Random Pulse Sequences: Theory & Applications

Updated 7 January 2026

Random Pulse Sequences are defined as temporal pulses with random or pseudorandom timing, enabling robust quantum control, communication, and computation.
They are analyzed via spectral and statistical reconstruction techniques that facilitate unique signal encoding, noise suppression, and effective synchronization.
Applications include quantum sensing, NMR selective excitation, and random pulse computing, offering enhanced fidelity, throughput, and noise immunity.

Random Pulse Sequences (RPS) are sequences of temporal events, typically voltage or field pulses, distributed according to specified random or pseudorandom statistics rather than strictly periodic timing. RPS arise in a variety of physical, engineering, and computational settings, including quantum control and sensing, communications, statistical reconstruction tasks, and biologically-inspired computing. Unlike deterministic pulse sequences, RPS can be designed to exploit their statistical properties—such as spectral coverage, robustness to noise, or information encoding—to achieve specific objectives unattainable by conventional methods.

1. Mathematical Structures and Statistical Properties

The fundamental mathematical object underlying RPS is the random sequence of event times or amplitudes, whose statistics can be adjusted from purely random (Poissonian, Bernoulli, or exponential waiting times) to structured pseudorandomness with imposed correlations.

For instance, in random pulse trains for encoding digital variables, arrival times are often modeled as independent random variables, exponentially distributed with rate parameter $\lambda$ in the continuous (Poisson) case: $f_{\Delta t}(t) = \lambda e^{-\lambda t},\quad t \geq 0.$ In clocked or “time-digitized” settings, one has equally spaced bins of duration $\Delta t$ and pulses appear with independent probability $p$ in each bin, resulting in a binomial process where the mean pulse frequency is $f = p / \Delta t$ (Stipčević et al., 2019).

For more elaborate RPS in quantum control, the sign sequence $\{U_i\}$ that determines the $\pm 1$ filter-function value within each time window of duration $\tau$ can be engineered to exhibit not just independence but prescribed stationary autocorrelations up to some lag $\lambda$ , specified by $\mathbb{E}[U_i U_{i+k}] = R(k)$ for $1 \leq k \leq \lambda$ (Huang et al., 2023).

2. Spectral and Information-Theoretic Analysis

The statistical properties of RPS manifest directly in their power spectral density (PSD) and capacity for unique signal encoding or processing.

For binary random pulse trains with i.i.d. amplitudes, the full PSD—which generally splits into continuous (broadband) and discrete (line) spectrum components—can be computed via renewal process theory. The non-periodic part is given by

$S_c(\omega) = \frac{2}{\omega^2 T_0} \Re\left\{ \frac{[1-\Phi_1(\omega)][1-\Phi_2(\omega)]}{1-\Phi_1(\omega)\Phi_2(\omega)} \right\},$

where $\Phi_1(\omega)$ and $\Phi_2(\omega)$ are the characteristic functions of the “on” durations and inter-pulse intervals, respectively. Discrete spectral lines at multiples of the symbol rate appear only with certain forms of temporal structure or symbol duration unbalance; their weights are analytically derived as

$W_k = \left(\frac{p(1-p)}{k\pi}\right)^2 2[1 - \cos(2\pi k \Delta/T_0)].$

These lines are critical for functions such as symbol-timing recovery, radar sensing, and synchronization (Stepanov et al., 2015).

In information-theoretic terms, the randomization in pulse sequences can grant unique identifiability of underlying pulse shapes from randomly shifted samples, as the distribution of sample vectors along the trajectory in $\mathbb{R}^d$ uniquely determines the generating waveform under mild conditions (Rupniewski, 2023).

3. Algorithms and Statistical Reconstruction

Recovery of signals or parameters from RPS relies on inversion of statistical mappings. For reconstructing a finite-length pulse $s(t)$ from short, randomly positioned sample sequences, the observations can be regarded as points lying along a one-dimensional curve $C \subset \mathbb{R}^d$ . The induced empirical density $p(x)$ along $C$ determines $s(t)$ , with reconstruction proceeding by:

Curve estimation via nearest-neighbor “curve-crust” or marching algorithms to order samples.
Quantile-based density estimation along arclength.
Pulse shape reconstruction using explicit inversion formulas, typically based on the correspondence between the sample quantiles and original time shifts (Rupniewski, 2023).

Sample complexity for density-based reconstruction follows $O(N^{-1/2})$ . Practical robustness can be achieved through smoothing or local principal component analysis, while computational costs are largely dictated by curve ordering and interpolation.

4. Randomness-Enhanced Quantum Sensing and Control

RPS play an essential role in the design of advanced quantum control protocols including noise spectroscopy and dynamical decoupling. Randomized pulse sequences can probe wide frequency intervals efficiently, or suppress systematic errors more effectively than their deterministic counterparts.

In qubit noise spectroscopy, RPS are tailored so that the average filter function $\mathbb{E}[W(\omega)]$ matches arbitrary target shapes $T(\omega)$ . This is achieved by solving a one-dimensional FIR phase-retrieval problem to impose the desired autocorrelation structure $R(k)$ in the sign sequence $\{U_i\}$ . Measurements under these filter functions yield direct linear functionals of the noise spectral density, enabling compressed sensing algorithms to recover sparse spectra with $O(s \log N)$ measurements—an order of magnitude fewer than required in conventional protocols (Huang et al., 2023).

In dynamical decoupling and quantum sensing, randomization—either by blockwise global phase randomization or by correlating random phases—suppresses the coherent accumulation of pulse imperfections and spurious spectral responses. For instance, fully random global phase shifts reduce spurious harmonic peaks and error accumulation as $1/\sqrt{M}$ with $M$ blocks, whereas correlated randomization (enforcing zero-sum phason blocks of size $G$ ) eliminates certain error terms entirely, with practical gains of %%%%27 $\Phi_1(\omega)$ 28%%%% in fidelity and suppression of unsolicited responses to below $10^{-3}$ of the main signal under realistic noise (Wang et al., 2020, Wang et al., 2019).

5. Deterministic vs. Pseudorandom Sequence Design in Selective Excitation

RPS with controlled pseudorandomness underpin high selectivity protocols such as p-DANTE in nuclear magnetic resonance (NMR). By imposing a temperature-matched phase ramp across pseudorandomly spaced pulses, excitation is confined to a prescribed resonance frequency, while off-resonant responses destructively interfere. Enhanced selectivity is achieved by averaging over $M$ independent pseudorandom realizations, suppressing random excitation artifacts at unwanted frequencies as $1/\sqrt{M}$ (Walls et al., 2010).

The mechanism centers on the average Hamiltonian: on-resonance, all phasors align for constructive excitation; off-resonance, randomization ensures that the effective drive averages out.

6. Random Pulse Sequences in Computation: Random Pulse Computing (RPC)

In the Random Pulse Computing paradigm, every variable is encoded as a frequency or probability within a random pulse train (RPT), either in continuous time (Poissonian) or in time-digitized (binomial) form (Stipčević et al., 2019). Arithmetic is implemented via combinational logic on RPTs:

AND gates yield multiplication: $p_z = p_0 p_1$ .
OR gates provide approximate addition.
Random MUX units and feedback counter architectures achieve exact scaled addition, division, and subtraction operations.
Comparisons and flow control are performed by combining subtractor circuits in latch configurations.

Quantum-random flip-flops (RFFs) provide the underlying stochasticity, outperforming classical pseudorandom number generators in statistical purity and eliminating long-period correlations.

This methodology offers high throughput, parallelism, and noise immunity. Feedback-based division and subtraction circuits achieve accuracy scaling as $O(2^{-N})$ for $N$ -bit counters. Notable use-cases of these architectures include stochastic filtering, vector inner-products, and probabilistic control in hardware-efficient, continuously-updating frameworks.

7. Applications, Limitations, and Practical Considerations

RPS are central to modern quantum sensing, spectrometry, communication, and unconventional computation:

In quantum devices, RPS-based protocols provide broadband noise characterization and robust signal discrimination, making them critical for advancing error-resilient solid-state quantum sensors.
In digital communications, closed-form spectral calculations enable optimal trade-offs for synchronization, spectral mask compliance, and channel estimation.
In computation, RPS and RPTs are foundational to efficient, error-tolerant, and massively parallel implementations of arithmetic and probabilistic algorithms.

However, leveraging RPS typically incurs added complexity in sequence generation (e.g., phase-retrieval algorithms), randomness requirements (true quantum sources for maximal statistical purity), and design optimization (balancing degree of correlation, block size, and hardware overhead). A plausible implication is that the trade-off between randomness-induced robustness and sequence design complexity remains an active area of research, particularly as RPS power new classes of quantum and neuromorphic technologies.

Table: Representative Random Pulse Sequence Paradigms

Application Domain	RPS Construction/Usage	Core References
Qubit Noise Spectroscopy	RPS with tailored autocorrelations, phase-retrieval, compressed sensing	(Huang et al., 2023)
Quantum Sensing (DD)	Global or correlated-random phase block modulations	(Wang et al., 2019, Wang et al., 2020)
Selective NMR Excitation	p-DANTE: pseudorandom delays + synchronized phase ramp	(Walls et al., 2010)
Pulse Shape Statistical Reconstruction	Random shifts yield sample curves in $\mathbb{R}^d$	(Rupniewski, 2023)
Random Pulse Computing (RPC)	Digital logic/gates on RPTs from quantum random flip-flops	(Stipčević et al., 2019)
Pulse Train PSD Analysis	Renewal-theoretic, explicit continuous/discrete spectrum	(Stepanov et al., 2015)