Estimating expectation values on near-term quantum computers often requires a prohibitively large number of measurements. One widely-used strategy to mitigate this problem has been to partition an operator's Pauli terms into sets of mutually commuting operators. Here, we introduce a method that relaxes this constraint of commutativity, instead allowing for entirely arbitrary terms to be grouped together, save a locality constraint. The key idea is that we decompose the operator into arbitrary tensor products with bounded tensor size, ignoring Pauli commuting relations. This method -- named
k-NoCliD (
k-local non-Clifford diagonalization) -- allows one to measure in far fewer bases in most cases, often (though not always) at the cost of increasing the circuit depth. We introduce several partitioning algorithms tailored to different Hamiltonian classes. For electronic structure, we numerically demonstrate the existence of threshold values of
k for which
k-NoCliD leads to the lowest shot counts, though we leave improved partitioning algorithms to future work. We focus primarily on three Hamiltonian classes -- molecular vibrational structure, Fermi-Hubbard, and Bose-Hubbard -- and show that
k-NoCliD reduces the number of circuit shots, often by a very large margin, and often even for
k as small as 2.
A
U(1) quantum rotor pairs a periodic angle with an integer-valued conjugate momentum, and occurs in molecular rotation, superconducting phase-charge circuits, and compact gauge fields. Coupling such a rotor coherently to qubits gives a hybrid register whose control structure is not inherited from either the oscillator-qubit or the qudit case. We develop a Clifford theory, together with a universal-control result, for registers of
n qubits and
r rotors. We classify all automorphisms of the hybrid phase space
F22n×Zr×Tr that preserve the Weyl commutation relations, and give an explicit finite Clifford circuit for each one. The classification is directional: rotor momentum parity may control qubit Pauli operations within the Clifford group, while every nonzero qubit-controlled rotor momentum shift is non-Clifford. It also yields normal forms for the mixed qubit-rotor couplings and the exact minimum number of elementary mixed gates needed to synthesize them. Adding a rotor cosine potential and one fixed qubit-rotor conditional phase to the local Clifford operations gives universal control on the full Hilbert space in the strong operator topology. We then apply this structure in three settings: an exact controlled-shift realization of gauge-covariant matter hopping, which is necessarily non-Clifford; rotor phase estimation with direct angle readout and probe optimization under momentum-support and energy constraints; and finite Fourier transforms on rotor momentum codes, where the one-rotor transform for
d=2s compiles into
O(s) momentum-selective and controlled-shift instructions and each cross-register Fourier factor is implemented by one quadratic rotor Clifford gate.
Quantum estimates of molecular energy require repeated state preparation and measurement. We bound the minimum repetition count for given measurements and processing rules from above with a practical schedule and from below with independently verified inequalities. For an H
4 chain at H-H distances of 0.90 and 1.80 Åwith 2 per cent uniform noise in the ground state, entangling Clifford measurements, which map Pauli products onto one another, cut leading-order repetitions by at least 65.9 and 61.0 per cent, respectively, relative to any schedule using only single-qubit Pauli measurements and Hamiltonian terms. With 69 unaltered measurement settings, each a basis change followed by qubit readout, processing further observables saves at least 12.04 and 6.70 per cent. For four lanthanide and actinide Hamiltonians with depolarized Hartree-Fock covariances, reducing the sum of absolute Hamiltonian coefficients does not proportionally reduce sampling. The bounds cover linear unbiased estimates from independent batches at small error.
Entangling measurements can reduce the cost of learning many-body correlations. We consider classical shadows in which a fixed unitary on a
k-qubit region is sandwiched between two layers of random single-qubit Clifford gates, and ask which middle unitary best estimates Pauli strings with full support on the region. Combining symplectic support counting with a constraint from operator-size parity, we prove that within this architecture the minimum squared shadow norm over all Clifford middle unitaries is
Ckopt=4⋅9k/(3⋅5k+3k−6) for odd
k and
Ckopt=4⋅9k/(3⋅5k−3k+2) for even
k. A dual-axis rotation circuit
U∗ attains this minimum for every
k. Asymptotically
Ckopt≃(4/3)⋅(9/5)k: the exponential base of Wu et al.'s contractive unitary is optimal, and the prefactor improves from
2 to
4/3. For
1≤k≤7, exact certificates show that
U∗ is optimal over the entire unitary group
U(2k), and we conjecture that this holds for every
k.
We show that qubit Hamiltonians derived from spectral graph theory lattices exhibit structurally favorable Pauli decompositions for quantum simulation. After Jordan–Wigner transformation, the GeoVac lattice Hamiltonian yields
O(Q3.15) Pauli string terms in the number of qubits
Q, compared to
O(Q4+) for conventional Gaussian-basis Hamiltonians (our own log-log fit of the published Pauli counts of Trenev et al. [ref] gives
Q4.25 for LiH and
Q3.92 for H
2O, spanning
Q3.9–
Q4.3). The advantage originates in angular momentum selection rules—enforced by Gaunt integrals at the two-electron integral level—which cause the electron repulsion integral (ERI) tensor density to decay as
∼1/M (empirically
≈M−0.85–
M−1) with the number of spatial orbitals
M, while Gaussian ERI tensors remain near-dense at
O(1). Beyond term counts, we quantify two operationally relevant cost metrics: the number of qubitwise-commuting (QWC) measurement groups scales as
O(Q3.36), and the Pauli 1-norm
λ=∑i∣ci∣—which controls both Trotter step counts and qubitization query complexity—scales as
O(Q1.69) with
R2=0.997. The sub-quadratic 1-norm scaling is the key result for fault-tolerant simulation: it implies that Trotter circuit depth grows far more slowly with system size than term counts alone would suggest. All Gaussian baselines are validated against published integral values (Szabo and Ostlund for H
2; analytical STO for He). For composed multi-center systems, the pseudopotential approximation of the composed natural geometry framework [ref] yields block-diagonal electron repulsion integrals with a universal Pauli scaling of
O(Q2.5), independent of molecular topology (exponent spread
0.02 across three molecules). At equal qubit counts, this represents two or more orders of magnitude fewer Pauli terms than Gaussian baselines (published counts of Trenev et al. [ref],
Q3.9–
Q4.3 span fitted by us)—an advantage that grows with system size due to the
∼1.5 exponent gap. These comparisons are at matched qubit counts, not matched accuracy; the composed basis carries classical equilibrium-geometry errors of 5–26% (LiH/BeH
2/H
2O; Sec. (ref)), making the framework optimal for fixed-geometry quantum simulation rather than geometry optimisation. Across the composed molecular library (35 molecules spanning
Z=1–56, H through Ba, including the first transition series ScH–ZnH), the composed Pauli count is exactly linear in the qubit count:
NPauli=11.10×Q (exact, non-identity terms;
d-orbital blocks give 9.23). These counts are realized under a pair-diagonal angular approximation (
ma=mc,
mb=md) adopted deliberately for sparsity; the exact global-
ML Coulomb rule (used in the framework's precision-physics paths, and considered but left on the table here) gives
≈2.5× more terms with the same scaling, under which the
d-block coefficient is in fact higher, not lower (Sec. (ref)). GeoVac Paper 14. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.
Preserving particle number and spin helps quantum circuits target molecular electronic states, but does not guarantee access to every state with the required quantum numbers. We determine which additional operations, combined with spin-independent orbital rotations connecting all spatial orbitals, generate every real state-space rotation within each complete subspace of fixed particle number
N, total spin
S, and spin projection
MS. We consider spin-free molecular calculations in real orbitals, without an additional spatial-symmetry restriction, and continuously tunable operations that preserve particle number, full spin symmetry, and real amplitudes. Below the maximal-spin limits set by the electron and hole numbers, repeated singlet-pair transfer between any two fixed spatial orbitals is sufficient. On nontrivial maximal-spin boundaries, pair transfer vanishes, and an additional generator built from one- and two-electron terms is sufficient exactly when its action in the target subspace is not a linear combination of orbital-rotation generators. The minimum number of spatial orbitals needed by one additional generator is two in the interior and three on nontrivial boundaries, even when terms involving more than two electrons are allowed. Both minima are attained using only one- and two-electron terms. The three-orbital optimum rotates two orbitals according to the occupation of a third, and its unitary factors exactly into eight commuting Pauli rotations. Molecular benchmarks show that this operation removes the observed boundary energy-error plateaus, while sparse interior constructions attain the prescribed energy accuracy with fewer compiled CNOT gates in selected fixed-orbital comparisons.