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.
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.
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.
Circuits built from gates that conserve the Hamming weight keep their state in one weight sector, so the shots that left the sector can be discarded at read-out: the qg filter of the open-source library qang [1], a form of symmetry verification [2, 3]. We make it usable on IBM devices through IBM Runtime and Qiskit, and measure it on the noise models of twenty IBM devices. The library adds a Qiskit transpiler pass that decides whether the filter applies, read-out correction from each device's published per-qubit errors, qg read-outs of IBM Runtime Sampler and Estimator results, zero-noise extrapolation by gate folding that stacks with the filter, a prediction of the filter's kept fraction from the calibration, qubit selection with that prediction, a shot planner, and the weight-conserving quantum neural network (QNN) of qang as one parameterized job and as a qiskit-machine-learning SamplerQNN. Every prediction was committed before its run and is reported as passed or failed. On ten five-qubit weight-1 circuits the filter lowers the error of the measured qgZ = ⟨Z⟩ from 0.065-0.104 to 0.018-0.023 on three devices (0.013-0.022 with read-out correction), and filter plus zero-noise extrapolation reaches 0.011-0.015. On seventeen further devices never used before, the filter cuts the error 2.4-8.9× on every one (median 3.1× with read-out correction). Choosing the qubits by the predicted kept fraction keeps 6-17 points more shots than Qiskit's default layout on chain circuits, avoids a qubit published as reading 1 always, and the shot planner met a 0.02 target precision on three devices. For classification the gain is small: the QNN recovers 5-6 of the 10 decisions noise costs it. Reading a known five-component principal-component spectrum out of the data's density matrix in the weight-1 sector, filter plus read-out correction lowers the error 4.6× (median over twenty-one devices) and orders the spectrum correctly on 18 of 21 (13 raw). In Trotter simulation of the XY chain the filter halves the best error and moves the optimal number of Trotter steps deeper (16 of 21 devices), and in a quantum walk it turns a raw spreading indistinguishable from a classical random walk (exponent 0.87) into ballistic spreading (1.84). With two or three interacting excitations (XXZ chain) the gain falls from 5.6× to 2.6×: larger sectors let more of the noise through. Two interoperability bugs were found, one in qiskit-machine-learning and one in the library. All results are on device noise models; no hardware run is reported.
A quantum neural network (QNN) built from gates that conserve the Hamming weight keeps its data in one weight sector, so the shots that left the sector can be discarded at readout. This is the qg filter of the qang library [1, 2]. We show that, with equal amplitude damping (T1) on every qubit, the filtered readout of such a network is exactly the noiseless one in any weight sector. The kept fraction of shots is (1 − γ)k d for weight k and depth d, and training under T1 with the filter gives exactly the parameters of noiseless training. We then measure what this is worth on four small classification datasets. Every result is reported with the filter and without it, and every prediction was committed before its run, failures included. Over 60 runs per model, a network trained on a simulator and run under T1 gains +2.5 points from the filter at weight 1 (95% CI [0.8, 4.1]) and +16.5 points at weight 2 ([11.8, 21.2]), recovering its noiseless accuracy exactly. A network trained under the calibrated noise without the filter catches up (diflerences of 0.1 points), and the filter corrects neither dephasing nor unequal T1 across qubits, although alone it loses less than one point up to a ±80% spread of 1/T1. With noise-aware training and a richer readout, the model without the filter was 1.2 points better in one study, but over 60 new runs that gap shrank to +0.5 points and was not significant. The weight-2 deficit comes from the pair-product encoding, not from the readout; with an encoding that fills the whole sector and a full-rank readout, weight 2 reaches the weight-1 accuracy. On noise models of IBM and IonQ devices the filter cuts the error of the measured qgZ by 2.7-3.6×; on the IonQ Forte-1 noise model over 189 inputs of four datasets the filtered model is 1.1 points more accurate than the raw one, and calibrating the errors the filter keeps with five echo circuits per model halves the remaining error of the decision value: decisions that difler from the noiseless model fall from 15 (raw) and 7 (filter) to 3 over 945 readings on five device noise models. With a faithful weight-2 encoding the advantage of the unfiltered model under noise-aware training disappears, and a classifier given both the filtered and the raw features gains nothing over the filter alone. Over 513 readout configurations, including kept fractions down to 2.6×10−7, unequal T1 and dephasing, the filter has the lower readout error whenever about 20 shots are kept, and a closed-form rule predicts the winner. These models are classically simulable and no quantum advantage is claimed. The filter is a way to make noise-free training valid under relaxation, at a shot cost known in advance. The models are released as qang. qml (version 0.6.13, pip install qang).
A quantum neural network (QNN) built from gates that conserve the Hamming weight keeps its data in one weight sector, so the shots that left the sector can be discarded at readout. This is the qg filter of the qang library [1, 2]. We show that, with equal amplitude damping (T1) on every qubit, the filtered readout of such a network is exactly the noiseless one in any weight sector. The kept fraction of shots is (1 − γ)k d for weight k and depth d, and training under T1 with the filter gives exactly the parameters of noiseless training. We then measure what this is worth on four small classification datasets. Every result is reported with the filter and without it, and every prediction was committed before its run, failures included. Over 60 runs per model, a network trained on a simulator and run under T1 gains +2.5 points from the filter at weight 1 (95% CI [0.8, 4.1]) and +16.5 points at weight 2 ([11.8, 21.2]), recovering its noiseless accuracy exactly. A network trained under the calibrated noise without the filter catches up (diflerences of 0.1 points), and the filter alone corrects neither dephasing nor unequal T1 across qubits, although it loses less than one point up to a ±80% spread of 1/T1. Unequal T1 is corrected by training UÏtl the filter under the calibrated decay rates: the kept state depends only on their ratios, so the model lands within 0.5 points of noiseless at a ±100% spread, ignores a 10% calibration error and, at weight 2, loses 0.7 points under a 1.5-fold drift of all T1 that costs the raw noise-aware model 10. With noise-aware training and a richer readout, the model without the filter was 1.2 points better in one study, but over 60 new runs that gap shrank to +0.5 points and was not significant. The weight-2 deficit comes from the pair-product encoding, not from the readout; with an encoding that fills the whole sector and a full-rank readout, weight 2 reaches the weight-1 accuracy. On noise models of IBM and IonQ devices the filter cuts the error of the measured qgZ by 2.7-3.6×; on the IonQ Forte-1 noise model over 189 inputs of four datasets the filtered model is 1.1 points more accurate than the raw one, and calibrating the errors the filter keeps with five echo circuits per model halves the remaining error of the decision value: decisions that difler from the noiseless model fall from 15 (raw) and 7 (filter) to 3 over 945 readings on five device noise models. With a faithful weight-2 encoding the advantage of the unfiltered model under noise-aware training disappears, and a classifier given both the filtered and the raw features gains nothing over the filter alone. Over 513 readout configurations, including kept fractions down to 2.6 ×10−7, unequal T1 and dephasing, the filter has the lower readout error whenever about 20 shots are kept, and a closed-form rule predicts the winner. The same holds beyond the first model: at 8 qubits a weight-2 network keeps its noiseless 0.965 with the filter and drops to 0.705 without it; a five-class classifier (one class per qubit) keeps its noiseless accuracy under T1 with the filter and loses 15-18 points without it on five-class digits; training from 100 or 1000 shots per step with the filter is 1.0-1.5 points ahead of training without it; and the filtered gradient equals the noiseless one, while the raw one is K times smaller at weight 1. On five device noise models the echo calibration turns class confusions back: flipped decisions of the five-class model fall from 51 (raw) and 24 (filter) to 15 over 500 readings. The echo helped only where the filtered error was well above shot noise, and it still removed a third of the filtered error with a 28 × 28 matrix at 8 qubits. These models are classically simulable and no quantum advantage is claimed. The filter is
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.
Qubits with strongly biased noise, in which phase-flip errors are orders of magnitude more frequent than bit-flips, arise both naturally, as in electron and nuclear spins, and by engineering, as in stabilized cat qubits. This noise structure holds the promise of reducing the daunting hardware overhead of fault-tolerant quantum computing, but exploiting it requires physical operations that do not convert frequent phase-flips into rare bit-flips. In this review, we analyze the most prominent fault-tolerant protocols for biased-noise qubits, organized according to the available set of such bias-preserving operations. When this set is restricted to the CZ gate together with preparation and measurement in the X basis, we show that the complexity of the required syndrome extraction gadgets essentially cancels the benefit of the noise bias: at experimentally relevant error rates, one may as well ignore the bias and rely on standard error correction designed for depolarizing noise. The situation changes drastically when a bias-preserving CX gate is available: the hierarchy of errors can then be reflected in the structure of the code, with frequent phase-flips corrected by a dedicated high-threshold code and rare bit-flips by concatenation with a high-rate code. The same hierarchy also enables hardware-efficient preparation of magic states. Finally, as a bias-preserving CX is forbidden in naturally biased platforms and challenging in engineered ones, we present a measurement-based architecture in which a high-fidelity quantum non-demolition readout of multi-qubit Pauli Z operators takes its place, extending these overhead reductions to a much broader range of physical platforms.