Basis transforms are a basic mechanism by which quantum algorithms extract hidden structure. For standard Gaussian inputs, Hermite functions and Fock states provide the spectral representation matched to the input measure. Recent work introduced an efficient quantum Hermite transform (QHT) and pointed to its possible uses in property testing and learning, but a clear model showing how QHT yields a provable readout advantage in a concrete Gaussian learning task has been missing. We introduce a high-frequency Gaussian ridge learning model. The target function is controlled by a small number of hidden directions, and binary labels are generated by passing a high-frequency periodic score through a weak noisy classification channel. Given coherent binary samples, the quantum procedure applies QHT to each copy, measures Fock labels, and uses shell postselection, low-order moment estimation, and tensor decomposition to recover the hidden-direction energy structure. We prove that this recovery uses polynomially many coherent sample calls and quantum gates. For the same candidate family and recovery target, any classical general statistical query (SQ) algorithm with inverse-polynomial tolerance requires exponentially many queries. This gives an exponential query separation between coherent quantum sample access and classical general SQ access. The result promotes QHT from a Gaussian spectral-transform primitive to a readout mechanism for learning, while making explicit the access models and recovery target underlying the separation.
Abstract A Floquet-driven red-sideband architecture is proposed and quantitatively analyzed that couples an Andreev spin qubit (ASQ) in a semiconductor Josephson junction to a collective nuclear spin-wave bright mode of the host, establishing a parameterized feasibility map and actionable design criteria for an ASQ–nuclear-spin quantum memory. The frequency-modulated exchange bridges the GHz ASQ transition and the MHz nuclear Larmor frequency with effective coupling g eff = g mg J 1 (β), where β = Ω d /ω d is the modulation index. An analytical model of a single-isotope 75 As (I = 3/2) bright-mode envelope with N eff = 10 5 –10 6 nuclei gives g eff /2π in the 9–367 kHz range for two explicit parameter sets, with SWAP times of 0.7–28 μs. A next-order Floquet expansion identifies, within the single-circular-component limit g− ≪ g+, the n = 0 Jaynes–Cummings dispersive dressing as the dominant off-resonant contribution, decomposing into a static (β = 0) component δ static ≃ 2g 2 mg /ω d and a drive-induced increment δ AC ≃ −g 2 mg β 2 /ω d that are distinguishable by modulation-on/off spectroscopy combined with an independent bare-reference calibration. A diagnostic M≡ 2π(g eff /2π)T 2,ASQ = 1 requires T 2,ASQ ∼ 0.4–17 μs, while a high-fidelity write–read cycle (FWR ≳ 0.9, M≳ 30) requires T 2,ASQ ∼ 13–520 μs. A two-mode stroboscopic Lindblad simulation independently reproduces the M≳ 30 crossover at fixed g eff /(2π) = 100 kHz and shows the direct counter-rotating g− contribution is sub-percent at the baseline resolved-sideband point (ε − = g −,eff /(2ω n ) = 0.025), while worst-case g − /g + = 1 at the optimistic coupling produces a 2.3-percentage-point penalty. Currently demonstrated T 2,ASQ ∼ 10–100 ns lie two to five orders of magnitude below the M≳ 30 regime; the analysis identifies sweet-spot operation, phase-bias engineering, and Floquet-dressed subspace protection as concrete engineering directions to close the gap.
Abstract Randomized compiling (RC) converts coherent errors into stochastic Pauli noise, but this conversion alone does not imply isotropic contraction of the output state. Here we use the generalized Bloch vector (GBV) representation to examine the cumulative effect of RC on state purity and orientation in multi-layer circuits. For the independent, isotropically distributed weak coherent errors considered here, the accumulated ensemble-averaged RC channel lies close to the depolarizing direction, producing a state-level near-isotropic contraction that sacrifices a small amount of purity to achieve a substantial suppression of orientational deviation. When dephasing coexists, RC suppresses the coherent-error contribution but leaves a residual deviation dominated by dephasing, setting a limit that we term the ``dephasing wall.'' We experimentally validate these findings on a superconducting quantum processor through GHZ state preparation and phase estimation in a simulation of topological braiding. The results confirm that RC's effectiveness stems from its ability to suppress the orientational bias of coherent errors—a structural transformation not resolved by fidelity alone but directly visible through GBV diagnostics.
Abstract Understanding the origins of trainability in variational quantum circuits remains one of the central challenges in quantum machine learning. Existing approaches based on dynamical Lie algebras characterize the reachable orbit generated by a variational ansatz and provide important information about controllability and expressivity. However, optimization is inherently a local differential process, and the geometric mechanisms governing the transmission of optimization information remain poorly understood. In this work, we develop a differential-geometric framework for trainability based on the variational Jacobian. The Jacobian is interpreted as the differential map connecting the parameter manifold with the tangent space of the reachable orbit. \textcolor{blue}{We derive an exact modal decomposition of gradient generation in terms of Jacobian singular directions and task-dependent modal information, showing that gradient suppression may arise from spectral attenuation, from a reduction of the task-gradient magnitude, from directional misalignment between the task gradient and the Jacobian-accessible subspace, or from combinations of these effects.} We further establish an explicit connection between the variational Jacobian and the underlying dynamical Lie algebra through effective commutator directions. \textcolor{blue}{In the numerical settings considered here, coarse task-independent spectral diagnostics such as the Jacobian rank, the largest singular value, and the total spectral weight exhibit little correlation with optimization performance, whereas normalized directional alignment and task-weighted spectral transmission retain significant predictive associations.} \textcolor{blue}{Finally, we show that Jacobian degeneration suppresses the task-Hessian component transmitted through the variational map, while the additional second-order map contribution need not vanish.} \textcolor{blue}{Taken together, these results provide a geometric framework in which local trainability can be viewed as emerging from the interaction between Lie-algebraic structure, Jacobian geometry, and the learning task, and as a problem of differential information transmission rather than accessibility alone.}
Abstract Quantum networks are expected to form a foundational infrastructure for quantum communication and distributed quantum computation, and their central challenge is reliable transmission of quantum states across noisy links. In classical networks this reliability is the job of the link layer, a role expected to be more critical for quantum networks, whose states cannot be copied or amplified. While quantum error correction (QEC) offers a powerful theory for protecting quantum information, the protocols that manage errors over quantum links, such as entanglement distillation, encoded-entanglement distribution, and direct encoded transmission, have evolved as separate families, obscuring their common coding structure and limiting cross-level optimization. Here we establish a unified, code-centric framework for error management over quantum network links. We prove that, with noise-free local operations, these protocol families are equivalent realizations of one underlying process: applying a quantum error-correcting code and extracting an error syndrome, differing only in what is encoded and how the syndrome is used. This equivalence induces a design space organized along two orthogonal axes, the encoding paradigm and the error management strategy, which exposes design choices and engineering optimizations, including localized preparation of logical entanglement and pipelined classical signaling. Within this design space, error detection and error correction emerge as two extreme modes of syndrome usage, naturally introducing hybrid error management strategies that interpolate between them. Through circuit-level simulations, we show that hybrid error management can reduce logical error rates by one to two orders of magnitude compared with pure error detection or pure error correction for small surface codes, while enabling explicit trade-offs between logical fidelity and acceptance probability. By reframing quantum-link error management as a quantum error correction problem, this work unifies previously disparate protocols, pushes the boundaries of performance, and lays a principled foundation for the quantum-network link layer.
