Papers

In the space $\mathbb R^3$, for Robin parameter $L>0$, it is shown that there is a family of Schrödinger operators with penetrable toroidal solenoid and variable conductivity that approximates the magnetic Aharonov-Bohm operator with a Robin boundary condition at the solenoid (border). It is also shown that approximations via smooth potentials and then a barrier in the solenoid interior give Dirichlet boundary conditions. The approximations are in the strong resolvent sense and obtained through the $\Gamma$-convergence technique, and they hold for the more general setting of smooth, closed and compact surfaces and continuous and bounded magnetic potentials.

We construct an explicit symplectic realization of the Barnes-Wall lattice that yields a family of multimode Gottesman-Kitaev-Preskill (GKP) codes with encoding rate $R = \frac{1}{2}\log_2 N$ and a deterministic $O(N\log^2 N)$ bounded-distance decoder. The recursive generator $G_{m+1} = \bigl(\begin{smallmatrix} G_m & 0 \\ G_m & R_m G_m \end{smallmatrix}\bigr)$ with $R_m = I + \Omega$ simultaneously guarantees symplectic integrality for valid quantum stabilizers and preserves the exact Barnes-Wall decoding structure through a chain of isometric isomorphisms. The code distance is constant at $\Delta^2 = 1$ (in units of $2\pi$), representing an explicit distance–rate tradeoff in which logarithmic encoding efficiency is achieved at the cost of non-scaling protection. This construction provides a deterministic, space-efficient paradigm for GKP error correction in platforms supporting non-local modular connectivity.

We report a previously overlooked general mechanism to obtain highly efficient spin selectivity in conventional time-reversal symmetric one-dimensional systems without invoking phase decoherence. We reveal that Rashba quantum wires featuring non-collinear spin states at the Fermi level inherently possess spin-selective transport properties. We show that this spin noncollinearity can be systematically designed and engineered by introducing an additional pseudospin degree of freedom - such as valley, sublattice, or orbital angular momentum - into spin-orbit coupled systems. By applying this framework to multi-subband semiconducting quantum wires and oxide nanowires, we establish a generalized route toward quantum-coherent spin selectivity up to 10 %. Our findings offer practical design principles for spin-selective transport devices.

Solid-state single organic molecules, such as dibenzoterrylene (DBT) in organic matrices, are prominent deterministic single-photon sources, usually modeled as effective two-level systems (TLS). We show that single DBT molecules in anthracene nanocrystals, under strong continuous-wave driving, depart from this picture: instead of the expected saturation, fluorescence is strongly suppressed at high resonant pump power, while the linewidth broadens beyond the TLS prediction – an anomalous saturation regime. Comparing coherent and incoherent excitation and independently calibrating temperature via phonon-induced dephasing rules out laser-induced heating and intersystem crossing. Instead, a model of intensity-dependent excited-state absorption (ESA) toward a short-lived dark state quantitatively reproduces both the fluorescence suppression and the linewidth broadening. We further show that matrix quality mitigates this quenching, and that pulsed excitation schemes are strategic toward full population inversion, with direct implications for quantum nanophotonics and molecular optomechanics.

We bridge two sides of singular perturbation theory: the classical theory of slow-fast systems and the semi-classical approach to quantum mechanical systems. For a specific but physically important class of dynamical systems, we show that purely classical and exotic objects, so-called canard solutions, are shadows of instantons in the corresponding quantum system. We demonstrate that canard solutions exist in a domain of parameter space whose boundaries are determined by an instanton action. We illustrate our statements analytically for the relevant example, the overdamped Josephson junction, and confirm them numerically. For the Josephson junction, the canard window is the exponentially narrow gap between consecutive Shapiro steps.

Electronic-structure and embedding workflows terminate in effective many-body Hamiltonians, whereas quantum eigensolver studies often start from hand-built qubit models. We present the Adaptive Clifford-Algebra Subspace Eigensolver (A-CASE), a single-reference, operator-generated Rayleigh–Ritz method. Overlap, Hamiltonian, observable, and response matrices are reconstructed from one shared Pauli-expectation bank, while adaptive growth scores overlap-aware local pencils and rejects symmetry leakage or near-linear dependence. A strict FCIDUMP adapter supplies the active-space boundary. For linear H$_4$ in STO-3G with CAS(4e,4o), the mapped sector agrees with independent determinant FCI to $3.1\times10^{-15}$ Ha. At a nine-vector budget A-CASE has a $3.019$ mHa error; replacing the determinant reference by a two-operator ADAPT-VQE state reduces it to $0.342$ mHa, without implying a matched total-cost advantage. Under a matched contract, the fixed-reference route uses one state preparation versus ADAPT-VQE's ninety but measures roughly an order of magnitude more Pauli words. Exact fixed-angle ADAPT-GCIM gives $13.364$ mHa at the nearest size match and $10.674$ mHa at the iteration match, with its transition-pair burden reported separately. Across a broader benchmark ladder, fixed Krylov bases are generally more accurate and often narrower but substantially less well conditioned. A grouped bootstrap propagates finite-shot variability through thresholding, diagonalization, root matching, spectral weights, susceptibility, and broadening; its bands are explicitly heuristic and conditional, not finite-sample confidence certificates. The work establishes an executable path from an interchange Hamiltonian to energies, correlations, and response, without claiming materials accuracy, favorable scaling, or quantum advantage.

As photons propagate through curved spacetime, gravitational effects become unavoidable. In particular, gravitational redshift can induce significant distortion in photon wave packets, making it es?sential to investigate parameter estimation within this context. While previous research has focused on single-parameter estimation using the quantum Cramer-Rao bound, the multiparameter scenario remains largely unexplored. In this work, we investigate multiparameter quantum estimation for a photon system subject to gravitational redshift under both amplitude-damping and Ohmic-like dephasing channels. Our analysis reveals that the quantum Cramer-Rao bound fails to provide a tight error bound for the two-parameter estimation involving the initial phase and weight parameters inboth types of noisy channels. To overcome this limitation, we numerically compute two tighter error bounds, i.e., the Holevo Cramer-Rao bound and the Nagaoka bound, when utilizing a semidefinite program. We demonstrate that the Nagaoka bound yields the tightest error bound among all considered bounds, consistent with the general hierarchy of multiparameter quantum estimation. Furthermore, for the three-parameter estimation, including the initial weight parameter, the phase parameter, and the strength of gravitational redshift, we observe significantly enhanced estimation precision in the strong-coupling regime compared to the weak-coupling regime under the amplitude-damping channel. Similarly, in the Ohmic-like dephasing channel, the sub-Ohmic regime consistently affords higher precision than the Ohmic and super-Ohmic regimes.

Integration and sampling in high dimensions are among central challenges in modern science and technology, underlying applications ranging from quantum many-body physics to Bayesian inference and artificial intelligence. Although conventional Monte Carlo methods are formally scalable, their efficiency deteriorates rapidly in the presence of strong correlations or sharp features in high-dimensional configuration space. Here we introduce a new framework, termed the Schr"odinger Generator, for integration and sampling based on the explicit optimization of coordinate transformations. The method decomposes the total Jacobian into two complementary components, including an adaptive map that minimizes estimator variance by learning the marginal structure in each dimension, and a normalizing-flow-based transformation that captures non-factorizable correlations in the target distribution. A final resampling step guarantees unbiased sampling even when the learned transformation is imperfect. We demonstrate stable and scalable performance for nuclear quantum many-body states in dimensions exceeding 600. Short-range correlations among nucleons in finite nucleus are faithfully reproduced. The framework offers a physically transparent approach to high-dimensional stochastic integration and sampling, opening new possibilities for simulations of complex quantum systems.

Cooperative platooning can reduce the energy use of Connected and Autonomous Vehicle (CAV) fleets, but the routing problem becomes difficult when vehicles must meet on the same road segments at compatible times while moving through unstable urban traffic. This paper develops an edge-assisted, closed-loop evaluation pipeline for platooning-aware vehicle routing. Roadside Units estimate local traffic kinematics from video, classify segment-level flow stability, and activate platooning rewards only on road segments where close-gap coordination is physically appropriate. The resulting multi-vehicle routing problem is written directly as a Quadratic Unconstrained Binary Optimization (QUBO) model, so pairwise platooning interactions are represented as native quadratic Ising terms instead of requiring auxiliary MILP linearization variables. We evaluate the framework using a 24-hour microscopic SUMO simulation of Troy, NY, together with localized IBM Quantum hardware benchmarks. The SUMO study shows an $18.5\%$ reduction in fleet tractive-energy demand relative to a non-cooperative baseline. On 25-active-qubit benchmark instances executed on $\texttt{ibm_boston}$, Linear-Chain QAOA reduces two-qubit CNOT depth by $66.7\%$ compared with dense QAOA and samples the exact classical ground state with $P_{\text{feas}} = 38.6\%$ and $P_{\text{opt}} = 14.2\%$ at $p=2$. These results suggest that edge perception and shallow quantum optimization can work together as a useful component of closed-loop CAV platoon dispatching.

It is rather surprising that modern quantum physics does not appear to have provided any clear answer to the simple question “what is electric charge?”. Even when the total charge operator $Q$ is well-defined, the non-locality of $Q$ implies that it is not an observable in the usual sense, which can be measured by a (local) experimental apparatus. A candidate for the “local version” of charge operator is the 4-current operator $j=(j^{\mu})_{\mu=0,1,2,3}$. However, it is known that the rigorous definition of $j$ is difficult in $(3+1)$-dimensional Minkowski space. Although it was found that the current can be defined in $(1+1)$-dimensions (Carey et al.), I argue that even when $j$ can be suitably defined, the interpretability of $j$ as the “local charge operator” is dubious. Instead I return to Araki and Wyss (1964), and propose the concept of “scope-local charge” $Q_{\zeta}(P)$ for a “scope” $P$, expressed by a finite-dimensional projection. I work in an abstract $C^{*}$-algebraic setting.

Imaginary-time evolution (ITE) is a foundational method for ground state preparation on quantum computers. However, because ITE is non-unitary, existing implementations incur a sample complexity and/or classical cost that scales exponentially with the target imaginary time $\beta$. Moreover, the state itself converges slower than the energy, making accurate estimation of arbitrary ground state observables even more expensive. In this work, we improve upon standard ITE by introducing twirled imaginary-time evolution (TITE), which pairs ITE with real-time evolution applied for a random duration drawn from a carefully designed distribution. We prove that this randomization quadratically suppresses the trace distance to the ground state, and thus also the error of arbitrary observables, which allows roughly half of the imaginary time to be replaced with real-time evolution ($\beta \mapsto \beta/2$) while maintaining the same level of accuracy. Because real-time evolution is unitary and does not incur an overhead in sample complexity or classical computation, this affords a quadratic reduction in the cost of any black-box ITE implementation, including Trotterization and quantum imaginary-time evolution. We demonstrate the efficiency of our algorithm in noisy circuit-level simulations of a non-integrable Ising chain, showing substantial improvements over standard ITE.

We study the fault-tolerant cost of preparing sparse quantum states, measured by $T$-count in the Clifford+$T$ model. Here an $n$-qubit state is called $s$-sparse if it is supported on at most $s$ computational-basis states. For arbitrary $n$-qubit states, the optimal $T$-count is $\Theta(\sqrt{2^n\log(1/\epsilon)}+\log(1/\epsilon))$, but for $s$-sparse states the best previous upper bounds remained linear in $s$. We show that any $n$-qubit $s$-sparse state can be prepared up to error $\epsilon$ using $\widetilde{O}(\min\{s,\ n^{3/4}\sqrt{s}\}+\sqrt{s\log(1/\epsilon)}+\log(1/\epsilon))$ $T$ gates, giving the first sublinear dependence on $s$ once the support is sufficiently large. Our approach is based on a support-aware synthesis theorem for sparse Boolean functions, which may be of independent interest. We also prove that, for every $0<\epsilon\le 1/6$ and $2\le s\le 2^{n/2}$, sparse-state preparation requires $\Omega(\min\{s,\sqrt{ns}\})$ $T$ gates, showing that linear dependence on $s$ is unavoidable in the small-support regime and substantially narrowing the gap between the known upper and lower bounds within this parameter range.

Interfacing integrated photonics with solid-state spin defects holds great promise for future quantum networks, but the scaling of spin-photon architectures is hindered by frequency mismatches arising from fabrication-induced variations in photonic cavity resonances and the inhomogeneous optical transition frequencies of individual spins. These challenges call for a photonic platform with deterministic and wide-range tunability. Here, we demonstrate a hybrid nanophotonic platform based on direct bonding of silicon carbide photonic crystal nanocavity arrays onto thin-film lithium niobate on insulator, enabling deterministic electrical tuning of multiple SiC nanocavities into mutual spectral resonance. By exploiting the strong electro-optic response of lithium niobate, we achieve continuous and wide-range cavity tuning of 380 GHz ($\sim$1.1 nm), sufficient to compensate both cavity disorder and spin inhomogeneity. The nanocavities balance strong optical confinement with electrical tunability, exhibiting a theoretical Purcell factor of approximately 400. This hybrid platform enables electrically reconfigurable spin-photon interfaces for large-scale integrated quantum photonics.

Classical objectivity requires an environment to witness a pointer value, not merely to decohere it. We derive microscopic Hamiltonian laws for this distinction. Exact QND product monitoring yields a certified two-edge strong-Darwinism window: sufficient witnesses cross the optimal record-discrimination edge while retaining an unobserved complement that crosses the residual-coherence edge. Under additional regularity assumptions, the supplement gives converse bounds at the same logarithmic scale. For controlled-phase monitors, environmental asymmetry gives a single-fragment information bound, a redundancy converse, and, in efficient i.i.d. streams, an achievable block-capacity law for redundant witnesses. A finite-temperature spin-phase bath shows the physical separation: hot probes can decohere efficiently while losing the local asymmetry needed for objective witnesses.

Strongly regular graphs, hereafter SRGs, are highly structured combinatorial objects of central interest in algebraic graph theory and quantum information, whose explicit construction becomes rapidly intractable as the number of vertices grows; Conway's still unresolved $99$-graph problem is an emblematic example. The matrix equation defining SRGs is quadratic in the binary edge variables, so the natural penalty enforcing it, a sum of squared residuals, is quartic and lies outside quadratic unconstrained binary optimization, known as QUBO. We prove a considerably stronger obstruction, namely that no quadratic function of the edge variables has the labeled SRGs as its exact set of global minimizers, except for three degenerate families given by the perfect matchings, the cocktail-party graphs and the pentagon. In particular, we formally establish that the $99$-vertex graph of Conway's problem cannot be obtained from any QUBO over the edge variables without auxiliary variables. Guided by this theorem we develop two complementary algorithms. The first operates within pure QUBO, solves the degenerate families exactly, and beyond them works as a generate–prune–verify pipeline whose output is a reduced set of candidates certified by a polynomial-time classical test. The second solves the full problem by quadratizing this quartic penalty through Rosenberg auxiliary variables, whose number grows as $\Theta(N^3)$ with the number of vertices $N$. We test the pure-QUBO route under simulated annealing on all feasible families with at most $18$ vertices and the auxiliary-variable route on representative targets up to $16$ vertices, and tabulate the algebraically feasible parameter sets for graphs with at most $49$ vertices.

Two quantum states may relax toward the same steady state, yet the one that starts farther away can overtake the closer one. This is the quantum Mpemba effect. The same pair can reverse order more than once, but these crossings may depend on the divergence used to compare them. We ask which crossings persist for every normalized operator-convex Petz divergence. Because each such divergence is a positive average of one-parameter $\chi^2$ kernels, the problem reduces to the sign of one profile across the full kernel range. Alternating sign margins guarantee repeated family-wide reversals, while finite dimension yields a polynomial positivity test. A simple real slow mode fixes the common late-time order; near stationarity, coherence between unequal-eigenvalue sectors makes the order divergence dependent. In a trapped-ion qutrit ideal model, the reported preparation shows divergence-selective crossings. A nearby preparation is a candidate for at least two family-wide reversals.

Grover search, amplitude amplification, and quantum counting all rely on the same reusable subroutine, a phase oracle, whose construction the algorithms literature takes as given: the classical predicate is assumed to be already encoded as a correct, resource-bounded circuit. We turn this assumption into a measured capability. QEncodeBench tasks large language models (LLMs) with encoding classical constraint problems as phase oracles and scores the generated circuits with an adversarially self-validated verifier that decides full solution-set equivalence up to a global phase, with ancillas restored and resource budgets enforced. Sampled basis-state tests, we show, systematically overestimate this ability. Measured this way, models separate sharply: code models without a reasoning mode solve essentially nothing, and enabling native reasoning on identical weights improves accuracy by an order of magnitude. The failures are overwhelmingly semantic rather than syntactic. Two architectures, a unit-verified constraint agent and a neuro-symbolic compilation pipeline, close most of the remaining gap by delegating correctness-critical composition to deterministic procedures. Ablations quantify the contribution of each component, and resource gating exposes an architecture-dependent trade-off between circuit width and depth. Finally, controlled difficulty escalation reveals architecture-specific responses to difficulty structure: different difficulty axes degrade different methods, while the neuro-symbolic pipeline passes every evaluated instance. Code and data are available at https://github.com/chexujun/QEncodeBench.

Information distance and circuit complexity are both obtained by minimizing lengths, but they minimize over different objects. We make this distinction explicit for faithful one-mode Gaussian states. First, invariant-form uniqueness implies that no positive-definite quadratic gate cost can be invariant under the full adjoint action of the noncompact symplectic group; a positive Cartan majorant necessarily introduces additional reference data. The Uhlmann purification quotient realizes the Bures metric, and the radial covariance direction requires a system-ancilla coupling because system-only Gaussian unitaries preserve the Williamson eigenvalue. We then minimize fixed right-invariant quadratic norms on the minimal two-mode Gaussian gate algebra \(\mathfrak{sp}(4,\mathbb R)\). For the unweighted Frobenius norm, the quotient coefficients for radial and traceless covariance tangents are $G_0=[\hbar^2(u-1)]^{-1}$ and $G_2=[\hbar^2(3u-1)]^{-1}$, where $u=(2\nu/\hbar)^2$. Their ratio does not equal the Bures ratio. The radial coefficient, however, reproduces the Bures value exactly at every $u$; the mismatch is confined to the traceless sector. More generally, a constant block-diagonal two-weight schedule gives $G_0/G_2=1+2(\beta/\alpha)u/(u-1)$; matching Bures throughout the isotropic family would require the state-dependent relation $\beta/\alpha=1/u$. At the Bures-Fisher determinant crossing \(u=\varphi\), pointwise matching is possible only by inserting $\beta/\alpha=\varphi^{-1}$. Thus the Bures purification quotient is an exact state-geometric cost, but it is neither an unweighted symplectic gate cost nor a member of this fixed two-weight Nielsen family. The existence of a more general fixed positive gate norm realizing the quotient remains open.

We study the recently developed theory of quantum convolution for discrete-variable quantum systems. We classify the discrete quantum convolutional channels generated by an invertible $2\times2$ coupling matrix and a fixed environmental state according to which entries of the coupling ma- trix vanish. Under role-preserving basis relabelings, matrices with all four entries nonzero form $d-2$ canonical cross-ratio classes, whereas matrices with exactly one vanishing entry split into four inequivalent branches. We determine the optimized one-shot Holevo information and the classical, quantum, and private capacities for all four one-entry-vanishing branches. If a diago- nal entry vanishes, the channel is an entanglement-breaking measure-and-prepare orbit channel: its classical capacity equals a relative entropy coherence of the environmental state, whereas its quantum and private capacities vanish. If an off-diagonal entry vanishes, the channel is a degradable generalized-dephasing channel: one complete orthonormal basis is transmitted without error, whereas the quantum and private capacities are determined by the entropy deficit of the environ-ental state after dephasing in the conjugate basis. Thus, the position of a single vanishing entry selects two qualitatively different communication regimes. When all four entries are nonzero, the channels are irreducibly Weyl covariant, reducing the classical-capacity problem to a regularized minimum-output-entropy problem. For the unique fully nonzero coupling class in the single-qutrit case, we further derive an exact Holevo formula for a depolarized one-parameter family and identify its entanglement-breaking threshold.

Quantum programs are difficult to validate: circuits are typically expressed as imperative gate sequences with limited inspectability, execution modalities must be selected manually, and outputs are inherently probabilistic. These challenges are compounded when programs must be portable across backend frameworks with differing internal conventions. We present a quantum domain-specific language (QDSL) that addresses these problems through three mechanisms: (i) intent-execution separation, where algorithmic constructs such as preparation, superposition, entanglement, and measurement are represented as inspectable objects compiled into backend-specific circuits only after pre-execution Intermediate Representation (IR) validation; (ii) first-class IR introspection, exposing circuit width, wire mapping, operation order, and measurement intent for developer inspection prior to execution; and (iii) assertion-guided modality inference, where the engine examines declared verification properties to automatically select sampling, statevector, or dual execution without user intervention. Results are logged in structured form to support reproducible regression testing. We evaluate our prototype on benchmark circuits spanning entanglement, oracle-based, structured-transform, and variational examples. In fault-injection experiments on Bell and 3-qubit GHZ circuits, a total-variation-distance detector identifies the injected faults across all evaluated shot budgets (True Positive Rate 1.0), with observed false positives (False Positive Rate 2.7-3.8%) only at 128 shots. Differential testing demonstrates numerical agreement across PennyLane and Qiskit compilation targets for the evaluated constructs after endianness canonicalization. IR generation costs are sub-millisecond.

Whether negative-partial-transpose entanglement can remain undistillable is a longstanding problem in quantum information theory. We analyze the first unresolved three-copy Werner endpoint using a sharp dimension-free inequality for complementary partial traces and an optimal exterior-square inequality for orthonormal tripartite vectors. The latter identifies local SWAP- parity statistics with metric data of a decomposable Plucker bivector. Together these inequalities prove endpoint nonnegativity for every positive semidefinite rank-two coefficient operator and for the complete normal rank-two sector in arbitrary finite local dimensions. For genuinely nonnormal operators, an exact crossed-Gram criterion proves nonnegativity when one local outpu-input support overlap is at most two, including every system with a qubit-sized factor, and when either support plane contains a product ray. Explicit anti-state and rank-boundary families establish optimality of the constants and the rank restriction.

Many-body quantum-optical systems, where a collection of emitters interacts through a common electromagnetic reservoir, exhibit rich out-of-equilibrium behavior and hold promise for applications in quantum technologies. However, exact numerical simulations of their dynamics quickly become unfeasible due to the exponential growth of the Hilbert space with system size. Semiclassical, phase-space approaches – such as the Truncated Wigner approximation (TWA) – provide computationally efficient alternatives by capturing leading-order quantum fluctuations. In this paper, we present a comprehensive overview of how to tackle problems in many-body quantum optics using phase-space methods. We derive the exact partial differential equation governing many-body dissipative evolution in any phase-space representation and discuss the approximations that yield the dissipative TWA proposed by Mink and Fleischhauer [SciPost Phys. 15, 233 (2023)]. We find that $P$ and $Q$ distributions are generally suboptimal for many-body quantum optics. Additionally, we extend the formalism to calculate multi-time correlation functions, thereby broadening the scope of phase-space simulations of open spin systems to include coherence and spectral properties, as well as directional correlations of collectively radiating emitters. These developments provide valuable tools for investigating exotic light sources driven by collective dissipation, driven-dissipative phase transitions, and a wealth of many-body phenomena arising in state-of-the-art experimental platforms.

Assuming the Unique Games Conjecture, we show it is NP-hard to approximate MAX-3-CUT within a multiplicative factor of $\alpha_3+\epsilon$ for every $\epsilon>0$, where $\alpha_3\approx.83600811464$ is the approximation ratio of Frieze-Jerrum's polynomial-time algorithm from 1995. That is, we prove sharp hardness of approximation for MAX-3-CUT. This result resolves a conjecture of Khot-Kindler-Mossel-O'Donnell from 2004 by proving the three candidate Plurality is Stablest Conjecture for correlations in $[-1/2,2/5]$ and generalizes the Majority is Stablest Theorem of Mossel-O'Donnell-Oleszkiewicz [Annals of Math, 2010]. With a similar strategy we prove: assuming the Unique Games Conjecture, it is NP-hard to approximate the product-state value of Quantum MAX-CUT within a multiplicative factor of $\alpha_{\rm BOV}+\epsilon$ for every $\epsilon>0$, where $\alpha_{\rm BOV}\approx 0.9563372685$ is the approximation ratio of the Briët-de Oliveira Filho-Vallentin algorithm. This sharp hardness result completes the conjectured hardness of Hwang-Neeman-Parekh-Thompson-Wright from 2021 by proving their $S^{k-1}$-valued Borell inequality for correlations in $[-.5843,.5843]$ for all $k\geq3$.

We study the problem of testing low-degree phase states, namely m-qudit quantum states of the form $q^{-m/2} \sum_{x \in \mathbb{F}_q^m} \omega^{f(x)} |x>$, where $f$ is a degree-$d$ polynomial. In contrast to the classical setting, where low-degree polynomials admit highly efficient classical testers, it is not known whether analogous quantum tests exist. We show that no such quantum low-degree test exists: any tester requires $\Omega(\binom{\lfloor m/2\rfloor}{\lfloor (d-1)/2 \rfloor})$ copies to determine whether a given state is a degree-$d$ phase state or is far from every such state. Our results follow from a general framework that relates quantum testing of codeword phase states to classical decoding properties of the dual code, which allows us to leverage known bounds on the tolerance of high-rate Reed–Muller codes to random errors.

Periodically driven open quantum systems are central to quantum thermodynamics and quantum control. These systems are typically described using Floquet-Born-Markov master equations, derived with the use of a full secular approximation, and whose thermodynamic implications are often overlooked. In this context, we show that such a strong secular approximation may lead to unphysical predictions for steady state energy currents. We then demonstrate that a coarse-grained formulation of the master equation can regularize these issues while yielding completely positive dynamics and consistent energy currents. The coarse-graining time has a clear physical interpretation, as it defines the temporal resolution at which a Markovian master equation can describe the evolution of the periodically driven system. We show the consistency and validity of our approach by comparing to an exact non-Markovian simulation in paradigmatic examples: a driven two-level system and a three-level maser coupled to hot and cold thermal reservoirs. Our work provides a practical framework for correctly applying the secular approximation in periodically driven-dissipative systems and for assessing the accuracy of master equations of the GKSL form.

Quantum marked vertex search algorithms are known to outperform their classical counterparts, yet their behavior in the presence of disorder remains largely unexplored. Here, we address this gap by studying marked vertex search on disordered random graphs. To introduce disorder, we implement the Rosenzweig-Porter (RP) model, a random matrix ensemble with tunable ergodic, non-ergodic extended, and localized phases, on Erdős-Rényi (ER) graphs. This produces a doubly random system where ER graph connectivity randomizes which interactions exist, while RP disorder controls their strength and `on-site' potentials, providing a two-parameter framework to study quantum dynamics on disordered networks. First, we show that the characteristic Wigner-Dyson-to-Poisson spectral crossover of the RP ensemble survives under graph constraints across the sparse-to-dense range, and we derive an analytical estimate for the finite-size localization boundary that shifts systematically with the graph edge probability $p$, consistent with a resonant-hybridization argument. Thereafter, using this disordered graph ensemble, we study the marked vertex search problem and find that search performance tracks the underlying quantum phase directly. Counterintuitively, the ergodic phase, despite supporting fast transport, yields lower success probability than the localized phase, which achieves high success probability at the cost of significantly longer search times. These results establish a direct and quantitative link between random matrix disorder on graphs and the performance of continuous-time quantum walk search, and suggest that disorder, rather than being merely an obstacle, can be exploited as a tunable parameter in quantum search protocols.

Quantum-circuit compilation aims at finding an optimized realization of a target circuit under given constraints, e.g., the minimization of hardware-induced errors and the unitary-equivalence of the circuit. We connect the compilation process with the thermodynamics of a many-body spin system: circuit infidelity plays the role of the energy function and low-temperature states correspond to compiled circuits. In the paradigmatic case where crosstalk between parallel gates is present, we find a phase transition between a disordered phase and an antiferromagnetic brick-wall phase, compatible with the Ising universality class. At larger crosstalk, we observe a $\mathbb{Z}_3$-ordered regime, suggesting that increasingly serial compiled circuits are associated with emergent $\mathbb{Z}_n$-ordered phases. When the unitary-equivalence constraint is removed, these phases disappear, showing that the equivalence between circuits underlies the emergent criticality and constitutes a source of complexity in quantum-circuit compilation and, more generally, in equivalence-constrained optimization. Finally, we observe that the Kolmogorov complexity of the circuit enhances the emergence of ordered phases.

Quantum sensors hold great promise for achieving better sensitivity in the measurement of physical quantities compared to their classical counterparts. However, the conditions under which quantum advantage in sensing can be achieved are rather restrictive, and most quantum enhancements in sensing are lost in the presence of noise, errors, or a poorly calibrated system. To overcome these limitations, we are motivated to import ideas from fault-tolerant quantum computing to quantum sensing. Specifically, we consider a qubit noise model where the probability of phase-flip errors is exponentially smaller (in qubit number) compared to the probability of bit-flip errors that occur with probability $p$. For this noise structure, we demonstrate that, given a total sensing time $T$, Heisenberg scaling can be attained for times up to $T\propto 1/p^{(N+1)/2}$ for a $N$-qubit repetition code, in contrast with $T\propto 1/p$ without using a fault-tolerant sensing protocol.

It has previously been shown by Rajput, Roggero, and Wiebe that $\mathbb Z_2$ Gauss's law constraints can be used to build efficient quantum error-correcting codes (QECCs) that are robust against arbitrary single-qubit errors. In this work, we generalize the construction to be robust against arbitrary $t$-qubit errors, where $t$ is any positive integer. This includes a derivation of the optimal Gauss's law code within the considered family by minimizing the number of physical qubits required for a given code distance. Finally, we compare our codes against other efficient QECCs on metrics such as the number of physical qubits, the locality of the encoded Hamiltonian, and the logical error rate in the code capacity setting. Compared to using a domain-agnostic code for every lattice degree of freedom, we find that the Gauss's law code primarily excels at reducing the locality of the encoded Hamiltonian. Moreover, the physical qubit overhead is also reduced for $t \le 3$ (distance $d \le 7$).

Topological pumps transfer charge or energy at quantized rates determined solely by band topology, independent of the details of the control fields. Photon pumps operating on this principle could enable the robust preparation of non-classical states in a quantum cavity, even in the presence of control imperfections, but have not been directly observed. We present the first experimental observation of a topological photon pump, using a transmon qubit coupled to a microwave cavity. The pump operates in the quantum regime, pumping up from the vacuum state up to $\approx 7$ photons, and produces demonstrably non-classical cavity states for the first few cycles.