Certifying the fidelity of a prepared state to a target stabilizer state is a fundamental task in quantum information processing. Ref. [Phys. Rev. A 99, 042337 (2019)] gave the optimal worst-case lower bound from one fixed stabilizer generator gauge, but gauge dependence can leave a large fidelity ambiguity. We develop an adaptive extension that reports the full certified fidelity interval. First, for a single gauge, we derive the complementary optimal worst-case upper bound. We then formulate gauge selection as an adaptive design problem in which each round solves exact endpoint linear programs and chooses a new gauge by a witness elimination policy. We prove monotonic tightening, exact recovery once all nontrivial stabilizers are covered, and the worst-case necessity of full coverage. Finally, we identify structured syndrome distributions for which adaptivity beats this exponential benchmark, and we numerically confirm faster concentration.
We give an exact-arithmetic case study of maximum-likelihood (ML) stabilizer decoding under uncertain dephasing, separating channel-adaptive, fixed deterministic, and fixed randomized recovery decisions. For the five-qubit [[5, 1, 3]] code, we enumerate syndrome–stabilizer-coset masses and certify the homogeneous dephasing success polynomial 1 − 10t³ + 15t⁴ − 6t⁵. On the full homogeneous interval, adaptive ML has robust success 1/2, every fixed deterministic syndrome decoder has worst-case success zero, and fixed randomization attains 1/2. For a heterogeneous but conditionally independent family with site rates (t, t, 2t, 2t, 2t), 0 ≤ t ≤ 1/2, we obtain four algebraic switching thresholds and an exact five-piece adaptive ML curve. The adaptive and optimal fixed randomized worst-case success both equal 3/4, whereas the best fixed deterministic decoder has success min₀≤ₜ≤₁⁄₂ (16t⁵ − 28t⁴ + 10t³ + 3t² − 2t + 1) ≈ 0.749591899. A finite exhaustive reduction and rational witness checks certify strict separation. A CSS-code theorem and independent Steane-code calculation establish a second, strictly separated heterogeneous minimax instance with R_det < 0.4999 < R_rand = R_ML = 1/2. We retain biased-noise orientation sensitivity and two counterexamples to invalid vertex shortcuts, and distinguish these instance-level certificates from general decoding algorithms and established enumerator or robust-optimization methods.
The apparent absence of a stabilizer-specific phase-space description for qubit systems is not fundamental, but arises from fixing a state-independent discrete Wigner map. Allowing the phase-space representation to adapt to the state through a simple optimization procedure, we prove that every qubit stabilizer state admits an exact delta-function discrete Wigner representation for any fixed phase-space partition. This optimization can be implemented directly at the level of measurement statistics by selecting maximal probabilities across mutually unbiased bases forming the partition.
This construction naturally leads to a non-stabilizer witness defined by the optimized value of the Wigner function at the origin. Stabilizer states saturate the universal lower bound of this witness for pure states, whereas a value strictly larger than the bound certifies non-stabilizerness. Although the witness is not a finite-size resource monotone, it becomes sharply concentrated and effectively invariant under random Clifford transformations in the large-qubit limit, suggesting an asymptotically linear growth law for broad families of non-stabilizer states. Our results restore a meaningful geometric phase-space characterization of qubit stabilizer structure and provide an operational diagnostic of non-stabilizerness.
Fermionic Gaussian states are the workhorse reference states for qubit-based quantum simulation of fermionic systems, yet existing certification protocols either proceed via fidelity estimation, leading to suboptimal sample complexity in the target precision
ε, only apply to Haar-typical states, or require adaptive measurements on a large fraction of qubits. We give an adaptive protocol that certifies any
d-mode pure fermionic Gaussian state using
O(d2ε−1) copies, single-qubit measurements with only one qubit measured adaptively per copy, and
O(d3) classical processing time per copy. The
ε−1 dependence is optimal for the certification problem, even among strategies using entangled measurements. The sample complexity is controlled by the spectral gap of a
d↔d−2 down-up walk---a Markov chain studied in the theory of high-dimensional expanders, equivalent to a two-site Glauber dynamics in statistical physics---relating certification efficiency to relaxation time to equilibrium. The worst-case bound is tight for this protocol and is attained by physically relevant states, including ground states of the fully dimerized Su-Schrieffer-Heeger (SSH) chain and BCS pair states. In contrast, numerics up to
d=14 suggest that
O(dε−1) copies suffice to certify Haar-typical Gaussian states, a factor of
d improvement over the worst-case bound. Finally, because the bound depends only on the target's computational-basis distribution, the protocol extends to efficiently phase-dressed states---states obtained by injecting an efficiently computable diagonal phase to Gaussian states---with the same sample complexity. The class includes a continuous family of four-mode non-Gaussian magic states for matchgate computation.
The task of quantum state discrimination provides an operational characterization of distinguishability and plays a central role in quantum information science. Since the achievable success probability in quantum state discrimination depends both on the states being discriminated and on the allowed measurements, it is natural to study discrimination under physically motivated measurement constraints. Here, we investigate minimum-error quantum state discrimination under measurements implementable by both fixed and adaptive stabilizer circuits. Given arbitrary stabilizer-state ancillas, we show that fixed stabilizer circuits provide no additional discrimination power, whereas adaptive circuits do. Then, focusing on single-qubit systems, we derive analytical expressions for the success probability across both circuit classes when supplied with arbitrary (non-stabilizer) ancillas, showing that the performance gap between fixed and adaptive circuits persists. We further consider discrimination with non-stabilizerness incorporated directly into the measurement operators. Here, the optimization is formulated as a semidefinite program, and analytical bounds interpolating between the stabilizer and Helstrom limits for qubits are derived. Finally, we illustrate applications in quantum random access codes and bounds on unitary synthesis fidelity with a finite number of magic states.
We study single-copy stabilizer learning, the problem of identifying a stabilizer group of dimension
n−t from an
n-qubit quantum state
ρ. We obtain two complementary results. First, in the average case, logarithmic-depth local Clifford circuits suffice to efficiently learn almost all stabilizer groups with
t=O(logn), instead of the linear-depth measurements required in previous approaches. We support this result with numerical simulations for systems of up to 100 qubits. Second, we show that, in the worst case, any adaptive single-copy measurement scheme requires a number of samples that scales exponentially in
t. Together with existing results on two-copy learning, our findings suggest that, for large
t, identifying Pauli symmetries of a quantum system exhibits a quantum advantage in the learning setting.
We propose a measurement scheme that validates the preparation of an
n-qubit stabilizer state. The scheme involves a measurement of
n Pauli observables, a priori determined from the stabilizer state and which can be realized using single-qubit gates. Based on the proposed validation scheme, we derive an explicit expression for the worst-case fidelity, i.e., the minimum fidelity between the stabilizer state and any other state consistent with the measured data. We also show that the worst-case fidelity can be certified, with high probability, using
O(n2) copies of the state.