We study a dimension-saturating fixed-core ansatz in which four copies of a fixed, non-tunable two-qutrit core $K\in SU(9)$ are interleaved with five adjustable local layers from $L=SU(3)\otimes SU(3)$. Since $\dim SU(9)=80$ and $5\dim L=80$, this is the shortest fixed-core architecture not excluded by parameter counting. We formulate the smooth map $Φ_K:L^5\to SU(9)$ and use its right-trivialized differential to give verifiable certificates for local universality. We construct an explicit Clifford-word core whose Pauli-label splitting makes the identity-point differential an exact isometry, and we classify all 2304 symplectic actions satisfying the same splitting criterion. We also prove a structural obstruction for an important symmetry class: every complex-symmetric core $K=K^{T}$, including every core generated by a time-independent real-symmetric Hamiltonian in the chosen computational basis, has identity-point differential rank at most 78; hence any full-rank certificate for such a core must occur away from that point. We then assess a hardware-motivated superconducting core generated by a noncommuting, temporally asymmetric drive. Direct calculation verifies $K_{\rm sc}\neq K_{\rm sc}^{\mathsf T}$, and the core achieves $F_{\rm avg}\ge 0.999$ for all 1000 Haar-random targets tested under the stated restart protocol. We also report favorable sampled Jacobian-rank, structured-target, and robustness diagnostics. These results establish local universality at the parameter-counting-minimal, dimension-saturating depth, with an exact Clifford certificate complemented by a hardware-motivated numerical case study. Throughout, we separate exact local certificates from numerical evidence for broader synthesis performance.
We study a dimension-saturating fixed-core ansatz in which four copies of a fixed, non-tunable two-qutrit core K∈SU(9) are interleaved with five adjustable local layers from L=SU(3)⊗SU(3). Since dimSU(9)=80 and 5dimL=80, this is the shortest fixed-core architecture not excluded by parameter counting. We formulate the smooth map ΦK:L5→SU(9) and use its right-trivialized differential to give verifiable certificates for local universality. We construct an explicit Clifford-word core whose Pauli-label splitting makes the identity-point differential an exact isometry, and we classify all 2304 symplectic actions satisfying the same splitting criterion. We also prove a structural obstruction for an important symmetry class: every complex-symmetric core K=KT, including every core generated by a time-independent real-symmetric Hamiltonian in the chosen computational basis, has identity-point differential rank at most 78; hence any full-rank certificate for such a core must occur away from that point. We then assess a hardware-motivated superconducting core generated by a noncommuting, temporally asymmetric drive. Direct calculation verifies Ksc=KscT, and the core achieves Favg≥0.999 for all 1000 Haar-random targets tested under the stated restart protocol. We also report favorable sampled Jacobian-rank, structured-target, and robustness diagnostics. These results establish local universality at the parameter-counting-minimal, dimension-saturating depth, with an exact Clifford certificate complemented by a hardware-motivated numerical case study. Throughout, we separate exact local certificates from numerical evidence for broader synthesis performance.
We present the \emph{first} analytical determination of the exact accessible dimension for every fixed architecture of arbitrary two-qubit gates on three qubits. A support word represents which pair of qubits each two-qubit gate acts on, and its reduced length is obtained by merging consecutive gates on the same pair. If the reduced length is r, then the set of implementable three-qubit unitaries has accessible dimension d(w)=min{63,9r+9}. Consequently, six arbitrary two-qubit gates are \emph{necessary and sufficient for local universality}: every fixed architecture reaches a nonempty open subset of SU(8) when its reduced length is at least six. The result is stronger than existence of one favorable architecture. Every reduced six-slot support word is locally universal, including the alternating nearest-neighbor line AB,BC,AB,BC,AB,BC. The upper bound follows from a standard parameter-counting argument. The matching lower bounds are proved by Jacobian certificates in the Pauli basis. Up to qubit relabeling and reversal, there are 22 reduced architectures of lengths two through six. For each one, a product of rational Pauli rotations yields a nonzero maximal minor modulo the prime 1,000,003. Thus, any obstruction to global universality with six two-qubit gates must go beyond parameter counting, connectivity, and differential rank.
A three-state object is not yet a qutrit. The claim requires a three-dimensional complex Hilbert sector, coherent state transformations, sufficient local control, a two-qutrit entangler, and an experimental route that distinguishes quantum coherence from classical population switching. The exact covering arithmetic of the pentachoron K5 fixes three as the minimal equal support size for a two-set cover of five vertices and gives the partition (2, 1, 2) on covering triangular faces; identification of such a face with a physical interface remains an explicit model bridge. On that declared support, the fixed-one sector and the antisymmetric fixed-two sector of three fermionic modes in one resolved internal channel are both 3-dimensional qutrit sectors and are related by particle–hole duality. Hard-core exclusion alone is not sufficient for the exterior-square gate lift. After removal of the uniform offset, the most general real, site-permutation-invariant effective Hamiltonian has the form H0 = cL(K3) and spectrum {0, 3c, 3c}; symmetry fixes neither the sign nor the scale of c. On the associated singlet–doublet decomposition, U△ = P0 + ω²PE, ω² + ω + 1 = 0, ω ≠ 1, is an exact balanced tritter: every transition probability equals 1/3. Diagonal Clifford factors make U△ projectively equivalent to the qutrit Fourier gate F3. The ideal logical generators AK3 = X3 + X3† and D01 = diag(1, −1, 0) generate a real control Lie algebra of dimension 8, isomorphic to su(3). A programmable rank-one phase compiles into CZ3 using 4 commuting factors, and CZ3 compiles into SUM3. Under the stated control assumptions this gives an ideal universal qutrit architecture. Exact counterexamples delimit every promotion: three sites do not force an oriented cycle, symmetry fixes neither the sign nor the scale of c, connected hopping plus a non-degenerate diagonal need not generate su(3), localization is not coherence, and a population-sensitive observable is not tomography. Oxo-centred mixed-valence Fe3O complexes therefore enter only as a falsifiable implementation route. Published structural, Mössbauer, IVCT, and polarization data establish relevant populations and observables, not coherent logical gates. No logical gate, effective-Hamiltonian parameter, or laboratory pulse duration is identified in this paper with a K5 tick, local generation, or commit.
We study the structure and transversal logical capabilities of stabilizer quantum error correcting codes. Among our results, we identify universal lower bounds on circuit depth to generate a full logical Clifford algebra, and develop novel constructions of logical transversal gates including a new depth-one transversal phase S gate in the rotated surface code and a depth-one intra-block CZ gate in the 2D-toric code that generalizes to all odd distances and all lengths L≥3, respectively. Finally, we construct a high-rate non-LDPC CSS code family with parameters [[n,n,Θ(nβ)]] where β≈0.2823 in one demonstrated case, that provably possesses a constant-depth complete 2-local transversal logical Clifford basis instruction set architecture (ISA) composed of all individually targeted S, SHS=X, and CZ gates. This ISA is depth-one for certain subfamilies that we design and generally constant-depth under certain conditions. The code family is built from a small code with parameters [[n0,2,d0]], and is tunable in the standard way: it tiles out to form utility-scale logical qubit counts, and it scales up through concatenation to achieve higher distances and error suppression. We show that this construction preserves the depth-one complete transversal logical Clifford basis ISA when composed with these commuting construction actions, inheriting structure from the core codes so that at scale the complete logical Clifford basis ISA remains depth-one up to depth-two addressable operations between tiled cores. We call these Quantum Logic Codes.
Can an encoded quantum state be controlled continuously without repeated measurements to prevent leakage? This study gives a constructive answer for a four-spin-2 AKLT plaquette, where a three-dimensional singlet code sits inside a 625-dimensional physical Hilbert space. The result combines an exact restriction with a coherent control scheme: strictly leakage-free two-site Hamiltonians cannot provide universal control of the full encoded qutrit, yet two explicitly constructed interactions enable universal single-qubit control without intermediate measurements or postselection within a two-dimensional subspace. The classification begins with the complete physical leakage map for all six spin pairs and exchange polynomials through degree four. Exact rational calculations establish rank L = 6 and dim ker L = 18 for the 255 × 24 matrix. Despite this large family of leakage-cancelling combinations, every accessible logical Hamiltonian obeys [A, R] = 0, where R is a fixed geometric involution. Its two-dimensional and one-dimensional eigenspaces therefore remain separate, obstructing universal SU(3) control. The restriction extends to arbitrary anisotropic two-site Hamiltonians that preserve the code. Within the rotationally invariant pair-polynomial model, interactions confined to the four plaquette edges act only as logical scalars; noncommuting protected controls require at least three pair supports and polynomial degree three. The constructive solution uses quadratic and cubic exchange Hamiltonians acting simultaneously on three pairs, including a diagonal link. Their coefficients cancel physical leakage exactly, so QH₂P = QH₃P = 0 throughout the full system, with P projecting onto the code and Q onto its complement. Their logical generators satisfy Lie(iA, iB) ≅ u(2) ⊕ u(1), allowing arbitrary coherent qubit gates while preserving the code at every intermediate time. The manuscript supplies exact certificates and verification procedures, together with operator-norm and calibration-error bounds that make the engineering requirements explicit. The practical opportunity is to remove intermediate code-check measurements from this control scheme; demonstrating a hardware advantage requires calibrated simultaneous interactions, including the diagonal channel, and measured gate fidelity and latency.
Consider exact scalar gate-sequence data consisting of one preparation, one binary effect, and the probability of that effect after every word over a fixed two-gate alphabet. This preprint constructs a single word law of exact noncommutative Hankel rank four that admits a control-irreducible unitary quantum realization on C^d for every finite d >= 2. In each realization the state is full rank, the binary effect is strict, and the two controls generate the full matrix algebra M_d(C), excluding direct-sum padding and invariant spectator sectors. The associated four-dimensional angular-momentum operator systems have the same first-level Lorentz cone in every dimension, while a level-2 matrix pencil is positive exactly when t >= (d+1)/(d-1). Thus matrix-level-two order distinguishes all members of the constructed family. The result is a scoped non-identifiability theorem for the Hilbert-space dimension of an arbitrary compatible irreducible realization. It does not claim that the minimum compatible dimension is unidentifiable, does not contradict informationally complete gate set tomography within a declared model dimension, and does not reconstruct a tensor-product structure. The record includes the preprint PDF, LaTeX source, bibliography, numerical verification script and output, citation metadata, and a targeted priority audit.
Gate synthesis and circuit optimization are usually studied separately, and evidence on their interaction is contradictory: ZX-calculus rewriting removes a stable fraction of Solovay-Kitaev circuits, yet almost nothing from number-theoretically synthesized circuits. We show both behaviours follow from a single bound. For any optimizer that preserves the implemented element exactly--including all sound ZX rewriting with extraction--the achievable T-count is bounded below by the denominator exponent of the synthesized ring element. This exactness barrier is computable per instance and separates exact post-processing from approximation-aware resynthesis by a certified factor reaching 101x at recursion depth five. The two behaviours are then the barrier operating at different distances from the floor. For Solovay-Kitaev circuits we prove that the local ZX simplification layer (spider fusion and identity removal) computes exactly the free-product normal form of Z_2 * Z_8, giving exact per-instance compression and, under a calibrated ergodicity hypothesis, a depth-independent limit law confirmed on two independently constructed nets. For number-theoretically synthesized circuits the floor is already saturated: on single-qubit words automated ZX simplification attains it exactly, via a closed-form formula for minimal T-count in terms of phase linkage through the Z-axis normalizer. At two qubits and beyond the same valuation yields unconditional rigidity certificates, which on the quantum-Shannon-decomposition plus gridsynth pipeline certify 99.4-99.9% of the synthesized T-count as incompressible, with rigidity strengthening as accuracy tightens. This explains, and predicts the size of, the near-null optimization recently reported for that pipeline.