Gate Set Tomography determines experimental probabilities only up to similarity transformations of the fitted gate set. In practice, reported error metrics are therefore evaluated after choosing a gauge representative, often with gauge fixing restricted to unitary transformations.This work studies the geometry of the larger physical gauge orbit and asks whether nonunitary but still physically admissible representatives can improve reported gate-quality metrics beyond unitary gauge fixing.Analytically, the paper identifies a class of unital common-isotropic models for which nonunitary gauge transformations provide no advantage, and shows how a quadratic advantage appears when common nonunitality is introduced. Exact results are also derived for pure-translation gauges, together with a gauge-invariant lower bound.The analytical mechanism is tested numerically in two complementary settings. First, a historical GST model is audited conditionally, with the model-mismatch limitation stated explicitly. Second, a synthetic CPTP benchmark is generated, refitted by maximum likelihood, and shown not to be rejected by its goodness-of-fit test. In that well-specified setting the nonunitary gauge advantage reappears and agrees with the analytical prediction to numerical precision.A paired parametric bootstrap shows that the absolute value of the reported error metric can fluctuate substantially under sampling while the difference between unitary and nonunitary gauge fixing is much more stable because the two estimates are strongly correlated. An analytical bound also quantifies the maximum relative size of the nonunitary advantage within the solvable model family.The archive contains the manuscript, numerical certificates, reproducibility scripts, provenance information, package manifests, and automated verification tools. Version 1.8.1 differs from version 1.8 only by a clarification in the smoke-test output; the manuscript and scientific results are unchanged.
Gate Set Tomography determines experimental probabilities only up to similarity transformations of the fitted gate set. In practice, reported error metrics are therefore evaluated after choosing a gauge representative, often with gauge fixing restricted to unitary transformations.This work studies the geometry of the larger physical gauge orbit and asks whether nonunitary but still physically admissible representatives can improve reported gate-quality metrics beyond unitary gauge fixing.Analytically, the paper identifies a class of unital common-isotropic models for which nonunitary gauge transformations provide no advantage, and shows how a quadratic advantage appears when common nonunitality is introduced. Exact results are also derived for pure-translation gauges, together with a gauge-invariant lower bound.The analytical mechanism is tested numerically in two complementary settings. First, a historical GST model is audited conditionally, with the model-mismatch limitation stated explicitly. Second, a synthetic CPTP benchmark is generated, refitted by maximum likelihood, and shown not to be rejected by its goodness-of-fit test. In that well-specified setting the nonunitary gauge advantage reappears and agrees with the analytical prediction to numerical precision.A paired parametric bootstrap shows that the absolute value of the reported error metric can fluctuate substantially under sampling while the difference between unitary and nonunitary gauge fixing is much more stable because the two estimates are strongly correlated. An analytical bound also quantifies the maximum relative size of the nonunitary advantage within the solvable model family.The archive contains the manuscript, numerical certificates, reproducibility scripts, provenance information, package manifests, and automated verification tools. Version 1.8.1 differs from version 1.8 only by a clarification in the smoke-test output; the manuscript and scientific results are unchanged.
This is a separate preprint record that develops the robustness layer of the outcome-resolved mid-circuit-measurement gauge analysis. It uses the exact complete-positivity benchmark of the preceding preprint as an input, but its main result concerns the composition of imperfect-control uncertainty, residual-gauge conditioning, and instrument-model error. The general ingredients—gate-set-tomography gauge freedom, commutant geometry, perturbation by singular-value gaps, quantum-instrument physicality, and mid-circuit-measurement learnability—are prior art. The paper does not claim a new general GST gauge principle, a dimension-free tomography theorem, or experimental validation. Related identifier ● Identifier: 10.5281/zenodo.23043202 ● Relation: Is derived from (isDerivedFrom) ● Resource type: Publication -> Preprint ● Title: Outcome-Resolved Mid-Circuit Measurements as Exact Gauge Constraints: Complete-Positivity Windows, Correlation Certificates, and Robustness Limits Description / Abstract Self-consistent quantum characterization is invariant under similarity transformations, so an experimentally fitted gate set is generally identified only up to gauge. Exact trusted controls can reduce this ambiguity, but realistic controls are imperfect; simultaneously, a mid-circuit measurement (MCM) instrument is itself only approximately described by a model. These two uncertainties act at different logical stages and should not be merged before the gauge geometry is reconstructed. This preprint gives an explicit finite-dimensional stability analysis. For a finite reference control family ({R_i}), define the commutator map (\mathcal C(B)=([B,R_i])_i) and let (\gamma) be its smallest positive singular value. If two exactly gauge-equivalent implemented gate families ({G_i}) and ({G’_i}) both lie within Frobenius errors (\delta_i) of the same references, then every trace-preserving gauge matrix (B) lies within [\operatorname{dist}{F}(B,\ker\mathcal C)\le \frac{2\Delta}{\gamma}|B|{\mathrm{op}},\qquad\Delta=\left(\sum_i\delta_i^2\right)^{1/2}.] For the two-qubit control set (H\otimes I), (S\otimes I), (I\otimes H), (I\otimes S), and CNOT, the exact commutant is (\operatorname{span}{\mathcal D,\mathrm{id}-\mathcal D}), so the residual scalar gauge is (B_c=\mathcal D+c(\mathrm{id}-\mathcal D)). In the declared unweighted Frobenius convention, direct computation gives (\gamma=0.830129650657\ldots). The main result closes two previously separate robustness budgets. Suppose an independent physical or calibration argument places the scalar projection (c) in a positive prior interval ([\ell,u]). If (\alpha=2\Delta/\gamma<1), a self-normalized estimate removes the otherwise unknown factor (|B|): [|B-B_c|_F\le \bar e=\frac{\alpha M_0}{1-\alpha},\qquad M_0=\max(1,u).] If (\bar e<m_0=\min(1,\ell)), the condition number of (B) is explicitly bounded. Composing this control-side bound with a Choi-Frobenius instrument-model error (\varepsilon) yields a closed outer complete-positivity condition for the symmetric noisy Lüders MCM benchmark, [(1-\eta)|1-2p|,|c-c^{-1}|\le \eta+2d\rho,] where (\rho) is an explicit function of (\Delta,\gamma,\ell,u), and (\varepsilon). Hence the imperfect-control problem again reduces to an explicit scalar gauge interval. The result separates transverse control uncertainty from instrument-model uncertainty and identifies the loss of robustness as the prior gauge envelope approaches singularity.
Self-consistent quantum tomography is invariant under similarity transformations that can leave all observed circuit probabilities unchanged. Mid-circuit measurements (MCMs) add a qualitatively different object to this problem: an outcome-resolved quantum instrument has both a classical label and a post-measurement quantum state, so each outcome branch must itself remain completely positive under any admissible representation change. We study an exactly solvable residual-gauge model in even dimension d = 2r ≥ 4. After a fixed two-qubit Clifford control set reduces the trace-preserving similarity freedom to the scalar family B_c = D + c(id − D), we consider a symmetric two-sector Lüders instrument with label-confusion probability p and isotropic state-and-label replacement probability η. For the transformed branches J_{s,c} = B_c J_s^{η,p} B_c^{-1}, we prove exact necessary-and-sufficient complete-positivity criteria for both positive and negative scalar gauges. For c > 0 the condition is (1 − η)|1 − 2p||c − c^{-1}| ≤ η, while for c < 0 it is (1 − η)|1 − 2p|(|c| + 2 + |c|^{-1}) ≤ η. The result follows from an explicit Choi-spectrum calculation. It yields a closed positive gauge interval, an exact threshold for the appearance of negative gauges, and a logarithmic gauge radius |log c| ≤ arsinh(K/2), with K = η/[(1 − η)|1 − 2p|]. If the outcome label is discarded, the averaged channel is invariant under every B_c, so the additional gauge information disappears exactly. Two lag correlations determine K within the assumed model and therefore give an observable conditional gauge interval; a Hoeffding bound gives a finite-sample outer interval. We also derive an explicit outer interval when each instrument branch is known only within a Choi-Frobenius error budget. Finally, a Lüders-versus-measure-and-prepare countermodel shows that repeated outcome strings alone do not validate the quantum state continuation. A targeted literature audit finds substantial prior art on gate-set-tomography gauge freedom, instrument tomography, MCM learnability, and gauge-aware physicality intervals, including a closely related 2026 three-measurement protocol. The contribution is therefore deliberately narrow: the exact higher-dimensional full-complete-positivity boundary for the stated instrument family and its derived consequences, not the general idea that physicality can restrict tomography gauge.
Gate set tomography (GST) is a protocol for detailed, predictive characterization of logic operations (gates) on quantum computing processors. Early versions of GST emerged around 2012-13, and since then it has been refined, demonstrated, and used in a large number of experiments. This paper presents the foundations of GST in comprehensive detail. The most important feature of GST, compared to older state and process tomography protocols, is that it is calibration-free . GST does not rely on pre-calibrated state preparations and measurements. Instead, it characterizes all the operations in a gate set simultaneously and self-consistently, relative to each other. Long sequence GST can estimate gates with very high precision and efficiency, achieving Heisenberg scaling in regimes of practical interest. In this paper, we cover GST's intellectual history, the techniques and experiments used to achieve its intended purpose, data analysis, gauge freedom and fixing, error bars, and the interpretation of gauge-fixed estimates of gate sets. Our focus is fundamental mathematical aspects of GST, rather than implementation details, but we touch on some of the foundational algorithmic tricks used in the pyGSTi implementation.
The quality of quantum information processing devices has been improving at an unprecedented speed. How to faithfully represent the quality of these devices has become an increasingly imminent problem. In this thesis we focus on two aspects in representing and characterizing quantum devices. First, we discuss why most conventional quality metrics are not in principle appropriate to quantify experimentally-determined representations of gate-set elements, due to a gauge degree of freedom in quantum experiments. We then propose an operational quality measure for a gate-set and discuss its usefulness in representing degree of errors and improving experimental control. Second, we develop a protocol that separately and unambiguously characterizes state and measurement errors, relying on high-quality quantum gates. By integrating a method called randomized compiling, we derive a favorable upper bound for the effects of gate errors on the estimated parameters, and numerically demonstrate its performance in the presence of an adversarial gate error.
Noisy logic operations on a quantum computational register -- e.g., one or more qubits -- can be described by transfer matrices (a.k.a. CPTP maps or superoperators) that act linearly on the density matrix representing the register's quantum state. Collectively, these operations form a gate set. Gate sets have a gauge freedom; many gate sets that appear different actually predict the same experimental outcomes. A property of a gate set can be observable (and thus physically relevant) only if it is gauge-invariant. Unfortunately, no good gauge-invariant parameterizations of gate sets are known. We introduce the next best thing, a perturbative gauge-invariant parameterization of small Markovian errors in gate sets. We construct vector spaces of properties that are first-order gauge-invariant (FOGI). We show how to construct and understand FOGI properties, how to use them as coordinates to parameterize gate sets without gauge freedom, and how to extract approximately gauge-invariant error metrics.
Abstract We develop Ω TRINITAS ENGINE v6, a mathematical and computational framework for constraint-preserving projected unitary dynamics, with particular emphasis on invariant physical subspaces, gauge-constraint leakage, local unitary evolution, and reproducible Hamiltonian lattice gauge theory. The central problem is to distinguish approximation error in discrete or product-formula time evolution from genuine violation of a prescribed physical constraint. For an orthogonal projector P defining the physical subspace and a unitary evolution U, the framework quantifies constraint leakage through ||(I-P)UP|| and establishes the exact identity ||PUx||^2 = ||x||^2 - ||(I-P)Ux||^2 for physical initial states. In finite-dimensional Hilbert spaces, exact preservation of the physical subspace is characterized by invariance under U, equivalently by vanishing leakage. For products of local unitaries, leakage admits an additive bound in terms of the individual factors, while Hamiltonian-generated evolution is controlled through commutators between the projector and the local generators. In particular, when every local Hamiltonian term preserves the Gauss-law physical subspace, finite-depth product-formula dynamics can preserve the gauge constraint exactly even when the approximation to the target continuous-time evolution remains imperfect. We investigate these results computationally using an explicitly constructed gauge-enriched successor model based on the 1+1-dimensional U(1) lattice Schwinger model with staggered fermions, periodic boundary conditions, finite electric-flux truncation, and Gauss-law reduction. Independent constructions of the physical basis and Hamiltonian agree exactly in the audited finite systems, with zero matrix and spectral discrepancies and no detected Gauss-violating hopping transitions. A symmetric local product formula exhibits approximately third-order single-step error and second-order fixed-time convergence while maintaining numerical unitarity near machine precision. Sparse low-energy calculations at fixed physical volume reproduce the massless Schwinger rest-gap benchmark: an even-power continuum extrapolation gives 0.5642377786 in units of the gauge coupling, compared with the exact continuum value 1/sqrt(pi) = 0.5641895835. In the representative electric-flux cutoff test, variations are below the reported numerical resolution. Momentum-resolved T2 sectors further reproduce the expected relativistic dispersion with strong numerical agreement at the lowest momenta and a larger, explicitly retained deviation at higher momentum. The numerical analysis constitutes finite-regulator calibration and continuum extrapolation rather than certification of a controlled continuum limit. Systematic uncertainties associated with fit family, fit range, finite physical volume, regulator ordering, and higher-momentum behavior remain distinct from the proved constraint-preservation results. This separation prevents numerical agreement with a known continuum theory from being promoted to a stronger mathematical or physical claim. The work explicitly distinguishes mathematical proof, finite computational verification, numerical observation, calibration, and physical interpretation. Agreement with the Schwinger model is treated as a nontrivial calibration of the gauge-enriched successor construction, not as a derivation of gauge theory from canonical Ω TRINITAS dynamics. The manuscript does not establish endogenous U(1) gauge emergence from canonical Ω TRINITAS v9 and does not claim correspondence between Ω TRINITAS and the microscopic dynamics of the actual universe. Negative results, failed constructions, limitations, and unresolved questions are retained as part of the epistemic record. Ω TRINITAS ENGINE v6 therefore provides a reproducible reference framework for projected quantum dynamics, invariant subspaces, Gauss-law preservation, constraint leakage, Hamiltonian lattice gauge theory, local unitary circuits, Trotter and product-formula evolution, sparse exact diagonalization, electric-flux truncation, continuum extrapolation, and momentum-sector spectroscopy. The canonical reference implementation and machine-readable numerical results are designed to preserve a strict distinction between proved statements, computationally verified finite cases, numerical evidence, open problems, and unsupported physical correspondence, providing an auditable foundation for subsequent mathematical and computational investigations.