Most of quantum computing today is written in the mathematics of quantum mechanics: complex vectors and the matrices that act on them. This mathematics has served physics well, yet it is not a Euclidean number system. In it, two states of one system cannot be multiplied and divided back, and a turn about a general axis is a matrix rather than a number. What would be an alternative Euclidean number system, possibly suitable for quantum mechanics? Recently, we proposed the Arrow Numbers, the complex numbers, quaternions and octonions, as such a system. Together with the real numbers, they are the normed division algebras. In them, lengths multiply, every nonzero arrow divides, and the Euler relation turns an arrow about any axis. What if we build a Boolean algebra on this Euclidean mathematics? With a bit written as a sign, ±1, the basis units multiply as the XOR logic function of their labels. Then a half turn is the NOT function, and a quarter turn is its square root. When an arrow moves through the zero of the arrow Argand sphere, it acts as a threshold. Thus the sign of a sum of three inputs gives the majority gate, and hence AND, OR and NAND. Reversibly, the quotient of the two orders of a quaternion product is the AND of two bits, AB(BA)⁻¹ = (−1)^(ab), which builds the Toffoli gate. Likewise, the two groupings of an octonion product give the AND of three bits. Thus each doubling gives up one rule of arithmetic, and the quotient that measures it carries one more degree of AND. These identities are known, and we read them as gates. In standard CMOS design, NOT and every product by a fixed basis unit are implemented as wiring. Moreover, the Toffoli gate takes fourteen transistors, and the product and the zero crossing make a full adder. SPICE simulations confirm every truth table. Accepted for publication in the Journal of Applied Mathematics and Physics (2026-10-22). The checks archive contains the Julia script of the algebra (59 tests), the switch-level circuit checks and the SPICE netlists and driver, each check with a control that must fail, and the logs of the runs.
Most of quantum computing today is written in the mathematics of quantum mechanics: complex vectors and the matrices that act on them. This mathematics has served physics well, yet it is not a Euclidean number system. In it, two states of one system cannot be multiplied and divided back, and a turn about a general axis is a matrix rather than a number. What would be an alternative Euclidean number system, possibly suitable for quantum mechanics? Recently, we proposed the Arrow Numbers, the complex numbers, quaternions and octonions, as such a system. Together with the real numbers, they are the normed division algebras. In them, lengths multiply, every nonzero arrow divides, and the Euler relation turns an arrow about any axis. What if we build a Boolean algebra on this Euclidean mathematics? With a bit written as a sign, ±1, the basis units multiply as the XOR logic function of their labels. Then a half turn is the NOT function, and a quarter turn is its square root. When an arrow moves through the zero of the arrow Argand sphere, it acts as a threshold. Thus the sign of a sum of three inputs gives the majority gate, and hence AND, OR and NAND. Reversibly, the quotient of the two orders of a quaternion product is the AND of two bits, AB(BA)⁻¹ = (−1)^(ab), which builds the Toffoli gate. Likewise, the two groupings of an octonion product give the AND of three bits. Thus each doubling gives up one rule of arithmetic, and the quotient that measures it carries one more degree of AND. These identities are known, and we read them as gates. In standard CMOS design, NOT and every product by a fixed basis unit are implemented as wiring. Moreover, the Toffoli gate takes fourteen transistors, and the product and the zero crossing make a full adder. SPICE simulations confirm every truth table. Accepted for publication in the Journal of Applied Mathematics and Physics (2026-10-22). The checks archive contains the Julia script of the algebra (59 tests), the switch-level circuit checks and the SPICE netlists and driver, each check with a control that must fail, and the logs of the runs.
Quantum computing (Q-computing) is a revolutionary technology that applies quantum mechanics (QM) to implement calculations significantly quicker than traditional computers. Q-computing employs Q-bits (qubits) rather than classical bits, which can take on a superposition state, making it possible to represent both 0 and 1 at once. This distinctive property allows qubits to perform multiple calculations simultaneously, making Q computers exponentially more powerful as more qubits are added, compared to the linear scaling of classical computers. However, challenges such as qubit instability and error rates remain significant obstacles to fully realizing this potential. A vital aspect of this technology is the development of Q arithmetic circuits, which act as the foundational blocks for efficiently executing Q algorithms. Q half-adder and full-adder circuits are integral to performing fundamental arithmetic operations within Q computers. In contrast to classical adders, Q adders rely on Q gates such as Controlled NOT (CNOT) and Toffoli gates. These gates exploit Q phenomena like entanglement and superposition, allowing for more efficient computations. The Q half-adder consists of two qubits and adds two single bits, producing a carry and sum output. It utilizes the CNOT gate to compute the sum and the Toffoli gate (controlled-controlled-NOT [CCNOT]) to compute the carry. The Q full-adder builds on this, adding three qubits to produce a sum and a carry output, further increasing computational capacity. Through the entanglement of qubits, these circuits allow for more parallelism and speed compared to classical adders. Q carry-look-ahead adders (QCLAs) and Q ripple-carry adders (QRCAs) are used to perform more complex arithmetic operations based on these half-adder and full-adder structures. Q arithmetic circuits have broad applications, boosting secure key generation in cryptography, speeding up matrix operations for faster model training in machine learning, and enabling more precise, efficient image transformations in Q image processing, benefiting fields like medical imaging and computer vision.
Every tool of quantum computing is one of three moves on two sliding rings marked with the sixty last digits of the Fibonacci numbers. A setting is a value read off one placement of the rings. A fold is a product turned into a sum by the half-angle law. A turn is a shift of position that carries a digit with it. Every gate is a setting, its entries cos(θ/2) and sin(θ/2) the half-distances of the rings at crossing separation θ, and the Hadamard, CZ and CNOT are quarter and half turns. Born's rule is the fold, amplitudes read at position k and expectation values at 2k. Every gradient is a quarter turn, because the parameter-shift rule is the ring's derivative law, its two shifts the ×7 and ×3 laws, with four eighth turns where a Givens gate needs them. A complete variational run is worked on a two-qubit number-conserving Hamiltonian. The energy is the ring shifted ten positions, the ground state the vesica setting at position 20, the optimizer three readings and a gnomon with nothing to tune, its convergence one line. Noise is the radius of the rings, and the square inside the ring, the stabilizer octahedron, says when the noise has cost a magic state. Phase estimation, Shor's classical end, Grover's search, Trotter chains, QAOA, Bell tests, teleportation, error mitigation, benchmarking and the [[4,2,2]] and [[5,1,3]] codes each reduce to the same three moves, forty tools on one page, and because every unitary is a finite product of rotation gates, the list is open and the three moves are closed. A checksum rides every classical number of the loop, the digits under the crossings on each setting, the addition law on each sum, the quarter-turn law on each gradient, the symmetry identity ⟨XX⟩ = ⟨YY⟩ on the measured data with its alarm threshold read off the setting, every one-, two- and three-bit upset of a 32-bit phase word and every burst of up to fifteen bits caught by three rings in lockstep, every corruption of any kind caught by the ladder, and the decoder table of the perfect code built on the four quarter turns of the net, any four simultaneous faults in an entry restored. The guard stands on every classical register of the loop, and the code space stands on the qubits. Thirty-one jobs on three IBM machines close the paper. A clean run on a 156-qubit machine had every compiled angle certified, the injected bit flip flagged at over a hundred standard errors, the clean checksum inside its line, the radius reading the noise, and the Bell value above 2, and one bit flipped in a phase word before submission, four ways, caught by the guard every time, turning the ground state into the top of its number sector and, from the entangler word, into the top of the whole spectrum when sent unguarded. The same words were corrupted up to all ten at once, by any change up to every bit inverted, and read both ways in four runs, unguarded and through the guard, and every setting word of a many-qubit tower was corrupted past any real error rate, caught and rebuilt in place, and read right on the machine. Every number is verified by the script in the appendix on explicit matrices and states, and the hardware readings are checked against the result files.
The mathematics behind the quantum Hamiltonian computing (QHC) approach of designing Boolean logic gates with a quantum system are given. Using the quantum eigenvalue repulsion effect, the QHC AND, NAND, OR, NOR, XOR, and NXOR Hamiltonian Boolean matrices are constructed. This is applied to the construction of a QHC half adder Hamiltonian matrix requiring only six quantum states to fullfil a half Boolean logical truth table. The QHC design rules open a nano-architectronic way of constructing Boolean logic gates inside a single molecule or atom by atom at the surface of a passivated semi-conductor.
A quantum gate is the quantum analog of the classical logic gate of digital computers. It is a basic structural unit of a quantum circuit which operates on a small number of qubits to deliver a predefined output. A quantum circuit is a graphical representation of the sequence of the quantum gates used and the measurements performed in a quantum computation. Various quantum logic gates, namely, the Hadamard, T phase, NOT, Pauli-X, controlled NOT (CNOT), controlled-Z, SWAP and controlled phase gates as well as higher-level Toffoli and Fredkin gates are described. Conventions of reading quantum circuit diagrams are explained. Operation of a quantum full adder consisting of registers and gates is discussed, elaborating the role of the NOT gate, the sum calculation with CNOTgates and the carry C-out calculation with Toffoli gates. A primary tool used for performing arithmetic operations on a quantum computer is the quantum Fourier transform (QFT). The QFT adder and multiplier circuits are presented.
We develop detailed quaternionic and octonionic frameworks for quantum computation grounded on normed division algebras. Our central result is to prove the polynomial computational equivalence of quaternionic and complex quantum models: Computation over H is polynomially equivalent to the standard complex quantum circuit model and hence captures the same complexity class BQP up to polynomial reductions. Over H, we construct a complete model—quaternionic qubits on right H-modules with quaternion-valued inner products, unitary dynamics, associative tensor products, and universal gate sets—and establish polynomial equivalence with the standard complex model; routes for implementation at fidelities exceeding 99% via pulse-level synthesis on current hardware are discussed. Over O, non-associativity yields path-dependent evolution, ambiguous adjoints/inner products, non-associative tensor products, and possible failure of energy conservation outside associative sectors. We formalize these obstructions and systematize four mitigation strategies: Confinement to associative subalgebras, G2-invariant codes, dynamical decoupling of associator terms, and a seven-factor algebraic decomposition for gate synthesis. The results delineate the feasible quaternionic regime from the constrained octonionic landscape and point to applications in symmetry-protected architectures, algebra-aware simulation, and hypercomplex learning.
Quantum computing represents a paradigm shift in computation by leveraging the principles of quantum mechanics such as superposition, entanglement, and interference to perform operations beyond classical limitations. Unlike classical computing systems that rely on irreversible logic operations and suffer from energy dissipation due to information loss, quantum systems utilize reversible unitary transformations that preserve information and improve computational efficiency. This paper presents a comprehensive framework for the design, simulation, and validation of quantum gates and reversible arithmetic circuits, including half adders and half subtractors. The proposed approach systematically maps classical arithmetic logic into quantum circuits using fundamental quantum gates such as Pauli-X, Hadamard, Controlled-NOT (CNOT), and Toffoli gates. The circuits are implemented and simulated using the Qiskit Aer framework, enabling detailed analysis of quantum state evolution and measurement outputs. The results obtained from simulation include program-level outputs, state vector representations, measurement counts, and probability distributions, all of which demonstrate strong agreement with classical truth tables. The analysis confirms the correctness, reversibility, and scalability of the proposed designs. Furthermore, a comparative performance evaluation highlights the advantages of quantum computing over classical approaches in terms of parallelism and energy efficiency. The proposed framework establishes a robust foundation for future implementation of complex Boolean logic and arithmetic operations in quantum systems.