Continuous Signal Processing for Josephson Junction Readout

author: Rowan Brad Quni-Gudzinas

ORCID: 0009-0002-4317-5604

ISNI: 0000000526456062

title: "Continuous Signal Processing for Josephson Junction Readout: Addressing the Discrete Collapse Tension via Parametric Amplification"

aliases:

- "Continuous Signal Processing for Josephson Junction Readout: Addressing the Discrete Collapse Tension via Parametric Amplification"

modified: 2026-01-12T11:49:15Z

Addressing the Discrete Collapse Tension via Parametric Amplification

Author: Rowan Brad Quni-Gudzinas

Contact: [email protected]

ORCID: 0009-0002-4317-5604

ISNI: 0000000526456062

DOI: 10.5281/zenodo.18221366

Date: 2026-01-12

Version: 1.0

Abstract

The prevailing architecture of superconducting quantum information processing relies heavily on the Josephson junction, yet its dual role as both a qubit-defining non-linear inductor and a latching readout element creates a fundamental tension in measurement dynamics. While the former enables the isolation of a two-level subspace, the latter—often implemented as a bifurcation amplifier or latching comparator—imposes a premature digitization that destroys the continuous time-series of the quantum state evolution. This study reframes the quantum measurement problem through the lens of classical signal processing, positing that the “collapse” observed in standard readout schemes acts as a hardware-level quantization error isomorphic to a 1-bit Analog-to-Digital Converter (ADC). By simulating the information recovery rates of Traveling Wave Parametric Amplifiers (TWPAs) against traditional latching readouts, we demonstrate that continuous monitoring coupled with Kalman filtering can recover phase trajectories with a mean squared error (MSE) of 0.015, preserving the “verb” of quantum evolution. Crucially, control simulations reveal that even under high-SNR conditions ($\sigma_{noise}=0.2$), the act of latching increases error by a factor of 6 (MSE 0.254) compared to continuous monitoring, isolating the discrete readout mechanism as the primary source of information loss. These findings suggest that replacing discrete “noun-based” storage architectures with “verb-based” continuous flow processors is essential for potentially enabling quantum-limited metrology and efficient quantum error correction.

Keywords

Josephson Junction, Traveling Wave Parametric Amplifier, Quantum Measurement, Kalman Filter, Signal Processing, Quantization Error, Superconducting Qubits

1.0 Introduction

1.1 The Myth of the Quantum Bit: Isomorphism Errors

The nomenclature of the “qubit” suggests a fundamental isomorphism with the classical binary digit, implying a system that exists natively in one of two discrete states. This linguistic shorthand, while convenient for logical abstraction in high-level algorithms, obscures the physical reality that a superconducting qubit is a continuous analog oscillator restricted to a specific energy manifold. The state vector $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ resides on the continuous surface of the Bloch sphere, possessing infinite precision in its amplitude and phase coordinates (Murch et al., 2013). To treat this system as isomorphic to a classical switch $\{0, 1\}$ is to confuse the basis vectors with the state itself, a category error that fundamentally limits readout architecture design.

The persistence of this discrete isomorphism in engineering literature stems from the dominance of projective measurement protocols, which force the continuous quantum state into a binary outcome. However, the underlying Hamiltonian evolution is unitary and smooth, governed by the Schrödinger equation which describes a deterministic flow of probability amplitudes rather than stochastic jumps. The “discreteness” is not an intrinsic property of the information carrier during its computational life cycle but is instead imposed at the boundary of the classical control system. By adhering to the “bit” metaphor, current architectures inadvertently discard the rich “verb” of the quantum evolution—the phase accumulation, the coherent drift, and the entanglement dynamics—in favor of a static “noun” (the collapsed state).

1.2 Historical Context of Josephson Junctions

The evolution of superconducting quantum circuits has been inextricably linked to the development of the Josephson junction (JJ) as a measurement device. Early implementations of superconducting qubits, such as the Cooper-pair box, relied on switching events to detect the quantum state, effectively using the JJ as a binary threshold detector. In these architectures, the readout process was destructive and inherently discrete; the junction would physically switch to a voltage state if the critical current was exceeded, a process isomorphic to a latching relay (Averin, 2000).

This “latching” paradigm was driven by the necessity of achieving high Signal-to-Noise Ratios (SNR) in an era where cryogenic amplification was limited. The macroscopic switching of the junction provided a robust, easily detectable signal that could drive room-temperature electronics without sophisticated pre-amplification. However, this robustness came at the cost of information dimensionality; the latching event erased all information regarding the superposition coefficients except for their projection onto the measurement basis. The historical reliance on these latching or bistable mechanisms has entrenched a “measure-and-collapse” methodology, creating a feedback loop in the literature where the observation of discrete outcomes reinforced the design of discrete readouts.

1.3 The Dual Nature: Inductor vs. Switch

The Josephson junction acts as a unique quantum component that exhibits a dual nature depending on its bias conditions and embedding circuit. Fundamentally, it acts as a non-linear inductor where the inductance $L_J$ depends on the superconducting phase difference $\phi$ across the junction. This non-linearity is the “hero” of superconducting circuits, as it creates the anharmonic potential well necessary to isolate the $|0\rangle$ and $|1\rangle$ energy levels from the rest of the harmonic ladder, thereby defining the qubit itself (Murch et al., 2013).

However, this same non-linearity becomes the “villain” in the context of latching readout. When the current through the junction approaches the critical current $I_c$, the potential landscape tilts such that the phase particle can escape the well, transitioning the junction into a voltage state. This “switching” behavior is highly non-linear and irreversible, effectively acting as a hard thresholding function. In this mode, the JJ functions as a comparator, discarding all information about how far the signal was above or below the threshold. Ideally, a readout chain would utilize the JJ’s non-linearity solely for parametric amplification—mixing the signal with a pump tone to transfer energy—while avoiding the bifurcation or switching regimes.

1.4 The Measurement Problem as Signal Processing

If we accept that the quantum state is fundamentally continuous until the moment of digitization, the “measurement problem” can be rigorously mapped to a signal processing problem. In this isomorphism, the wavefunction collapse is equivalent to the quantization error introduced by an Analog-to-Digital Converter (ADC) with insufficient bit depth. A projective measurement, which yields a binary outcome, is structurally isomorphic to a 1-bit ADC that thresholds a continuous signal (Averin, 2000).

While the “collapse” involves fundamental stochasticity inherent to the Born rule—distinct from deterministic classical rounding—the information theoretic consequence is identical: the reduction of a continuous signal to a discrete symbol. Standard signal processing theory dictates that reducing a continuous signal to 1 bit destroys the vast majority of its information content, specifically the amplitude and phase nuances that constitute the signal’s “texture.” In the quantum context, this corresponds to the loss of the superposition coefficients and the relative phase information. The “collapse” is thus not a mystical discontinuity but a severe instance of data compression—a lossy compression algorithm implemented in hardware.

1.5 Current State of Readout Technology

The current landscape of superconducting readout technology is bifurcated between high-fidelity but slow projective schemes and emerging continuous monitoring architectures. The standard industry approach utilizes High-Electron-Mobility Transistors (HEMTs) at the 4K stage, which introduce significant thermal noise, typically 10-20 times the quantum limit (White et al., 2015). To overcome this noise floor, the signal must be integrated over a long period or latched using a JBA, both of which preclude real-time trajectory tracking.

In contrast, the Traveling Wave Parametric Amplifier (TWPA) represents a paradigm shift. By embedding a sequence of Josephson junctions in a transmission line, TWPAs achieve near-quantum-limited amplification over a wide bandwidth (Macklin et al., 2015). This device acts as the “hardware of the verb,” processing the traveling wave continuously without forcing a collapse. The TWPA boosts the signal power sufficiently to overcome the HEMT noise, preserving the delicate phase information of the microwave photons. Recent advancements have introduced Kinetic Inductance TWPAs (KI-TWPAs), which offer simpler fabrication and higher saturation powers, though potentially with different noise characteristics (Castellanos-Beltran et al., 2025).

1.6 Research Objectives and Scope

This study aims to bridge the gap between the physics of Josephson junction readouts and the mathematics of signal processing. Specifically, we seek to validate the hypothesis that the “collapse” in superconducting circuits is a technological artifact of latching readouts rather than a fundamental limit of observation. To this end, we define three primary research objectives:

1. RQ1: To rigorously map the structural isomorphism between quantum measurement dynamics and classical signal quantization, quantifying the information loss associated with latching versus continuous readout architectures.

1. RQ2: To demonstrate, via simulation, that continuous monitoring using TWPAs coupled with Kalman filtering can reconstruct single-qubit trajectories with fidelity exceeding the standard quantum limit for projective measurement.

1. RQ3: To propose a “Verb-based” readout architecture that integrates analog pre-processing and feedback to minimize quantization error and enable real-time quantum control.

1.7 Thesis: Quantization as Readout Artifact

We argue that the discrete “quantum jump” observed in standard superconducting qubit readout is primarily a readout artifact resulting from the use of latching comparators (1-bit ADCs) early in the amplification chain. By replacing these “noun-based” discrete elements with “verb-based” continuous amplifiers (TWPAs) and applying optimal estimation theory (Kalman filtering), the quantum state can be observed as a continuous, differentiable process. This reframing shifts the engineering challenge from “detecting the state” to “estimating the trajectory,” unlocking the potential for Heisenberg-limited metrology and continuous-variable error correction. The “collapse” is not an end, but a failure of resolution; with the correct lens, the quantum world remains a continuous flow.

2.0 Theoretical Framework: Quantum Signal Processing Isomorphism

2.1 Foundations of Quantum Measurement Theory

The orthodox description of quantum measurement, codified by Von Neumann, relies on the concept of Projective Valued Measures (PVMs), where the measurement operator is a projector onto an eigenbasis of the observable. In this framework, the interaction between the quantum system and the measuring apparatus is treated as instantaneous and irreversible, resulting in the discontinuous update of the state vector $|\psi\rangle \to |n\rangle$ with probability $P_n = |\langle n|\psi\rangle|^2$. This “collapse” postulate serves as a convenient mathematical abstraction for calculating outcome probabilities in ideal scenarios (Hacohen-Gourgy & Martin, 2020). However, it fails to capture the temporal dynamics of realistic experimental setups, particularly in superconducting circuits where the measurement timescale is comparable to the system’s dynamical timescales.

To address the limitations of the PVM formalism, modern quantum information theory employs Positive Operator-Valued Measures (POVMs), which describe generalized measurements that may be weak, continuous, or incomplete. A POVM consists of a set of operators $\{E_m\}$ such that $\sum E_m = I$, where the probability of outcome $m$ is $\text{Tr}(E_m \rho)$. The mechanism of continuous measurement can be derived by taking the continuum limit of a sequence of weak POVMs. In this limit, the state evolution is governed by a Stochastic Master Equation (SME), which includes both the unitary Hamiltonian dynamics and a stochastic term representing the back-action of the measurement. The “collapse” then emerges not as a postulate, but as the asymptotic behavior of the system under continuous monitoring.

2.2 The Signal Processing Isomorphism

To operationalize the continuous nature of quantum measurement, we propose a structural isomorphism between the formalisms of quantum mechanics and classical signal processing. This mapping, detailed in Table 1, posits that the quantum wavefunction $|\psi(t)\rangle$ is functionally equivalent to a complex analytic signal in communication theory, carrying information in both its amplitude (population) and phase (coherence). Within this framework, the Hamiltonian $\hat{H}$ acts as the system transfer function, governing the linear time-invariant (or time-varying) evolution of the signal vector (Mastriani, 2018).

Table 1: The Quantum-Signal Processing Isomorphism

The core of this isomorphism lies in the reinterpretation of the measurement event. In signal processing, the conversion of a continuous analog signal into a discrete digital value is governed by quantization theory. A projective measurement, which forces the continuous quantum state into a binary basis, is mathematically isomorphic to a 1-bit Analog-to-Digital Converter (ADC) or a hard thresholding comparator. This operation introduces quantization noise, which is the difference between the continuous input and the discrete output.

2.3 Wavefunction as Complex Carrier Signal

In the domain of superconducting circuit quantum electrodynamics (cQED), the abstract quantum state vector finds a concrete physical realization in the microwave field quadratures. The information of the qubit is encoded in the complex amplitude of the probe tone reflected from or transmitted through the readout resonator. This signal can be represented in the In-Phase ($I$) and Quadrature ($Q$) plane, forming a phasor $A(t) = I(t) + iQ(t)$ (Guarcello et al., 2024). The magnitude of this phasor corresponds to the measurement strength (photon number), while its angle encodes the qubit state information relative to the dispersive shift.

The trajectory of this phasor in the $IQ$-plane is the physical manifestation of the quantum trajectory. As the measurement proceeds, the phasor accumulates a phase shift conditional on the qubit state. For a qubit in a superposition $\alpha|0\rangle + \beta|1\rangle$, the signal is not a single phasor but an entangled state of the field and qubit. However, from the perspective of the amplification chain, it appears as a noisy complex voltage signal whose mean evolves stochastically. This representation underscores the analog nature of the information carrier.

2.4 Projective Measurement as 1-Bit ADC

The operation of a latching readout, such as a Josephson Bifurcation Amplifier (JBA) or a standard comparator in an SFQ circuit, can be modeled as a 1-bit quantization process. Mathematically, this is represented by the signum function applied to the noisy signal $S(t)$: $Y = \text{sgn}(S(t) - V_{th})$, where $V_{th}$ is the threshold voltage. This operation maps the continuous domain $\mathbb{C}$ of the complex signal to the discrete set $\{-1, 1\}$ (Hacohen-Gourgy & Martin, 2020).

The quantization error $E_q = S(t) - Y$ represents the information discarded by the readout. In a 1-bit system, this error is massive, effectively equal to the signal magnitude itself minus the sign. Standard quantization theory states that the Signal-to-Quantization-Noise Ratio (SQNR) increases by approximately 6 dB for every additional bit of resolution. A 1-bit ADC thus has the worst possible SQNR, fundamentally limiting the resolution of the state estimation. Furthermore, the latching process is often hysteretic. Once the junction switches to the voltage state, it remains there until the bias current is reset, introducing a “dead time” during which the system is blind.

2.5 Noise Models: Quantum vs. Classical

In the signal processing isomorphism, noise is the fundamental limit to information recovery. However, the nature of noise in quantum systems differs from classical thermal noise. Classical noise is typically modeled as additive white Gaussian noise (AWGN) arising from thermal fluctuations ($k_B T$). In contrast, quantum noise arises from vacuum fluctuations and is subject to the Heisenberg uncertainty principle, which imposes a lower bound on the noise power added by any phase-preserving amplifier (Mastriani, 2018).

Additionally, quantum measurement introduces “shot noise” in the measurement record. This is not due to technical imperfections but is intrinsic to the stochastic nature of the quantum state collapse (or diffusion). The distinction is critical: thermal noise is uncorrelated with the signal and can be averaged out. Quantum back-action noise is correlated with the system’s evolution. The measurement record $I(t)$ contains both the signal (the qubit state expectation value $\langle \sigma_z \rangle$) and the noise ($\xi(t)$).

2.6 The Role of Back-Action in Estimation

Back-action in quantum measurement is the phenomenon where the act of observing the system perturbs its state. In the context of the signal processing isomorphism, this is functionally equivalent to a control system with a stochastic feedback loop. The measurement outcome $z_t$ is not just an observation; it is an input that drives the system state $x_{t+1}$ (Guarcello et al., 2024).

The “back-action” is dictated by the term $\sqrt{\eta} \mathcal{H}[L]\rho dW$ in the stochastic master equation. It dictates how the state density matrix $\rho$ deforms in response to the information gain. If we know the measurement record $dW$ (which we do, from the TWPA output), we can calculate exactly how the state has changed. This implies that back-action is not “noise” in the sense of information loss; it is “process noise” in the control theory sense—a random force whose value is known after the fact (a posteriori). By preserving the continuous record $I(t)$, we preserve the history of the back-action, enabling optimal estimation.

2.7 Theoretical Limits of Information Recovery

The ultimate limits of this continuous recovery are set by the Heisenberg uncertainty principle and the Standard Quantum Limit (SQL). The efficiency of the measurement is defined by the quantum efficiency $\eta = \Gamma_{meas} / (\Gamma_{meas} + \Gamma_{loss})$. An ideal TWPA approaches $\eta = 1$. In this limit, the state remains pure throughout the trajectory (for a pure initial state), and the evolution is described by a “quantum trajectory” on the surface of the Bloch sphere (Hacohen-Gourgy & Martin, 2020).

The SQL represents the balance point where the measurement back-action noise equals the intrinsic projection noise. Continuous monitoring allows us to operate at or near this limit, extracting the maximum allowable information per unit time. In contrast, a latching readout operates far from these limits. By discarding the trajectory, it effectively sets $\eta \approx 0$ for the duration of the latch, yielding only a single bit of information at the end.

3.0 Methodology

3.1 Simulation Environment Setup

To rigorously quantify the information loss associated with discrete latching readouts versus continuous parametric amplification, we developed a numerical simulation environment using Python. The simulation framework models the time-evolution of a single superconducting qubit’s measurement record under varying readout architectures. The “ground truth” quantum trajectory was generated by modeling the qubit’s phase evolution as a stochastic process, specifically a Wiener process, representing the diffusive evolution of the quantum state under weak measurement back-action (White et al., 2015).

The simulation discretized the time evolution into $N=1000$ steps with a time step $dt=0.01$. The true signal $S_{true}(t)$ was constructed as a complex phasor with constant amplitude $A=1.0$ and the stochastic phase $\phi(t)$: $S_{true}(t) = A \cdot e^{i(\omega t + \phi(t))}$. This continuous complex signal serves as the reference “verb” that the readout architectures attempt to capture.

3.2 Modeling the Latching Readout (Comparator)

The latching readout architecture, typical of Josephson Bifurcation Amplifiers (JBAs) or SFQ-based comparators, was modeled as a non-linear thresholding device operating in a high-noise environment. This reflects the physical reality where the signal is amplified by a High-Electron-Mobility Transistor (HEMT) at the 4K stage, which introduces significant thermal noise before the signal reaches the digitizer.

We simulated the HEMT noise environment by adding high-variance Gaussian white noise to the true signal: $S_{HEMT}(t) = S_{true}(t) + \xi_{thermal}(t)$, where $\xi_{thermal} \sim \mathcal{N}(0, \sigma_{HEMT}^2)$ with $\sigma_{HEMT} = 2.0$. The latching mechanism was implemented as a hard thresholding function applied to the real component of the noisy signal: $Y_{latch}(t) = \text{sgn}(\text{Re}(S_{HEMT}(t)))$. This operation maps the continuous, noisy signal to a discrete set $\{-1, 1\}$, simulating the voltage state switching of the Josephson junction (Marceaux & Young, 2023).

3.3 Modeling the TWPA (Continuous Amplifier)

In contrast, the Traveling Wave Parametric Amplifier (TWPA) was modeled as a linear, phase-preserving amplifier operating near the quantum noise limit. The TWPA output was simulated by adding low-variance Gaussian noise to the true signal, representing the minimum added noise required by quantum mechanics (vacuum fluctuations): $S_{TWPA}(t) = S_{true}(t) + \xi_{quantum}(t)$, where $\xi_{quantum} \sim \mathcal{N}(0, \sigma_{TWPA}^2)$ with $\sigma_{TWPA} = 0.2$. This represents an order-of-magnitude improvement in noise performance compared to the HEMT model, consistent with experimental characterizations of Josephson TWPAs (White et al., 2015).

3.4 Control Simulation Protocol (High-SNR Latching)

A key critique of comparing “HEMT-Latching” vs. “TWPA-Continuous” is the confounding of two variables: the noise floor (SNR) and the quantization method (Readout). To disentangle these, we implemented a specific Control Simulation: “Latching at High SNR”. In this scenario, we applied the 1-bit latching threshold to the high-quality TWPA signal ($\sigma = 0.2$). This isolates the error introduced purely by the discrete readout method, independent of the amplifier’s noise performance. By comparing the TWPA continuous output against the TWPA latched output, we can rigorously quantify the information loss attributable to the “collapse” artifact itself.

3.5 Kalman Filter Design and Linearity Considerations

To reconstruct the continuous trajectory from the TWPA output, we implemented a linear Kalman filter. The state vector was defined as the In-Phase and Quadrature components of the signal: $\mathbf{x} = [I, Q]^T$. The system dynamic model assumed a random walk (identity transition matrix) with process noise covariance $\mathbf{Q}$.

It is important to note that while the phase $\phi$ is periodic on $[0, 2\pi)$, for the small incremental phase drifts simulated ($d\phi \approx 0.1$ rad/step), the linear approximation holds locally. In a practical deployment involving large rotations or Rabi oscillations, an Extended Kalman Filter (EKF) would be required to handle the coordinate transformation from the $IQ$-plane to angular coordinates and manage phase wrapping. Our simulation assumes the dispersive shift keeps the state within a local region of the phase space, valid for weak measurement tracking (Marceaux & Young, 2023).

4.0 Results I: Quantization Error and Information Loss

4.1 Baseline Dimensionality Analysis

The simulation produced a continuous complex phasor $S_{true}(t)$ representing the state of a qubit undergoing diffusive trajectory evolution. As illustrated by the trajectory data, the signal explores a continuous manifold within the $IQ$-plane, characterized by time-varying amplitude and phase coordinates. This “true” signal possesses infinite bit depth in the analog domain, limited only by the Heisenberg uncertainty principle. The trajectory visualization reveals a rich “texture” of phase accumulation, where the qubit state drifts stochastically due to back-action. This continuous path constitutes the “verb” of the quantum evolution—the process of becoming—rather than the static “noun” of the final eigenstate (Macklin et al., 2015).

4.2 Quantization Noise in Latching Systems

The application of the latching readout model resulted in a catastrophic reduction of signal fidelity. The continuous input trajectory was mapped to a binary output stream $Y_{latch}(t) \in \{-1, 1\}$. Quantitatively, this loss manifested as a Mean Squared Error (MSE) of 1.089 relative to the true signal. Given that the signal amplitude was normalized to $A=1.0$, an MSE exceeding unity indicates that the readout error is larger than the signal itself. This result confirms that the latching readout acts as a dominant noise source, introducing quantization noise that completely obscures the subtle diffusive dynamics of the qubit. The output resembles a random telegraph signal rather than the smooth diffusion of the wavefunction (Castellanos-Beltran et al., 2025).

4.3 SNR Comparison and Control Case Analysis

The TWPA provided a ~20 dB improvement in SNR over the HEMT model. The raw MSE of the TWPA output was 0.042, a factor of 25 improvement over the latching readout. However, the most critical finding arises from the control simulation.

When we applied the latching readout to the high-quality TWPA signal (High-SNR Latching), the MSE increased from 0.042 to 0.254. This represents a six-fold increase in error solely due to the act of quantization. This result explicitly disproves the notion that latching readouts are “good enough” if the amplifier is good. Even with quantum-limited amplification, the hard thresholding of the latching junction destroys 83% of the recoverable information. This confirms our thesis: the quantization step is a fundamental bottleneck, independent of the thermal noise environment (Wang et al., 2025).

4.4 Information Loss Heatmaps

To visualize the interplay between noise and quantization, we generated Information Loss Heatmaps sweeping across SNR and Threshold parameters. The heatmap reveals a distinct “Dead Zone” at SNRs below 1.0, where the information loss exceeds 0.8 regardless of the threshold setting. This confirms that in the HEMT-limited regime (low SNR), no amount of optimization of the latching threshold can recover the lost information. A “Valley of Fidelity” emerges only in the high-SNR regime enabled by TWPAs, but as shown by the control case, this valley is only accessible if the readout remains continuous (Wang et al., 2025).

5.0 Results II: Reconstructing the Verb

5.1 Kalman Filter Convergence

To extract the clean quantum trajectory from the noisy TWPA record, we applied a linear Kalman filter. The filter operates by recursively updating its state estimate based on the incoming measurement data and a model of the system dynamics. The application of the Kalman filter reduced the MSE from the raw value of 0.042 to 0.015, a nearly threefold improvement in estimation fidelity. This reduction demonstrates the power of using a dynamical model to distinguish between measurement noise (which is uncorrelated in time) and process noise (the qubit’s diffusive drift, which is correlated). The filter effectively “learns” the trajectory, smoothing out the vacuum fluctuations while tracking the genuine evolution of the quantum state (Hacohen-Gourgy & Martin, 2020).

5.2 Recovering Phase Information

The most critical metric for “verb-based” processing is the recovery of phase information, as the relative phase encodes the coherence of the superposition. The latching readout, by projecting onto the real axis, destroys this phase information entirely. In contrast, the Kalman-filtered TWPA output maintained a tight lock on the qubit’s phase evolution. Our analysis yielded a mean phase tracking error of 0.118 radians (approximately 6.8 degrees). This level of precision implies that the observer maintains a high degree of knowledge regarding the “latitude” of the state vector on the Bloch sphere, sufficient for feedback control protocols (Mastriani, 2018).

5.3 Handling Back-Action in Estimation

A unique feature of quantum estimation is the role of measurement back-action. In the signal processing isomorphism, back-action manifests as “process noise” that is correlated with the measurement outcome. In our simulation, the filter’s ability to track the trajectory relied on the correct calibration of the Kalman gain, which balances the trust between the system model and the measurement record. Because the measurement record contains information about the back-action kicks, the filter uses the noisy measurement not just to estimate the current state, but to infer how the state has been perturbed. This closes the information loop, converting the “destructive” back-action into a known control input (Hacohen-Gourgy & Martin, 2020).

6.0 Discussion: Engineering the Hardware of the Verb

6.1 Reframing the Readout Stack

The results of this study necessitate a fundamental reframing of the quantum readout stack. The traditional architecture—comprising a HEMT, a room-temperature amplifier, and a thresholding digitizer—is designed for “noun” extraction. To enable the continuous processing demonstrated in Section 5, the stack must be redesigned as a “verb” processor. This new architecture prioritizes flow preservation over state determination. It requires a quantum-limited pre-amplifier (TWPA) to establish a high SNR immediately at the quantum-classical boundary. Following this, the signal chain must remain linear and continuous up to a high-resolution ADC. The “readout” is no longer a discrete event at the end of a pulse but a continuous stream of $IQ$ data (Castellanos-Beltran et al., 2025).

6.2 Scalability and Data Rate Analysis

A critical engineering challenge is scaling this continuous readout to multi-qubit processors. While latching readouts are lossy, they are bandwidth-efficient. Continuous monitoring requires significantly more data throughput. We analyzed the data rate for a 100-qubit system. Assuming each qubit is sampled at 100 MHz (Nyquist for a 50 MHz cavity linewidth) with 14-bit resolution for both I and Q channels, the aggregate data rate is:

While substantial, 280 Gbps is within the I/O capabilities of modern high-performance FPGAs (e.g., Xilinx Versal Premium series), which offer Terabit-per-second transceiver bandwidths. The bottleneck shifts from the physical readout line to the real-time processing logic, necessitating optimized, parallelized Kalman filter kernels on the FPGA fabric (White et al., 2015).

6.3 Thermodynamic Implications

The shift to continuous processing also has thermodynamic implications. Landauer’s principle states that information erasure is the source of energy dissipation. A latching readout, by collapsing the state and erasing the superposition information, incurs a thermodynamic cost proportional to the information lost. In contrast, a reversible parametric amplifier and a continuous estimation process (which can be viewed as a unitary transformation on the joint system of qubit and controller) theoretically approach thermodynamic reversibility. By maintaining the information flow and avoiding the “hard” erasure of the collapse, the “verb” processor is potentially more energy-efficient per bit of extracted information (Castellanos-Beltran et al., 2025).

6.4 Comparison with Ensemble Averaging

It is acknowledged that latching readouts are robust and are often used in ensemble averaging to recover expectation values. By averaging thousands of “single-shot” latched outcomes, one can reconstruct the diagonal elements of the density matrix. However, this approach inherently fails for real-time feedback on a single instance of the system. Ensemble averaging reconstructs the average verb, not the current verb. The continuous readout architecture enables single-shot trajectory tracking, a capability that ensemble averaging cannot provide. This distinction is critical for quantum error correction, which requires correcting specific errors on specific qubits in real-time, not correcting the average error of the ensemble.

7.0 Conclusion

This study rigorously validates the structural isomorphism between quantum measurement dynamics and classical signal quantization. By controlling for noise levels, we demonstrated that the discrete latching mechanism itself introduces a six-fold increase in error compared to continuous monitoring (MSE 0.254 vs 0.042). This confirms that the “collapse” observed in standard readouts is largely a technological artifact of using 1-bit comparators. We recommend the adoption of “verb-based” readout architectures—comprising TWPAs, high-resolution ADCs, and FPGA-based Kalman filters—to enable the next generation of quantum control. This architecture transforms the qubit from a fragile two-state switch into a robust continuous information source, potentially enabling Heisenberg-limited metrology.

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Appendix (Supplementary Materials)

A. Formal Derivations (Stochastic Master Equation)

The evolution of the density matrix $\rho$ under continuous measurement is described by the Stochastic Master Equation (SME):

The term $\sqrt{\eta} \mathcal{H}[L]\rho dW$ represents the stochastic back-action update, where $dW$ is the innovation derived from the measurement record. This maps directly to the Kalman update step $\hat{x}_{k|k} = \hat{x}_{k|k-1} + K_k (z_k - H \hat{x}_{k|k-1})$.

B. Summary of Simulation Data