Stabilizer Expectation Values

• Stabilizer expectation values are measurements of a state's alignment with Pauli operators, forming a probability distribution central to quantifying quantum resources.
• They enable the computation of stabilizer Rényi entropy, providing rigorous benchmarks for nonstabilizerness and insights into resource costs in fault-tolerant quantum systems.
• Their experimental accessibility via randomized Clifford measurements connects theoretical magic measures to practical diagnostics in quantum chaos and many-body physics.

Stabilizer expectation values play a central role in quantum information theory, particularly in the analysis of many-body quantum states, resource theories of magic, and measurement protocols for quantum devices. The expectation value of a stabilizer or generalized Pauli operator in a given quantum state quantifies the state's alignment with that operator and forms the basis for constructing quantitative measures such as the stabilizer Rényi entropy. These quantities enable rigorous characterizations of nonstabilizerness, experimental diagnostics of magic, and efficient computational protocols in both commuting and non-commuting stabilizer frameworks.

1. Formal Definition and Construction

Let $\mathcal{P}_n$ denote the $n$-qubit Pauli group, and $|\psi\rangle$ a pure quantum state on $n$ qubits with Hilbert space dimension $d=2^n$. For each $P \in \mathcal{P}_n$, the stabilizer expectation value is defined as $\langle \psi|P|\psi\rangle$. The squared, normalized expectation values

$$\xi_P(\psi) = d^{-1} |\langle \psi|P|\psi\rangle|^2$$

constitute a probability distribution over $\mathcal{P}_n$ satisfying $\sum_P \xi_P(\psi) = 1$ (Leone et al., 2021). For operator-valued measures of non-stabilizerness, such as the stabilizer Rényi entropy, this probability distribution serves as the foundational object.

2. Stabilizer Rényi Entropy and Magic Measures

The stabilizer $\alpha$-Rényi entropy for a pure state $|\psi\rangle$ is defined by

$$S^{(r)}_\alpha(|\psi\rangle) = \frac{1}{1-\alpha} \log \left(\sum_{P \in \mathcal{P}_n} |\langle\psi|P|\psi\rangle|^{2\alpha}\right) = M_\alpha(\psi) + \log d,$$

where $$M_\alpha(\psi) \equiv \frac{1}{1-\alpha} \log \sum_P \xi_P(\psi)^\alpha - \log d$$, and $M_2$ is particularly relevant for experimental protocols. The entropy $S^{(r)}_\alpha$ is a faithful magic monotone, vanishing if and only if $|\psi\rangle$ is a stabilizer state, invariant under Clifford unitaries, and additive with respect to tensor product states (Leone et al., 2021).

These quantities are tightly bounded by known resource measures of magic, such as the stabilizer nullity and robustness of magic. Explicitly, $M_\alpha(\psi) \leq \nu(\psi)$ for all $\alpha > 0$, where stabilizer nullity $$\nu(\psi) = \log d - \log |\operatorname{Stab}(\psi)|$$. For the robustness of magic $\mathcal{R}(\psi)$, $M_\alpha(\psi) \leq 2 \log \mathcal{R}(\psi)$ for $\alpha \geq 1/2$ (Leone et al., 2021).

3. Computation and Experimental Measurement

A salient advantage of stabilizer expectation value distributions is that their associated Rényi entropies are computable directly from measured Pauli string expectations, with no requirement for convex or minimization procedures (Leone et al., 2021). This tractability extends to experimental protocols: $M_2(\psi)$ can be obtained via randomized Clifford measurements. The protocol consists of the following steps:

• Prepare many copies of $|\psi\rangle$.
• For each measurement, apply a randomly chosen Clifford $C \in \operatorname{Cliff}(2^n)$ and measure in the computational basis.
• Estimate the outcome distribution $p(s|C)$.
• Using four-point empirical correlators and averages over measurement outcomes, compute

$$M_2(\psi) = -\log \left[ \sum_{s_1,...,s_4} (-2)^{-|s_1\oplus s_2\oplus s_3\oplus s_4|}\; \mathcal{Q}(s_1,s_2,s_3,s_4) \right] - \log d$$

where $\mathcal{Q}$ is the averaged four-point function over Clifford draws.

Sampling complexity to achieve error $\epsilon$ scales as $O(\epsilon^{-2})$ (Leone et al., 2021).

4. Connection to Out-of-Time-Order Correlators and Quantum Chaos

Nonstabilizerness as captured by the stabilizer Rényi entropy is deeply connected to the growth of operator complexity and quantum chaos, as quantified by out-of-time-order correlators (OTOCs). The average magic produced by a unitary $U$, defined as

$$\mathcal{M}_\alpha(U) = \frac{1}{|\operatorname{Stab}|} \sum_{|\psi\rangle \in \operatorname{Stab}} M_\alpha(U|\psi\rangle),$$

for the linear entropy case (denoted $M_{\text{lin}}$), admits a formula in terms of two-point and eight-point OTOCs of Pauli operators. The saturation of $\mathcal{M}_{\text{lin}}(U)$ to its maximal value parallels the approach to the universal OTOC plateau characteristic of Haar-random unitaries and chaotic dynamics, indicating that maximal nonstabilizerness is necessary for quantum chaos (Leone et al., 2021).

5. Distinction Between Commuting and Non-Commuting Stabilizer Formalisms

In the standard Pauli stabilizer formalism, all stabilizer operators commute, and the expectation value $\langle\psi|P|\psi\rangle$ is nonzero only if $P$ is in (or equivalent to) the stabilizer group. In this setting, for stabilizer states, the nonzero expectation values are restricted to $\pm 1/d$ on $d$ strings, resulting in $\xi_P$ supported on $d$ points (Ni et al., 2014).

The XS-stabilizer formalism introduces a non-commutative extension by allowing local group generators from $P = \langle \alpha I, X, S \rangle$ with $\alpha = e^{i\pi/4}$ and $S = \operatorname{diag}(1,i)$, leading to operators of the form $O = \alpha^s X(\mathbf{a}) S(\mathbf{b})$. Here, expectation value calculation involves a sum over phase factors and matching constraints, and may require circuit conjugation to reduce the observable to a Pauli operator, after which the standard tableau method applies with polynomial complexity (Ni et al., 2014).

6. Examples and Operational Behavior

Stabilizer expectation value distributions exhibit characteristic behavior for important classes of states (Leone et al., 2021):

• Stabilizer States: Concentrated support; $M_\alpha = 0$ for all $\alpha$.
• Tensor-Powers of Magic States: E.g., for $|H\rangle^{\otimes n}$, $M_\alpha$ grows linearly in $n$.
• Haar-Random States: $\sum_P |\langle\psi|P|\psi\rangle|^4 \approx 2/d^2$, so $M_2 \approx \log d - \log 2$.
• States from Chaotic Circuits: Rapid interpolation of $M_{\text{lin}}$ from 0 (Clifford) to maximal values as chaos develops.

In non-commuting stabilizer frameworks, explicit calculation of expectation values (e.g., in “cubic” XS states or topological models) requires handling non-trivial phase structures and possible non-commutativity, but remains tractable via conjugation and tableau-based computation (Ni et al., 2014).

7. Significance and Applications in Quantum Information

Stabilizer expectation values and their distributions underpin:

• Quantification and benchmarking of non-Clifford resource states and operations,
• Experimental diagnostics for magic buildup, scrambling, and many-body localization vs. thermalization,
• Lower bounds for T-counts or magic-state cost in fault-tolerant architectures,
• Analytical connections between quantum chaos, OTOC growth, and the resource theory of magic.

Their mathematical properties—computability, operational interpretability, tightness with other resource measures, and experimental accessibility—make them a pivotal tool in the study of quantum resource theories and nonclassicality in quantum devices (Leone et al., 2021, Ni et al., 2014).