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Z-basis metrics for parameterized quantum circuits: exact identities, structural blind spots, pole-damped optimization, and a relaxation-aware entropy benchmark

Vicente Humberto Monteverde

Published Sep 30, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.23063424

Related papers

Z-basis metrics for parameterized quantum circuits: exact identities, structural blind spots, pole-damped optimization, and a relaxation-aware entropy benchmark

Vicente Humberto Monteverde

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.

•Sep 23, 2026•0 citations

The register-mean polar bias as a symmetry witness: error mitigation, constrained optimization and qubit characterization in qg units

Vicente Humberto Monteverde

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.

•Sep 29, 2026•0 citations

The Qang (qg): A Unified Angular-Probability Unit and Metric for Parametric Quantum Circuit Design

Vicente Humberto Monteverde

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

•Sep 18, 2026•0 citations

The local Bloch radius as an entanglement readout: what the qg symmetry filter restores and what it does not

Vicente Humberto Monteverde

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).

•Oct 5, 2026•0 citations

What the polar bias can and cannot do for cryptography: telling drift from eavesdropping in BB84, and certified quantum randomness

Vicente Humberto Monteverde

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.

•Sep 30, 2026•0 citations

When Does Zero-Noise Extrapolation Help? A Controlled Bias–Variance Analysis of Error Mitigation for Variational Quantum Chemistry

Sahibjot Singh

Variational Quantum Eigensolvers (VQE) promise molecular ground-state energies on today's noisy intermediate-scale quantum (NISQ) processors, but device noise biases the estimated energy severely — under a device-calibrated noise model, the hydrogen molecule's energy error grows from 0.000 mHa (noiseless) to 303.6 mHa (noisy), roughly 190× the "chemical accuracy" threshold of 1.6 mHa that computational chemistry requires. Error-mitigation techniques repair such biases in software, but published results usually report each technique in isolation, making it impossible to know which mitigation to buy with a fixed shot budget. This work answers that question with a controlled comparison. Under identical circuits, noise profiles and shot budgets, I measure the bias and variance of four estimator stacks — raw sampling, twirled-readout extinction (TREX-style calibrated readout inversion), zero-noise extrapolation (ZNE), and ZNE+TREX — across independent random seeds. Three findings emerge. (1) ZNE is a bias–variance tradeoff, not a free lunch: a 7-point Richardson extrapolant reduces the deterministic bias 3200-fold (303.6 → 0.094 mHa) but amplifies statistical noise by a measured factor of up to 25× per run, while TREX removes readout bias with no variance penalty. (2) ZNE has a finite operating window in circuit depth: error recovery falls 99.97% → 99.1% → 95.5% as CX count grows 56 → 112 → 168, and is exhausted entirely for a 1616-CX lithium-hydride circuit at near-term error rates. (3) A noise-bias-tailored repetition code corrects the dominant error axis with logical error rates matching theory (3p², 10p³, 35p⁴ for distances 3, 5, 7) at strictly linear gate overhead (gates = 6d − 6). The results reframe mitigation selection as budget allocation on an empirically measured bias–variance frontier and provide open-source tooling to reproduce every number.

•Sep 30, 2026•0 citations
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.
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
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).
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.
Variational Quantum Eigensolvers (VQE) promise molecular ground-state energies on today's noisy intermediate-scale quantum (NISQ) processors, but device noise biases the estimated energy severely — under a device-calibrated noise model, the hydrogen molecule's energy error grows from 0.000 mHa (noiseless) to 303.6 mHa (noisy), roughly 190× the "chemical accuracy" threshold of 1.6 mHa that computational chemistry requires. Error-mitigation techniques repair such biases in software, but published results usually report each technique in isolation, making it impossible to know which mitigation to buy with a fixed shot budget. This work answers that question with a controlled comparison. Under identical circuits, noise profiles and shot budgets, I measure the bias and variance of four estimator stacks — raw sampling, twirled-readout extinction (TREX-style calibrated readout inversion), zero-noise extrapolation (ZNE), and ZNE+TREX — across independent random seeds. Three findings emerge. (1) ZNE is a bias–variance tradeoff, not a free lunch: a 7-point Richardson extrapolant reduces the deterministic bias 3200-fold (303.6 → 0.094 mHa) but amplifies statistical noise by a measured factor of up to 25× per run, while TREX removes readout bias with no variance penalty. (2) ZNE has a finite operating window in circuit depth: error recovery falls 99.97% → 99.1% → 95.5% as CX count grows 56 → 112 → 168, and is exhausted entirely for a 1616-CX lithium-hydride circuit at near-term error rates. (3) A noise-bias-tailored repetition code corrects the dominant error axis with logical error rates matching theory (3p², 10p³, 35p⁴ for distances 3, 5, 7) at strictly linear gate overhead (gates = 6d − 6). The results reframe mitigation selection as budget allocation on an empirically measured bias–variance frontier and provide open-source tooling to reproduce every number.