The Qang (qg) framework [1] expresses single-qubit rotations through two measurement-level quantities, the polar bias qgZ = ⟨σz⟩ and the outcome entropy qgS = H(p0). We report results obtained with the open-source reference implementation qang. First, the two metrics are linked by an exact, branch-free identity, qgS = H (1 + qgZ)/2 , which holds for any single-qubit state. Consequently, any new multi-qubit information must come from the joint outcome distribution. We name the resulting Z-basis total correlation and give its bounds and its scope: it is not an entanglement measure. Second, we identify exact blind spots: graph-state entanglement and readout-only dephasing are invisible to every Z-diagonal statistic, and qg-space updates cannot reach optima outside [0, π] because of the arccos range. On H2 they recover none of the correlation energy. Third, we study a pole-damped gradient step whose damping schedule is |dqgZ/dθ|. On a 300-landscape benchmark, H2, and a 4-qubit LiH Hamiltonian with a mixed Ry/Rx ansatz, it trades 3-20× more iterations at well-tuned learning rates for convergence where plain gradient descent diverges. Fourth, as a benchmark, qgS tracks heavy output probability (r = −0.93) and needs no calibration, unlike linear XEB. However, it is ÙOÍ monotonic under amplitude damping: under device-calibrated noise with added idle relaxation it falls below its own noiseless value. Pairing qgS with the register-averaged qgZ removes the ambiguity in 23 of 24 circuits tested. Every number is reproduced by a script and pinned by a regression test in the public repository.
The Qang (qg) framework [1] expresses single-qubit rotations through two measurement-level quantities, the polar bias qgZ = ⟨σz⟩ and the outcome entropy qgS = H(p0). We report results obtained with the open-source reference implementation qang. First, the two metrics are linked by an exact, branch-free identity, qgS = H (1 + qgZ)/2 , which holds for any single-qubit state. Consequently, any new multi-qubit information must come from the joint outcome distribution. We name the resulting Z-basis total correlation and give its bounds and its scope: it is not an entanglement measure. Second, we identify exact blind spots: graph-state entanglement and readout-only dephasing are invisible to every Z-diagonal statistic, and qg-space updates cannot reach optima outside [0, π] because of the arccos range. On H2 they recover none of the correlation energy. Third, we study a pole-damped gradient step whose damping schedule is |dqgZ/dθ|. On a 300-landscape benchmark, H2, and a 4-qubit LiH Hamiltonian with a mixed Ry/Rx ansatz, it trades 3-20× more iterations at well-tuned learning rates for convergence where plain gradient descent diverges. Fourth, as a benchmark, qgS tracks heavy output probability (r = −0.93) and needs no calibration, unlike linear XEB. However, it is ÙOÍ monotonic under amplitude damping: under device-calibrated noise with added idle relaxation it falls below its own noiseless value. Pairing qgS with the register-averaged qgZ removes the ambiguity in 23 of 24 circuits tested. Every number is reproduced by a script and pinned by a regression test in the public repository.
The Qang () framework [1] expresses single-qubit rotations through two measurement-level quantities, the polar bias and the outcome entropy . We report results obtained with the open-source reference implementation qang. First, the two metrics are linked by an exact, branch-free identity, , which holds for any single-qubit state. Consequently, any new multi-qubit information must come from the joint outcome distribution. We name the resulting Z-basis total correlation and give its bounds and its scope: it is not an entanglement measure. Second, we identify exact blind spots: graph-state entanglement and readout-only dephasing are invisible to every Z-diagonal statistic, and qg-space updates cannot reach optima outside because of the range. On H unsigned qg updates recover none of the correlation energy. Keeping the sign of as a branch label removes this limit and reaches the FCI energy, but the best signed update reduces to -space descent, so qg coordinates bring no optimization gain. Third, we study a pole-damped gradient step whose damping schedule is . On a 300-landscape benchmark, H, and a 4-qubit LiH Hamiltonian with a mixed ansatz, it trades 3–20 more iterations at well-tuned learning rates for convergence where plain gradient descent diverges. Fourth, as a benchmark, tracks heavy output probability () and needs no calibration, unlike linear XEB. However, it is not monotonic under amplitude damping: under device-calibrated noise with added idle relaxation it falls below its own noiseless value. Pairing with the register-averaged removes the ambiguity in 23 of 24 circuits tested. A summary section lists fifty-two follow-up studies (QML encoding, control quantization, noise-type detection, circuit knitting, chemistry, error mitigation, thermal states, error correction, Hubbard dynamics, constrained QAOA, qubit characterization, syndrome tracking, coherence, few-shot estimation, IonQ noise models, shot-noise propagation, neural quantum states, coherent gate errors, symmetry checks in rotated bases, Heisenberg-chain dynamics, classical shadows, noisy Grover search, GHZ metrology, circuit-level surface codes, a lattice gauge theory, and the choice between filter and ZNE by shot budget), each with its classical or standard control; for the qg quantities, direct measurement is 3–41 cheaper than classical shadows, and the per-qubit reading loses to the outcome histogram when the answer is a bitstring; the symmetry-witness results are developed in a companion paper. Every number is reproduced by a script and pinned by a regression test in the public repository.
This repository hosts the manuscript and companion code for: "The Qang (qg): A Unified Angular-Probability Unit and Metric for Parametric Quantum Circuit Design." Abstract:We introduce the Qang (qg), a unified angular-probability metric designed to bridge the gap between physical rotation angles in the Bloch sphere and measurement probability distributions in parametric quantum circuits. We formalize two complementary operational modes:1. The Polar Bias Qang (qg_Z), defined as qg_Z(θ) = cos(θ) = <σ_z>, mapping directly to the computational basis expectation value in [-1.0, 1.0].2. The Measurement-Outcome Entropy Qang (qg_S), defined as qg_S(θ) = H(cos²(θ/2)), quantifying the binary Shannon entropy of Born-rule outcomes in [0.0, 1.0]. We analyze the domain limitations, phase decoupling, and scale invariance properties of these metrics, as well as the gradient singularity at the poles (|qg_Z| -> 1). Numerical implementations, closed-form verification, and an interactive explorer notebook are included. Repository & Companion Code:GitHub: https://github.com/Viny2030/quangORCID: 0000-0001-8884-4811
In the qg framework [1] a qubit is described by its polar bias . When a circuit conserves a Hamming weight, the average of over the register is fixed in advance: for excitations on qubits. This one number is a reference-free witness of noise that changes the conserved quantity, and its shot-level version is the known symmetry-verification filter [2], [3]. We measure when the witness and the filter help, and when they do not, on six problems with classical controls and on four molecules for the rotated-basis extension, using calibration-based and vendor noise models. (i) On H the filter lowers the energy error from 18 to 5.5 mHa, below Hartree–Fock (20.3 mHa), and it stays below Hartree–Fock along the dissociation curve. (ii) Across noise channels the witness predicts the outcome: it moves under T1 and the filter removes 94–95% of the error for free, while under dephasing it stays at its ideal value and the filter does nothing. Zero-noise extrapolation is complementary and fails on idle decoherence; the combination reaches mHa. (iii) With separate spin-up and spin-down witnesses, a second filter pays off only when a spin-leak witness is nonzero. Qubit routing on a heavy-hex device creates such leak in 1D Hubbard dynamics. (iv) In QAOA with an “exactly ” constraint, the filter doubles the success probability of the constraint-preserving ansatz. Penalty QAOA at depth does not beat a random feasible guess, and a greedy algorithm beats all quantum variants. (v) The witness relies on calibrated readout. A heralded sweep separates thermal population from asymmetric readout error, which the standard calibration conflates, and gives an unbiased effective temperature ( mK) together with and . (vi) On IonQ’s trapped-ion noise models the filter doubles the constraint-preserving QAOA success and halves the H error in every run, while zero-noise extrapolation, which must fold native gates because CX folding is undone by the compiler, is nearly unbiased but too noisy at the 2000-shot limit. (vii) Beyond symmetries, every stabilizer expectation of the repetition and Leung codes is a power of one single-qubit qg, so the syndromes track noise drift for free and choose between codes by the rule . (viii) Z-basis filtering cannot reach Hamiltonian terms measured in rotated bases, which hold most of the LiH error. Parity and mod 4 checks on two ancillas do reach them: they cut the LiH error (below Hartree–Fock on a generic model, on IonQ’s forte-1 model) and approach the full number-projection ceiling up to 12 qubits (H, HO, H). That ceiling shrinks with depth, because number-conserving errors grow. Coherent over-rotation of the entangling gate passes through native-gate ZNE untouched, while the filter removes 94% of it. We also report where the method stops: Hubbard circuits beyond 190 two-qubit gates, where the witness correctly reports unital scrambling. No result is a quantum advantage over classical computation. Every number is reproduced by a script and pinned by a regression test.
Each qubit’s local qg values are a point in the Bloch ball. Its squared distance from the centre, , is the radius of the sphere on which the qubit lives (area ). For a pure register the deficit is the known one-tangle of the qubit with the rest; with noise it mixes entanglement and impurity. We show that circuits which conserve the Hamming weight remove this ambiguity under one noise model. In a state of fixed weight on every qubit, so and the deficit is read from Z-basis shots alone; under equal amplitude damping the qg symmetry filter returns the noiseless state exactly, so the filtered deficit is the noiseless one-tangle. In a pre-registered simulation (90 configurations) the filtered deficit equals it to , and on product states it is exactly zero, while the unfiltered readout reports a false deficit of 0.60–0.95. On five device noise models (three IBM backends, two IonQ) the filter cuts the deficit error of trained circuits 3.2–4.0 times, but it removes only 15–33% of the false deficit of a product state: errors that move an excitation inside the sector survive the filter, and near a pole the deficit amplifies them as . The radius also separates the polar angle in radians from the length of the vector, : this angle is exact under depolarizing noise, where $\arccosqg_Z$ errs by up to 0.7 rad, and it is 5–9 times more accurate on the device noise models once the noise gates are actually executed. Three of nineteen predictions failed and are reported. Everything is reproduced by the open-source library qang (version 0.6.8, pip install qang).
Qiskit and Cirq both parameterize single-qubit rotations by a raw angle in radians (RYGate(theta), cirq.ry(theta)), with no built-in notion of the physically meaningful quantity that angle controls: the Born-rule population bias of the resulting state. The qang unit (qg), published in Monteverde (2026), gives that quantity a name and a closed algebra: qg_Z = cos(theta) = <sigma_z> (the "Polar Bias Qang") and qg_S = H(p0) (the "Entropic Qang", binary Shannon entropy of the outcome distribution), with explicit, tested inversion formulas back to theta. A reference implementation already exists and runs today: native RQangGate/FullRQangGate classes for Qiskit, an equivalent native gate for Cirq, and a quang.transformations module (Householder reflections, the Grover diffuser, and Born-rule transition probabilities, all written directly in terms of qg_Z with no runtime trigonometry beyond the algebra's own square root). This has been validated locally against both simulators and, for the Qiskit gate, against IonQ's cloud simulator via qiskit-ionq, with results matching theory within ordinary shot noise. This RFC proposes upstreaming a qang module into each library: a first-class way to parameterize rotations, transition probabilities, and (per Section 6 below) calibration/estimation by qg value instead of by raw angle, alongside -- not replacing -- the existing angle-based API.
We aim to rigorously address how local noise affects classical simulability of quantum dynamics benchmarked by tensor-network methods. Using operator entanglement entropy (OEE), we prove the following: (1) For single-qubit depolarizing noise on arbitrary circuits, tensor networks with poly(n) bond dimension suffice for fixed absolute Hilbert-Schmidt error after $\order{1}$ depth, while relative error demands $\order{\log n}$ depth; and this bound is optimal. (2) For single-qubit depolarizing noise on 1D local circuits, the existence of whole-trajectory error-bounded matrix product operator (MPO) of poly(n) bond dimension at all depths. (3) For general single-qubit noise in 1D brickwall circuits, random two-design gates with contraction coefficient c<1/3 yield an $\order{1}$ OEE plateau with probability 1−Te−Ω(n), while arbitrary gates with c<1/48 give $\order{\log n}$ OEE in the worst case. (4) In higher dimensions, these bounds yield uniform-in-depth poly(n) average boundary-bond dimensions for projected entangled pair operators~(PEPO) across every cut -- under depolarizing noise at either absolute or relative accuracy, and under general noise with strong contraction at absolute accuracy. Our results establish a rigorous connection between certain noise models, circuit types, and their classical simulability.