Papers

A finite reversible model can approximate individual quantum gates while retaining freedom in its transition tables and preparation contexts. We formulate the simultaneous certification problem in which one stationary controller and one preparation law must reproduce an entire collection of circuit probabilities. For a specified frame catalogue, accuracy graph and calibration constraints, we give an exact shared-table feasibility formulation and distinguish certificates for a fixed table from certificates for the complete model class. A four-frame qubit example exhibits a strict separation between fitting two circuits individually and fitting them jointly. A larger exact benchmark uses 464 frames and a phase symmetry that removes preparation-prior dependence. We independently reconstruct its admissible graph and certify a sharp repeated-circuit horizon: one controller stays within probability error 1/3 through 192 blocks, whereas every allowed controller violates that threshold by block 193. A separate controller reproduces all 8,190 nonempty Hadamard-phase words of length at most twelve within the same tolerance. We also derive a finite-window spectral obstruction and finite-shot confidence tests, illustrated by a reproducible sampling experiment. The conclusions concern the stated finite reversible model classes; they do not establish a universal memory bound or a discrepancy for physical quantum hardware.

The Guided Local Hamiltonian Problem (GLHP) is the problem of ground state energy estimation and preparation given a guiding state having good overlap with the true ground state. The problem is known to be BQP-complete for small error but is classically tractable for larger error. In this work, we identify an orthogonal regime in which the problem is classically tractable, namely when the true ground state $\psi$ has polynomially bounded $\ell_1$-norm. In particular, we show that the Truncated Power Method (TPM), a classical algorithm introduced by Kirby et al. in the quantum context, has a runtime that is polynomially comparable to quantum algorithms for GLHP in all relevant parameters, except for an extra factor of $\|\psi\|_1^2$. In other words, if $\|\psi\|_1^2$ is polynomially bounded, then TPM dequantizes quantum algorithms for GLHP.

A Hilbert space is usually described using an orthonormal basis, but a more general description can be defined using a tight frame. In the case of continuous-variable Gaussian states, the coherence is usually described using the Hermite-Gauss basis. We present an alternative description using the tight frame of all canonical coherent states. The description of Gaussian states in terms of coherent states is simpler than in terms of Hermite-Gauss basis, and this allows us to obtain an explicit formula describing the frame-dependent coherence of a mixed Gaussian state as a function of the determinant and the trace of covariance matrix. For each mixed Gaussian state, we construct a sequence of discrete-variable quantum states rapidly convergent to it (as concern the purity and coherence) when the dimension of Hilbert space becomes large enough (15 or more). In the discrete case, the coherence is measured with respect to the tight frame of all discrete coherent states.

A many-body wavefunction exponential in R, where the square of R is the sum of the squares of the interparticle distances, is shown to be the solution of Schrodingers equation for a potential proportional to the reciprocal of R, for N particles. A set of translationally invariant rotational eigenstates of angular momentum is found.

Charging a quantum battery through a pumped charger can lock a substantial part of its stored energy inside the bipartite correlations that grow between the constituent cells, and this correlation ergotropy, being the energy that no local single-cell operation can reach and that only collective operations can recover, is the object of the present study. Following the charging trajectory of two coupled three-level cells, we find that the correlation ergotropy closely tracks the logarithmic negativity, which shows that the locked energy is carried by genuine bipartite entanglement rather than by coherent collective dynamics alone, and that an equal-strength coherent non-entangling drive produces no such locking at all. For a pure two-cell battery we establish an exact, basis-independent closed form in which the locked energy depends only on the sorted eigenvalues of the reduced cell and the cell energy levels, revealing that the locked fraction rises from zero at the separable corners of the Schmidt simplex to unity at the symmetric maximally entangled state and remains invariant under local unitaries. Decoherence and dephasing, which destroy the correlations, do not release this energy but erase it, so extraction must be collective and must take place while the correlations persist, while cold damping leaves the effect intact and thermal occupation suppresses it. These results give a resource-theoretic reading of correlations in quantum batteries, sharpen the line between genuine entanglement and coherent collective dynamics, and point to experimentally accessible conditions for correlated-charge and collective-extraction protocols.

The study of quantum memories out of thermal equilibrium is motivated both by the need for memories that withstand realistic noise in fault-tolerant quantum computers and by the fundamental question of when quantum memory can emerge in autonomous many-body systems. We establish a general equivalence between CSS-based quantum memories subject to Pauli noise and classical memories via two associated probabilistic processes. In the asynchronous setting, where the quantum dynamics is generated by a Lindbladian, we identify a broad class of syndrome-based Lindbladians for which the quantum memory problem is equivalent to memory in continuous-time classical Markov dynamics. In the synchronous setting, we introduce syndrome-based probabilistic cellular automata (SPCAs) and establish a one-to-one correspondence between their classical memories and CSS-based quantum memories under local quantum channels. This framework provides a systematic route for deriving and constructing quantum memories from classical probabilistic processes, and recovers known results, including the fact that the 4D toric code is a quantum memory in the presence of thermal noise. As an application, we give the first rigorous proof that the 4D toric code with Toom's rule is a quantum memory in the presence of local Pauli noise, provided the error is below a constant threshold. We prove this both for the corresponding synchronous, PCA-type quantum dynamics and for a Floquet Lindbladian. More generally, we show that classical memory in an SPCA can be lifted to a quantum memory under Floquet Lindbladian dynamics; whether this can be done with a time-independent Lindbladian remains open. Our results provide a general framework for relating classical and quantum memory, shed light on the conditions for fully autonomous quantum memories, and may guide the construction of provable autonomous quantum memories in low dimensions.

We establish optimal quantum-classical separations in communication complexity for search problems. We introduce a total search problem called Pelagic Fourier Fishing and show that it admits an $n$-qubit quantum one-way protocol, whereas every randomized two-way protocol requires $\Omega(2^n)$ bits of communication. We then introduce a variant of this problem whose solutions can be verified in polynomial time. This variant also admits an $n$-qubit quantum one-way protocol, while every randomized one-way protocol requires $\Omega(2^n)$ bits of communication. We also construct a family of efficiently verifiable total search problems achieving an $n$ versus $\Omega_d(n^d)$ separation between quantum one-way and randomized one-way communication for every fixed $d \ge 2$. In the quantum protocol, Alice prepares her message using a single unitary from the $d$th level of the Clifford hierarchy, and Bob performs a Clifford measurement. This separation is asymptotically optimal under this restriction on Alice's message. For $d = 2$, Alice's message is a stabilizer state and Bob's measurement is Clifford, so the protocol uses no magic, yet achieves an optimal quadratic quantum advantage. Finally, we discuss how these separations can be adapted to near-term quantum advantage experiments in which the demonstrated advantage is both unconditional and efficiently verifiable.

Variational quantum algorithms turn choices of Hamiltonian, ansatz, and parameterization into a classical nonconvex optimization problem. We study how this objective function can be visualized and characterized in ways that help explain optimizer behavior. We distinguish two properties of the objective: the number of local minima encountered along sampled directions and the differences in quality among local-search endpoints. We then ask how a local optimizer, represented by BFGS, compares with adaptive differential evolution, represented by jSO. Rather than comparing jSO with a single local run, we allow BFGS multiple starts within the same function-evaluation budget. This gives local search repeated opportunities to explore different basins and provides a stronger baseline for asking when a global evolutionary solver is useful. We study these questions using VQE and QAOA, focusing on how frustra- tion, circuit depth, parameter tying, mixed locality, and nonlinear repa- rameterization change the Hamiltonian expectation-value objective seen by the classical optimizer. Increasing independent circuit depth raises the sampled local-minimum count without making global search more effective. Parameter tying, by contrast, produces both more repeated local structure and much larger differences in quality among local-search endpoints, and in this regime adaptive differential evolution outperforms function-evaluation-matched multistart BFGS. The comparison shows that the number of local minima alone does not determine whether global search is advantageous: the important distinction is whether different basins lead to similarly good solutions or to substantially different objective values. These results provide a practical workflow for connecting model construction, objective- function geometry, empirical diagnostics, and optimizer choice.

We introduce a distinct route for extending classical atomistic spin dynamics into the quantum regime for arbitrary spin $s$, based on a variational product-state ansatz that includes both pure and mixed local states. By construction, intersite entanglement – expected to remain negligible even on very short time scales in many real materials – is excluded, while local quantum effects are retained through the nonclassical structure of $s\geq 1$ density operators, and nonlinear dynamics arise from both the mean-field interdependence of the local states and the state-dependent dissipative terms. This framework clarifies the variational and geometric origin of the inequivalence between quantum Landau-Lifshitz and Landau-Lifshitz-Gilbert dynamics, provides a criterion for their equivalence under a common time rescaling, establishes connections with the classical theory through two qualitatively distinct limits ($s\to\infty$ and a large number of quantum spins $N$), and allows realistic $\mathcal{O}(N)$ simulations of quantum spin dynamics in materials.

We show that quasi-paraxial fluids of light propagating in nonlinear Kerr media naturally experience an effective gauge field arising from the intrinsic spin-orbit interaction of light encoded in Maxwell equations. By deriving the leading nonparaxial corrections to the nonlinear Schrödinger equation governing the fluid evolution, we find that, for circularly polarized beams, these corrections generate an effective Lorentz force that modifies the fluid dynamics. Remarkably, the corresponding gauge field is not externally imposed but self-consistently generated by the fluid itself, with its sign controlled by the light helicity and its magnitude governed by the spatial variations of the nonlinear refractive index. We illustrate the physical consequences of this gauge field in two configurations: the chiral deformation of normal and superfluid flows past an obstacle, accompanied by a transverse spin Hall force, and the cumulative precession of an optical wave packet trapped in a focusing fluid.

We develop a perturbative framework for systems coupled quadratically to bosonic environments with quadratic Hamiltonians and linear Lindblad operators. The method applies to both single-mode and multimode environments whose initial states may be non-Gaussian and nonstationary. By combining linear evolution in a doubled operator space with a characteristic-function representation, we express ordered multitime environmental correlation functions in terms of derivatives of the initial characteristic function. These correlation functions enter a recursive construction of the time-convolutionless generator at arbitrary finite order. We illustrate the method with a qubit coupled to a damped mode via a two-photon interaction. Initial Fock and thermal states with identical first and second field moments produce different second-order generators through their fourth moments.

Quantum conference key agreement (QCKA) enables multiple spatially separated users to establish a common key with composable information-theoretic security by exploiting multipartite quantum correlations. Greenberger–Horne–Zeilinger (GHZ) states are a natural resource for this task, but their distribution over optical fiber is degraded by transmission loss, polarization decoherence, and detector imperfections. We investigate the finite-key performance of an $N$-party GHZ-based QCKA system using the $N$-BB84 protocol, where each optical link is modeled by extending the bipartite erasure–Pauli channel to incorporate detector dark counts. We characterize the impact of this quantum channel on the shared GHZ state and derive closed-form expressions for the error rates. The analytical results reveal that the local bit-error rate depends only on transmissivity and dark counts, while the global phase-error rate compounds over all $N-1$ links and therefore degrades faster with both distance and the number of parties. These error rates are integrated into a finite-key secret-key-rate (SKR) framework, in which we optimize the parameter-estimation probability via Brent's derivative-free method. Numerical results show that increasing fiber distance and network size reduce the achievable SKR. Furthermore, the optimized parameter-estimation probability consistently increases the SKR and extends the maximum secure distance relative to a fixed-allocation baseline, with comparison to the asymptotic limit quantifying the residual finite-key penalty.

We establish a local Konno-Sato correspondence for reflected random and Szegedy walks on a finite path graph. A generalized reflection principle, using explicit involutions rather than point reflections in the asymmetric case, yields exact zeta factorizations into a boundary factor and local factors indexed by the infinite dihedral group. After symmetrization, the classical and quantum local factors correspond for each group element. The identity contribution determines the normalized thermodynamic limits. In the symmetric case, diffusive and ballistic scalings recover the Gaussian Poisson summation formula and the Dirac comb identity, respectively. Under a double scaling of the asymmetry and spectral parameter, the quantum factorization yields a massive Dirac-type limit: nonidentity even words produce the periodic-orbit factor, the identity produces the free-field factor, and odd words together with the boundary produce a finite boundary correction.

In the quantum state distinguishing problem, an algorithm is given copies of a state $|\psi\rangle$ or $|\phi\rangle$ separated by trace distance $\varepsilon$, and tasked with deciding which is the case. It is a standard result that $\Theta(\varepsilon^{-2})$ copies are required, while having access to a unitary state preparation oracle and its inverse improves this to $\Theta(\varepsilon^{-1})$ queries. We study the intermediate case with forward-only access, capturing scenarios where the oracle is a physical process that cannot be reversed. Tang and Wright recently showed that distinguishing a known state from an unknown close state requires $\Omega(\varepsilon^{-2})$ forward queries in high dimension; we strengthen this result by proving the same bound even when both states are known, as in the state distinguishing setting. Moreover, our lower bound uses a direct hybrid argument, avoiding advanced compressed oracle techniques used in the prior work. In contrast, we give a dimension-independent algorithm solving a continuous-time variant of state distinguishing with only $O(\varepsilon^{-1})$ total forward evolution time. The algorithm uses quantum Zeno dynamics to confine evolution to a known two-dimensional subspace. Thus, a quadratic advantage survives without inverse access for a natural continuous-time analogue of state distinguishing.

We study quantum state tomography when the unknown state lies in a trace-distance ball of radius $R$ around a known reference state $\rho$. For arbitrary collective measurements, we determine the minimax expected risk and confidence radius up to universal constants, uniformly over the dimension, center spectrum, radius, and sample size. Write $\Gamma=d(\mathrm{Tr}\sqrt\rho)^2+d^2R$. For $d\ge2$, $0<R\le1$, and $n\ge1$ copies, the expected risk is $\Theta(\min\{R,\sqrt{\Gamma/n}\})$. At failure probability $0<\delta\le1/3$, the minimax confidence radius is \[ \Theta\!\left(\min\left\{R,\sqrt{\frac{\Gamma+\log(1/\delta)}{n}}\right\}\right). \] For $\varepsilon\le c_0R$ with a universal $c_0>0$, the copy complexity is $\Theta((\Gamma+\log(1/\delta))/\varepsilon^2)$. For the lower bound we use a weighted trace-norm witness to retain all spectral scales in a square-root perturbation family. For the upper bound, we give a direct spectral and concentration analysis of a modified existing purification-based procedure.

Open-system dynamics govern decoherence, transport, and engineered dissipation in quantum many-body systems. We propose a quantum algorithm for simulating time-independent, finite-range Lindblad dynamics with bounded local strength on a constant-dimensional lattice of $n$ sites, allowing noncommuting jump operators. For time $T\ge1$ and diamond-norm error at most $\epsilon$, it has depth $O(T\operatorname{polylog}(nT/\epsilon))$ and gate count $O(nT\operatorname{polylog}(nT/\epsilon))$. These results extend the HHKL spatial decomposition to open systems. The main obstacle is that the reversible overlap corrections used for unitary dynamics have no direct analogue for dissipative channels. We overcome this by retaining the environment in a shared unitary dilation, keeping information lost from the system coherently available and making the spatial corrections reversible. We prove exponential occupation bounds for the environmental states generated by the decomposition, allowing compact representation and coherent simulation of local factors on the inputs that arise. The construction reduces each short-time evolution to a constant number of parallel layers on regions of logarithmic diameter, using nearest-neighbour gates and polylogarithmic auxiliary density. We also prove unconditional worst-case lower bounds using time-independent, purely dissipative processes that transport quantum information. For circuits with local quantum and classical communication, the lower bounds are $\Omega(nT)$ for the expected gate count and $\Omega(T)$ for depth on time scales up to the lattice diameter. They match the upper bounds' leading time and volume dependence up to polylogarithmic factors. At fixed time and sufficiently small fixed error, we further prove a depth lower bound of $\Omega(\log n/\log\log n)$, showing that global accuracy can require increasing depth as the lattice grows.

Code surgery measures a logical operator of a quantum LDPC code with $O(d)$ rounds of syndrome extraction. Lifted surgery amortises this cost on Abelian group-algebra codes by measuring an orbit of logical operators under a translation symmetry in one merged code. We extend it to two-block group-algebra codes over non-Abelian groups and ask whether non-commutativity lets one merged code measure more operators than any Abelian group of its symmetries. We classify the block-affine automorphisms of these codes and find that, measured against this full group, most apparent non-Abelian gains disappear. We prove that the gain is at most the index of a largest Abelian subgroup. We find codes with exact distance up to $13$ where the gain is two, including codes in which one merged code reads out every logical qubit, and codes over products of $A_4$, $S_4$ and SL(2,3) with gain three for a best logical, up to $[[336,26,12]]$. A merged-distance lemma gives a simple condition for the gadget to preserve the code distance. In circuit-level simulations with Relay-BP decoding, at the error rates we can resolve, the non-Abelian gadget is as reliable as, or more reliable than, the Abelian gadgets it replaces within statistical error, while using two to three times fewer rounds.

Quantum hardware advances now make it practical to explore quantum approaches to scientific problems. Applied quantum computing research sits at the intersection of domain knowledge, algorithms, theoretical physics, device physics, and quantum and classical computational methods. The breadth of expertise required and the inherent project complexity challenge not only any individual researcher, but also generic AI systems. Here, we introduce the Quantum Agentic Operating System (QAOS), a structured agentic research system designed to address this challenge for quantum and computational R&D projects. QAOS is built around a nonlinear workflow graph of heterogeneous multi-agent nodes, with a controlled flow of context and computational resources. It decomposes a complex research project into sandboxed multi-agent tasks executed under constraints from the project graph, task-relevant methodologies, and necessary verification steps. We demonstrate QAOS on a quantum chemistry prototyping task, in which it is asked to set up a minimal model for a proton-transfer problem and to implement its solution using sample-based quantum diagonalization (SQD), together with classical reference methods such as FCI (exact diagonalization) and density functional theory (DFT). We compare the QAOS solution to those obtained with baseline agents (Claude Sonnet 5 with the Claude Code harness). In contrast to QAOS, the baseline agents silently ignore explicit instructions, and the runs show large variability.

We ask whether a neural network can learn a structure-preserving map from a compressed latent space to a Hilbert-space representation. Using an 8-dimensional autoencoder bottleneck on MNIST and $n$-qubit product-state targets from PCA-based angle encoding, we report four findings. Although the target angles are generated by a nonlinear sigmoid transformation of the latent projections, the resulting mapping is well approximated by a linear function over the observed latent distribution: linear regression from $z$ to the true target angles achieves $R^{2} = 0.98$, while regression to the MLP's recovered angles achieves $R^{2} = 0.91$. The learned map's primary direction is strongly aligned with the target-induced direction, with cosine similarity $0.989$, while remaining nearly orthogonal to the input's principal direction, with cosine similarity $0.002$. The map is genuinely rank-4: removing any singular direction degrades inner-product preservation by $2.5$–$5.9\times$ despite a singular-value spectrum with two dominant and two small values. The learned subspace does not coincide with the PCA basis used to construct the target, and different random seeds recover the same primary direction but diverge in higher ranks. Finally, kernel ridge regression with an RBF kernel outperforms a tuned MLP (IP error $0.0144$ vs.\ $0.0197$), suggesting that for approximately linear structure-preserving mappings, classical kernel methods may be a simpler and more effective alternative.

Randomized product formulas provide simple and flexible methods for quantum simulation. While qDRIFT-type randomized simulation has been extended to Lindbladian dynamics, higher-order randomized compilers have primarily been developed for Hamiltonian simulation. In this work, we develop first- and second-order randomized sampling algorithms for Lindbladian simulation in both the time-independent and time-dependent settings. Building on first-order qDRIFT-type Lindbladian simulation, we introduce a qSWIFT-type second-order correction for time-independent Lindbladians with a finite local decomposition. For $\lambda t \geq 1$, where $\lambda$ is the sum of the Hamiltonian coefficients and dissipative rates, the resulting approximation achieves normalized diamond-norm error at most $\varepsilon$ using $r=O(\lambda^2 t^2/\sqrt{\varepsilon})$ time slices, improving the $\varepsilon^{-1}$ precision dependence of the first-order qDRIFT-type approximation. We then develop first- and second-order continuous-time sampling algorithms for general time-dependent Lindbladians, with the second-order construction likewise achieving an $\varepsilon^{-1/2}$ dependence on the target precision in the number of time slices. We further extend the time-dependent construction to a linear-combination input model over local Hamiltonian and dissipative generators, allowing the randomized sampler to be formulated in terms of local evolutions rather than the full time-dependent Lindbladian. Because the second-order corrected maps need not be CPTP, we use them to estimate observable expectation values rather than to deterministically prepare output states.

The quantum $n$-coloring problem, given graph $G$, asks whether there exists a perfect quantum strategy for the $n$-coloring game of $G$. Previously it was known that quantum $n$-coloring is undecidable only for $n=3$. We extend this to all $n\geq 3$ with one elementary reduction. Letting $G'=(G\square K_3)\lor K_{n-3}$, we show that $G$ is quantum $3$-colorable if and only if $G'$ is quantum $n$-colorable.

Dynamical quantum phase transitions (DQPT) provide a fundamental framework for characterizing nonequilibrium critical phenomena, and have recently been extended to non-Hermitian systems. Half-integer jumps of the dynamical topological order parameter can arise in both Hermitian and non-Hermitian systems, yet their origin in non-Hermitian settings remains unclear. Here, we investigate DQPTs in a two-band non-Hermitian model and show that these fractional jumps originate from the discontinuity in the complex energy spectrum. We demonstrate that this behavior is intrinsically linked to the steady-state phase diagram of the quench Hamiltonian, and also classify the occurrence of DQPT. We further extend our analysis to the multiband Hamiltonian and the two-dimensional model, providing a unified understanding of fractional dynamical topological signatures.

Suppose Alice and Bob can transform one shared mixed state into another by local operations and classical communication, and can also recover the first state from the second. What can change in such a conversion? We give a complete classification of reversible conversions between two-qutrit states. We consider exact deterministic LOCC transformations of a single copy, allowing finite protocols and the countable protocols specified here. Local flags are known to allow a separable part to be replaced by local preparation while an entangled block is retained. We prove that such a decomposition is necessary for two mutually convertible entangled states that are not locally unitarily equivalent. The two parts are distinguishable by local measurements. The entangled blocks have the same weight and are locally unitarily equivalent after normalization. Each entangled state has a unique smallest retained block. Two entangled states are reversibly convertible if and only if these unnormalized blocks are locally unitarily equivalent. For a state with negative partial transpose, the smallest block can be found using the negative eigenspace of its partial transpose and at most three matrix zero tests.

Fidelity estimation is a fundamental tool for assessing quantum resource states across quantum information science. For small infidelity $\epsilon$, tomography and direct fidelity estimation (DFE) have worst-case sample complexity $O(1/\epsilon^2)$ at fixed relative accuracy and confidence. Direct target projections achieve $O(1/\epsilon)$ but generally require non-Clifford operations whose reliability depends on the resource states that are typically the target of benchmarking. Here, we introduce the Bell–coherence fidelity estimation (BCFE) with optimal sample complexity $\Theta(1/\epsilon)$ using only Clifford gates and Pauli measurements. We prove that BCFE is applicable to arbitrary pure targets, possibly multi-qubit, with the statistical error of the single-copy coherence part explicitly controlled. To evaluate BCFE in a practical setting, we apply BCFE to analog rotation resource states in the candidate early-FTQC architecture called the Space-Time Efficient Analog Rotations (STAR) architecture. In the simulations using surface codes, BCFE is estimated to require up to approximately $27$ times fewer resource states than DFE for the same standard error in the infidelity estimate, while its estimates remain close to reference values despite circuit-level noise in the additional Bell measurement operations. These results support efficient resource benchmarking within fault-tolerant quantum computing.

In Part I, we introduced a general framework for constructing logical gates in quantum CSS codes from cohomology invariants, focusing on cup products and their generalizations from topological to qLDPC codes. In particular, we produced a family of $C^{k-1}Z$ logical gates that we called copy-cup on $k$ copies of the same quantum code implemented by finite-depth circuits. Here, we vastly extend this framework by introducing twists incorporating code automorphisms, higher twisted cohomology operations, and folding operations in relative cohomology. The framework includes operations lifting coefficients from $\mathbb{Z}/2$ to $\mathbb{Z}/2^k$, enabling all diagonal logical gates such as single-qubit rotations $\operatorname{R}(2\pi/2^k)$. Composing these operations with maps that select cohomology classes yields addressable logical gates. Examples include addressable $T$ gates on three- and four-tori and gates in qLDPC codes beyond topological ones. We introduce cohomology operations, such as the higher twisted Pontryagin powers, that to our knowledge have not previously appeared in the mathematics literature. These operations arise from $E_\infty$-structures on cochains, with explicit constructions using cyclic resolutions and the surjection operad. We expect these results to be of independent interest in homotopy theory. Our formalism applies much more broadly to chain complexes carrying suitable operadic structures, and we expect this operadic perspective to uncover further connections between homotopy theory and fault-tolerant quantum gates.

Numerical studies suggest that quantum reservoir computing performs optimally near the edge of chaos, a behavior commonly attributed to a balance between memory retention and nonlinear information processing. Here we develop an explicit quantum reservoir model and rigorously analyze its chaotic regime and integrable point. In the chaotic regime, we show that Gaussian-unitary-ensemble statistics lead to rapid memory loss: learning performance decays exponentially with the storage time required by a temporal task. This decay limits the ability of chaotic reservoirs to learn long-range temporal dependencies. At the integrable point, described by a free-fermion model, we prove that amplitude encoding and a linear readout of fermionic two-point correlations cannot learn a representative class of nonlinear tasks coupling present and past inputs. Introducing interactions alleviates this obstruction. By identifying distinct limitations on memory retention in the chaotic regime and nonlinear processing at the integrable point, our analysis provides a theoretical basis for the observed edge-of-chaos advantage and guides the design of quantum reservoirs for temporal learning.

Electron cotunneling in semiconductor quantum dots is a higher-order tunneling mechanism that connects initial and final charge states through energetically inaccessible virtual intermediate configurations. In quantum-dot arrays, such processes can produce simultaneous charge motion across multiple tunnel junctions and enable transitions that would otherwise be suppressed by Coulomb blockade. Here, building on an earlier proof-of-principle demonstration, we investigate radio-frequency (rf)-driven correlated cotunneling and its application to dispersive quantum-dot readout. Using both theoretical modeling and experimental measurements in two different foundry-fabricated silicon devices, we show that an external rf drive can induce correlated cyclic single-electron transitions, resulting in enhanced charge-polarization of the system. We develop a general theoretical framework for rf electron-cascade readout, which exploits cotunneling events in quantum-dot systems with three or more dots to produce a relative signal enhancement, and extend its experimental demonstration to a gate-sensed device architecture. Our results establish the operating principles of rf electron-cascade readout and provide a route to increased dispersive readout fidelity in semiconductor quantum computing architectures.

The standard Langevin noise formalism (LNF) is a well-established approach for macroscopic quantum electrodynamics in unbounded lossy media, and accordingly, its application to finite objects requires endowing the surrounding vacuum with a fictitious absorption that is taken to asymptotically vanish. Existing justifications for such a singular limiting procedure have been mainly restricted to phenomena governed by vacuum fluctuations and medium-induced interactions (Casimir-Polder forces, thermal radiation, quantum emitter dynamics, etc.), thereby strictly excluding scenarios that involve the scattering of external quantum radiation. Conversely, the modified Langevin noise formalism (MLNF) has recently been established as the complete canonical quantum theory for arbitrary macroscopic systems and electromagnetic processes, entirely bypassing the need for limiting procedures, treating incident quantum radiation fields as independent degrees of freedom, and exactly reducing to the LNF for unbounded media. In this work, we demonstrate that the LNF limiting procedure is universally valid by showing that it recovers any physical prediction of the MLNF across all regimes, including arbitrary quantum scattering processes. Specifically, we introduce a suitable mapping associating any MLNF state and observable with a corresponding entity within the LNF augmented by fictitious vacuum absorption. We then rigorously prove that, as the fictitious absorption tends to zero, every LNF matrix element constructed via this mapping exactly converges to its MLNF original, so that the limiting procedure establishes the asymptotic convergence of the LNF to the MLNF.

Code switching moves an encoded qubit between two codes whose transversal gates together are universal, but the switching operation itself is usually not transversal and can spread a single fault into an uncorrectable error. We study switching between two permutation-invariant codes on the same qubits, one with transversal Clifford gates and one with transversal gates from the non-Clifford binary icosahedral group. Both are compatible with transversal $X$, $Z$, and $F = HS^\dagger$, and the encoders with this property form a small linear space. We show that its codes of distance at least three are cut out by at most two quadratic equations, and that all of them have the same error geometry in the sense that their single-qubit error states have the same inner products. It follows that a switching unitary commuting with qubit permutations and with the shared transversal gates at every instant, and tolerating one single-qubit fault at an unknown time, exists if and only if the two codes are joined by a continuous path of distance-three codes. Such a switch can be made error transparent, so that a fault at any moment reaches the target code as the same single-qubit error. For real encoders, the distance-three codes are the solutions of a single quadratic equation, and on thirteen qubits, they form an explicit circle through two Clifford codes and an icosahedral code. We construct the switching unitary along this circle as a collective operator, leaving its synthesis into physical control open, and verify that its recovery infidelity per fault is at machine precision, whereas two natural alternatives leave average infidelities of $0.13$ and $0.85$. A survey over odd qubit numbers up to $29$ lists where such switches exist.

Can quantum mechanics be complete without being fundamental? Under stated source, detector, and statistical conditions, we reconstruct the quantum representation and Born probabilities and prove complete algebraic closure of the admitted quantum predictions without requiring a collapse postulate. We further show that many familiar quantum paradoxes arise from conflating coherent processes with retained records or probabilities with realized outcomes. We also ask whether mathematics as a whole, or the source architecture itself, emerges from a deeper structure.