Bit-Phase Representations in Digital Systems

• Bit-phase representations are methods that encode digital data by mapping binary states to discrete phase values, leveraging oscillator stability and noise resilience.
• They are implemented across diverse platforms such as quantum RNG, analog oscillators, and neural network quantization, ensuring robust and high-speed data processing.
• Advanced algorithms utilize bit-phase encoding for precise phase retrieval and binary estimation, enabling efficient signal reconstruction and parameter measurement.

A bit-phase representation encodes digital information not in amplitude or voltage level but in the discrete phase relationships—or phase states—of oscillatory, wave-like, or complex-valued signals. This paradigm spans physical, algorithmic, and information-theoretic domains, from oscillator-based digital logic, quantum random number generation, and binary phase estimation, to 1-bit phase retrieval and phase-aware quantization in deep learning. Central to these systems is the mapping of binary or quantized bits to phase values (often 0 and π or their complex analogues), thereby exploiting phase’s robustness, geometry, and noise-immunity in both analog and digital settings.

1. Fundamental Physical Realizations of Bit-Phase Encoding

In self-sustained oscillatory systems, bit-phase representation is achieved by injection-locking oscillators into distinct, bistable phase relationships. In the canonical mechanical demonstration, two 1 Hz metronomes, each modeled via phase-reduction as weakly nonlinear oscillators, are subharmonically injection-locked by a shared 2 Hz mechanical drive. Averaging theory yields the Adler equation for phase difference $\phi$,

$\frac{d\phi}{dt} = \Delta\omega - K \sin(2\phi),$

which admits exactly two stable fixed points at $\phi = 0$ and $\phi = \pi$ for nearly zero detuning, corresponding to logic “0” and logic “1,” respectively. These phase states are robust against perturbation: upon disturbance, the system naturally latches back into one of its bistable states, demonstrating memory-like restoration. The relative phase—measured, for example, from Lissajous plots of metronome motion—gives a direct, physically encoded bit (Wang, 2017).

This mechanism is universal to electronic oscillators (ring, LC), MEMS, photonic, spintronic, and even synthetic biological oscillators. Bit-phase encoding leverages the topology of phase space and threshold physics to create latching and noise-immune logic, in contrast to conventional amplitude latches (Wang, 2017).

2. Binary Phase States in Quantum and Optical Systems

In degenerate optical parametric oscillators (OPOs), the nonlinearity and phase-sensitive gain allow only two possible steady-state solutions for the generated signal: $0$ or $\pi$ difference relative to the pump. Writing the OPO signal as $A = Re^{i\theta_s}$, maximal gain occurs only at $\theta_s = 0$ or $\pi$. Each oscillation cycle thus randomly locks into one of these two phases due to quantum noise, providing a fundamentally random bit generator. Two such OPOs, excited in sequence and interfered via a Michelson interferometer, create a high-contrast, directly readable random bit stream—no post-processing is required, and statistical randomness is confirmed by NIST tests and entropy metrics (Marandi et al., 2012).

Shrinking OPOs to the microscale enables on-chip, ultrafast quantum random number generators, as the cavity decay time and threshold pump power scale favorably with $(V,Q)$, increasing bit rates to GHz and fitting the bit-phase principle into CMOS-compatible platforms (Marandi et al., 2012).

3. Bit-Phase Algorithms in Signal Processing and Communications

Bit-phase representations are also core to algorithms that operate on quantized, phase-derived measurements. In 1-bit phase retrieval, only binary comparisons between intensity measurements (or quantized phase values) are available: $y_i = \operatorname{sign}(|a_i^T x| - \tau),$ where $y_i \in \{-1,1\}$, and $x$ is the unknown vector. Here, the bit representation is the sign of whether the intensity passes a threshold. Fundamental limits show exact scaling laws for recovery error: for $n$-dimensional signals, the information-theoretic error is $\Theta((n/m)\log(m/n))$, and for $k$-sparse signals, $\Theta((k/m)\log(mn/k^2))$, with efficient algorithms (spectral initialization plus gradient descent) achieving these bounds (Chen et al., 8 May 2024). The “bit-phase” here is manifest not in physical phase but in binary quantized phase information, forming the backbone of robust, low-bitrate measurement and recovery systems.

Similarly, in continuous-phase modulation under 1-bit quantization with oversampling, bit-phase mapping is tailored such that two bits per symbol are reliably detected in high SNR, while coding is focused on the third, error-prone bit—a design that maximizes spectral and quantization efficiency in communication receivers (Alencar et al., 2019).

4. Binary Phase Estimation and Dyadic Representation Protocols

Bit-phase encoding extends to parameter estimation, notably in quantum interferometry. An unknown phase $\varphi\in[0,2\pi)$ can be written in binary expansion,

$$\varphi = 2\pi \sum_{k=1}^{\infty} b_k 2^{-k},\quad b_k \in \{0,1\},$$

with the task of extracting the first $n$ bits to a desired precision. Architectures employing sequential or parallel interferometers, where each photon undergoes a controlled number of unknown phase shifts, are engineered (via square-wave response interferometry) so that each photon output reveals a single bit, up to the $n$th, saturating the Heisenberg limit in phase sensitivity, $\Delta\varphi\sim1/N$, with no statistical averaging (Roncallo et al., 15 Jul 2024). Each extracted bit is a “bit-phase,” corresponding directly to a slice of the dyadic expansion, enabling exponentially precise parameter estimates.

5. Bit-Phase Structure in Classical and Quantum State Spaces

In state-space approaches, especially in analogues of qubits, the bit-phase representation acquires geometric and topological significance. The classical two-level elastic “bit,” constructed from two coupled oscillators, gives rise to a Bloch sphere description: $$|\psi(\theta, \varphi)\rangle = \cos(\theta/2)|E_1\rangle + e^{i\varphi}\sin(\theta/2)|E_2\rangle,$$ where $\theta$ encodes amplitude (population) and $\varphi$ the relative phase bit. Logic operations correspond to rotations on the Bloch sphere, and adiabatic cycles yield a quantized Berry phase, $\gamma_B = \pi(1 - \cos\theta)$. This geometric phase (or “bit-phase”) is directly observed and manipulated in classical systems, bridging the classical–quantum correspondence and underlining the universality of phase-bit encoding across platforms (Mahmood et al., 10 Jul 2024).

6. Phase-Aware, Low-Bit Representations in Deep Learning

Bit-phase representation is an emerging theme in neural network quantization. In the FAIRY2I framework, real-valued linear layers are exactly recast as widely-linear complex maps, and each complex weight is quantized to one of four fourth roots of unity ($\pm1$, $\pm i$)—a 2-bit phase codebook. This “phase-aware” quantization, with additional recursive residual codes, yields $1$–$2$ bits per real parameter and enables efficient multiplication-free inference (direct add/subtracts with sign-swaps). Empirically, this approach enables restoration of large models such as LLaMA-2 7B to near-full-precision performance at $2$ bits, outperforming traditional binary QAT/PTQ methods (Wang et al., 2 Dec 2025). The bit-phase nature here refers to both the algebraic representation and the codebook structure.

Platform / Model	Bit-Phase Representation	Characteristic States
Mechanical metronome, OPO, etc.	Physical phase of oscillator	$\phi = 0$ (bit 0), $\pi$ (bit 1)
Quantum RNG (OPO)	Phase difference in pulses	$\Delta\phi = 0, \pi$
Phase retrieval/CPM/Comms	Quantized sign or bit comparisons	$y_i\in\{0,1\}$
Neural quantization	Codebook of $e^{i k\pi/2}$	$k=0,1,2,3$ (2-bit phase)

7. Significance, Robustness, and Generality

Bit-phase representations harness the inherent bistability or quantization of phase relationships for encoding, processing, and communicating digital information. Notable properties include:

• Noise immunity: Phase-encoded bits require finite energy/perturbation (over a threshold $K$) to flip, conferring superior resilience over level-based latches in both physical and algorithmic domains (Wang, 2017).
• Universality: The phase bit paradigm is realized in diverse physical systems (mechanical, electronic, photonic, spintronic, chemical, biological), as well as in abstracted computational frameworks (phase retrieval, quantization) (Wang, 2017).
• Scalability: Phase-based schemes scale efficiently in hardware (microscale OPOs, on-chip interferometers) and have parallel in software/hardware co-design (neural network phase quantization) (Marandi et al., 2012, Wang et al., 2 Dec 2025).
• Geometric/topological attributes: Phase bit representations naturally encode logical and computational gates as geometric operations, supporting a topological view of computation (e.g., Berry phase quantization in elastic bits) (Mahmood et al., 10 Jul 2024).

A plausible implication is that future computation, communications, and random number generation architectures may further converge toward exploiting discrete phase logic, leveraging both its physical and informational advantages.