Exceptional points (EPs) occur in non-Hermitian systems where eigenvalues and eigenvectors coalesce. In open quantum systems, EPs are typically studied via non-Hermitian effective Hamiltonians or Liouvillians, but these omit the full generality of quantum dynamics, which requires description using quantum channels. Here, we introduce a general strategy for generating quantum-channel EPs in the single-qubit setting. We show that quantum channels naturally fall into two distinct phases, purely real or complex-conjugate eigenvalues, without the need of enforcing additional symmetries, and interpolating between different phases induces a phase transition where EPs can emerge. We experimentally simulate such channels on a nuclear magnetic resonance quantum computer, observing second-order quantum-channel EPs with 93% fidelity, and extend the approach to three channels to uncover exceptional lines and a third-order quantum-channel EP. Our results promise potential applications from enhanced quantum sensing to quantum control, such as Jordan-chain mediated asymmetric conversion.

Exact quantum codes detecting a prescribed set of Pauli errors are approached through algebraic constructions–stabilizer, codeword-stabilized, permutation-invariant, topological, and related families. Geometrically, exact Pauli detection is governed by joint higher-rank numerical ranges of these Pauli operators, whose structure for rank ≥ 2 is largely uncharted. From this viewpoint, we show that such codes often form connected continuous families rather than collections of disjoint solution regions. These families are characterized by a single scalar derived from the Knill-Laflamme conditions: denoted λ^*, it is the Euclidean norm of the signature vector of Pauli expectation values on the maximally mixed code state, and provides a one-parameter summary of the code's joint Pauli variance profile. Within these continuous landscapes, stabilizer codes occupy only discrete, measure-zero subsets of the attainable λ^*-spectrum, exposing a largely unexplored continuum of genuinely nonadditive exact codes. We establish this picture by analyzing the geometry of higher-rank operator compressions, and extend it to symmetry-restricted settings where cyclic and permutation symmetries are imposed on both the error model and the code projector. Small-system cases reveal interval, singleton, and empty regimes through eigenvalue interlacing and symmetry-sector decompositions; larger systems are treated numerically via Stiefel-manifold optimization and symmetry-adapted parameterizations. In every unrestricted and symmetry-compatible case analyzed, the attainable λ^*-spectrum forms a single closed interval whenever nonempty–although a general proof remains open. These results place stabilizer, symmetric, and nonadditive code families within a unified higher-rank variance framework, suggesting a continuous geometric perspective on the landscape of exact quantum codes.

Introduction: Quantum error-correcting codes (QECCs) are essential for fault-tolerant quantum computation, but standard algebraic and graph-state descriptions each expose only part of the encoder structure.Materials and methods: We introduce the ZX encoder diagram, a graph-like Clifford ZX representation that keeps the encoding map, the code geometry, and the extraction of stabilizers, logical operators, and encoder circuits within a single formalism. We apply this representation to two modular code-construction tasks: concatenation of graph and stabilizer codes, and tensor-network-style encoder contraction. The cost model counts local rewrites and binary matrix reductions.Results: For concatenation, the ZX language recovers generalized local complementation as a special case and remains effective when overlapping neighborhoods obstruct pure generalized local complementation. For holographic constructions, it yields a deterministic pipeline for building the 4-to-12 HaPPY instance and, more generally, bounded-degree tensor-network families. Under this cost model, the contraction stage uses a constant number of local rewrites per contracted leg, and the full extraction pipeline is polynomial in the final diagram size.Conclusions: ZX-calculus provides a code-construction-native framework for concatenated and tensor-network quantum codes, with complexity governed by the number of encoder blocks rather than by an exponential search over local Clifford equivalents.

We present a multi-agent, human-in-the-loop workflow that co-designs quantum error-correcting codes with pre-scribed transversal diagonal gates. It builds on the Subset-Sum Linear Programming (SSLP) framework, which partitions basis strings by modular residues and enforces $\mathbf{Z}$-marginal Knill-Laflamme (KL) equalities via small LPs. The workflow is powered by GPT-5 and implemented within TeXRA, a multi-agent research assistant platform where agents collaborate in a shared LATEX-Python workspace synchronized with Git/Overleaf. Three specialized agents formulate constraints, sweep and screen candidate codes, exactify numerical solutions into rationals, and independently audit all KL equalities and induced logical actions. Focusing on distance-two codes with nondegenerate residues, we catalogue new nonadditive codes for dimensions $K \in\{2,3,4\}$ on up to six qubits, including high-order diagonal transversals, yielding 14, 116 new codes. From these data, the system abstracts closed-form families and constructs a residue-degenerate ((6, 4, 2)) code implementing a transversal controlled-phase $\operatorname{diag}(1,1,1, i)$, illustrating how AI orchestration can drive rigorous, scalable code discovery.

Quantum error correction (QEC) is essential for protecting quantum information against noise, yet understanding the structure of the Knill-Laflamme (KL) coefficients {λ_ij} from the condition PE_i^†E_jP=λ_ijP remains challenging, particularly for nonadditive codes. In this work, we introduce the signature vector λ(P) , composed of the off-diagonal KL coefficients {λ_ij} , where each coefficient corresponds to equivalence classes of errors counted only once. We define its Euclidean norm λ*(P) as a scalar measure representing the total strength of error correlations within the code subspace defined by the projector P. We parameterize P on a Stiefel manifold and formulate an optimization problem based on the KL conditions to systematically explore possible values of λ*. Moreover, we show that, for ((n, K, d)) codes, λ* is invariant under local unitary transformations. Applying our approach to the ((6, 2, 3)) quantum code, we find that λ_min^* =√(0.6) and λ_max^* =1 , with λ* = 1 corresponding to a known degenerate stabilizer code. We construct continuous families of new nonadditive codes parameterized by vectors in ℝ^5 , with λ* varying over the interval [√(0.6),1] . For the ((7, 2, 3)) code, we identify λ_min^* =0 (corresponding to the non-degenerate Steane code) and λ_max^* =√(7) (corresponding to the permutation-invariant code by Pollatsek and Ruskai), and we demonstrate continuous paths connecting these extremes via cyclic codes characterized solely by λ*. Our findings provide new insights into the structure of quantum codes, advance the theoretical foundations of QEC, and open new avenues for investigating intricate relationships between code subspaces and error correlations.

Transversal gates play a crucial role in suppressing error propagation in fault-tolerant quantum computation, yet they are intrinsically constrained: any nontrivial code encoding a single logical qubit admits only a finite subgroup of SU(2) as its transversal operations. We introduce a systematic framework for searching codes with specified transversal groups by parametrizing their logical subspaces on the Stiefel manifold and minimizing a composite loss that enforces both the Knill-Laflamme conditions and a target transversal-group structure. Applying this method, we uncover a new ((6,2,3)) code admitting a transversal Z(2π5) gate (transversal group C_10), the smallest known distance 3 code supporting non-Clifford transversal gates, as well as several new ((7,2,3)) codes realizing the binary icosahedral group 2I. We further propose the Subset-Sum-Linear-Programming (SS-LP) construction for codes with transversal diagonal gates, which dramatically shrinks the search space by reducing to integer partitions subject to linear constraints. In a more constrained form, the method also applies directly to the binary-dihedral groups BD_2m. Specializing to n=7, the SS-LP method yields codes for all BD_2m with 2m≤ 36, including the first ((7,2,3)) examples supporting transversal T gate (BD_16) and √(T) gate (BD_32), improving on the previous smallest examples ((11,2,3)) and ((19,2,3)). Extending the SS-LP approach to ((8,2,3)), we construct new codes for 2m>36, including one supporting a transversal T^1/4 gate (BD_64). These results reveal a far richer landscape of nonadditive codes than previously recognized and underscore a deeper connection between quantum error correction and the algebraic constraints on transversal gate groups.

Quantum resource theories (QRTs) provide a comprehensive and practical framework for the analysis of diverse quantum phenomena. A fundamental task within QRTs is the quantification of resources inherent in a given quantum state. In this work, we introduce a unified computational framework for a class of widely utilized quantum resource measures, derived from convex roof extensions. We establish that the computation of these convex roof resource measures can be reformulated as an optimization problem over a Stiefel manifold, which can be further unconstrained through polar projection. Compared to existing methods employing semi-definite programming (SDP), gradient-based techniques or seesaw strategy, our approach not only demonstrates satisfying computational efficiency but also maintains applicability across various scenarios within a unified framework. We substantiate the efficacy of our method by applying it to several key quantum resources, including entanglement, coherence, and magic states. Moreover, our methodology can be readily extended to other convex roof quantities beyond the domain of resource theories, suggesting broad applicability in the realm of quantum information theory.

Exceptional points (EPs) are special points in non-Hermitian systems where both eigenvalues and eigenvectors coalesce. In open quantum systems, these points are typically analyzed using effective non-Hermitian Hamiltonians or Liouvillian superoperators. While quantum channels offer the most general framework for describing state evolution in such systems, the existence and properties of EPs within this setting remain largely unexplored. In this work, we present a general strategy for generating quantum EPs for a single-qubit setting. We show that quantum channels can be separated into two distinct phases, with the transition between them marked by the presence of an EP. Based on this, we propose a systematic method to realize EPs by interpolating between quantum channels representing different phases. Experimentally, we implement these interpolated channels on a nuclear magnetic resonance (NMR) quantum computer and confirm the emergence of second-order EPs with high fidelity. Extending the interpolation to three channels further reveals third-order EPs. Our results establish quantum channel interpolation as a versatile framework for generating EPs and provide a general description of EPs in open quantum systems.

Quantum state tomography (QST) is crucial for understanding and characterizing quantum systems through measurement data. Traditional QST methods face scalability challenges, requiring O(d2) measurements for a general d-dimensional state. This complexity can be substantially reduced to O(d) in pure state tomography, indicating that full measurements are unnecessary for pure states. In this paper, we investigate the conditions under which a given pure state can be uniquely determined by a subset of full measurements, focusing on the concepts of uniquely determined among pure states (UDP) and uniquely determined among aall states (UDA). The UDP determination inherently involves nonconvexity challenges, while the UDA determination, though convex, becomes computationally intensive for high-dimensional systems. To address these issues, we develop a unified framework based on the mugmented Lagrangian ethod (ALM). Specifically, our theorem on the existence of low-rank solutions in QST allows us to reformulate the UDA problem with low-rank constraints, thereby reducing the number of variables involved. Our approach entails parametrizing quantum states and employing ALM to handle the constrained nonconvex optimization tasks associated with UDP and low-rank UDA determinations. Numerical experiments conducted on qutrit systems, generalized Greenberger-Horne-Zeilinger states and four-qubit symmetric states not only validate theoretical findings but also reveal the complete distribution of quantum states across three uniqueness categories: (a) UDA, (b) UDP but not UDA, and (c) neither UDP nor UDA. This work provides a practical approach for determining state uniqueness, advancing our understanding of quantum state reconstruction.

Recent advancements in quantum hardware and classical computing simulations have significantly enhanced the accessibility of quantum system data, leading to an increased demand for precise descriptions and predictions of these systems. Accurate prediction of quantum Hamiltonian dynamics and identification of Hamiltonian parameters are crucial for advancements in quantum simulations, error correction, and control protocols. This Letter introduces a machine learning model with dual capabilities: it can deduce time-dependent Hamiltonian parameters from observed changes in local observables within quantum many-body systems, and it can predict the evolution of these observables based on Hamiltonian parameters. Our model's validity was confirmed through theoretical simulations across various scenarios and further validated by two experiments. Initially, the model was applied to a nuclear magnetic resonance quantum computer, where it accurately predicted the dynamics of local observables. The model was then tested on a superconducting quantum computer with initially unknown Hamiltonian parameters, successfully inferring them. Our approach aims to enhance various quantum computing tasks, including parameter estimation, noise characterization, feedback processes, and quantum control optimization.

In the 1970s, Wiesner introduced the concept of quantum money, where quantum states serve as currency, offering physical-level unforgeability through quantum mechanics. Yet, traditional proposals often unrealistically assume personal quantum computing access for each user. To address these issues, we propose a cloud-based semi-quantum money (CSQM) scheme. This approach only requires semi-honest third-party quantum clouds, while the rest of the system, including transactions and banks, remains fully classical. We also estimate the computational power required by the quantum cloud and provide a thorough security analysis. Our approach significantly reduces the quantum resource demands on local users and facilitates seamless integration with current classical systems.

In pure-state tomography, the concept of unique determinedness (UD), the ability to uniquely determine pure states from measurement results, is crucial. This study presents a variational approach to examining UD, offering a robust solution to the challenges associated with the construction and certification of UD measurement schemes. We put forward an effective algorithm that minimizes a specially defined loss function, enabling the differentiation between UD and non-UD measurement schemes. This leads to the discovery of numerous optimal pure-state Pauli measurement schemes across a variety of dimensions. Additionally, we discern an alignment between uniquely determined among pure states and uniquely determined among all states in qubit systems when utilizing Pauli measurements, underscoring its intrinsic robustness under pure-state recovery. We further interpret the physical meaning of our loss function, bolstered by a theoretical framework. Our study not only propels the understanding of UD in quantum state tomography forward, but also delivers valuable practical insights for experimental applications, highlighting the need for a balanced approach between mathematical optimality and experimental pragmatism.

Recent advancements in quantum hardware and classical computing simulations have significantly enhanced the accessibility of quantum system data, leading to an increased demand for precise descriptions and predictions of these systems. Accurate prediction of quantum Hamiltonian dynamics and identification of Hamiltonian parameters are crucial for advancements in quantum simulations, error correction, and control protocols. This study introduces a machine learning model with dual capabilities: it can deduce time-dependent Hamiltonian parameters from observed changes in local observables within quantum many-body systems, and it can predict the evolution of these observables based on Hamiltonian parameters. Our model's validity was confirmed through theoretical simulations across various scenarios and further validated by two experiments. Initially, the model was applied to a Nuclear Magnetic Resonance quantum computer, where it accurately predicted the dynamics of local observables. The model was then tested on a superconducting quantum computer with initially unknown Hamiltonian parameters, successfully inferring them. Our approach aims to enhance various quantum computing tasks, including parameter estimation, noise characterization, feedback processes, and quantum control optimization.

Determining whether a subspace spanned by certain quantum states is entangled and its entanglement dimensionality remains a fundamental challenge in quantum information science. This paper introduces a geometric measure of r-bounded rank, Er(S), for a given subspace S. Derived from the established geometric measure of entanglement, this measure is specifically designed to assess the entanglement within S. It not only serves as a tool for determining the entanglement dimensionality but also illuminates the subspace's capacity to preserve such entanglement. By employing developed nonconvex optimization techniques utilized in machine learning area, we can accurately calculate Er(S) within the manifold optimization framework. Our approach demonstrates notable advantages over existing hierarchical methods, PPT relaxation techniques, and the seesaw strategy, particularly by combining computational efficiency with broad applicability. More importantly, it paves the way for high-dimensional entanglement certification, which is crucial for numerous quantum information tasks. We showcase its effectiveness in validating high-dimensional entangled subspaces in bipartite systems, determining the border rank of multipartite pure states, and identifying genuinely or completely entangled subspaces.

As quantum technology rapidly advances, the need for efficient scalable methods to characterize quantum systems intensifies. Quantum state tomography and Hamiltonian learning are essential for interpreting and optimizing quantum systems, yet a unified approach remains elusive. Such an integration could enhance our understanding of the complex relationship between quantum states and Hamiltonians, contributing to the development of more efficient methodologies. In this paper, we present a method that integrates quantum state tomography and Hamiltonian learning, drawing inspiration from machine translation in the field of natural language processing (NLP). We demonstrate the effectiveness of our approach across a variety of quantum systems, successfully learning the complex relationships between quantum states and Hamiltonians. Furthermore, the scalability and few-shot learning capabilities of our method could potentially minimize the resources required for characterizing and optimizing quantum systems. Our research provides valuable insights into the relationship between quantum states and Hamiltonians, paving the way for further studies on quantum systems and advancing quantum computation and related technologies.

Quantum query complexity is pivotal in the analysis of quantum algorithms, encompassing well-known examples like search and period-finding algorithms. These algorithms typically involve a sequence of unitary operations and oracle calls dependent on an input variable. In this study, we introduce a variational learning approach to explore quantum query complexity. Our method employs an efficient parameterization of the unitary operations and utilizes a loss function derived from the algorithm’s error probability. We apply this technique to various quantum query complexities, notably devising a new algorithm that resolves the 5-bit Hamming modulo problem with four queries, addressing an open question from Cornelissen et al (2021 arXiv: 2112.14682 ). This finding is corroborated by a semidefinite programming (SDP) approach. Our numerical method exhibits superior memory efficiency compared to SDP and can identify quantum query algorithms (QQAs) that require a smaller workspace register dimension, an aspect not optimized by SDP. These advancements present a significant step forward in the practical application and understanding of QQAs.

This paper outlines an alternative approach to teaching quantum computing at the high school level, tailored for students with limited prior knowledge in advanced mathematics and physics. This approach diverges from traditional methods by building upon foundational concepts in classical computing before gradually introducing quantum mechanics, thereby simplifying the entry into this complex field. The course was initially implemented in a program for gifted high school students under the Hong Kong Education Bureau and received encouraging feedback, indicating its potential effectiveness for a broader student audience. A key element of this approach is the practical application through portable NMR quantum computers, which provides students with hands-on experience. The paper describes the structure of the course, including the organization of the lectures, the integration of the hardware of the portable nuclear magnetic resonance (NMR) quantum computers, the Gemini/Triangulum series, and detailed lecture notes in Additional file 1. The initial success in the specialized program and ongoing discussions to expand the course to regular high schools in Hong Kong and Shenzhen suggest the viability of this approach for wider educational application. By focusing on accessibility and student engagement, this approach presents a valuable perspective on introducing quantum computing concepts at the high school level, aiming to enhance student understanding and interest in the field.

In the realm of quantum information theory, the detection and quantification of quantum entanglement stand as paramount tasks. The relative entropy of entanglement (REE) serves as a prominent measure of entanglement, with extensive applications spanning numerous related fields. The positive partial transpose (PPT) criterion, while providing an efficient method for the computation of REE, unfortunately, falls short when dealing with bound entanglement. In this study, we propose a method termed "pure bosonic extension" to enhance the practicability of $k$-bosonic extensions, which approximates the set of separable states from the "outside", through a hierarchical structure. It enables efficient characterization of the set of $k$-bosonic extendible states, facilitating the derivation of accurate lower bounds for REE. Compared to the Semi-Definite Programming (SDP) approach, such as the symmetric/bosonic extension function in QETLAB, our algorithm supports much larger dimensions and higher values of extension $k$.

Quantum simulators offer the potential to utilize the quantum nature of a physical system to study another physical system. In contrast to conventional simulation, which experiences an exponential increase in computational complexity, quantum simulation cost increases only linearly with increasing size of the problem, rendering it a promising tool for applications in quantum chemistry. The variational-quantumeigensolver algorithm is a particularly promising application for investigating molecular electronic structures. For its experimental implementation, spin-based solid-state qubits have the advantage of long decoherence time and high-fidelity quantum gates, which can lead to high accuracy in the ground-state finding. This study uses the nitrogen-vacancy-center system in diamond to implement the variationalquantum-eigensolver algorithm and successfully finds the eigenvalue of a specific Hamiltonian without the need for error-mitigation techniques. With a fidelity of 98.9% between the converged state and the ideal eigenstate, the demonstration provides an important step toward realizing a scalable quantum simulator in solid-state spin systems.

Quantum variational algorithms (VQAs) are highly promising to realize quantum advantages on near-term quantum devices. Existing VQAs based on a manually fixed quantum Ansatz are computationally inefficient due to noise and the limited coupling maps of these devices. Previous work considers various quantum architecture search (QAS) algorithms to autodesign a quantum Ansatz based on specific questions to improve the performance of VQAs. Compared to manual design, autodesign can more efficiently explore the large space of a possible Ansatz and achieve better performance. However, two main challenges in utilizing QAS to design quantum circuits efficiently are the tremendous amount of space required for candidate quantum circuits, and the disconnection between quantum devices and autodesign in terms of qubit mapping and quantum noise. To address these issues, we propose an adaptive diversity-based quantum Ansatz search algorithm to efficiently generate the optimal quantum circuit based on device qubit mapping and noise. By considering the diversity among different candidate circuits and adaptively adding circuit depths, our approach only needs to focus on a small optimization space at each iteration step. In addition, the synchronization of optimizing circuit structure and aligning qubit mapping enables us to generate quantum circuits while avoiding additional mapping overhead. We evaluate the performance of our algorithm on simulators and real quantum devices for quantum eigenvalue problems and classification tasks. Results demonstrate that quantum circuits generated by our method outperform both a fixed hardware-efficient Ansatz and randomly generated quantum circuits in terms of final performance and resource-saving. Our algorithm provides a flexible way to efficiently generate excellent quantum circuits for significantly improving the performances of VQAs on near-term quantum devices.