Quantum Computing on the Sixty-Ring: Every Gate a Setting, Every Gradient a Quarter Turn, and a Checksum on the Classical Half

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.