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.
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.
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
We write fifteen standard quantum gates and fourteen standard quantum algorithms in qg units [1]. An -qubit state is described by its qg values, for every Pauli string , and a gate acts as . With this one rule each gate becomes a table of qg transformations and each algorithm a readout identity: which qg values hold the answer. The formulation is the Pauli (Heisenberg) representation and adds no new physics. It gives three practical classifications in the units used by the rest of the qang project: which gates conserve Hamming weight, and therefore admit the qg symmetry witness and filter; which algorithms are read out from a few qg values and which need the full outcome histogram; and where reading qg values is strictly worse than the standard readout. Three readout identities are, as far as we know, not stated in this form elsewhere: the per-qubit polar bias after Grover iterations, Simon’s secret as the unique non-trivial Z-parity equal to one, and quantum counting as a single . Every identity is checked numerically and pinned by tests in the open-source library qang (version 0.5.1, pip install qang), module qang.formulation.
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.
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.
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.
We ask where the qg description of a qubit, and its companions [1], is useful for cryptography. Post-quantum cryptography (lattice-based key encapsulation such as ML-KEM [2]) is classical and has no place for it. Quantum cryptography does, because its raw data are qg values. (i) In BB84 [3] over a qubit channel with relaxation, the sum of the received polar biases is a T1 witness. Natural relaxation raises it, while intercept-resend adds symmetric errors that leave it unchanged. A likelihood-ratio monitor built on it detects intercept-resend as well as the standard QBER monitor, but raises no false alarms when the memory’s T1 degrades, where the QBER monitor fires in 48–100% of windows. By construction it misses an attacker who mimics T1, which the QBER monitor catches, so the two are complementary and neither changes the secret-key rate. With finite keys [4] the diagnosis comes free from the error-corrected block: at key bits it attributes every block correctly, including a small attack hidden under drift. (ii) For a qubit random-number generator with a trusted measurement, the certified min-entropy is , independent of . The usual output-bias estimate is unsafe whenever the state is mixed: it certifies 0.97 bits per shot where 0.015 are private. A qg estimate from -rotation test rounds is safe in every run, and a readout calibration recovers 10–50% more certified bits. On IonQ trapped-ion noise models, with an environment ion that learns the output, the naive estimate stays near 1 bit while the qg estimate remains safe and captures 82–90% of the private randomness. All results are simulations, reproduced by scripts and pinned by regression tests.