Continuous Signal Processing for Josephson Junction Readout
Continuous
Signal Processing for Josephson Junction Readout
Addressing
the Discrete Collapse Tension via Parametric Amplification
Author: Rowan Brad Quni-Gudzinas
Contact: rowan.quni@outlook.com ORCID:
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:
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.
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.
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 \(\{Em\}\) such that \(\sum Em = I\), 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**
Quantum Construct |
Signal Processing Isomorphism |
Physical Mechanism |
Wavefunction (\(\Psi\)) |
Complex Carrier Signal
(I/Q) |
Microwave photon amplitude/phase |
**Projective
Measurement** |
**1-Bit ADC /
Quantization** |
Latching Comparator (Hysteresis) |
**Hamiltonian
Evolution** |
**System Transfer
Function** |
Unitary rotation / Filter dynamics |
Back-Action |
**Correlated Feedback
Noise** |
Measurement imposing state change |
Weak Measurement |
**Noisy Continuous
Monitoring** |
Low-SNR sampling without latching |
Quantum Trajectory |
**Stochastic
Time-Series** |
Path of state vector under noise |
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:
\[ \text{Rate} = 100 \text{ qubits} \times
100 \text{ MHz} \times 2 \text{ channels} \times 14 \text{ bits} \approx
280 \text{ Gbps} \]
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.
References
Averin, D. V. (2000). Quantum computing and quantum
measurement with mesoscopic Josephson junctions. *Fortschritte der
Physik, 48*(9–11), 1055–1074.
https://doi.org/10.1002/1521-3978(200009)48:9/11<1055::AID-PROP1055>3.0.CO;2-1
**Castellanos-Beltran, M. A., Howe, L., Giachero, A., Vissers,
M., Labranca, D., Ullom, J., & Hopkins, P.** (2025).
Measurable improvement in multi-qubit readout using a kinetic inductance
traveling wave parametric amplifier. *IEEE Transactions on Applied
Superconductivity, 35*(5), 1–5.
https://doi.org/10.1109/TASC.2024.3525451
**Guarcello, C., Ahrens, F., Avallone, G., Barone, C.,
Borghesi, M., Callegaro, L., Carapella, G., Di Gioacchino, D., Falferi,
P., Fasolo, L., Faverzani, M., Ferri, E., Filatrella, G., Gatti, C.,
Giachero, A., Giubertoni, D., Granata, V., Greco, A., Ligi, C., …
Zannoni, M.** (2024). Nonlinear behavior of Josephson traveling
wave parametric amplifiers. *IEEE Transactions on Applied
Superconductivity, 34*(3), 1–5.
https://doi.org/10.1109/TASC.2024.3367615
Hacohen-Gourgy, S., & Martin, L. S. (2020).
Continuous measurements for control of superconducting quantum circuits.
Advances in Physics: X, 5(1), 1813626.
https://doi.org/10.1080/23746149.2020.1813626
**Macklin, C., O’Brien, K., Hover, D., Schwartz, M. E.,
Bolkhovsky, V., Zhang, X., Oliver, W. D., & Siddiqi, I.**
(2015). A near-quantum-limited Josephson traveling-wave parametric
amplifier. Science, 350(6258), 307–310.
https://doi.org/10.1126/science.aaa8525
Marceaux, J. P., & Young, K. (2023). Streaming
quantum gate set tomography using the extended Kalman filter. *2023
IEEE International Conference on Quantum Computing and Engineering
(QCE)*, 1401–1411. https://doi.org/10.1109/QCE57702.2023.00159
Mastriani, M. (2018). Optimal estimate of quantum
states. Journal of Applied Mathematics and Physics,
06(11), 2177–2196. https://doi.org/10.4236/jamp.2018.66114
**Murch, K. W., Weber, S. J., Macklin, C., & Siddiqi,
I.** (2013). Observing single quantum trajectories of a
superconducting quantum bit. Nature, 502(7470),
211–214. https://doi.org/10.1038/nature12539
**Wang, J., Peng, K., Knecht, J. M., Cunningham, G. D., Lombo,
A. E., Yen, A., Zaidenberg, D. A., Gingras, M., Niedzielski, B. M.,
Stickler, H., Sliwa, K., Serniak, K., Schwartz, M. E., Oliver, W. D.,
& O’Brien, K. P.** (2025). High-efficiency, low-loss
Floquet-mode traveling wave parametric amplifier. arXiv.
https://doi.org/10.48550/arXiv.2503.11812
**White, T. C., Mutus, J. Y., Hoi, I.-C., Barends, R.,
Campbell, B., Chen, Y., Chen, Z., Chiaro, B., Dunsworth, A., Jeffrey,
E., Kelly, J., Megrant, A., Neill, C., O’Malley, P. J. J., Roushan, P.,
Sank, D., Vainsencher, A., Wenner, J., Chaudhuri, S., … Martinis, J.
M.** (2015). Traveling wave parametric amplifier with Josephson
junctions using minimal resonator phase matching. *Applied Physics
Letters, 106*(24), 242601.
https://doi.org/10.1063/1.4922348
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):
\[ d\rho = -i[H, \rho]dt +
\mathcal{D}[L]\rho dt + \sqrt{\eta} \mathcal{H}[L]\rho dW \]
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 \(\hat{x}{k|k} = \hat{x}{k|k-1} + Kk (zk - H
\hat{x}_{k|k-1})\).
B. Summary of Simulation
Data
Metric |
Latching Readout (HEMT) |
TWPA (Raw Continuous) |
TWPA + Kalman |
High-SNR Latching (Control) |
Noise Std (\(\sigma\)) |
2.0 |
0.2 |
0.2 |
0.2 |
Readout Type |
Discrete |
Continuous |
Filtered |
Discrete |
MSE |
1.089 |
0.042 |
0.015 |
0.254 |