The Pythagorean Angle Lattice in Quantum Mechanics: Exact Spin Rotations, Gate Synthesis Obstructions and Arithmetic No-Go Theorems for Quantum Computation
The Pythagorean Angle Lattice produces exact spinor-norm and continuity no-go theorems for single-qubit gate synthesis, free and dense groups of difference gates, and exponent walls at positive distance from the Clifford group. Let θ be the smaller acute angle of a primitive Pythagorean triangle (a, b, c) and z = e^{iθ} = (b + ia)/c. The rational norm-one phases form the Pythagorean angle lattice P ⊂ ℚ(i). We study the exact arithmetic of the associated single-qubit rotations and of the difference angle Δ = π/2 − 2θ, whose spin rotation is U_n(Δ) = (A I − i B n · σ)/(√2 c) with A = a + b, B = b − a. Within the sector of rational rotation matrices, Gaussian rational eigenphases occur exactly for rational Bloch axes, and the obstruction to an exact rational spin lift is the spinor squareclass, which on these phases is the halving class [N(1 + z)]. Every Pythagorean difference gate has spinor norm [2] and factors exactly as a quarter-turn times the rational spin gate defined by the triple. This gives a spinor-norm obstruction to exact synthesis with forced length parity, an obstruction to synthesising the rotation by θ from difference and Clifford gates unless c is a perfect square, and an exact criterion for rational fourth roots; difference gates about orthogonal axes generate a free group, and about non-parallel axes a dense group with approximating words of logarithmic length. For pairs of gates we prove exact commutator identities, including a relation between the commutator scalar and the Born overlap of PAL-internal axis states; nested commutators obey an exact logistic recurrence for their transverse intensity, with a transition at θ = π/12 and a 2-adic filtration of their spectral discriminants. In odd prime-power dimensions, suitable Weil operators admit exponent intertwiners U ↦ U^{−2} at projective distance at least 1/(2√2) and diamond distance at least √(1 − p^{−k}) from the Clifford group, with Weyl spreading at least 1/360. Finally, density of the dyadic classes gives a continuity obstruction: without an a priori height bound, no fixed finite-copy continuous protocol separates them with a uniform margin. All results concern exact arithmetic structures inside ordinary quantum mechanics; no modification of its postulates is proposed.
The Pythagorean Angle Lattice produces exact spinor-norm and continuity no-go theorems for single-qubit gate synthesis, free and dense groups of difference gates, and exponent walls at positive distance from the Clifford group. Let θ be the smaller acute angle of a primitive Pythagorean triangle (a, b, c) and z = e^{iθ} = (b + ia)/c. The rational norm-one phases form the Pythagorean angle lattice P ⊂ ℚ(i). We study the exact arithmetic of the associated single-qubit rotations and of the difference angle Δ = π/2 − 2θ, whose spin rotation is U_n(Δ) = (A I − i B n · σ)/(√2 c) with A = a + b, B = b − a. Within the sector of rational rotation matrices, Gaussian rational eigenphases occur exactly for rational Bloch axes, and the obstruction to an exact rational spin lift is the spinor squareclass, which on these phases is the halving class [N(1 + z)]. Every Pythagorean difference gate has spinor norm [2] and factors exactly as a quarter-turn times the rational spin gate defined by the triple. This gives a spinor-norm obstruction to exact synthesis with forced length parity, an obstruction to synthesising the rotation by θ from difference and Clifford gates unless c is a perfect square, and an exact criterion for rational fourth roots; difference gates about orthogonal axes generate a free group, and about non-parallel axes a dense group with approximating words of logarithmic length. For pairs of gates we prove exact commutator identities, including a relation between the commutator scalar and the Born overlap of PAL-internal axis states; nested commutators obey an exact logistic recurrence for their transverse intensity, with a transition at θ = π/12 and a 2-adic filtration of their spectral discriminants. In odd prime-power dimensions, suitable Weil operators admit exponent intertwiners U ↦ U^{−2} at projective distance at least 1/(2√2) and diamond distance at least √(1 − p^{−k}) from the Clifford group, with Weyl spreading at least 1/360. Finally, density of the dyadic classes gives a continuity obstruction: without an a priori height bound, no fixed finite-copy continuous protocol separates them with a uniform margin. All results concern exact arithmetic structures inside ordinary quantum mechanics; no modification of its postulates is proposed.
This paper maps the quadratic node sequence r_n = n²r₀ onto the two rotation classes of quantum states. The transition intervals between consecutive nodes are the odd numbers Δr_n = (2n+1)r₀, and they accumulate to the nodes. The interval count 2n+1 is the dimension of the ℓ = n representation, the sum of (2ℓ+1) over ℓ = 0 to n−1 is the orbital degeneracy n² of shell n, and the two spin-1/2 components give the shell capacity 2n². The planar coordinate u = √r e^{iφ/2} has the half-integer rotation character m = 1/2 and changes sign under a 2π rotation, and u² has the integer character m = 1. The half-integer and integer classes are tabulated against the observed particle classes with Particle Data Group spins. The spin-statistics theorem is applied as established physics and is not derived here. The paper does not claim that the geometry causes particle spin, generates the particle classes or produces exchange statistics. The construction rests on Euclid’s Proposition II.5, a² − b² = (a+b)(a−b), which carries the node intervals and the rational rotations through t = tan(ψ/2). Version 2 adds a computational section in which state preparation, gates, measurement and error correction are run along three paths on identical inputs: a complex-matrix path, a scalar path with Taylor-evaluated exponentials, and a Euclid-native path in exact rational arithmetic. For the gate set tested, the Euclid-native amplitudes equal the matrix-path amplitudes up to a positive factor exactly, with zero norm residual and exact outcome probabilities. Real angles enter through continued-fraction convergents with an error of at most 2/q² per gate, and the arithmetic after the convergent is chosen has no rounding. The error-correction tests include the bit-flip, phase-flip and Shor nine-qubit codes, and for the Shor code arbitrary single-qubit errors at real angles with every outcome of nonzero probability followed, with recovery fidelity exactly 1 on every branch. The paths agree to 4×10⁻¹⁵ or better on the original tests and to 8×10⁻¹⁵ or better on the real-angle and arbitrary-error tests. The paper is a map between rotation classes and their algebra. It is not an encoding of fermionic operators onto qubits, and it does not rank the complex-amplitude grid and the discrete algebraic grid, which carry the same relationships on the operations and inputs tested.
Fault-tolerant quantum computation requires compiling arbitrary one-qubit unitaries into short words over a fixed gate library. For arithmetic libraries such as the p=5 Lubotzky--Phillips--Sarnak, or Clifford+V, gate set, this problem is governed by quaternionic lattice points on $S^3\cong \SU(2)$. We study the complete norm shells \[ P_k=\{x/5^k\in S^3:x\in\ZZ^4,\ |x|^2=5^{2k}\} \] and the associated projective gate set T⊂PSU(2). The main worst-case quantity is Sarnak's covering exponent K(T), for which the classical range is 4/3≤K(T)≤2. We show that any positive localized cap-kernel certificate using only the Deligne--LPS square-root spectral estimate reaches only the volume-squared threshold ∣VT(t)∣≫μ(B(ε))−2, hence only exponent 2. Thus any unconditional improvement requires arithmetic cancellation in localized off-diagonal counting. We also record the conditional benchmark that twisted Linnik gives K(T)=4/3, matching Harman's obstruction. We prove the shell-to-gate implication ρ(Pk)≤C5−αk⇒K(T)≤4/(3α), so α>2/3 is exactly the threshold for improving the unconditional bound. Finally, exact enumeration of P1,P2,P3,P4 and Haar-random tests show median trace-defect error at the optimal N−2/3 scale, while high quantiles remain separated. This gives a sharp distinction between strong typical-case performance and rare arithmetic holes controlling worst-case synthesis.
In this paper we study single qutrit circuits consisting of words over the Clifford+D cyclotomic gate set, where D=diag(±ξa,±ξb,±ξc), ξ is a primitive 9-th root of unity and a,b,c are integers. We characterize classes of qutrit unit vectors z with entries in Z[ξ,χ1] based on the possibility of reducing their smallest denominator exponent (sde) with respect to χ:=1−ξ, by acting an appropriate gate in Clifford+D. We do this by studying the notion of `derivatives mod 3' of an arbitrary element of Z[ξ] and using it to study the smallest denominator exponent of HDz where H is the qutrit Hadamard gate and D. In addition, we reduce the problem of finding all unit vectors of a given sde to that of finding integral solutions of a positive definite quadratic form along with some additional constraints. As a consequence we prove that the Clifford+D gates naturally arise as gates with sde 0 and 3 in the group U(3,Z[ξ,χ1]) of 3×3 unitaries with entries in Z[ξ,χ1]. We illustrate the general applicability of these methods to obtain an exact synthesis algorithm for Clifford+R and recover the previous exact synthesis algorithm in \cite{kmm}. The framework developed to formulate qutrit gate synthesis for Clifford+D extends to qudits of arbitrary prime power.
We generalize an efficient exact synthesis algorithm for single-qubit unitaries over the Clifford+T gate set which was presented by Kliuchnikov, Maslov, and Mosca [Quantum Inf. Comput. 13(7,8), 607–630 (2013)]. Their algorithm takes as input an exactly synthesizable single-qubit unitary—one which can be expressed without error as a product of Clifford and T gates—and outputs a sequence of gates which implements it. The algorithm is optimal in the sense that the length of the sequence, measured by the number of T gates, is smallest possible. In this paper, for each positive even integer n, we consider the “Clifford-cyclotomic” gate set consisting of the Clifford group plus a z-rotation by πn. We present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of πn z-rotations. For the Clifford+T case n = 4, the group of exactly synthesizable unitaries was shown to be equal to the group of unitaries with entries over the ring Z[eiπn,1/2]. We prove that this characterization holds for a handful of other small values of n but the fraction of positive even integers for which it fails to hold is 100%.
In this paper we study single qutrit circuits consisting of words over the Clifford +D cyclotomic gate set, where D=diag(±ξa,±ξb,±ξc) , ξ is a primitive 9 -th root of unity and a,b,c are integers. We characterize classes of qutrit unit vectors z with entries in Z[ξ,1χ] based on the possibility of reducing their smallest denominator exponent (sde) with respect to χ:=1–ξ, by acting an appropriate gate in Clifford +D . We do this by studying the notion of `derivatives mod 3 ' of an arbitrary element of Z[ξ] and using it to study the smallest denominator exponent of HDz where H is the qutrit Hadamard gate and D . In addition, we reduce the problem of finding all unit vectors of a given sde to that of finding integral solutions of a positive definite quadratic form along with some additional constraints. As a consequence we prove that the Clifford +D gates naturally arise as gates with sde 0 and 3 in the group U(3,Z[ξ,1χ]) of 3×3 unitaries with entries in Z[ξ,1χ] . We illustrate the general applicability of these methods to obtain an exact synthesis algorithm for Clifford +R and recover the previously known exact synthesis algorithm of Kliuchnikov, Maslov, Mosca (2012). The framework developed to formulate qutrit gate synthesis for Clifford +D extends to qudits of arbitrary prime power.
Finite-dimensional closed-system gate synthesis is a geometric optimal-control problem on a compact Lie group, but global solutions require careful treatment of logarithm branches, determinant-one constraints, drift, anisotropic penalties, and amplitude limits. This paper assembles and extends a self-contained family of exactly solved benchmarks under explicit hypotheses. For a positive right-invariant quadratic metric, smooth stationary curves obey the Euler–Arnold equation and possess Lax invariants. Under fully actuated isotropic control on SU(n), the fixed-time minimum action is the squared Hilbert–Schmidt distance divided by twice the gate time and is generated by a minimum-norm skew-Hermitian logarithm. A finite eigenphase-unwrapping rule computes that logarithm, while the affine spectral-width cut wall and a branch-gap criterion distinguish stable selection from set-valued behavior. Drift is removed isometrically whenever the metric is invariant under the drift adjoint action. Weighted action and box-constrained time on maximal tori and arbitrary closed commuting subtori reduce to explicit period-lattice problems; the latter gives distinct strict determinant-one and projective period lattices for an ideal fSim sector. Numerical validation uses 1506 Haar-random targets through dimension 128, a complete cost-bounded dynamic-program cross-check for every target, 300 direct Cartesian branch enumerations, 120 random-conjugation tests with analytically prescribed spectra, 240 independently cross-checked random weighted-torus instances, a separate SU(256) stress target, large-cluster selector regressions, repeated timing trials, GRAPE–L-BFGS, a first-order sequential Krotov-type update with exact discrete propagation and Fréchet derivatives, randomized and drift-adapted CRAB–Powell bases, and cut-locus tests that exercise the implemented Schur/logarithm solver. When sparse actuation, decoherence, leakage, or uncertainty invalidates the ideal hypotheses, the exact results are positioned as ideal-model reference values, branch-aware seeds, and regression tests rather than as certificates for the enlarged objective.