Oracle filters, noise thresholds, boxcar readout (10/02/26)

TL;DR For fixed finite-impulse-response (FIR) filters—digital convolvers with frozen taps—summing the filtered dispersive-readout samples equals a boxcar (uniform-window) sum times one complex scale factor, up to a small edge correction (kernel collapse), so state discrimination cannot beat the boxcar limit. On FPGA hardware and offline replays of 2D-transmon and 3D-ancilla shots, least-mean-squares (LMS) adaptive filtering denoises traces without resolved fidelity gains, while window tuning, per-sample weighting, and early sequential assignment are the levers that move assignment accuracy.

• Boxcar limit: For the fixed FIR kernels tested, the filter sum is the boxcar sum multiplied by one complex number aside from a small edge correction (kernel collapse), which scales state separation and noise equally and therefore cannot raise discrimination.
• LMS on hardware: An adaptive FIR trained in real time by the least-mean-squares (LMS) algorithm on an FPGA controller freezes into a filter that cuts trace noise, yet yields no resolved fidelity gain at the tested readout lengths; offline replay on 2D transmon Q1 at $8\mu \mathrm{s}$ keeps $99.4\%$ of the boxcar shot's discrimination signal-to-noise ratio.
• Window and weights: On Q1, retuning the readout window's start and duration raises contrast readout fidelity by $+2.4\%$ to $+5.6\%$ across four lengths; per-sample weighting adds up to $+3.0\%$ over full-window integration and only $+0.33\%$ beyond the tuned $8\mu \mathrm{s}$ window.
• Early decision: Hardware sweeps place the early-decision crossover near $1\mu \mathrm{s}$; 3D ancilla Q2 sits above it and Q1 below, and an offline sequential test decides most shots early with fidelity within $0.5$ percentage points of the boxcar.
• Template cost: About $30$ labeled shots per class build a matched-filter template—weights matched to the mean $|0⟩$–$|1⟩$ waveform difference—nearly as good as one from hundreds, separating waveform denoising from decision improvement.

TL;DR Multivariate Quantum Signal Processing designs matrix-valued transforms of multiple noncommuting, nonnormal block-encoded oracles by giving quantum transducers the analytic structure of multiport Infinite-Impulse-Response (IIR) filters—recursive filters with infinite memory via feedback—and a transfer-function recipe for query algorithms. The framework yields near-unity-success circuits with logarithmic error dependence and resolves or improves per-oracle costs for multi-oracle simulation, linear systems, differential equations, and eigenvalue tasks.

• Main idea: Multivariate Quantum Signal Processing (a multivariate extension of Quantum Signal Processing / Quantum Singular Value Transformation) builds modular multiport IIR transducers whose achievable transfer functions are fully characterized, with uniform analytic bounds across the oracle promise.
• Simulation cost: For $H={\sum }_j{H}_j$, the leading query cost is the weighted quasinorm $t⟨\vec{\lambda },\vec{C}{⟩}_{1/2}+\mathcal{O}(\Vert \vec{C}{\Vert }_1\log (1/ϵ))$ in block-encoding normalizations $\vec{\lambda }$ and oracle costs $\vec{C}$; sparse simulation reaches the optimal time–norm $\Theta (\sqrt{d}\Vert H{\Vert }_{1\to 2}t)$ with additive $ϵ$ dependence.
• Linear algebra: Linear systems and differential equations are solved under minimal matrix assumptions with optimally weighted state-preparation and multi-term costs; ground-state preparation, energy estimation, and generalized eigenvalue problems inherit the same modular IIR library.
• Applications: Exact simulation of downfolded Hamiltonians costs $t⟨{\vec{\lambda }}_{\mathrm{eff}},{\vec{C}}_{\mathrm{eff}}{⟩}_{1/2}$, further cut by fractional dynamic correlation from coarse-grained terms; Kron reduction (Schur-complement coarsening of graphs) gives analogous classical-data input reduction for multiscale graph classification.
• Online extension: Anisotropic Kalman filtering of streaming quantum data supplies the online generalization of the same transfer-function methodology.

TL;DR For geometrically local circuits of unital gates (channels that fix the maximally mixed state) with single-qubit depolarizing noise of strength $p$, approximate classical sampling becomes polynomial-time once depth reaches $\Omega (p^{-1}\log p^{-1})$ — a threshold independent of system size. Prior general-circuit samplers needed logarithmic depth so noise could globally trivialize the state; here local noise accumulation alone already kills quantum advantage for sampling.

• Main result: Any geometrically local unital circuit with per-qubit depolarizing noise of strength $p$ admits approximate classical sampling in polynomial time after depth $\Omega (p^{-1}\log p^{-1})$, with no system-size dependence in the depth threshold.
• Shift vs prior work: Earlier classical samplers for general noisy circuits required depth scaling as $\log$ of system size so the global output approach a trivial state; this proof shows local noise buildup at constant-in-$n$ depth already suffices.
• Mechanism: The output distribution is mapped to a polymer model in statistical mechanics (a dilute gas of interacting clusters), then controlled by a convergent cluster expansion (series over connected polymer groups) together with hypercontractivity of the depolarizing channel (noise contracts higher moments faster than the mean).
• Scope: The argument assumes geometric locality, unital gates, and single-qubit depolarizing noise of fixed strength $p$; it delineates when noisy sampling leaves the regime of plausible quantum advantage.