QSQS: Quantum-Inspired Randomness System

• QSQS is a quantum-inspired framework that unifies deterministic algorithmic processes with physical non-deterministic fluctuations, enabling an uncertainty-like relationship between conjugate observables.
• It employs modulo reduction to collapse skewed raw distributions into nearly uniform ones, thereby amplifying entropy and ensuring statistical robustness in output.
• The QSQS foundation underpins the QPP-RNG implementation, which integrates hybrid seeding and empirical certification to produce high-entropy random numbers suitable for post-quantum cryptography.

A Quasi-Superposition Quantum-inspired System (QSQS) is a framework that unifies deterministic algorithmic processes with physical, non-deterministic fluctuations by formally treating two observables—typically one algorithmic and one physical—as if they were conjugate variables in a quantum system. This analogy allows systems to exhibit an “uncertainty-like” relationship between these observables, resulting in quasi-superposed states whose macroscopic measurement outcomes manifest key features reminiscent of quantum measurement, such as the collapse and internal degeneracy of microstates into uniformly distributed outputs. QSQS provides both a conceptual foundation and a practical implementation for generating true randomness and high-entropy numbers utilizing system-level physical entropy in software, as realized in the QPP-RNG post-quantum random number generator (Kuang, 1 Aug 2025).

1. The QSQS Physical-Algorithmic Duality

QSQS is constructed atop the distinction between two types of observables:

• Algorithmic Deterministic Observable: For example, the permutation count $n_p$ in a sorting or shuffling process. This variable is fully determined by the execution of the algorithm and the initial state, mapping to a well-defined position in an algorithmic state space.
• Physical Non-Deterministic Observable: For example, the real-time sorting duration $t$, which is fundamentally influenced by system-level phenomena such as CPU pipeline jitter, cache miss latency, and OS scheduling. This observable is inherently unpredictable, varying due to microphysical and environmental effects even for repeated runs.

By treating these two observables as conjugate in the quantum sense, a QSQS enforces an “uncertainty-like” constraint: precise specification or measurement of one reduces the determinacy of the other, paralleling the position-momentum uncertainty in quantum mechanics. The operational state of the system prior to measurement is then a quasi-superposition of all possible internal configurations characterized by $(n_p, t)$ pairs.

2. Modulo Reduction and Internal Degeneracy

A distinctive mathematical operation in QSQS is the use of modulo reduction:

• $\widetilde{n} = n_p \mod 2^n$
• $\widetilde{t} = t \mod 2^n$

This mapping collapses many microstates (e.g., all permutation counts that are identical modulo $2^n$) into a single output symbol. The outcome is an internal degeneracy, with multiple internal processes indistinguishable at the output. The effect is analogous to the quantum measurement process, where a wavefunction collapses and only certain eigenvalues of the observable appear. In QSQS, this mechanism is leveraged to transform skewed or non-uniform raw distributions into flat, nearly uniform ones in the output space.

Table: Raw vs. Collapsed Output Distribution

Observable	Raw Distribution	Distribution after Modulo
Permutation Count	Right-skewed, heavy tail	Nearly uniform
Sorting Time	Right-skewed, heavy tail	Nearly uniform

The collapse process enhances entropy and uniformity even when initial distributions are far from ideal for cryptographic purposes.

3. QPP-RNG Implementation

QPP-RNG is a realization of QSQS in software, designed as a true random number generator (TRNG) suitable for post-quantum cryptography. The system’s operation involves:

1. Permutation Sorting: Arrays are scrambled using a Fisher–Yates shuffle, driven by a PRNG.
2. Measurement of Conjugate Observables: In each cycle, both $n_p$ (algorithmic count) and $t$ (physical time taken) are recorded.
3. Hybrid Seeding: Real-time measurements of $t$ (from qQRNG mode) dynamically reseed the PRNG controlling the permutation process, ensuring injection of fresh physical entropy into the sequence.
4. Modulo Reduction: Both observables are reduced modulo $2^n$, resulting in outputs with high entropy and uniformity.

Empirical results presented in the paper show that as the number of repetitions $m$ (i.e., number of sorting cycles used to produce a block of random bits) increases, statistical measures such as Shannon entropy and min-entropy converge to theoretical maxima (e.g., $\simeq 8$ bits for $n=8$ bit outputs), and chi-squared statistics approach values expected for ideal uniformity. This demonstrates entropy amplification and statistical robustness conferred by the core QSQS mechanisms.

4. Quantum Analogy and Uncertainty-Like Principle

Drawing from quantum mechanics, QSQS analogizes:

• Conjugate observables $(n_p, t)$ to quantum pairs (e.g., position and momentum).
• State collapse via modulo reduction to quantum measurement collapsing a wavefunction.
• Internal degeneracy to the degeneracy of quantum energy levels, where observationally indistinguishable macrostates are realized from multiple distinct microstates.
• Uncertainty-like constraint: The deterministic evolution (tracked by $n_p$) contrasts and intertwines with the statistical irreducibility embodied in random $t$, such that the product of structural bias and measurement-based randomness remains bounded—there is always irreducible unpredictability in at least one observable.

This formal structure allows QSQS to bridge purely algorithmic randomness extractors (dQRNG) and purely physical or quantum randomness sources (qQRNG) within a single protocol.

5. Empirical Entropy Amplification and Statistical Certification

The flattening of output distributions and convergence to maximal entropy are evidenced through experimental results:

• Entropy Metrics: Both Shannon entropy and min-entropy approach 8 bits for $n=8$-bit outputs as $m$ increases.
• Chi-squared Uniformity: The test statistic converges to values characteristic of i.i.d. uniform output, and visual inspection reveals bell curves transitioning from right-skewed to flat distributions.
• Hybrid Mode: When physical entropy is injected by reseeding, any algorithmic pattern bias is eliminated, and the independence and unpredictability properties required for cryptographic randomness are certified.

This behavior demonstrates that QSQS is effective in transforming both deterministic and non-deterministic, skewed input data into outputs suitable for cryptographic applications even under post-quantum adversarial assumptions.

6. Cryptographic and Foundational Implications

QSQS provides a blueprint for future post-quantum secure entropy sources:

• System-embedded entropy: No hardware TRNG is required, as the system leverages unavoidable physical fluctuations inherent in commodity computing platforms.
• Auditability and Ease of Deployment: Since the process is software-implemented, the design is transparent and reproducible.
• Resilience to Quantum Adversaries: The mechanism is not reliant on computational hardness but on physical unpredictability, safeguarding against advances in quantum algorithmic attacks.
• Physics-based randomness engineering: By mapping complexity-theoretic unpredictability onto a formally quantum-inspired framework, QSQS opens avenues for rigorous analysis and certification of software-based randomness generators.

In foundational terms, QSQS demonstrates that a union of deterministic and physically noisy observables—when mapped through an appropriate collapse mechanism—can achieve the practical goals of quantum randomness while being entirely compatible with classical system implementations.

7. Summary and Future Perspectives

QSQS operationalizes a quantum-inspired uncertainty principle at the intersection of deterministic algorithmic evolution and non-deterministic system-level fluctuations. By leveraging the collapse of internal degeneracy through modulo reduction, and the continual infusion of physical entropy, QSQS-based systems such as QPP-RNG achieve high-entropy, uniformly distributed outputs provably suitable for post-quantum cryptographic use (Kuang, 1 Aug 2025). This concept not only addresses the entropy gap for software-based random number generation but also contributes to a broader understanding of how quantum-inspired design can unify structural uniformity with fundamental unpredictability in information systems.