Quaternionic Computing

Jose M. Fernandez; William A. Schneeberger

Home/Archive/arXiv

arXiv · quant-ph/0307017

Quaternionic Computing

Jose M. Fernandez·William A. Schneeberger

View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce a model of computation based on quaternions, which is inspired on the quantum computing model. Pure states are vectors of a suitable linear space over the quaternions. Other aspects of the theory are the same as in quantum computing: superposition and linearity of the state space, unitarity of the transformations, and projective measurements. However, one notable exception is the fact that quaternionic circuits do not have a uniquely defined behaviour, unless a total ordering of evaluation of the gates is defined. Given such an ordering a unique unitary operator can be associated with the quaternionic circuit and a proper semantics of computation can be associated with it. The main result of this paper consists in showing that this model is no more powerful than quantum computing, as long as such an ordering of gates can be defined. More concretely we show, that for all quaternionic computation using n quaterbits, the behaviour of the circuit for each possible gate ordering can be simulated with n+1 qubits, and this with little or no overhead in circuit size. The proof of this result is inspired of a new simplified and improved proof of the equivalence of a similar model based on real amplitudes to quantum computing, which states that any quantum computation using n qubits can be simulated with n+1 rebits, and in this with no circuit size overhead. Beyond this potential computational equivalence, however, we propose this model as a simpler framework in which to discuss the possibility of a quaternionic quantum mechanics or information theory. In particular, it already allows us to illustrate that the introduction of quaternions might violate some of the ``natural'' properties that we have come to expect from physical models.

Explore related subjects

quant-ph cs.CC

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Jose M. Fernandez, William A. Schneeberger. 2004-11-05. Quaternionic Computing. https://arxiv.org/abs/quant-ph/0307017

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Everlasting secure key agreement from a quantum computational hybrid security model

We introduce the Quantum Computational Timelock (QCT) model, combining time-lock encryption with a quantitative noisy-quantum-storage assumption. Time-lock encryption gives Bob immediate access to the encoding seed and gives Eve the same final information as access to that seed only after a protected interval. Eve may then process her retained information without computational restriction. We state this delayed-access encryption assumption explicitly and construct MUB-QCT from a complete set of mutually unbiased bases in dimension $d$ and an affine pair-wise independent permutation family. For single-copy encoding, every seed-independent immediate measurement has trace-distance advantage at most $1/(2\sqrt{d+1})$ and mutual information at most $1/(d+1)$. Effective residual storage at diamond distance at most $δ$ from replacement adds at most $δ$ to the individual-attack distinguishing bound. Under i.i.d. individual attacks and entropy-preserving acceptance, we obtain an asymptotic key-rate bound and formulate final-key secrecy in composable trace distance. For $m$ independently randomized signals sharing one bit and seed, the optimal leakage under non-adaptive product measurements is $\min\{1,m/(d+1)\}$, allowing a linear-in-$d$ detection enhancement in that attack model. We also construct a seed-independent joint measurement that recovers the bit after disclosure with error at most $\varepsilon$ from $O(\log d+\log(1/\varepsilon))$ signals, retaining only classical information after measurement. This gives a quantitative separation between product and joint measurements. Bob uses two logical detection outcomes independently of $d$.

quant-ph↗

arXiv2026

Linear optics and the problem of Bell-like state discrimination

A linear optics-based scheme to implement various quantum information processing tasks is of paramount importance due to its ease of implementation and low noise. Many information-theoretic tasks depend on the successful discrimination of Bell states. A no-go theorem states that it is not possible to perfectly discriminate among the four Bell states when the measurement apparatus is restricted to linear optical elements. The success probability is only $50\%$. Using extra resources such as hyperentanglement, ancillary entanglement, and even a minimum amount of non-linearity, complete Bell-state discrimination can be achieved. The success probability for Bell-like state discrimination is only $25\%$. Bell-like states are non-maximally entangled two-qubit states that resemble Bell states in structure but have unequal Schmidt coefficients. We find that this can be boosted to $50\%$ using hyperentanglement, where extra ancillary degrees of freedom (DOF) of photons, such as polarization, momentum, or OAM, are also entangled along with the system DOF. This contrasts with Bell-state discrimination, where $100\%$ can be achieved. Furthermore, we find that using correlations in the photons' arrival times, all four Bell states can be distinguished with $100\%$ success probability, while for Bell-like state discrimination, it increases only to $37.5\%$ from $25\%$. We observe a similar contrast when using ancillary entangled photons. While the success probability for discriminating all four Bell states increases as $1-\frac{1}{2^N}$, where $N$ is the number of ancillary photon pairs, for Bell-like states, it depends on the state parameters and can be less than $25\%$ in some cases. Also, adding further ancillary photons decreases the success probability. We then show that using non-linear gadgets, namely SFG, $100\%$ success probability can be achieved even for Bell-like state discrimination.

quant-ph↗

arXiv2026

Memory-Augmented Hybrid Quantum Reservoir Computing

Reservoir computing (RC) is an effective method for predicting chaotic systems by using a high-dimensional dynamic reservoir with fixed internal weights, while keeping the learning phase linear, which simplifies training and reduces computational complexity compared to fully trained recurrent neural networks (RNNs). Quantum reservoir computing (QRC) uses the exponential growth of Hilbert spaces in quantum systems, allowing for greater information processing, memory capacity, and computational power. However, the original QRC proposal requires coherent injection of inputs multiple times, complicating practical implementation. We present a hybrid quantum-classical approach that implements memory through classical post-processing of quantum measurements. This approach avoids the need for multiple coherent input injections and is evaluated on benchmark tasks, including the chaotic Mackey-Glass time series prediction. We tested our model on two physical platforms: a fully connected Ising model and a Rydberg atom array. The optimized model demonstrates promising predictive capabilities, achieving a higher number of steps compared to previously reported approaches.

quant-ph↗