A compact abelian group acts unitarily on a finite-dimensional space. This paper proves that the overlap of two states, one shifted by a group element, is a finite Fourier series whose coefficients are positive semi-definite kernels indexed by the isotypic components of the conjugation representation; the coefficient of the trivial character is the inner product of the twirled states, and the sum of the squared moduli of the other coefficients is the variance of the overlap over the group. For the circle the series is a trigonometric polynomial whose degree is bounded by the total charge, so it is determined exactly by its values at finitely many equally spaced shifts. The same decomposition is applied to a reset channel, a fixed unitary that absorbs encoded inputs one at a time into a hidden register and discards the input between steps. Its readout under a common shift of the inputs is a finite Fourier series whose zero mode is the readout of the jointly twirled input; this need not equal the readout of inputs twirled one at a time, and an explicit Clifford channel separates the two. The degree of the readout is at most the charge bound times the number of steps, and for every charge bound an explicit channel attains it. Every coefficient can be estimated from overlaps measured at finitely many shifts, with a number of runs quadratic in the inverse error and logarithmic in the number of coefficients, and a charge sector of an encoded state can be read at the amplitude level by exact phase estimation of the charge followed by a Hadamard test. Sector kernels of product encodings and Fourier sectors of Clifford channels with Pauli readout are shown to be classically computable in polynomial time. The paper claims no quantum advantage. A final section carries the Fourier argument to the signature of a piecewise-linear path, read as a vector in a Fock space, and decomposes the signature kernel under rotation of the increments into harmonics that are themselves kernels.
A compact abelian group acts unitarily on a finite-dimensional space. This paper proves that the overlap of two states, one shifted by a group element, is a finite Fourier series whose coefficients are positive semi-definite kernels indexed by the isotypic components of the conjugation representation; the coefficient of the trivial character is the inner product of the twirled states, and the sum of the squared moduli of the other coefficients is the variance of the overlap over the group. For the circle the series is a trigonometric polynomial whose degree is bounded by the total charge, so it is determined exactly by its values at finitely many equally spaced shifts. The same decomposition is applied to a reset channel, a fixed unitary that absorbs encoded inputs one at a time into a hidden register and discards the input between steps. Its readout under a common shift of the inputs is a finite Fourier series whose zero mode is the readout of the jointly twirled input; this need not equal the readout of inputs twirled one at a time, and an explicit Clifford channel separates the two. The degree of the readout is at most the charge bound times the number of steps, and for every charge bound an explicit channel attains it. Every coefficient can be estimated from overlaps measured at finitely many shifts, with a number of runs quadratic in the inverse error and logarithmic in the number of coefficients, and a charge sector of an encoded state can be read at the amplitude level by exact phase estimation of the charge followed by a Hadamard test. Sector kernels of product encodings and Fourier sectors of Clifford channels with Pauli readout are shown to be classically computable in polynomial time. The paper claims no quantum advantage. A final section carries the Fourier argument to the signature of a piecewise-linear path, read as a vector in a Fock space, and decomposes the signature kernel under rotation of the increments into harmonics that are themselves kernels.
This preprint is the second part of a series on Fourier harmonics of shifted overlaps under a U(1) symmetry. The first part (Fourier sectors of shifted overlaps under a compact abelian symmetry, doi:10.5281/zenodo.23137889) introduced the framework: the overlap of two states, one shifted by a group element, decomposes into a finite Fourier series whose coefficients are positive semi-definite sector kernels, read exactly from 2n+1 shifts. That paper made no advantage claim. This paper determines the copy complexity of the diagonal of those kernels for an unknown n-qubit mixed state: the harmonics κ_m(ρ) = ‖A_m ρ‖₂² of the self-overlap Tr(ρ e^{iΔN} ρ e^{−iΔN}), with N the number operator and A_m the projection onto relative charge m. For a reduced state graded by its subsystem charge, these are the replica-two charged moment of the entanglement asymmetry and, normalised, the sector weights of symmetry-resolved operator entanglement. The setting is the model of learning with and without quantum memory of Chen, Cotler, Huang and Li. The results are unconditional. With a quantum memory of one copy, swap tests between a copy and a shifted copy at 2n+1 shifts return every harmonic within ε from O(ε⁻² log(n/δ) + n) pairs of copies. Without quantum memory, and whatever the classical memory and adaptivity, a single harmonic needs Ω(max{ε⁻², √D_{n,m}/ε}) copies, where D_{n,m} = C(n, ⌊(n−m)/2⌋) is the dimension of the charge sectors at distance m; O((2n+1)(ε⁻² + 2^{n/2}ε⁻¹) ln((2n+1)/δ)) single-copy measurements suffice. The separation between learning with and without quantum memory therefore survives symmetry resolution, with its lower bound fixed by the sector dimension rather than by 2ⁿ. The paper locates the boundaries of this separation. It is absent for pure states, where charge statistics suffice, and for n−m bounded, where polynomially many single-copy measurements suffice. It survives unital phase-covariant noise, including charge dephasing of any strength and depolarising noise up to a fixed rate. The bounded-memory bounds of Gong, Haferkamp, Ye and Zhang transfer to each harmonic. A dimension cap shows how far reductions affine in the purity can reach. A final corollary carries the lower bound to the normalised sector weights and to the Rényi-2 entanglement asymmetry: at constant error, estimating the entanglement asymmetry of an n-qubit mixed state from single copies costs Ω(2^{n/2}/n^{1/4}) copies, while two swap tests between pairs of copies return it from O(1) pairs for states of bounded purity. The lower-bound technique is that of Chen, Cotler, Huang and Li (purity testing) and Anshu, Landau and Liu (distributed inner-product estimation), applied through an embedding of a 2^k-dimensional problem across two charge sectors. The purity-testing bound is reproved in an appendix with an explicit constant. No advantage is claimed for states given with a classical description, for which every harmonic is classically computable in polynomial time. The repository contains a numpy implementation, 65 tests tied to the numbered statements, and the script that regenerates every figure and number of the paper from a single seed. Keywords: quantum learning theory; copy complexity; quantum memory; symmetry-resolved entanglement; entanglement asymmetry; modes of asymmetry; swap test; purity estimation; U(1) symmetry
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.
For two pure states ψ, φ of an n-qubit register, the overlap after a common phase shift, g(θ) = ⟨ψ|e^{iθN}|φ⟩ with N the number operator, is a trigonometric polynomial of degree at most n. When g does not vanish on the circle, its winding number W(ψ, φ), the shift winding of the pair, counts the zeros in the unit disk of the polynomial whose coefficients are the overlaps within each Hamming-weight sector. W is invariant under independent global phases, under common shifts and under every joint unitary that commutes with N. Along a path of pairs it can change only where some common shift makes the two states orthogonal. The sector kernels are blind to W. These kernels are the Fourier coefficients of |g|², measured by swap tests on a shift grid; their diagonal values are the multiple-quantum-coherence intensities of NMR. An antiunitary mirror preserves every kernel and sends W to n − W. Product states: W counts the qubits with h_q + h′_q > 1. Spin-coherent reference: W counts the Majorana stars of the partner's symmetric part beyond a circle of latitude. The kernels are then the Fourier coefficients of the partner's Husimi function on the reference's circle of latitude. Haar-random partner: averaged over the partner, W(ψ, ·) equals the mean excitation number ⟨ψ|N|ψ⟩. For the reference |+⟩^⊗n, its variance is ζ(3/2)/(8√π)·√n to leading order. With one control qubit and copies of the two states, a certified protocol returns W exactly or abstains, using O(n log²n log(n/δ)/ν²) copy pairs. The margin ν is inverse-polynomial exactly for pairs that stay close under every common shift, and exponentially small for random partners. Classically, W is efficiently computable for product, matrix-product and computationally tractable states. Deciding W for circuit-prepared pairs at constant margin is BQP-complete. We claim no quantum advantage. 13 pages, 3 figures, 2 tables. The deposit includes the LaTeX source and a reference implementation in numpy. Its tests cover every theorem, proposition, lemma and corollary, and all figures and numbers are regenerated from a fixed seed.
We give an explicit treatment of the third tensor power of the Clifford representation on n qutrits, for n≥2. This representation has five pairwise inequivalent irreducible constituents, with multiplicities 1,1,1,1,2. The decomposition reduces the Clifford twirl of a tensor cube to four real parameters. Two of them are spectral moments of the input. The other two are cubic invariants expressed as sums of cubes of the Weyl characteristic function and the discrete Wigner function. On trace-one Hermitian matrices, the invariant polynomials of degree at most three are spanned by the constant function and four algebraically independent functions. These functions do not separate Clifford orbits. For pure inputs a single parameter suffices. We recover its range and the characterization of its maximum by stabilizer states. We bound the stabilizer fidelity, determine full separability across the three copies in the symmetric family, and recover the criterion for a complex projective 3-design. We classify the jointly invariant quantum Boolean functions. For stabilizer testing and for testing unknown observables we give explicit bounds on copies and queries. We also give character formulas for the finite projective Clifford groups and record the complete character table for two qutrits.
We prove an exact, unconditional identity in the finite Weil (oscillator) representation of the Heisenberg group Heis3(Z/qZ), q an odd prime: for every complete (inversion-symmetric) BFS shell of the Cayley graph Cay(Heis3(Z/qZ),Sq), the orbit vectors generated by conjugate Weil sectors c and q−c span identical subspaces of Cq, for the special character block and generic character triples alike. The proof reduces every orbit vector to a single pure Fourier mode and shows the reachable frequency set is closed under negation whenever the generating shell is inversion-symmetric — unconditional, coherence-independent, verified independently by 1,548 exact integer-arithmetic checks across six primes with zero failures, plus a negative control on a deliberately truncated shell. We characterise precisely which diagnostics this forecloses: not only Fourier support, rank, projectors, dimensions, and principal angles, but any quantity invariant under independent rephasing and reindexing of the frame — including normalised Bargmann invariants, proved trivial by an explicit frame-orbit argument. A bibliography search confirms ρq−c=ρc is standard; no matching result was found for the subspace-equality identity or its no-go corollaries. This paper corrects an earlier version (v1.1.2): independent review found its central-theorem proof unsound and its separate Born-rule proof independently defective; both are withdrawn, and the singlet correlator and Tsirelson bound survive only as a standard, conditional remark. What remains is the rigidity result above and its precisely scoped no-go corollaries. Read as a methodological instance for the wider programme (interpretive, not a further result): a numerical near-zero-residual observation can be forced by a construction's own structural degeneracy rather than signal an intended physical mechanism.
One should not assume that reconstructing an electronic observable with a smaller set of measurement settings will cut the number of repetitions in proportion. Each setting is simply a basis change and qubit readout, but the savings are not guaranteed to be commensurate. Take two electrons of opposite spin in two spatial orbitals: the orbital degree of freedom of each electron acts as a logical qubit and rotations are independent in the spin sectors. From the observable's one- and two-qubit coefficients we derive an exact floor for the setting count and a decomposition to meet it in no more than three settings. With fewer settings, the minimum error in the two-qubit correlation coefficients is set by the singular values left out. A case in point is an exchange with equal nonzero couplings on the three Pauli axes. This demands three orbital settings yet only one joint measurement. We offer a Clifford circuit for four physical qubits to effect the necessary mapping of Pauli products under conjugation. In the case of a one-parameter family of logical singlet-triplet mixtures, the orbital-to-Clifford shot cost ratio is three at maximum mixing and trends toward 3/2 near the logical singlet. Where spin sectors may be mixed, isotropic exchange needs one orbital setting. For spin-conserving Hamiltonians with N spatial orbitals encoded via Jordan-Wigner, the X-rank, the binary rank of the X parts of its strings, of any commuting Pauli string family is no more than 2(N−1). Searches with the Robust Adaptive Nature-inspired Global Explorer (RANGE) reach this limit for CH4, with the rank of the Hamiltonian support vouching for the maximality of the search values. And for all nonidentity Pauli strings on n qubits, semidefinite witnesses from the Lovász theta number give a lower bound on the Clifford setting count that agrees with the integer cover and counting bounds at 2n+1.