Homodyne Gradient Extraction in Physical Systems

• Homodyne Gradient Extraction is a method that uses small sinusoidal perturbations to directly measure local gradients without an analytical model.
• It leverages frequency multiplexing and lock-in demodulation to isolate gradient components, ensuring scalability and robustness to noise.
• The technique is implemented in hardware, enabling real-time in situ gradient descent across diverse physical platforms.

Homodyne gradient extraction (HGE) is a methodology for direct in situ estimation of gradients in physical systems, leveraging the lock-in (homodyne) detection principle. Unlike digital, model-based optimization, HGE enables gradient-based learning in “black-box” physical substrates—such as dopant-network processing units—without requiring an analytical model of the input–output transformation. By simultaneously applying small, frequency-multiplexed perturbations to hardware-controlled parameters, HGE demodulates the physical system’s response to acquire the local Jacobian needed for gradient descent. This approach generalizes across a range of material-based computing platforms and is scalable, robust to noise, and compatible with analog or co-located electronic implementations (Boon et al., 2021).

1. Theoretical Foundation and Detection Principle

HGE is fundamentally based on the superposition of small sinusoidal parameter perturbations, each at a distinct frequency, onto controllable device parameters:

$$\theta_i(t) = \theta_i^0 + A_i \sin(\omega_i t + \phi_i)$$

for $i=1,\ldots,p$ control variables. The device output,

$$y(t) \approx y_0 + \sum_{i=1}^p g_i A_i \sin(\omega_i t + \phi_i)$$

with $$g_i \equiv (\partial y/\partial \theta_i)_{\theta^0}$$, encodes the partial derivatives in the amplitude of the corresponding frequency components. Because sinusoids at distinct frequencies are orthogonal over a suitable time window, each $g_i$ can be isolated via synchronous (homodyne) demodulation. The method requires no analytic description of $y(\theta)$; it extracts the Jacobian $\partial y/\partial \theta$ directly from measurements.

2. Mathematical Formalism and Gradient Estimation

The core of HGE is the lock-in detection process. By multiplying the measured response by $\sin(\omega_j t+\phi_j)$ (and its quadrature), then integrating over an interval $T \gg 1/\min|\omega_j-\omega_i|$, the $j$th gradient component is obtained:

$i=1,\ldots,p$0

$i=1,\ldots,p$1

The amplitude $i=1,\ldots,p$2 gives $i=1,\ldots,p$3. The sign of $i=1,\ldots,p$4 is determined from the measured phase, leading to

$i=1,\ldots,p$5

After estimating the vector $i=1,\ldots,p$6, a standard gradient descent update is performed:

$i=1,\ldots,p$7

where $i=1,\ldots,p$8 is analytically available from the specified loss function (Boon et al., 2021).

3. Hardware Realization and Workflow

HGE can be realized entirely in hardware. In the context of a dopant-network processing unit (DNPU), the workflow is as follows:

• The system is cooled in a cryostat and interfaced via a PCB carrying the DNPU.
• Multi-channel DACs apply DC voltages $i=1,\ldots,p$9 plus small sinewave perturbations $$y(t) \approx y_0 + \sum_{i=1}^p g_i A_i \sin(\omega_i t + \phi_i)$$0 to the $$y(t) \approx y_0 + \sum_{i=1}^p g_i A_i \sin(\omega_i t + \phi_i)$$1 terminals.
• The device output current $$y(t) \approx y_0 + \sum_{i=1}^p g_i A_i \sin(\omega_i t + \phi_i)$$2 is converted to a voltage and digitized via a high-sample-rate ADC ($$y(t) \approx y_0 + \sum_{i=1}^p g_i A_i \sin(\omega_i t + \phi_i)$$3).
• For each parameter, lock-in demodulation is performed digitally or in analog hardware to extract $$y(t) \approx y_0 + \sum_{i=1}^p g_i A_i \sin(\omega_i t + \phi_i)$$4.
• Gradients $$y(t) \approx y_0 + \sum_{i=1}^p g_i A_i \sin(\omega_i t + \phi_i)$$5 are reconstructed, and in situ gradient descent is iterated.

This procedure has been demonstrated extracting up to seven gradients in parallel for Si:As and Si:B multi-terminal dopant networks (Boon et al., 2021).

4. Performance, Accuracy, and Scalability

HGE’s estimation error decays inversely with integration time $$y(t) \approx y_0 + \sum_{i=1}^p g_i A_i \sin(\omega_i t + \phi_i)$$6; ten cycles typically yield $$y(t) \approx y_0 + \sum_{i=1}^p g_i A_i \sin(\omega_i t + \phi_i)$$7 error. The perturbation amplitudes $$y(t) \approx y_0 + \sum_{i=1}^p g_i A_i \sin(\omega_i t + \phi_i)$$8 must be optimized: values too small degrade SNR, while excessive amplitude exits the linear regime, introducing distortion. Orthogonality of drive frequencies is critical to suppress crosstalk. Empirical validation with “single-channel” and “multi-channel” perturbations shows agreement within measurement error, confirming frequency-orthogonal isolation of gradients. Scalability is achieved since each channel (frequency) is processed independently; for $$y(t) \approx y_0 + \sum_{i=1}^p g_i A_i \sin(\omega_i t + \phi_i)$$9 parameters, $$g_i \equiv (\partial y/\partial \theta_i)_{\theta^0}$$0 oscillators and demodulators are required and the per-iteration time remains $$g_i \equiv (\partial y/\partial \theta_i)_{\theta^0}$$1, conditioned on fixed hardware resources.

5. Generalization Across Physical Platforms

HGE does not require an analytic model and is thus applicable to any system where (1) parameter perturbations can be superimposed with distinct frequencies, (2) the output can be measured with sufficient bandwidth, and (3) lock-in demodulation is feasible. Candidate systems include quantum-dot arrays, integrated photonic circuits, spintronic or memristive networks, and metasurfaces with tunable analog elements. All lock-in and gradient-extraction steps may be implemented with on-chip analog electronics, enabling “learning in materio” modules with no external computation (Boon et al., 2021).

6. Comparison with Quantum Homodyne Gradient Extraction

In quantum parameter estimation, similar homodyne data acquisition principles are adopted, for example in continuous monitoring of quantum trajectories. There, the stochastic master equation (SME) governs the evolution of the conditional state $$g_i \equiv (\partial y/\partial \theta_i)_{\theta^0}$$2 under continuous homodyne detection, and the log-likelihood score for a parameter $$g_i \equiv (\partial y/\partial \theta_i)_{\theta^0}$$3 is built from the innovation process. The Fisher information quantifies the sensitivity of the continuous record to $$g_i \equiv (\partial y/\partial \theta_i)_{\theta^0}$$4, and its gradient can be accumulated along a quantum trajectory via the SME and a derivative filter $$g_i \equiv (\partial y/\partial \theta_i)_{\theta^0}$$5:

$$g_i \equiv (\partial y/\partial \theta_i)_{\theta^0}$$6

with $$g_i \equiv (\partial y/\partial \theta_i)_{\theta^0}$$7 the measured quadrature mean (Kiilerich et al., 2016). While conceptually related, this quantum formalism emphasizes statistical inference and estimation theory, and does not directly yield parameter updates for learning in physical hardware, but rather underpins measurement-based parameter estimation.

7. Applications and Prospects

Demonstrations of HGE in dopant networks have achieved parallel extraction of multiple gradients, real-time optimization, and hardware instantiation of Boolean logic via physical learning (Boon et al., 2021). All signal-injection, detection, and update steps can be implemented in analog circuits and, in principle, co-integrated with material-computing substrates. HGE thus offers a physically grounded, scalable route to self-contained “learning in materio” devices—where parameter inputs, in situ gradient computation, and memory reside entirely within a compact, non-von Neumann system.

Aspect	Lock-In (HGE)	Quantum Homodyne Score
Physical Basis	Frequency demodulation	Stochastic master equation
Application Domain	Black-box material ML	Quantum parameter estimation
Update Mechanism	Hardware parameter step	Bayesian, gradient-of-log-likelihood

HGE’s main significance lies in enabling gradient-based optimization and learning directly in physical materials, bypassing the energy inefficiency and model dependence of digital backpropagation, and generalizing across electronic, photonic, and quantum-compatible architectures.