Stabilizer Nullity in Quantum Systems

• Stabilizer nullity is a measure of non-stabilizerness defined as the difference between the number of qubits and the logarithm of the stabilizer group's size.
• It acts as a magic monotone, guiding resource estimation by providing lower bounds on non-Clifford gate counts in fault-tolerant quantum circuits.
• Algorithms compute nullity through tensor-network methods, group actions, and combinatorial approaches, with applications in quantum many-body physics and digraph invariants.

Stabilizer nullity quantifies the extent to which a quantum state, operator, matrix, or group element fails to be stabilized by the actions of a particular subgroup, such as the Pauli group or a symmetry group. It serves as a nonstabilizerness monotone in quantum information theory, bounding resource usage in fault-tolerant quantum computation and constraining classical simulability of quantum systems. Across contexts—including quantum many-body systems, fault-tolerant gate synthesis, and group actions on matrix spaces—stabilizer nullity is always a function of the dimension or rank of a stabilizer subgroup or algebra.

1. Stabilizer Nullity: Definitions and Basic Concepts

For an $n$-qubit pure quantum state $|\psi\rangle$, the Pauli stabilizer group is

$$\mathrm{Stab}(|\psi\rangle) = \{P \in \mathcal{P}_n : P|\psi\rangle = \pm |\psi\rangle\}$$

where $\mathcal{P}_n$ is the $n$-qubit Pauli group. The stabilizer nullity $\nu(|\psi\rangle)$ is defined as

$$\nu(|\psi\rangle) = n - \log_2|\mathrm{Stab}(|\psi\rangle)|$$

Equivalently, $\nu=0$ iff $|\psi\rangle$ is a stabilizer state. For general group actions on vector spaces, such as orthogonal similarity on symmetric matrices, the stabilizer nullity at $y$ is the dimension of the group-theoretic stabilizer:

$\mathrm{nullity}(y) = \dim \,\mathrm{Stab}(y)$

where $\mathrm{Stab}(y)$ is the subgroup leaving $y$ invariant under the relevant action (Starčič, 2020).

In the context of unitary operators, the unitary stabilizer nullity $v(U)$ for an $n$-qubit unitary $U$ extends the definition:

$$v(U) = 2n - \log_2 |U\mathcal{P}_nU^\dagger \cap \mathcal{P}_n|$$

capturing the nonstabilizerness of dynamical transformations (Jiang et al., 2021).

2. Quantum Computational Significance

Stabilizer nullity is a pivotal nonstabilizerness monotone: it strictly decreases under stabilizer operations and is invariant under Clifford unitaries (Lami et al., 2024). Notably, the unitary stabilizer nullity lower-bounds the $T$-count, i.e., the minimum number of non-Clifford $T$ gates required in a fault-tolerant synthesis:

$t(U) \geq v(U)$

with $v(T) = 1$ and $v(C) = 0$ for any Clifford $C$ (Jiang et al., 2021). For classes of gates and circuits, explicit families have been constructed where the unitary stabilizer nullity scales as $2n$, strictly surpassing the state stabilizer nullity by a factor of two and providing tighter resource lower bounds.

Comparison theorems guarantee that the unitary bound always equals or outperforms the state-based lower bound, and explicit constructions (e.g., $U_n = C^{n-1}Z \cdot H^{\otimes n} \cdot C^{n-1}Z$) illustrate strict separation between the two (Jiang et al., 2021).

3. Nullity in Many-Body Quantum Physics

For many-body quantum systems, stabilizer nullity quantifies the deviation of quantum states from the stabilizer formalism—a regime where efficient classical simulation is possible. When Hamiltonians are perturbed by $k$ arbitrary Pauli terms, all eigenstates possess stabilizer nullity at most $k$. Small nullity thus bounds the algorithmic complexity of eigenstate sampling, time evolution, thermalization, and entanglement entropy computation, with runtime scaling exponential only in the nullity parameter (Gu et al., 2024).

States of bounded nullity exhibit a "Clifford-compression:" they can be mapped via a Clifford unitary $C$ to the form

$$C|\psi\rangle = |x\rangle_{n-\nu} \otimes |\phi\rangle_{\nu}$$

where the nonstabilizerness is concentrated on a $\nu$-qubit subsystem. This structure enables classical algorithms whose complexity is polynomial in $n$ and exponential in $\nu$, e.g., $O(n^3+2^{3\nu})$ for random eigenstate sampling (Gu et al., 2024).

In perturbed stabilizer models ("doped stabilizer states"), this approach enables efficient simulation of out-of-equilibrium and highly entangled systems as long as the perturbation rank (and thus nullity) remains bounded.

4. Algorithms for Learning Stabilizer Nullity

In tensor network representations, particularly matrix product states (MPS) of high bond dimension, stabilizer nullity can be efficiently estimated with classical algorithms that identify the full stabilizer group by biased sampling in the Pauli (or Bell) basis. The number of independent generators found yields the stabilizer rank $k$, hence the nullity $\nu = n-k$ (Lami et al., 2024).

A key property is that, with high probability and $O(1)$ algorithmic sweeps, all generators are identified even in regimes of high entanglement. The per-sweep runtime is $O(n\chi^3)$, where $\chi$ is the MPS bond dimension. This bridges tensor-network simulability and Clifford-based classical algorithms, enabling systematic study of "magic" in 1D quantum systems.

5. Stabilizer Nullity in Matrix Group Actions

For the action of complex orthogonal groups on spaces of symmetric and Hermitian matrices by similarity or $*$-conjugation, stabilizer nullity is given by the dimension of the stabilizer subgroup:

$$\mathrm{Stab}(y) = \{Q \in O_n(\mathbb{C}): Q^T A Q = A\} \quad \text{or} \quad \{Q \in O_n(\mathbb{C}): Q^* A Q = A\}$$

Closed-form and recursive algorithms exist to compute these dimensions, based on block diagonalization, Toeplitz block decomposition, and solving Sylvester-type equations (Starčič, 2020). The nullity depends on the Jordan or canonical block structures, with explicit dimension formulas for each irreducible block type:

Matrix type	Stabilizer Nullity formula	Notes
Symmetric (similarity)	$$\displaystyle \sum_{r=1}^N a_r m_r ((m_r-1) + \sum_{s<r} m_s)$$	$a_r$ block size, $m_r$ multiplicity
Hermitian ($*$-conjugacy)	Structured by block: exact or lower bounds	Depends on eigenvalue type (see below)

For example, real-zero eigenvalue blocks lead to explicit dimension counts, while unit or complex eigenvalues may only result in lower bounds (Starčič, 2020).

6. Nullity and Digraphs

In combinatorics, the stable maximum nullity $\overrightarrow{\nu}(D)$ of a digraph $D$ is the largest nullity among matrices in a prescribed pattern class satisfying the Asymmetric Strong Arnold Property (ASAP). Here, the nullity reflects the dimension of the kernel of a matrix constrained by the digraph's adjacency and diagonal pattern (Arav et al., 2024).

A fundamental result is that a digraph and its reverse have stable maximum nullity at most one if and only if both are partial $1$-DAGs (spanning subdigraphs of $1$-DAGs), or, equivalently, have Kelly-width at most two. Monotonicity under directed minors and subdigraphs is established, and small examples illustrate the achievable nullity as a sharp graph invariant.

7. Properties, Monotonicity, and Operational Applications

Key structural properties of stabilizer nullity include:

• Faithfulness: Nullity is non-negative; zero if and only if the object is stabilized by the full group (e.g., Clifford for unitaries) (Jiang et al., 2021).
• Clifford invariance: Stabilizer nullity is invariant under conjugation by Clifford unitaries or group symmetries.
• Tensor additivity: For composite systems, nullity is additive, e.g., $v(U\otimes V) = v(U) + v(V)$ (Jiang et al., 2021).
• Subadditivity under composition: For unitaries, $v(UV) \leq v(U) + v(V)$.
• Lower bounds on resource cost: Nullity gives explicit lower bounds for non-Clifford resource counts in fault-tolerant synthesis (Jiang et al., 2021).
• Monotonicity under stabilizer operations: Nullity does not increase under measurement, Clifford evolution, or addition of ancillas (Lami et al., 2024).

Applications encompass efficient simulation of perturbed many-body Hamiltonians, tight lower bounds on quantum circuit resources, explicit magic monotone computation for highly entangled quantum states, and classification of matrix or graph invariants in pure mathematics and combinatorics.

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References:

• (Jiang et al., 2021) J. Jiang & X. Wang, "Lower bound for the T count via unitary stabilizer nullity"
• (Lami et al., 2024) "Learning the stabilizer group of a Matrix Product State"
• (Gu et al., 2024) "Doped stabilizer states in many-body physics and where to find them"
• (Arav et al., 2024) "The stable maximum nullity of digraphs and $1$-DAGs"
• (Starčič, 2020) "The stabilizers for the action of orthogonal similarity on symmetric matrices and orthogonal $*$-conjugacy on Hermitian matrices"