete charging, along with accessing the full daemonic band, is possible for the single-mode battery using all of the other initial states by repeating the charging cycle and maximizing ergotropy in every round. We extend the battery-charger design to a multi-mode resonator battery and a qudit charger, and demonstrate, for a double-mode battery and a qutrit charger, that repeating the charging cycle is vital for simultaneously charging individual modes of the collective battery.

A spin precessing in a magnetic field is often used as a quantum clock, for example in tunneling-time measurements. We show that such a clock can exist in a coherent superposition of opposite temporal orientations, treating the arrow of time as a quantum two-level system. A Mach-Zehnder interferometer with equal and opposite magnetic fields provides a simple implementation, enabling interference between forward and backward evolution and Bell-type tests of temporal coherence. Extending the framework to relativistic dynamics reveals that the Dirac Hamiltonian describes an intrinsic coupling between a spin qubit and a time qubit, with matter and antimatter corresponding to opposite poles on the temporal Bloch sphere.

In this work, we introduce the Quantum-Classical Hybrid Physics-Informed Neural Network with Multiplicative and Additive Couplings (QPINN-MAC): a novel hybrid architecture that integrates the framework of Physics-Informed Neural Networks (PINNs) with that of Quantum Neural Networks (QNNs). Specifically, we prove that through strategic couplings between classical and quantum components, the QPINN-MAC retains the universal approximation property, ensuring its theoretical capacity to represent complex solutions of ordinary differential equations (ODEs). Simultaneously, we demonstrate that the hybrid QPINN-MAC architecture actively mitigates the barren plateau problem, regions in parameter space where cost-function gradients decay exponentially with circuit depth, a fundamental obstacle in QNNs that hinders optimization during training. Furthermore, we prove that these couplings prevent gradient collapse, ensuring trainability even in high-dimensional regimes. Thus, our results establish a new pathway for constructing quantum-classical hybrid models with theoretical convergence guarantees, which are essential for the practical application of QPINNs.

Matrix-product state (MPS) skeletons are connected networks of Hamiltonians with exact MPS ground states that underlie a phase diagram. Such skeletons have previously been found in classes of free-fermion models. For the translation-invariant BDI and AIII free-fermion classes, it has been shown that the underlying skeleton is dense, giving an analytic approach to MPS approximation of ground states anywhere in the class. In this paper, we partially expose the skeleton in certain interacting spin chains: the $N$-state Onsager-integrable chiral clock families. We construct MPS that form a dense MPS skeleton in the gapped regions surrounding a sequence of fixed-point Hamiltonians (the generators of the Onsager algebra). Outside these gapped regions, these MPS remain eigenstates, but no longer give the many-body ground state. Rather, they are ground states in particular sectors of the spectrum. Our methods also allow us to find further MPS eigenstates; these correspond to low-lying excited states within the aforementioned gapped regions. This set of MPS excited states goes beyond the previous analysis of ground states on the $N=2$ free-fermion MPS skeleton. As an application of our results, we find a closed form for the disorder parameter in a family of interacting models. Finally, we remark that many of our results use only the Onsager algebra and are not specific to the chiral clock model representation.

We investigate optimal metrological protocols for phase estimation in the presence of correlated dephasing noise, including spin-squeezed state sensing strategies as well as parallel and adaptive protocols optimized using tensor-network-based numerical methods. The results are benchmarked against fundamental bounds obtained either via the latest quantum comb extension method or an optimized classical simulation method. We find that the spin-squeezed states offer practically optimal performance in the regime where phase fluctuations are positively correlated but can be outperformed by tensor-network-optimized strategies for negatively correlated fluctuations.

Relativistic time dilation implies that an accelerating excited atom would have its lifetime prolonged in the lab frame. In this paper, we demonstrate a complementary effect: Longer-lived excited atoms turn out to have been accelerated. We propose the following experiment. An excited atom is prepared in a superposition of momenta. Due to relativistic time dilation, in the lab frame, the decay rate of components with higher momentum is diminished. As time passes without the atom emitting a photon, higher-momentum components become more probable. This results in a time-dependent shift of the expectation value of momentum. Taking into account first-order relativistic corrections, we calculate the force and acceleration undergone by the atom due to the fact that it did not emit a photon.

Magnetotransport underlines many important phenomena in condensed matter physics, such as the Hall effect and magnetoresistance (MR) effect. Thus far, most magnetotransport studies are based on bulk resistance measurements without direct access to microscopic details of the spatial transport pattern. Here, we report nanoscale imaging of magnetotransport using a scanning single-spin quantum magnetometer, which is demonstrated in a graphene-metal hybrid device at room temperature. By visualizing the current flow at elevated magnetic fields (~0.5 T), we directly observe the Lorentz deflection of current near the graphene-metal interface, which is a hallmark of magnetotransport. Combining the local current distribution with global resistance measurements, we reveal that transport properties of the hybrid are governed by a complex interplay of intrinsic MR around the Dirac cone, carrier hydrodynamics, interface resistance, and the nanoscale device geometry. Furthermore, accessing the local transport pattern across the interface enables quantitative mapping of spatial variations in contact resistance, which is commonly present in electronic devices made from two-dimensional materials yet non-trivial to characterize. Our work demonstrates the potential of nanoscale current imaging techniques for studying complex electronic transport phenomena that are difficult to probe by resistance-based measurements.

In this work, we introduce a method for deriving exact master equations from the dynamical map for finite open quantum systems coupled to (in)finite reservoirs, using the principle of minimal dissipation. The exact dynamics of the central spin model, which models a finite-bath open quantum system, is developed for two interaction types: Heisenberg and stochastic pure-dephasing interactions. The Heisenberg interaction yields a novel phase-covariant quantum channel in the strong-coupling regime, offering a new platform for studying a range of quantum information protocols. The stochastic pure-dephasing interaction provides the microscopic derivation of the paradigmatic non-Markovian random telegraph noise (RTN) channel, establishing its quantum foundation and offering insight into stochastic couplings. We derive the closed-form master equations for both models. As a demonstration, we explore the thermodynamic performance of these systems as quantum batteries. A direct relationship between quantum heat current and charging power is revealed, and RTN quantum batteries are shown to have advantages in charge storage.

We investigate the planar Dirac equation with the most general time-independent contact (singular) potential supported on a circumference. Taking advantage of the radial symmetry, the problem is effectively reduced to a one-dimensional one (the radial), and the contact potential is addressed in a mathematically rigorous way using a distributional approach that was originally developed to treat point interactions in one dimension, providing a physical interpretation for the interaction parameters. The most general contact interaction for this system is obtained in terms of four physical parameters: the strengths of a scalar and the three components of a singular Lorentz vector potential supported on the circumference. We then investigate the bound and scattering solutions for several choices of the physical parameters, and analyze the confinement properties of the corresponding potentials.

Self-sustained oscillators play a central role in the stabilization and synchronization of complex dynamical systems. A number of different physical systems are currently being investigated to clarify the importance of such active components in the quantum realm. Here we explore the properties of a driven dissipative electron-photon hybrid system based on superconducting microwave resonators coupled resonantly to a voltage-biased double quantum dot (DQD). First, we establish a Hopf bifurcation at a critical value of the electron-photon coupling, beyond which an effective negative friction sustains steady limit-cycle oscillations of individual resonators. Second, we show that two such limit-cycle resonators coupled via the same voltage-biased DQD synchronize for small enough frequency detuning. A nonlinear photon Keldysh action is derived by perturbation theory in the effective circuit fine-structure constant, and the limit-cycle dynamics is analyzed in terms of resulting saddle-point, and Fokker-Planck equations. In the Markovian limit of infinite bias voltage, these results are shown to agree well with the solution of a corresponding Lindblad master equation for the DQD resonator system.

Quantum batteries (QBs), acting as energy storage devices, have potential applications in future quantum science and technology. However, the QBs inevitably losses energy due to their interaction with environment. How to enhance the performance of the QBs in the open-system case remains an important challenge. Here we propose a scheme to realize the driven-dissipative QBs in atom-waveguide-QED systems and demonstrate significant improvements in both the stored energy and extractable work (ergotropy) of the QBs via feedback control. For a single-atom QB, we show that combining the measurement and coherent feedback controls enables nearly perfect stable charging under the weak coherent driving. For the QB array, the measurement-based feedback allows us to control different dynamical phases in the thermodynamic limit: (i) a continuous boundary time-crystal phase, where persistent periodic energy charge-discharge oscillations emerge despite the presence of the dissipation into the waveguide, and (ii) two stationary phases – one reaches full charge while the other maintains only small energy storage. This work broadens the scope of driven-dissipative QBs and provides practical strategies for enhancing their performance.

Hybrid photonic circuits, harnessing the complementary strengths of multiple materials, represent a key resource to enable compact, scalable platforms for quantum technologies. In particular, the availability of bright sources of tunable biphoton states is eagerly awaited to meet the variety of applications currently under development. In this work we demonstrate a heterogeneously integrated device that merges biphoton generation and on-chip quantum state engineering, combining an AlGaAs photon-pair source with a CMOS-compatible silicon-on-insulator (SOI) circuit. Photon pairs are generated in the C telecom band via spontaneous parametric down-conversion and transferred to the SOI chip through a multimode evanescent coupling scheme. This design achieves a pair generation rate above 10$^{6}$ s$^{-1}$mW$^{-1}$ and a coincidence-to-accidental ratio up to 600. Crucially, the coupling design induces strong and predictable transformations of the biphoton joint spectral amplitude, enabling complex quantum state engineering entirely on-chip in a compact device compliant with electrical pumping.

The geometric phase stands as a foundational concept in quantum physics, revealing deep connections between geometric structures and quantum dynamical evolution. Unlike dynamical phases, geometric phases exhibit intrinsic resilience to certain types of perturbation, making them particularly valuable for quantum information processing, where maintaining coherent quantum operations is essential. This article provides a review of geometric phases in the context of universal quantum gate construction, i.e., the geometric quantum computation (GQC), with special attention to recent progress in nonadiabatic implementations that enhance gate fidelity and/or operational robustness. We first review a unified theoretical framework that can encompass all existing nonadiabatic GQC approaches, then systematically examine the design principles of nonadiabatic geometric gates with a particular focus on how optimal control techniques can be leveraged to improve the accuracy and noise resistance. In addition, we conducted detailed numerical comparisons of various nonadiabatic GQC protocols, offering a quantitative assessment of their respective performance characteristics and practical limitations. Through this focused investigation, our aim is to provide researchers with both fundamental insights and practical guidance for advancing geometric approaches in quantum computing.

The pursuit of practical quantum utility on near-term quantum processors is critically challenged by their inherent noise. Quantum error mitigation (QEM) techniques are leading solutions to improve computation fidelity with relatively low qubit-overhead, while full-scale quantum error correction remains a distant goal. However, QEM techniques incur substantial measurement overheads, especially when applied to families of quantum circuits parameterized by classical inputs. Focusing on zero-noise extrapolation (ZNE), a widely adopted QEM technique, here we devise the surrogate-enabled ZNE (S-ZNE), which leverages classical learning surrogates to perform ZNE entirely on the classical side. Unlike conventional ZNE, whose measurement cost scales linearly with the number of circuits, S-ZNE requires only constant measurement overhead for an entire family of quantum circuits, offering superior scalability. Theoretical analysis indicates that S-ZNE achieves accuracy comparable to conventional ZNE in many practical scenarios, and numerical experiments on up to 100-qubit ground-state energy and quantum metrology tasks confirm its effectiveness. Our approach provides a template that can be effectively extended to other quantum error mitigation protocols, opening a promising path toward scalable error mitigation.

Noisy intermediate-scale quantum computers hold the promise of tackling complex and otherwise intractable computational challenges through the massive parallelism offered by qubits. Central to realizing the potential of quantum computing are perfect entangling (PE) two-qubit gates, which serve as a critical building block for universal quantum computation. In the context of quantum optimal control, shaping electromagnetic pulses to drive quantum gates is crucial for pushing gate performance toward theoretical limits. In this work, we leverage reinforcement learning (RL) techniques to discover near-optimal pulse shapes that yield PE gates. A collection of RL agents is trained within robust simulation environments, enabling the identification of effective control strategies even under noisy conditions. Selected agents are then validated on higher-fidelity simulations, illustrating how RL-based methods can reduce calibration overhead when compared to quantum optimal control techniques. Furthermore, the RL approach is hardware agnostic with the potential for broad applicability across various quantum computing platforms.

One of the main residual limitations of inertial sensors based on atom interferometry stems from laser beam distortions, which cause parasitic phase shifts and non-homogeneous matter-light couplings. Here we present numerical simulations, accompanied by analytical calculations, which quantify the impact of these effects in a cold atom gradiometer. We demonstrate that the propagation of interferometric laser beam aberrations, combined with initial asymmetry and significant time-of-flight expansion of the the two atomic sources, limit the common-mode rejection of phase noise in a differential configuration. The resulting deviations in gravitational acceleration and its gradient are within reach of current experimental devices. Our study allows us to evaluate the surface quality requirements for retroreflective optics in cold-atom gradiometers of various baselines, and can be extended to other sensors based on different interferometer geometries.

We study, classify, and explore the mathematical properties of finite-dimensional Lie algebras occurring in the quantum dynamics of single-mode and self-interacting bosonic systems. These Lie algebras are contained in the real skew-hermitian Weyl algebra $\hat{A}_1$, defined as the real subalgebra of the Weyl algebra $A_1$ consisting of all skew-hermitian polynomials. A central aspect of our analysis is the choice of basis for $\hat{A}_1$, which is composed of skew-symmetric combinations of two elements of the Weyl algebra called monomials, namely strings of creation and annihilation operators combined with their hermitian conjugate. Motivated by the quest for analytical solutions in quantum optimal control and dynamics, we aim at answering the following three fundamental questions: (i) What are the finite-dimensional Lie subalgebras in $\hat{A}_1$ generated by monomials alone? (ii)~What are the finite-dimensional Lie subalgebras in $\hat{A}_1$ that contain the free Hamiltonian? (iii) What are the non-abelian and finite-dimensional Lie subalgebras that can be faithfully realized in $\hat{A}_1$? We answer the first question by providing all possible realizations of all finite-dimensional non-abelian Lie algebras that are generated by monomials alone. We answer the second question by proving that any non-abelian and finite-dimensional subalgebra of $\hat{A}_1$ that contains a free Hamiltonian term must be a subalgebra of the Schrödinger algebra. We partially answer the third question by classifying all nilpotent and non-solvable Lie algebras that can be realized in $\hat{A}_1$, and comment on the remaining cases. Finally, we also discuss the implications of our results for quantum control theory. Our work constitutes an important stepping stone to understanding quantum dynamics of bosonic systems in full generality.

Quantum algorithms have demonstrated provable speedups over classical counterparts, yet establishing a comprehensive theoretical framework to understand the quantum advantage remains a core challenge. In this work, we decode the quantum search advantage by investigating the critical role of quantum state properties in random-walk-based algorithms. We propose three distinct variants of quantum random-walk search algorithms and derive exact analytical expressions for their success probabilities. These probabilities are fundamentally determined by specific initial state properties: the coherence fraction governs the first algorithm's performance, while entanglement and coherence dominate the outcomes of the second and third algorithms, respectively. We show that increased coherence fraction enhances success probability, but greater entanglement and coherence reduce it in the latter two cases. These findings reveal fundamental insights into harnessing quantum properties for advantage and guide algorithm design. Our searches achieve Grover-like speedups and show significant potential for quantum-enhanced machine learning.

Quantum entanglement and coherence are crucial resources in quantum information theory. In some scenarios, however, it is not necessary to directly estimate entanglement or coherence measures to quantify the capabilities of a state in quantum information processing. Instead, fully entangled fraction and coherence fraction are two alternatives for entanglement and coherence in specific quantum tasks. Here, we establish a link between the coherence fraction and the Bernstein-Vazirani algorithm, which has several potential applications including cryptography and database search. We show that the success probability of the generalized Bernstein-Vazirani algorithm depends only on the coherence fraction of the initial state rather than its entanglement or coherence. Moreover, we discuss the coherence fraction dynamics and establish a relation between the operator's coherence fraction and the algorithm's success probability. Our findings highlight how quantum coherence fraction influences the efficiency of quantum algorithms.

Open systems feature a variety of phenomena that arise from non-Hermitian physics. Recent theoretical studies have offered much insights into these phenomena through the non-Bloch band theory, though many of the theory's key features are experimentally elusive. For instance, the correspondence between complex momenta and non-Hermitian bands, while central to non-Bloch band theory, has so far defied direct experimental observation. Here we experimentally study the non-Hermitian spectral deformation in complex-momentum space, by implementing a non-Hermitian lattice with long-range couplings in the synthetic orbital-angular-momentum (OAM) dimension of photons inside a degenerate cavity. Encoding the complex momenta in the phase and amplitude modulations of the OAM modes, and devising a complex-momentum-resolved projective detection, we reconstruct the spectral deformation in momentum space, where the eigenspectrum on the complex plane morphs through distinct geometries. This enables us to experimentally extract key information of the system under the non-Bloch band theory, including exceptional points in the complex-momentum space, the open-boundary spectra, and the generalized Brillouin zone. Our work demonstrates a versatile platform for exploring non-Hermitian physics and non-Bloch band theory, and opens the avenue for direct experimental investigation of non-Bloch features in the complex-momentum space.