Papers

The impossibility of simultaneously cloning non-orthogonal states lies at the foundations of quantum theory. Even when allowing for approximation errors, cloning an arbitrary unknown pure state requires as many initial copies as needed to fully learn the state. Rather than arbitrary unknown states, modern quantum learning theory often considers structured classes of states and exploits such structure to develop learning algorithms that outperform general-state tomography. This raises the question: How do the sample complexities of learning and cloning relate for such structured classes? We answer this question for an important class of states. Namely, for $n$-qubit stabilizer states, we show that the optimal sample complexity of cloning is $\Theta(n)$. Thus, also for this structured class of states, cloning is as hard as learning. To prove these results, we use representation-theoretic tools in the recently proposed Abelian State Hidden Subgroup framework and a new structured version of the recently introduced random purification channel to relate stabilizer state cloning to a variant of the sample amplification problem for probability distributions that was recently introduced in classical learning theory. This allows us to obtain our cloning lower bounds by proving new sample amplification lower bounds for classes of distributions with an underlying linear structure. Our results provide a more fine-grained perspective on No-Cloning theorems, opening up connections from foundations to quantum learning theory and quantum cryptography.

We introduce a systematic framework to construct nonlocal observables with extensive quantum Fisher information (QFI) density in stabilizer codes. The construction maps stabilizer generators to dual Ising spins whose correlators equal string order parameters, converting hidden nonlocal order into a metrologically accessible observable. Applying this to monitored cluster codes and the toric code, we identify transitions in the QFI scaling from an extensive regime, where long-range string order prevails, to an intensive one driven by competing single-site measurements.

We develop a quantum Gibbs-sampling approach to free-energy estimation for trapped systems of distinguishable particles with Coulomb interactions in dimensions $d\in\{2,3\}$. The singular infinite-dimensional problem is reduced to a finite-rank low-energy truncation of the interaction, the truncated Gibbs state is prepared by a quantum Markov semigroup, and thermodynamic integration gives the free energy. We obtain explicit truncation rates, quantitative spectral-gap estimates, and a finite-dimensional circuit implementation. In particular, every fixed interaction truncation has a positive gap, while weak truncated interactions admit a particle-number-independent lower bound. The circuit complexity is conditional on a certified gap lower bound at the accuracy-dependent interaction cutoff.

Photonic architectures are one of the leading platforms for demonstrating quantum computational advantage, with Boson Sampling and Gaussian Boson Sampling as the primary schemes. Yet, we lack for these photonic primitives a systematic theoretical understanding of linear cross-entropy benchmarking (LXEB), which is a central tool for testing quantum advantage proposals. In this work, we develop a representation-theoretic framework for the classical computation of average LXEB scores and second moments of output probability distributions, covering a range of quantum advantage experiments based on scattering $n$-photon states through $m$-mode Haar-random interferometers. Our methods apply in any regime, including the saturated regime, where the (expected) number of photons is comparable to the number of optical modes. The same second-moment techniques also allow us to prove anticoncentration for traditional Fock-state Boson Sampling in the saturated regime. Interestingly, for Gaussian Boson Sampling second moments are not sufficient to establish a meaningful anticoncentration statement. The technical core of our approach rests on decomposing two copies of the $n$-particle bosonic space $\mathrm{Sym}^n(\mathbb{C}^m)$ into irreducible representations of $\mathrm{U}(m)$. This reduces two-copy Haar averages to computing purities of initial states after partial traces over particles, highlighting the role that particle entanglement plays for LXEB and anticoncentration.

The $2$-Forrelation problem provides an optimal separation between classical and quantum query complexity and is also the problem used for separating $\mathsf{BQP}$ and $\mathsf{PH}$ relative to an oracle. A natural question is therefore to ask what are the minimal quantum resources needed to solve this problem. We show that $2$-Forrelation can be solved using Instantaneous Quantum Polynomial-time ($\mathsf{IQP}$) circuits, a restricted model of quantum computation in which all gates commute. Concretely, signed $2$-Forrelation can be solved by a classical random choice between two one-query $\mathsf{IQP}$ circuits, while the absolute-value variant uses two independent executions of this randomized procedure. This answers a recent open question of Girish (STOC 2026) on the power of commuting quantum computations. For the Raz-Tal distribution, this randomization is unnecessary. We use this to show that there is an oracle $O$ such that $\mathsf{IQP}^O \not\subseteq \mathsf{PH}^O$, strengthening the result of Raz and Tal (STOC 2019). It also yields an oracle separation between $\mathsf{IQP}$ and $\mathsf{DQC}_1$. We prove Fourier growth bounds for multi-query $\mathsf{IQP}$ circuits, including bounds in terms of the size of their accepting set. Our results suggest a possible route toward decision-based quantum advantage within the restricted $\mathsf{IQP}$ model. The key ingredient is an algebraic identity of the quadratic function $Q(x) = \sum_{i < j} x_ix_j$ that allows extracting inner-product phases within an $\mathsf{IQP}$ circuit.

Pure quantum states are a fundamental resource for quantum information processing, but unavoidable noise degrades their purity and limits the performance of quantum tasks. Quantum state purification attempts to restore purity by jointly processing several noisy copies of an unknown state, typically in a probabilistic way. Whether purification is possible, and how much it can achieve, depends strongly on the set of operations that are physically allowed. In this work, we study universal purification under energy-preserving operations, which cannot exchange energy with an external environment. We identify exactly when the energy-preserving constraint forbids any universal purification advantage, and we provide concrete examples in which an advantage available to unrestricted operations disappears under energy conservation. When energy-preserving purification remains possible, we determine its optimal fidelity and success probability and construct the corresponding protocols. The framework includes unrestricted purification as a special case and extends naturally to purification assisted by an external energy resource.

Quantum kernel methods are among the leading candidates for achieving quantum advantage in supervised learning. A key bottleneck is the cost of inference: evaluating a trained model on new data requires estimating a weighted sum $\sum_{i=1}^N \alpha_i k(x,x_i)$ of $N$ kernel values to additive precision $\varepsilon$, where $\alpha$ is the vector of trained coefficients. The standard approach estimates each term independently via sampling, yielding a query complexity of $O(N\lVert\alpha\rVert_2^2/\varepsilon^2)$. In this work, we identify two independent axes for improvement: (1) How individual kernel values are estimated (sampling versus quantum amplitude estimation), and (2) how the sum is approximated (term-by-term versus via a single observable), and systematically analyze all combinations thereof. The query-optimal combination, encoding the full inference sum as the expectation value of a single observable and applying quantum amplitude estimation, achieves a query complexity of $O(\lVert\alpha\rVert_1/\varepsilon)$, removing the dependence on $N$ from the query count and yielding a quadratic improvement in both $\lVert\alpha\rVert_1$ and $\varepsilon$. We prove a matching lower bound of $\Omega(\lVert\alpha\rVert_1/\varepsilon)$, establishing query-optimality of our approach up to logarithmic factors. Beyond query complexity, we also analyze how these improvements translate into gate costs and show that the query-optimal strategy is not always optimal in practice from the perspective of gate complexity. Our results provide both a query-optimal algorithm and a practically optimal choice of strategy depending on hardware capabilities, along with a complete landscape of intermediate methods to guide practitioners. All algorithms require only amplitude estimation as a subroutine and are thus natural candidates for early-fault-tolerant implementations.

We explore the expected performance of a semiconducting spin cQED quantum processor for Multiple Hypothesis Tracking (MHT) algorithm via a quantum annealing procedure. From two different benchmarking scenarios we evaluate this type of quantum annealer on a quantum emulator in which we incorporated both dynamical coherent errors and incoherent errors. From estimate of the reset, measurement and annealing time of the processor, we find that cQED-spin processors could reach a total run time of around 50 ms. This makes this technology promising for potential real time application such as radar tracking.

We investigate the generation of quantum states for precision metrology in noisy two-level systems. These states are obtained by optimizing a variational quantum circuit to maximize the quantum Fisher information (QFI) of the output state for a given decoherence rate and interaction Hamiltonian. The circuit architecture, inspired by twist-and-turn schemes, features a sequence of $n$ entangling layers, each consisting of entangling gates followed by a global rotation. We observe notable improvements in the QFI as the circuit layer depth increases, even for appreciable noise rates, demonstrating that our entangle-rotate architecture expands the accessible state space under realistic noise conditions. Our approach thus provides a general and efficient framework for generating quantum-enhanced sensing states. Our analysis extends to systems of power-law interactions spanning from all-to-all to nearest-neighbor interactions. We also analyze the capabilities of our circuit to prepare states for system sizes greater than $8$ qubits.

The Metropolis-Hastings algorithm is a cornerstone of Markov Chain Monte Carlo methods, underpinning a wide range of applications in computational physics, Bayesian inference, and machine learning. Quantum variants of Metropolis-Hastings promise accelerated mixing through quantum walks, but their practical realisation remains challenging. In this work, we construct and simulate an explicit circuit level implementation of a quantum Metropolis-Hastings algorithm based on the framework introduced by Claudon et al. (arXiv:2506.11576). We present the full quantum workflow required to prepare a stationary distribution, including a number of modifications required to make the algorithm implementable in a realistic quantum circuit model. Our results demonstrate that these modifications are essential to recover the correct stationary behaviour and highlight both the potential and current limitations of quantum Metropolis-Hastings algorithms, which are expected to become practically relevant in the fault tolerant quantum computing regime.

In the context of waves in space time modulated materials, we ask two questions how much energy does it cost to break time reversal symmetry and transition to a PT symmetric state. and can a PT symmetric system have a ground state in the sense that no photons are present. Our model system is a periodic metamaterial set in virtual motion with velocity cg to become a space-time crystal. We find that the expectation of energy content is always increased on breaking symmetry. At the same time breaking T symmetry introduces photon-pairs even when we start from a T symmetric ground state empty of photons, except in certain pathological examples which we describe. For a range of velocities, PT symmetry is broken so that energy must be continuously invested to preserve motion, creating a trail of photon pairs. Here energy must be continuously invested to preserve motion. We make an analogy with acoustic radiation generated from breaking the sound barrier.

We propose a coherent control scheme for engineering quantum correlations in a cavity optomechanical (COM) system consisting of a driven optical cavity with an embedded nonlinear medium and a membrane, assisted by a coherent feedback loop. The nonlinear medium and the membrane are pumped to implement optical and mechanical parametric amplifications with controllable modulation frequencies and pump amplitudes. Within the stable regime identified by Floquet analysis, the feedback-induced enhancement of optomechanical entanglement persists when the period-averaged mean-field intracavity photon number is matched, showing that the enhancement cannot be attributed simply to an increase in this quantity. A systematic comparison of the four configurations further reveals a nonadditive interplay between dual parametric amplification and coherent feedback. Varying the amplitude reflectivity of the beam splitter and the feedback loop phase enables effective control of optomechanical entanglement and directional EPR steering, including a transition from one-way to two-way steering. The combined scheme also improves the robustness of both correlations against thermal noise, providing a potential route toward robust quantum-state engineering in COM systems.

We present a minimal implementation of SU($N$) pure Yang-Mills theory in $3+1$ dimensions for digital quantum simulation, designed to enable quantum advantage. Building on the orbifold lattice simulation protocol with logarithmic scaling in the local Hilbert-space truncation, we introduce further simplified Hamiltonians. Furthermore, we test simple methods that improve the convergence to the infinite mass limit, thereby removing the requirement of a large scalar mass to obtain the Kogut-Susskind Hamiltonian. For the SU(2) theory, we can cut the resource requirement further by utilizing the embedding of $\mathrm{SU}(2)\cong\mathrm{S}^3$ into $\mathbb{R}^4$. Monte Carlo simulations of the Euclidean path integral were used to benchmark the accuracy of these new analytical improvements to the theory. These results provide further support for the noncompact-variable-based approach as a practical framework for quantum simulation of non-Abelian gauge theories.

Executing a logical quantum circuit fault-tolerantly incurs a large spacetime overhead. Recent work has proposed and investigated phantom codes, defined by the property that every in-block logical $\mathrm{CNOT}$ circuit can be implemented with a physical permutation, a property that has the potential to greatly reduce the depth of compiled circuits. Here we show that phantomness comes at the cost of low encoding rate. Specifically, we prove that any binary phantom code encoding $k$ logical qubits into $n$ physical qubits with distance $d\geq 2$ obeys the bound $k\leq \log_2(n+1)$ for all $k\neq 4$. For $k=4$ we explicitly construct a nonstabiliser $(\!(8, 2^4, 2)\!)$ phantom code that violates the bound and has a transversal non-Clifford gate. We further show that, within the class of nontrivial CSS phantom codes with $k\neq 4$, there is a unique family of codes saturating this bound. In addition, we prove that this logarithmic ceiling cannot be circumvented by permitting additional local unitary gates, or by making use of subsystem codes: any subspace or subsystem code admitting a $\mathrm{SWAP}$-transversal implementation of every logical $\mathrm{CNOT}$ circuit is constrained to satisfy the same bound. These bounds follow from a general theorem relating the length of a quantum code to the structure of its automorphism group, a result which may find applications beyond phantom codes.

We present a systematic study of static solutions to the source-free SU(2) Yang-Mills equations, in which the gauge potential explicitly depends on spin operators. By employing the vector potential extraction approach – which requires the total angular momentum operator (orbital plus spin) to satisfy the standard angular momentum algebra – we derive the most general form of the spin vector potential $\vec{A}$. This leads to the static ansatz $\{ \vec{A} = [k_1(\hat{r}\times\vec{\Gamma}) + k_2\vec{\Gamma} + k_3(\vec{\Gamma}\cdot\hat{r})\hat{r}]/r, \varphi = f_1(r)\,(\vec{\Gamma}\cdot\hat{r}) + f_2(r)\}$, parametrized by three constants $\{k_1, k_2, k_3\}$ and two radial functions $\{f_1(r), f_2(r)\}$. After substituting this static ansatz into the Yang-Mills equations we obtain a set of consistency equations. Solving these equations provides a complete classification of the exact static solutions, including both real and complex families. The known simple SU(2) static solution $\{\vec{A}=\tilde{k} (\hat{r}\times\vec{\Gamma})/r, \varphi=\kappa/r \}$ is recovered as a special case. Our classification reveals new static configurations that could be valuable for non-perturbative studies and for models where the internal spin couples to non-Abelian gauge fields.

We present a framework based on the determinantal geometry of two-qubit gates. Combining the Weyl chamber representation with operator Schmidt theory, we interpret gate synthesis as a distance problem to determinantal varieties. This gives an operational geometry to the Weyl chamber, quantifying nonlocal complexity. We show that the square root iSWAP gate is the closest perfect entangler to the variety of local operations, and that no perfect entangler can be approximated by a local gate with average gate fidelity above 79.8%. The three different determinantal costs form a synthesis-adapted coordinate system that encodes nonlocal complexity and generally reconstructs the Weyl chamber.

Toward the large-scale, practical realization of quantum computing, quantum error correction is essential. Among various quantum error-correcting codes, the surface code stands out as a leading candidate, and lattice surgery based on surface codes has emerged as a promising technique for fault-tolerant quantum computation (FTQC). However, implementing quantum algorithms using lattice surgery introduces both resource and time overhead. Existing approaches typically focus on large layout designs, with compiler passes aimed primarily at optimizing time overhead. This often overlooks the trade-off between rotation bottlenecks and movement distance, which leads to inefficient resource utilization and prevents further reduction of the quantum computation failure rate. To address these challenges, we introduce O3LS, a framework for optimizing lattice surgery through automatic layout search and loose scheduling. O3LS achieves an optimal balance by automatically generating squeezed data layouts to reduce space requirements and employing loose scheduling algorithms combined with circuit synthesis techniques to reduce time overhead, thereby effectively minimizing overall logical error rates. Numerical results indicate that O3LS can reduce space overhead by 28.0% over standard layouts and 46.7% over sparse layouts without increasing the number of time steps, leading to suppression of logical error rates by up to 16% relative to larger data layout designs. O3LS can also achieve time overhead reductions of 36.07% and 24.76% in compact and standard data layout designs, respectively. It suppresses logical error rates by up to an order of magnitude compared to prior compilers that focus primarily on maximizing parallelism.

To address the urgent need in the NISQ era for high-performance, scalable quantum compilers and to advance the integration of classical and quantum computing, we present QLLVM, an advanced Quantum-Classical co-compilation framework built on LLVM. To our knowledge, QLLVM delivers an end-to-end, LLVM-based compilation workflow that unifies the build of classical high-performance programs, including CUDA, MPI, and C++, together with quantum programs into a single executable. For quantum program compilation, QLLVM adopts a three-stage design: high-level optimizations are implemented in the MLIR Quantum dialect and then lowered to QIR, an LLVM IR-based representation, for low-level optimization and hardware mapping. Its extensible architecture and seamless interoperability with classical high-performance computing provide an efficient, flexible, industrial-grade compilation infrastructure for future quantum software development. Experimental results show that, on the MQTBench benchmark suite, QLLVM reduces circuit depth and gate counts compared with state-of-the-art compilers and demonstrates clear advantages in compiling hybrid classical-quantum programs.

Collective spin systems – driven-dissipative spin ensembles whose unitary and dissipative processes act uniformly on all spins, allowing their dynamics to be described as a single macrospin – play an important role in atomic and solid-state physics. Their intrinsic nonlinearity gives rise to multiple long-lived metastable states that ultimately relax to a unique most probable state. This dominant state can change with a control parameter, leading to first-order phase transitions. We develop a real-time instanton approach based on quantum quasiprobability dynamics that captures the stationary state in the large-spin limit and the asymptotic scaling of relaxation rates. We further show that these features are not accurately described by the previously applied semiclassical Wigner approach due to its neglect of non-Gaussian fluctuations.

We report a hardware validation of the DAGI (Directed Acyclic Graph Information) framework on IBM Quantum hardware using a small, controlled experiment whose ideal output distribution is constrained to a low-dimensional modular manifold (a "ridge"). For two $n$-bit registers $(u,v)$ with $n=4$ (modulus 16), each key instance $k$ induces an ideal relation $v \equiv k \cdot u \pmod{16}$, producing a visually distinct ridge in the joint $(u, v)$ distribution. Executed on ibm\_torino in a single Sampler V2 job (8 keys, 1024 shots/key, $N=8192$ total shots), the ridge persists under hardware noise with ridge-hit probability $p_{hit} = 0.1830$ (uniform baseline $1/16$), corresponding to a ridge contrast of $2.93\times$ (95% bootstrap CI [2.80, 3.06]). Key recovery exceeds chance: per-shot accuracy 0.1689 (chance 0.125, 95% Wilson CI [0.1610, 0.1772]), and per-group dictionary recovery 0.375 (chance 0.125). To test the central DAGI hypothesis – that recoverable key information is predominantly high-order/synergistic rather than visible in low-order marginals – we compute a Möbius-based information decomposition of $I(K;D_S)$ over detector-bit subsets $S$ via a Möbius inversion pipeline and evaluate targeted positive synergy $CPS_K$ at order $k_{max}=3$. We observe $CPS_K(k=3) = 0.08788$ with significance under label-shuffle permutation tests (accuracy $p=0.001996$, $CPS_K$ $p=0.004975$). Uniformity diagnostics show near-uniform single-bit marginals while correlation concentrates in specific low-order pairs, and a bootstrap reliability sweep confirms order-3 targeted synergy remains statistically reliable at the full 1024-shot target budget. These results support the claim that DAGI detects and quantifies nontrivial, hardware-resilient, higher-order information structure associated with a known global algebraic constraint.

Hybrid Quantum Neural Networks (HQNNs) combine classical learning with parameterized quantum circuits, but their practical performance is often limited by (i) the noise of Noisy Intermediate-Scale Quantum (NISQ) devices and (ii) the large, discrete design space of quantum circuit architectures. Moreover, HQNNs are commonly trained using a fixed circuit and a single backend, even though deployment frequently targets heterogeneous backends where compilation and execution characteristics may differ. To address these challenges, we propose GAT-QNN, a genetic algorithm (GA)-based framework that trains a macroCircuit (search space) by iteratively sampling microCircuits (subcircuits), training them, and reintegrating their learned parameters into the macroCircuit. After training, we run an independent GA-driven inference stage that evaluates candidate microCircuits using the trained macroCircuit weights and selects top-performing architectures for deployment. This two-stage approach enables backend-aware microCircuit selection without retraining each candidate architecture and can also reduce computational resources (gate count) by deploying smaller microCircuits derived from the macroCircuit. We validate the approach on MNIST classification (four classes) and report consistent 22-23% test accuracy gains for GA-driven inference across multiple backends.

The classical simulation of universal quantum circuits is crucial both fundamentally and practically for quantum computation. We propose SyQMA, a simulator with several convenient features, particularly suited for quantum error correction (QEC). SyQMA simulates universal quantum circuits with incoherent Pauli noise and computes exact expectation values and measurement probabilities as symbolic functions of circuit parameters: rotation angles, measurement outcomes, and noise rates. This simulator can sample measurement outcomes, enabling the simulation of dynamic quantum programs where circuit composition depends on prior measurement outputs. For QEC, it performs circuit-level maximum-likelihood decoding, provides exact symbolic expressions for logical error rates, and verifies the fault distance of fault-tolerant (FT) stabiliser and magic state preparation protocols. These features are enabled by an intuitive extension of stabiliser simulators, where each non-Clifford Pauli rotation and incoherent Pauli channel is compactly represented via auxiliary qubits and a modified trace. Representing the state requires only polynomial memory and time, while computing expectation values and measurement probabilities takes exponential time in the number of non-Clifford rotations and deterministic measurements, but only polynomial memory. The FT preparation of stabiliser and magic states, including the first stage of magic state cultivation, is analysed without approximations. We also exactly convert the disjoint error probabilities of a general multi-qubit Pauli channel to independent ones, a key step for creating and sampling from detector error models. The code is publicly available and open-source.

We present a novel lackadaisical alternating quantum walk (LAQW) algorithm whose circuit depth scales as $\mathcal{O}(n^2+nt)$ for a $n\times n$ lattice over $t$ time steps. We show that this is a significant depth reduction compared to the existing controlled alternating quantum walk (CAQW) model, which has a circuit depth that scales as $\mathcal{O}(n^2t)$ (Li et al., 2017, arXiv:1707.07389). This makes the implementation of the LAQW viable for Noisy Intermediate-scale Quantum (NISQ) devices. We then showcase the applicability of the LAQW algorithm by proposing a chaos-based symmetric-key generation scheme. Our approach uses the LAQW as a quantum entropy source from which reproducible random bitstring sequences are generated using the underlying probability distribution and subsequent post-processing methods. We provide a comprehensive evaluation of the LAQW algorithm and demonstrate the reproducibility of 128-bit keys under simulated quantum noise provided by IBM's FakeTorino backend. A direct comparison with the CAQW model, which has been used in image encryption and hash function schemes (Li et al., 2017, arXiv:1707.07389; Abd EL-Latif et al., 2020, ScienceDirect; Abd El-Latif, Abd El-Atty, and Venegas-Andraca, 2020, ScienceDirect), highlights the potential and usefulness of the LAQW model in cryptographic applications.

The certification of quantum systems is essential for emerging quantum technologies, particularly in quantum communication, networks, and distributed computing, where maintaining a common reference frame across distant nodes poses significant challenges. Reference frame independent approaches, such as randomized measurement schemes, offer a promising route by reducing experimental demands while granting access to basis-independent quantities, including entanglement. However, the efficiency of such schemes in measuring such local invariants has remained unclear. In this work, we determine the minimal number of measurement settings required to access all two-qubit invariants. We further demonstrate that entanglement certification necessarily involves the most demanding invariants, establishing it as a maximally difficult task. Our results reveal a fundamental hierarchy among invariants, with direct implications for experimental feasibility and theoretical understanding of quantum certification. Finally, we extend our analysis beyond bipartite systems by applying it to the Kempe invariant in three-qubit systems, improving known measurement protocols and providing a first step toward uncovering similar hierarchies in higher dimensions.

Decoded quantum interferometry (DQI) offers an approach to a class of combinatorial optimization problems by reducing quantum state preparation to a decoding task. However, the existing performance bound for imperfect decoding uses the maximum failure rate over error-weight layers to bound the performance loss uniformly, which may overestimate the effect of decoding failures in layers with low probability. Building on the existing analysis, we combine the failure rate of each layer with its probability weight in the chosen state to derive tighter performance lower bounds. For binary linear constraint problems with uniformly random target values, we obtain a finite-size lower bound on the average satisfaction ratio for fixed state parameters and a fixed decoder, and establish its improvement over the existing bound. In part of the parameter region where the existing bound falls to the random-assignment baseline, the weighted bound still provides a guarantee above that baseline. We further extend the analysis to general finite fields under a model of uniformly random translations of allowed sets of equal size. Trials in individual layers using a fixed belief propagation decoder illustrate the size of the improvement and provide confidence guarantees after accounting for sampling uncertainty. These results reveal the role of retaining the correspondence between the failure distribution and the state probabilities, and suggest directions for more refined performance analyses of other DQI constructions.

Partially error-corrected logical executions are expected to become available before fully fault-tolerant quantum computation, but such executions usually take much longer runtime than physical, unencoded ones. We formulate zero-noise extrapolation in this regime as a statistical resource-allocation problem in which the physical/logical execution mode is itself an extrapolation design variable. In the proposed mixed strategy, one or a few logical circuits provide low-noise anchor points, while cheaper folded physical circuits provide a larger extrapolation lever arm. Within an effective error suppression model $p_L=\gamma p$, we derive Richardson variance prefactors for all-logical and mixed data sets, include folded-circuit runtime accounting, obtain the optimal shot allocation for a prescribed target variance, and state the bias–variance criterion determining when the mixed estimator improves finite-runtime accuracy. We illustrate the mixed-data strategy via simulating dynamics of transverse-field Ising model. For error suppression factor $\gamma\lesssim 0.1$ the mixed strategy can significantly(orders-of-magnitude) reduce the runtime needed to reach a fixed estimator variance, as well as provide better mean-square-error estimators in reasonable parameter regions.

While the geometry of equilibrium states and driven non-equilibrium processes is clearly understood, a geometric description for relaxation towards equilibrium is still lacking. Here, we propose a thermo-geometric measure based on the Rayleigh quotient, reformulating relaxation as a fundamental competition between entropic stiffness and frictional dissipation. Taking a van der Waals gas with two dissipation channels as an example, we explicitly demonstrate its relaxation landscape. Particularly, we find that upon approaching the critical temperature $T_c$, the slow-mode relaxation rate vanishes linearly as $\lambda_s \propto (T-T_c)/T_c$, indicating critical slowing down. This study completes the thermodynamic geometry framework, providing a general tool for characterizing the relaxation dynamics of complex systems.

Mechanically isolated defect centers in hexagonal boron nitride are promising coherent quantum emitters, yet spectral instabilities persist, and their spin-related nature remains unclear. Here we investigate a single mechanically isolated quantum emitter in hBN integrated onto a coplanar waveguide. The emitter exhibits exceptionally bright resonant fluorescence with saturation count rates exceeding $10\,\mathrm{Mc/s}$. High-resolution spectroscopy reveals two closely spaced zero-phonon-line transitions originating from the same defect complex. Time-resolved spectroscopy shows that these transitions exhibit markedly different spectral diffusion dynamics, consistent with distinct donor-acceptor-pair-like recombination pathways with different sensitivities to local electrostatic fluctuations. Off-resonant blue illumination redistributes emission between the two transitions and increases the emission duty cycle without significantly modifying the dominant spectral diffusion rates at low temperature, indicating repumping from long-lived shelving states. Magnetic-field-dependent photoluminescence, optically detected magnetic resonance, and pump-probe measurements reveal millisecond-scale relaxation dynamics and magnetic-field-dependent fluorescence contrast, demonstrating spin-dependent population dynamics in the metastable shelving state. These results clarify how charge-driven spectral fluctuations and spin-dependent shelving jointly shape the optical cycling dynamics.

We study the time it takes for all states of a finite quantum system to return simultaneously to their original configuration. In particular, we define the recurrence time for a quantum system to be the time at which all time-evolved states are close to their initial configuration, and at least one state has deviated significantly during this interval. Considering finite-dimensional quantum systems evolving unitarily, we find bounds on this notion of recurrence time, for continuous time and discrete time, by using Dirichlet's approximation theorem. We show how the problem of finding a bound on recurrence time can be related to approximating the difference of real numbers by rationals. We present a mathematical result on the latter, which we then use to obtain tighter bounds on recurrence time.

Near-infrared (NIR) emitters operating in the 650-900 nm range are highly attractive for imaging and sensing in turbid media; however, cadmium-free InP-based quantum dots (QDs) often suffer from limited brightness due to nonradiative pathways and inefficient photon outcoupling. In particular, heterostructured InP QDs can exhibit band alignments that induce partial spatial separation of charge carriers, leading to reduced electron-hole wavefunction overlap. This modifies intrinsic recombination dynamics and enhances the sensitivity of their emission to the surrounding photonic environment. Here, we investigate silver toroidal plasmonic nanoantenna dimers (Ag TPNDs) through finite-difference-time-domain (FDTD) simulations as a geometry-tunable platform for enhancing NIR emission of heterostructured InP-based QDs. The coupled toroidal geometry supports strongly confined bonding modes that generate intense nanogap hotspots, while its resonance can be systematically tuned through the toroid aspect ratio. By spectrally aligning the antenna response with QD emission bands (675-845 nm), we achieve large Purcell enhancements together with high quantum efficiencies, demonstrating efficient conversion of enhanced decay rates into radiative emission. We further show that nanometer-scale variations in emitter-antenna separation strongly modulate the radiative rates and spectral response. These results establish toroidal plasmonic nanodimers as a topology-driven platform for controlling emission in NIR quantum emitters and for advancing NIR nanophotonic applications.