Papers

Electronic Hamiltonian simulation is commonly formulated by mapping fermionic operators to qubit operators and subsequently expanding the resulting ladder-operator products into Pauli strings. While general, this procedure obscures the higher-level fermionic structure and can hide opportunities for circuit optimization. Building on ladder-string-pair (Lasp) diagonalization originally developed for Hamiltonian simulation of partial differential equations, we construct time-evolution circuits for the second-quantized electronic Hamiltonian without performing a Pauli expansion. For the most general case of complex-valued coefficients, we present a time-evolution circuit for a two-body fermionic Lasp operator that corresponds to 16 Pauli strings in the Pauli-expansion approach but does not suffer from Trotter error at this stage. Expanding the optimization scope from a single operator to a triad of three fermionic Lasp operators sharing the same four spin-orbital indices enables systematic cancellation of CX gates, reducing the CX-gate count from 36 to 12 in the example considered, still without introducing Trotter error at this stage. For $n$ spin orbitals, further expanding the optimization scope to a sequence of $O(n)$ suitably ordered triads enables a cascade of CX-gate reductions across triad boundaries, reducing the CX-gate count from $O(n^2)$ to $O(n)$. The Lasp-based approach also naturally accommodates controlled time evolution and yields further optimizations for real-valued Hamiltonians. These results demonstrate that the Lasp-based approach enables more efficient time-evolution circuits by preserving high-level circuit structures and thereby expanding the scope of optimization, providing a systematic route toward more efficient electronic Hamiltonian simulation.

Appeals to typicality are common in physics, but it is often unclear what it means for a physical state to be typical relative to a probability measure, and correspondingly unclear what a law that appeals to typicality asserts. Here we consider how one might characterize physical typicality using ideas from the theory of algorithmic randomness. As a concrete example, we show how taking a physical state to be typical relative to a computable measure when it is Martin-Löf random allows one to formulate the distribution postulate in Bohmian mechanics as a statistical constraining law of the theory. Using a toy model, we show how this constraint guarantees the standard Born statistics for computable experimental protocols. Algorithmic Bohmian mechanics (aBM) thus illustrates how algorithmic randomness may be used to provide precise content to a statistical law.

We numerically implement full quantum circuits of optimal tapered Quantum Phase Estimation (in the form of an implementable approximate bandwidth-limited DPSS taper) for the purpose of minimum eigenvalue computation, where the Hamiltonian time-evolution unitary is implemented using standard Trotterization, and the taper state is constructed using an optimized bandwidth-limited taper. We illustrate, on representative quantum Hamiltonians, with tractable exact classical numerical quantum circuit simulations, how tapered QPE performs with respect to optimal phase sampling rate, absolute error, Shannon entropy of the optimal phase distribution, when the tQPE algorithm parameters are changed. Those algorithm parameters are Trotter error, finite sampling, imperfect initial states, total evolution time, and digitization error (number of phase qubits). The reported numerical experiments include up to $\ell=10$ phase qubits of precision with $m=3$ additional phase-register error suppression qubits, on a $4$-qubit Heisenberg quantum magnet model Hamiltonian, using up to 12th-order Trotterization. We numerically show i) the steady-state optimal phase sampling rate is determined by the initial state overlap with the ground-state, ii) with respect to phase sampling probability, or overall eigenvalue sampling error rate, there is no strong evolution time dependence for tQPE unless there is high Trotter error, iii) the approximated time-evolution unitaries in tQPE, like in standard QPE, can result in substantial ringing, which leads to non-physical eigenvalue estimates.

Precisely timed, multi-device control in atomic, molecular and optical physics is provided by specialized real-time systems, which impose their own terms on the description of the experiment: a program over a global clock, in which every stage boundary is an absolute time computed from everything preceding it. Such descriptions are tightly coupled – changing one stage affects all later ones – and are correspondingly hard to reuse, inspect, or move elsewhere. We introduce Wigner Time, a Python package in which the experimental procedure is instead represented as data: a table of timed updates, assembled by composing functions and referred to named points in the experiment rather than to absolute instants. Hardware enters only at a final conversion step, leaving the description readable and back-end agnostic. Wigner Time has run two cold-atom setups for more than two years, where replacing a hand-written real-time program improved the achievable temporal resolution five-fold; the design applies to any domain requiring precise multi-device timing.

We develop a manifestly gauge and Lorentz invariant perturbative calculus to study the finite time and distance evolution of the electrically charged states created by Wilson-line-dressed scalar field operators in scalar quantum electrodynamics. By an Osterwalder-Schrader reflection and Euclidean gluing, the overlap of two dressed states is expressed as a Euclidean 2 point function of dressed operators, which is then evaluated in renormalized perturbation theory with a Stueckelberg photon mass as a gauge, BRST and Lorentz invariant infrared cutoff. This renders explicit and computable in closed form several structural features of the charged sector that have so far been accessible mainly through non-perturbative or algebraic arguments. We find that states dressed by non-parallel Wilson lines are “cloud orthogonal". Their overlap vanishes as a power of the infrared regulator, with an exponent that depends only on the angle between the two dressings and is an infrared analogue of the cusp anomalous dimension with the difference that it is one-loop exact whenever the charged matter is mass-gapped. Charged states are thereby superselected by the orientation of their dressing. Within a superselection sector the 2 point function is infrared finite but exhibits the infraparticle. The mass shell pole is replaced by the edge of a branch cut. As a result, the asymptotic spread of a dressed charge is a power law in proper time with a one-loop-exact computable exponent depending on the dressing and on the direction of motion. We show that although cloud orthogonality and the infraparticle are each extremely sensitive to fundamental infrared cutoffs, the infraparticle scaling law persists across a wide window of proper time.

Optical tweezers are a versatile tool in the domain of cytology, enabling trapping and manipulation of individual cells. However, the incidence of laser light causes photodamage to biological matter, even when operating at low optical powers and over short periods of time. Here, we demonstrate stable trapping of single living Saccharomyces cerevisiae yeast cells in vitro using a dark optical trap operating in the repulsive regime of light-matter interactions. In contrast to standard tweezers, our dark optical trap confines cells for hours with negligible disruption to their morphology and reproduction cycle, opening up new possibilities for long duration experiments with living organisms such as the observation of cell reproduction under laser trapping.

We introduce an operational framework for heat-exchange statistics between a quantum field and an Unruh-DeWitt detector in Minkowski spacetime. Using an interferometric protocol, we define the characteristic function of heat without projective measurements, consistently with relativistic causality. We derive perturbative expressions for this function and the heat distribution. In the vacuum, the statistics coincides with the low-intensity expansion of a unidirectional Poisson law, while for Kubo-Martin-Schwinger (KMS) states it has a bidirectional Poisson structure up to second order in perturbation theory. We study how fluctuation relations emerge from detector properties and field correlations, identifying regimes where they reduce to known heat-exchange fluctuation relations. We derive the entropic Landauer inequality as a specialization of the algebraic entropy-balance theorem of Jakšić and Pillet and verify it numerically within the perturbative detector-field interaction. We compare its tightness with a complementary Landauer bound derived from the full counting statistics of heat, analyzing their dependence on the detector's initial state and interaction time. We also formulate the first and second laws for the explicitly switched detector-field dynamics, identifying the external switching work that reconciles the field-energy gain with the detector-energy change. In the gapless and delta-switching regimes, where the dynamics admits a non-perturbative treatment, the characteristic function coincides with that of a bidirectional Poisson process, yielding an exact fluctuation theorem determined by the KMS condition. These results provide a consistent framework for heat statistics in relativistic quantum field theory and clarify the emergence of fluctuation relations.

The quantum lattice Boltzmann method (QLBM) holds promise for efficient fluid simulations, yet the treatment of realistic boundary conditions, particularly solid wall boundary with arbitrary geometry, remains a critical open challenge. We propose a QLBM algorithm that incorporates wall boundaries as well as inlet/outlet velocity conditions. Building upon the existing QLBM with ensemble transformations, we adopt a density-matrix encoding that offers greater flexibility in circuit design and enables the implementation of non-unitary operations through Kraus operators. For boundary enforcement, we design a quantum implementation of the half-way bounce-back scheme and introduce a component exchange step prior to the streaming step, which naturally enforces the no-slip condition without modifying the transport operation. Inlet and outlet conditions are imposed via controlled SWAP operations combined with ancilla registers prepared in the prescribed velocity states. Our QLBM is quantitatively validated through simulations of two-dimensional flows past a backward-facing step, a cylinder, and an obstacle with complex geometry, demonstrating both accuracy and extensibility of our boundary algorithm.

Nonstabilizerness (magic) is the quantum resource that, together with stabilizer operations, makes universal quantum computation possible. However, quantifying nonstabilizerness is difficult. The standard measure sums over exponentially many Pauli operators, and for quantum codes no quantitative theory has been available. A structural observation removes this obstacle: the Clifford sector carries no magic and can be removed by a Clifford transformation, leaving an order-three object whose magic is an exactly evaluable fourth moment. Removing this inert background yields closed forms for several code families: a formula for all Dicke states, reducing a 100-qubit case from $4^{100}$ terms to a short binomial sum; a bound for cubic-phase codes (twisted quantum doubles and non-Abelian topological order), saturated only by the $D_4$ code; and a cyclic/zero criterion for group multiplication states. For codeword-stabilized (CWS) codes, the companion paper \cite{Liu26arXiv} carries this reduction to its conclusion: the most magical codes are exactly the Sidon sets, and the nonstabilizerness bounds the code's transversal non-Clifford power. Together these results reduce the computation of nonstabilizerness for a broad class of codes to finite classical counting problems, and identify the maximally magical codes among them.

I recently showed how objective Everettian probability can be defined over the records borne by an observer's later continuations, without unique actualization or additional premeasurement subjects \citep{Baxter2026}. Here I extend that model to all decoherent physical record states, including records that support no such continuation. The observer-indexed model is then a restriction of an exhaustive physical-record model.

This review develops a control-theoretic perspective on measurement-based feedback for continuously monitored open quantum systems, with the main analysis focused on finite-dimensional systems governed by diffusive stochastic master equations. We introduce the relevant state-space, invariant-subspace, and quantum non-demolition structures, interpret quantum filtering as nonlinear observer dynamics, and review open-loop asymptotics, filter stability, state-feedback stabilization, robustness, and reduced-order observer-based control. Particular emphasis is placed on a recurrence–contraction framework, in which Hamiltonian feedback removes non-target invariant obstructions while measurement-induced dynamics provide local exponential contraction. Although the detailed analysis is developed for finite-dimensional diffusive models, the underlying measurement–estimation–feedback architecture is relevant across a broad range of quantum platforms. We further discuss implementation challenges and open problems involving scalable estimation, sampling and delay, adaptation, hybrid and non-Markovian dynamics, practical stability, optimal control, and learning-based design. By organizing these developments around invariance, estimation, recurrence, and contraction, the review provides a tutorial bridge between measurement-based quantum feedback and nonlinear stochastic control.

Bounds on the charging power of quantum batteries in direct charging protocols are often interpreted as charging-side constraints. Although the locality of the charging Hamiltonian is known to restrict attainable power in certain protocols, whether battery locality is itself operationally necessary has remained unresolved. Here we show that it is necessary for spin batteries whose interactions do not pairwise commute. We construct commuting and noncommuting high-locality batteries with identical interaction supports and local energy scale, driven by the same charging. The commuting battery saturates the battery-locality-independent bound, whereas the noncommuting battery exceeds it by a parametrically growing factor, demonstrating that this bound cannot apply generally. The enhanced norm of the commutator is dynamically attained from a suitable joint state supported entirely in the battery ground-energy sector, although this state may be correlated with the auxiliary fermionic degrees of freedom. Our results establish battery locality as an operational resource enabled by noncommuting battery interactions.

We present a combined experimental and Floquet-theoretical study of avoided crossings in the AC Stark-shifted electromagnetically induced transparency (EIT) spectra of Rydberg atoms. Observed avoided-crossing structures cannot be fully explained by conventional AC Stark maps alone, which provide the energy shifts but do not reveal the underlying state composition and coupling pathways. Using the Shirley method, we analyze the Floquet eigenstates to establish a direct connection between the measured spectra and the underlying dressed-state dynamics. By tracking the evolution of state mixing, bare-state composition, and dominant higher-order Floquet coupling pathways, we reveal the physical mechanisms responsible for the formation of avoided crossings across different principal quantum numbers and RF frequencies. Our results demonstrate that Floquet-eigenstate analysis provides a powerful framework for interpreting complex AC Stark-shifted Rydberg-EIT spectra and the underlying higher-order Floquet coupling interactions.

Metastability in open quantum systems is usually inferred from spectral separation in the Liouvillian, which governs unconditional, ensemble-averaged dynamics. We show that this diagnosis is incomplete at the level of individual quantum trajectories: conditioned realizations of the same unconditional dynamics can bypass, transiently access, or operationally preserve a metastable memory, depending on the monitored channel and the observed record. We demonstrate these mechanisms in a driven-dissipative nonlinear oscillator realizing a quantum associative memory, by comparing spectra, target fidelities, and phase-space distributions. The non-Hermitian dynamics obtained by post-selecting on the absence of detected events supports metastable retrieval, but with a distinct long-time fate: normalization selects the least-decaying mode of the non-Hermitian spectrum rather than the addressed memory branch. In contrast, stochastic jump trajectories can preserve retrieval over extended times when the post-measurement update remains compatible with the coherent memory structure. Thus trajectory-level metastability is not determined by the averaged generator alone; it requires compatibility between the monitored channel, the measurement record, and the metastable manifold.

Neural quantum states (NQSs) have emerged as a powerful ansatz for quantum dynamics. Their high entanglement capacity promises to overcome the entanglement barrier. However, existing time-dependent NQS methods are ill-equipped to propagate sparsely supported initial states due to the inherent mismatch between the supports of such states and their time derivatives. To address this issue, we introduce the interpolation sampling method as a novel form of importance sampling, where samples are drawn from a distribution that interpolates between the wave function and its time derivative. Combined with a global-in-time variational principle, we demonstrate with the example of the two-dimensional transverse-field Ising model that interpolation sampling allows for accurate propagation of extremely peaked states that standard wave function sampling could not tackle. We further improve sampling efficiency by introducing (i) a configuration-time joint sampling scheme where spin configurations and time are both treated as random variables, and (ii) a hybrid strategy for sample proposal that incorporates the knowledge of a Krylov subspace. Our work extends the scope of time-dependent NQS methods to a wider range of physically relevant scenarios, while challenging the prevailing reliance on the Born distribution and its close variants as the default basis for sampling.

Quasicrystals occupy a distinctive position between the translational order of crystals and the disordered amorphous matter. Simulating this physics has remained challenging, since quasiperiodic structures lack the translational symmetry exploited for crystals and generally require costly diagonalization of large finite approximants. Quasicrystalline order admits two equivalent descriptions, a cut-and-project scheme from a higher-dimensional periodic crystal, and a discrete set of substitution rules acting on a finite alphabet. We show that the latter, written in a numeration system adapted to the substitution, defines a deterministic finite automaton with output, the digits of a site index are fed, and the automaton returns the letter occupying that site. We further exploit another equivalence to a different construction, the transition matrices are exactly the tensors of a matrix product state, whose bond dimension is the number of automaton states and is independent of system size. This allows efficient representation of extremely large tight-binding Hamiltonians in the tensor-train language, thereby yielding an exact matrix product operator for the quasicrystal Hamiltonian at any system size. We show how this framework works for two families of one-dimensional quasicrystals, the metallic-mean and $k$-bonacci families and we explicitly construct the Fibonacci, silver-mean, and Tribonacci quasicrystals. By leveraging efficient tensor-network compression and the kernel polynomial method, we compute spectral densities for chains with more than $10^9$ sites and directly resolve the hierarchical structure of the spectrum.

The $I_{3322}$ inequality is one of the simplest bipartite Bell inequalities, with three possible questions and two possible answers per party. Yet, despite its simplicity, it has long been conjectured by Pál and Vértesi (Physical Review A, 2010) to possess the following quirk: no finite-dimensional quantum strategy can attain its maximal violation, but an infinite-dimensional strategy can. A Bell correlation, with five questions and three answers per party, that provably possesses the same property has since been discovered by Coladangelo and Stark (Nature Communications, 2020). However, proving that the same property holds for $I_{3322}$, which lives in the simplest Bell scenario in which such a phenomenon could occur, has remained elusive. Here, we provide a proof of this conjecture. An approximate proof, containing all the main ideas, was produced by GPT 5.5 Pro after various rounds of interaction. The presentation of this proof was revised substantially by the author for completeness, correctness, and clarity.

A static deformation of graphene can act on its Dirac electrons as a valley-odd magnetic field. In sublattice-gapped graphene this field makes the two valleys add in the parity-odd response rather than cancel. We derive the equilibrium thermodynamics of this effect and separate it from the finite-frequency transport response. At fixed electrochemical potential, reversing the pseudomagnetic field removes every nonzero pseudo-Landau level in the continuum theory. The remaining grand-potential difference is fixed by the spectrally asymmetric zeroth level. The static charge response is a thermally broadened plateau confined to the gap and has no metallic $m/|\mu|$ tail. Near a band edge, pseudofield reversal transfers $\mathrm{B}\ln2$ of entropy per unsplit zero-mode state in the low-temperature window. The fixed-$\mu$ heat capacity has two side lobes per edge and a universal peak $0.439229\,D_B k_{\mathrm B}$. We then formulate a definite constant-gate-voltage circuit and show that the measured sheet heat capacity depends on the electrical boundary condition. The full massive-Dirac density of states and an exact finite-field pseudo-Landau-level calculation give the same gate crossover in their common limit. At fixed carrier number, the low-temperature edge value is $-2(\ln2)^2 D_B k_{\mathrm B}$, rather than a node. A finite geometric capacitance gives a continuous and experimentally tunable interpolation. Finally, we give a trace-free triaxial strain geometry, a disorder–interaction window, and realistic calorimetric and quantum-capacitance scales. The field-reversal protocol isolates an equilibrium electromechanical anomaly without a real magnetic field.

The quest for autonomous alignment and maintenance of complex free-space optical systems is increasingly urgent for large-scale neutral-atom quantum processors. Leveraging a high-finesse optical cavity as a sensitive probe, we introduce a data-driven framework for structural diagnostics and closed-loop control that achieves cold-start convergence within tens of seconds. These results establish a rapid, scalable diagnostic-control methodology for autonomous alignment and continuous maintenance of complex free-space optical architectures.

Sample-based Krylov quantum diagonalization (SKQD) is a paradigmatic example of a quantum-centric supercomputing workflow that combines the convergence structure of Krylov quantum diagonalization with classical sampling-based post-processing. It provides convergence guarantees assuming that the important computational-basis configurations can be sampled from a set of Krylov states with sufficient probability. We analyze this assumption under depolarizing noise, deriving shot-count resource estimates that expose an exponential depth penalty, and perform experiments on current noisy hardware with the one-dimensional single-impurity Anderson model on a 20-site (40-qubit) instance. Device noise breaks the practical convergence predicted by the noiseless SKQD analysis, but approximate compilation techniques can compress the Krylov time-evolution circuits before execution. The compressed circuits accumulate less hardware noise, recover the expected energy convergence, and reduce both the quantum shot budget and the classical subspace dimension, remaining beneficial even when classical configuration recovery is applied.

Diffusion sampling can be viewed as imaginary-time annealing of probability densities. From a forward/backward Euclidean Schrödinger pair, we show that fixing the noising drift forces the reverse score term, and that the same logarithmic force is the one-sided imaginary-time counterdiabatic connection of a supersymmetric Score Hamiltonian. The correspondence turns score-learning error into a Hamiltonian perturbation and yields a schedule principle: if the first $r$ terminal derivatives vanish, Morita–Nishimori boundary cancellation suppresses the residual Hellinger error from $T^{-2}$ to $T^{-2r-2}$ until score or sampling floors are reached. Reverse-SDE benchmarks, including two-dimensional learned-score tests, confirm the predicted improvement.

High-resolution, high-sensitivity microwave (MW) magnetic field imaging is indispensable for non-destructive integrated circuit (IC) testing, radio-frequency device characterization, and spintronic research. Yet, the practical utility of these techniques is severely constrained by the pervasive challenge of isolating weak magnetic signatures from intense optical and electronic noise, which fundamentally limits both acquisition speed and detection sensitivity. Here, we overcome this barrier by introducing a wide-field imaging scheme based on an ensemble of diamond nitrogen-vacancy (NV) centers, synergistically combined with digital lock-in amplification (DLA). By exploiting digital demodulation, the DLA precisely extracts the MW-field response at a specific modulation frequency from background noise (e.g., laser intensity fluctuations), dramatically improving the signal-to-noise ratio (SNR). Consequently, our system attains a magnetic field sensitivity of 126 nT/$\sqrt(Hz)$. Critically, the unprecedented SNR permits a pixel dwell time of under one millisecond, allowing full-field images to be acquired within seconds-more than an order of magnitude faster than state-of-the-art NV-based wide-field techniques. This combination of speed, sensitivity, and micron-scale spatial resolution (1.6 $\mu$m) paves the way for quasi-real-time, non-invasive diagnostics of dynamic MW devices and integrated circuits.

We give an exact translation between fat-point interpolation in complex projective space and a dark-state problem for fixed-particle-number bosons. Under the unitary Fischer–Bargmann realization, homogeneous degree becomes particle number, the Veronese lift becomes a bosonic symmetric product state, and order-$m$ vanishing becomes orthogonality to all coherent jet excitations of orders below $m$. The initial degree of a symbolic power is therefore the first particle number admitting a dark state of a positive constraint Hamiltonian, and the Waldschmidt constant is the asymptotic particle cost per suppression order. After developing both sides of this dictionary from first principles, we prove a finite-scale no-dark-state amplification theorem. Frobenius amplifies the associated full-rank certificate in one good positive-characteristic fiber; a nonzero maximal minor returns it to characteristic zero. Its asymptotic consequence is Demailly's conjectured inequality. The paper is written so that the projective statement, the quantum reformulation, and the arithmetic part of the solution can be read independently and then identified exactly.

Belief propagation with quantum messages (BPQM) is a quantum algorithm that decodes classical codes transmitted over classical–quantum channels. It realizes optimal decoding on tree factor graphs over pure-state classical-quantum channels. However, this tree-based analysis does not ensure vanishing block-error probability for LDPC Tanner graphs with cycles. In this work, we construct a two-stage BPQM decoder for random $q$-ary LDPC codes over symmetric $q$-ary pure-state channels, where $q$ is prime, and prove that its ensemble-average block-error probability vanishes as the blocklength $N$ tends to infinity. For regular ensembles with $d_v\geq3$, fidelity bounds yield double-exponential decay of the average symbol-error probability throughout the BPQM success region. We apply depth-$\ell$ BPQM to coordinates with tree neighbourhoods and treat the remaining coordinates as erasures. With a suitable $\ell=\Theta(\log\log N)$, a noncommutative union bound controls the BPQM decoding errors, while the minimum-distance property guarantees erasure recovery. We also extend the analysis to finite-support irregular ensembles. These results are relevant to quantum algorithms based on Regev's reduction, where coherent decoding uncomputes a codeword register. Decoded quantum interferometry (DQI) uses a closely related Fourier-based framework that reduces sparse max-LINSAT optimization problems to LDPC decoding problems on pure-state channels. Our results justify the use of BPQM in the decoding step of DQI and of coding-theoretic algorithms based on Regev's reduction whenever the code is drawn from one of the random LDPC ensembles analyzed here and the induced memoryless symmetric pure-state channel lies in the BPQM success region. Curiously, our numerical results indicate that DQI+BPQM achieves a satisfaction ratio that closely matches that of simulated annealing.

Majorization inequalities encode spectral comparisons through leading partial sums, whereas complete Horn inequalities capture all admissible selected-eigenvalue constraints associated with sums of Hermitian matrices [Fulton, 2000]. We develop a general framework for lifting majorization inequalities to this complete Horn level, which is equivalent to unitary-orbit inequalities. The main ingredients are a generalized Hersch–Zwahlen variational formula for coordinatewise monotone functions of selected eigenvalues and a Schubert-geometric lifting principle that reduces the desired Horn inequalities to the concavity or convexity of suitable scalarizer functions on matrix compressions. As examples we apply the framework to three families of operator inequalities. First, we obtain a complete Horn extension of the weak-majorization inequality for convex functions established by Aujla and Silva [Aujla and Silva, 2003]. Second, we lift the concavity and convexity results for two-variable positive-power functions due to Carlen, Frank, and Lieb and Zhang [Carlen, Frank, and Lieb, 2016; Zhang, 2020]. Third, we establish multiplicative Horn inequalities for multivariable geodesic means, including the weighted Karcher mean and the two-variable geometric mean, extending determinant-level inequalities of Bourin and Hiai [Bourin and Hiai, 2014]. The resulting statements yield unitary-orbit inequalities that generalize familiar weak-majorization, trace, and determinant inequalities.

Reinforcement learning with verifiable rewards enables large language models to think slowly, but the same training can induce policy collapse: probability concentrates onto a few successful trajectories and exploratory diversity erodes. We ask whether quantum AI can realize slow thinking differently. We formulate slow thinking as coherent dynamics over reasoning trajectories, a discrete path integral in which action sequences coexist in superposition and recombine before measurement. In our trainable realization, an exact verifier partitions the ensemble into collective accepted and rejected components that interfere under Grover amplitude amplification. A finite Grover evolution is maximized when the pre-amplification success probability lies at an analytically determined value below one, so inference itself defines an interior training target and removes the monotonic pressure toward unit success. In exact statevector simulations of a 2x3 sliding puzzle, Grover training reaches accuracy 0.95 on a 32-question training set at one round, against 0.73 for the strongest classical control. On held-out questions specialization has a cost: an untrained uniform policy read out through the same amplification remains the strongest reference on this solution-dense benchmark, and quantum training preserves far more held-out accuracy than classical training - at four rounds with matched circuit applications the two quantum models reach 3.2 and 3.9 times the strongest classical controls. The number of training questions supported by fixed-size policies trained at each amplification budget also grows faster with the budget than with matched classical repetition. These results establish a Grover-based realization of path-integral slow thinking: the interior target preserves exploratory path diversity, and ensemble-level interference converts it into verified performance.

The pursuit of high-resolution eigenspectroscopy on noisy intermediate-scale quantum devices is often hindered by the trade-off between circuit depth and coherence time. In this work, we introduce and experimentally demonstrate a dynamical eigenspectroscopy protocol that extracts fine-grained energy structures from time-dependent survival amplitudes. To overcome the finite coherence window of current superconducting processors, we employ Variational Trotter Compression (VTC) as a practical means to extend the duration of high-fidelity unitary evolution. Using a multi-connected 9-qubit superconducting processor, we reconstruct the time-domain autocorrelation signal via quantum state tomography for the H2 molecule at different bond lengths and for the Fermi-Hubbard model across different correlation regimes. Through multi-frequency fitting and Fourier analysis, the extracted eigenenergies agree with the exact-diagonalization values to within 2x10^-3, including the near-degenerate levels in the strongly interacting regime. Our results establish experimental dynamical spectroscopy as a robust and generalizable framework for simulating both quantum chemistry and strongly correlated lattice systems, bridging weak- and strong-coupling regimes on near-term quantum hardware.

We formulate the generalized hidden shift problem over arbitrary finite groups and obtain upper and lower bounds for the sample complexity in terms of group theoretic data.

Topological insulators host protected boundary states that are robust to disorder, making them attractive for applications and motivating their realization in engineered systems. Discrete-time quantum walks, which have been implemented in a variety of experimental platforms, can exhibit topologically nontrivial phases and their associated boundary states. Here, we introduce a class of topological quantum walks with variable step lengths that offer theoretical insights into the topological classification of such walks. For example, these walks can access higher winding numbers and support multiple edge states. They also provide context for the reduction of topological protection from a $\mathbb{Z}$- to $\mathbb{Z}_2$-valued invariant when time-reversal symmetry is broken. The topologically protected edge states are investigated using a transfer-matrix approach that describes their spatial profiles and spin structures. These predictions are in excellent agreement with numerical results and, where applicable, with the Jackiw–Rebbi zero-mode solution of the Dirac equation. Taken together, our analysis provides a roadmap for designing quantum walks with control over the number, spatial profile, and spin structure of their topological edge states.

Quantum applications increasingly execute as multi-stage quantum-classical pipelines, interleaving QPU computation with classical stages like circuit generation, transpilation, layout mapping, quantum error mitigation (QEM), and post-processing. These stages have diverse resource requirements and exhibit stochastic behavior under drifting hardware noises, yet existing workflow frameworks treat them as static, isolated components. We present LLQM (Low-Level Quantum Machine), a profiling-driven meta-framework for quantum-classical pipelines. LLQM decomposes pipelines into fine-grained tasks and continuously profiles their CPU/GPU, memory, QPU, and queue dependencies alongside real-time hardware states. This unified runtime abstraction captures cross-stage resource dependencies and reveals how classical and quantum decisions interact, enabling characterization of their impact on fidelity and resource consumption. We evaluate LLQM using QEM as a representative pipeline stage, on IBM 156-qubit Heron r2 processors with circuits up to 100 qubits and 1e7 transpiled gates. Our results show that continuous profiling exposes runtime bottlenecks and enables hardware-, fidelity-, and workload-aware optimizations.