Optimal Stabilizer Ground State

• Optimal stabilizer ground state is defined as the stabilizer state that minimizes the energy expectation of a Pauli Hamiltonian, with fidelity used to resolve degeneracy.
• The approach uses closed maximally-commuting subsets and dynamic programming to efficiently reduce the search space, particularly in 1D and periodic systems.
• Applications span quantum simulation, state preparation, and efficient verification methods, serving as a certified reference in many-body physics.

An optimal stabilizer ground state is a stabilizer state selected relative to a Hamiltonian expressed in Pauli operators. In the basic variational sense, it is the stabilizer state minimizing $\langle \psi|H|\psi\rangle$ over all stabilizer states; in the refined convention introduced for degenerate stabilizer-ground-state manifolds, it is the member of that manifold with the highest fidelity with the true ground state. The notion turns the stabilizer formalism from a language for Clifford circuits and error correction into a ground-state ansatz with exact energy certificates for Pauli Hamiltonians, an exact linear-scaled algorithm for 1D local sparse systems, periodic extensions, and several roles in quantum simulation, state preparation, and imaginary-time refinement (Sun et al., 2024, Mao et al., 6 Mar 2026).

1. Definition and formal setting

For $n$ qubits, let the set of Hermitian Pauli operators be $\mathcal{P}_n=\pm\{I,X,Y,Z\}^{\otimes n}$. A stabilizer group $S\subset \mathcal{P}_n$ is a commuting subgroup with $-I\notin S$. If $g_1,\ldots,g_m$ are independent commuting generators, one writes $S=\langle g_1,\ldots,g_m\rangle$. A full stabilizer group has $n$ independent generators, and a stabilizer state $|\psi_S\rangle$ is the unique pure state satisfying $g_i|\psi_S\rangle=|\psi_S\rangle$ for all generators. Equivalently,

$n$0

and a state is a stabilizer state iff $n$1 is a full stabilizer group (Sun et al., 2024).

For a Pauli Hamiltonian

$n$2

the stabilizer ground state is defined as the stabilizer state minimizing the energy expectation. In this usage, “optimal” means the energy-minimizing stabilizer state for $n$3. If the true ground state of $n$4 happens to be a stabilizer state, then the optimal stabilizer ground state coincides with the true ground state; otherwise it is the best stabilizer approximation within the stabilizer ansatz (Sun et al., 2024).

A second convention arises when the minimizer is not unique. Over the set $n$5 of all $n$6-qubit stabilizer states, one defines

$n$7

and the stabilizer-ground-state manifold

$n$8

The optimal stabilizer ground state is then

$n$9

where $\mathcal{P}_n=\pm\{I,X,Y,Z\}^{\otimes n}$0 is the true ground state. In this convention, “optimal” resolves degeneracy by fidelity with the exact ground state rather than by energy alone (Mao et al., 6 Mar 2026).

This terminological split is substantial rather than merely stylistic. It separates exact minimization within the stabilizer ansatz from the additional problem of choosing, among energy-degenerate stabilizer minimizers, the one most useful as a proxy for the physical ground state.

2. Energy evaluation and algebraic characterization

The basic energy rule is unusually simple. For any stabilizer state $\mathcal{P}_n=\pm\{I,X,Y,Z\}^{\otimes n}$1 and Pauli $\mathcal{P}_n=\pm\{I,X,Y,Z\}^{\otimes n}$2,

• if $\mathcal{P}_n=\pm\{I,X,Y,Z\}^{\otimes n}$3, then $\mathcal{P}_n=\pm\{I,X,Y,Z\}^{\otimes n}$4;
• if $\mathcal{P}_n=\pm\{I,X,Y,Z\}^{\otimes n}$5, then $\mathcal{P}_n=\pm\{I,X,Y,Z\}^{\otimes n}$6;
• if $\mathcal{P}_n=\pm\{I,X,Y,Z\}^{\otimes n}$7, then $\mathcal{P}_n=\pm\{I,X,Y,Z\}^{\otimes n}$8.

Hence, for $\mathcal{P}_n=\pm\{I,X,Y,Z\}^{\otimes n}$9, only Hamiltonian terms contained in $S\subset \mathcal{P}_n$0 contribute to the energy; all others contribute $S\subset \mathcal{P}_n$1 (Sun et al., 2024).

This leads to the stabilizer-group energy functional

$S\subset \mathcal{P}_n$2

and

$S\subset \mathcal{P}_n$3

If $S\subset \mathcal{P}_n$4, then $S\subset \mathcal{P}_n$5 (Sun et al., 2024).

The central structural result is formulated in terms of closed commuting subsets. Let $S\subset \mathcal{P}_n$6 and define

$S\subset \mathcal{P}_n$7

Each $S\subset \mathcal{P}_n$8 generates a valid stabilizer group built only from Hamiltonian-supported Pauli terms. The theorem states

$S\subset \mathcal{P}_n$9

and the minimizing $-I\notin S$0 is the closed maximally-commuting subset (CMCS). Any stabilizer state stabilized by $-I\notin S$1 is a stabilizer ground state (Sun et al., 2024).

The CMCS formalism replaces brute-force search over all stabilizer states, whose count is of order $-I\notin S$2, by a search over subsets induced by the sparse Pauli support of the Hamiltonian. If $-I\notin S$3 acts on at most $-I\notin S$4 qubits, then

$-I\notin S$5

For sparse Pauli Hamiltonians with $-I\notin S$6, this scales as $-I\notin S$7 rather than with the full stabilizer-state count (Sun et al., 2024).

A noteworthy methodological point is that this formalism does not rely on the binary symplectic representation. By contrast, other large-scale OSGS procedures do use tableau and symplectic linear algebra to test membership, commutation, and generator independence in polynomial time (Sun et al., 2024, Mao et al., 6 Mar 2026).

3. Exact algorithms in one dimension and periodic extensions

For 1D $-I\notin S$8-local sparse Pauli Hamiltonians, there is an exact algorithm that computes the stabilizer ground state in linear time in the chain length $-I\notin S$9, with a sub-exponential dependence on locality $g_1,\ldots,g_m$0. The starting point is a decomposition of $g_1,\ldots,g_m$1 by the last non-identity site:

$g_1,\ldots,g_m$2

For any closed commuting subset $g_1,\ldots,g_m$3, one writes $g_1,\ldots,g_m$4 with $g_1,\ldots,g_m$5, and the stabilizer energy becomes

$g_1,\ldots,g_m$6

where

$g_1,\ldots,g_m$7

This localizes energy evaluation by last-site index (Sun et al., 2024).

The dynamic-programming construction uses a state machine

$g_1,\ldots,g_m$8

with

$g_1,\ldots,g_m$9

$S=\langle g_1,\ldots,g_m\rangle$0

and

$S=\langle g_1,\ldots,g_m\rangle$1

The crucial decoupling property is

$S=\langle g_1,\ldots,g_m\rangle$2

which turns the search into a Markov-like dynamic program (Sun et al., 2024).

The conditioned recurrence is

$S=\langle g_1,\ldots,g_m\rangle$3

with initial condition $S=\langle g_1,\ldots,g_m\rangle$4. After iterating from $S=\langle g_1,\ldots,g_m\rangle$5 to $S=\langle g_1,\ldots,g_m\rangle$6,

$S=\langle g_1,\ldots,g_m\rangle$7

and the minimizing CMCS is reconstructed by backtracking (Sun et al., 2024).

The candidate-state bound is

$S=\langle g_1,\ldots,g_m\rangle$8

where $S=\langle g_1,\ldots,g_m\rangle$9 for all $n$0. For local sparse Hamiltonians with $n$1, the overall runtime is

$n$2

linear in system size and sub-exponential in locality (Sun et al., 2024).

The same framework extends to infinite 1D periodic local sparse Hamiltonians of period $n$3 by imposing periodic boundary conditions on a supercell of size $n$4. The associated state machines satisfy $n$5 for some $n$6, and the energy density is obtained from a periodic dynamic program with $n$7. In higher-dimensional periodic settings, the analogous search is restricted to periodic closed commuting subsets $n$8, whose minimizer is the closed maximally-commuting periodic subset (CMCPS) (Sun et al., 2024).

Because the algorithm returns the explicit generator set $n$9, the output is not only an energy value but also a certificate of optimality within the stabilizer ansatz. If $|\psi_S\rangle$0 has $|\psi_S\rangle$1 independent generators, the degeneracy is $|\psi_S\rangle$2 (Sun et al., 2024).

4. Degeneracy resolution, overlap criteria, and state selection

When many stabilizer states attain the same minimum energy, a purely energetic definition is insufficient for simulation tasks that need large overlap with the true ground state. The refined OSGS program therefore adds an overlap criterion on top of the stabilizer minimum. Energy is still evaluated through stabilizer expectations,

$|\psi_S\rangle$3

which is computable in time polynomial in $|\psi_S\rangle$4 using the stabilizer tableau, because membership $|\psi_S\rangle$5 is a symplectic linear-algebra query and the eigenvalue $|\psi_S\rangle$6 is read off from the tableau phase bit (Mao et al., 6 Mar 2026).

For small to medium scale problems, one route uses a robustness-of-magic-inspired linear algebra. Writing the Hamiltonian coefficients as a vector $|\psi_S\rangle$7 under the normalized Pauli trace inner product and using the stabilizer-incidence matrix $|\psi_S\rangle$8, one obtains

$|\psi_S\rangle$9

which yields the SGS energy and the minimizing maximally commuting subgroups. Degeneracy is then resolved by a two-step filter: first form the common subgroup $g_i|\psi_S\rangle=|\psi_S\rangle$0 across the minimizers and a reduced Hamiltonian $g_i|\psi_S\rangle=|\psi_S\rangle$1; then retain generator sets satisfying

$g_i|\psi_S\rangle=|\psi_S\rangle$2

The selection is rerun on $g_i|\psi_S\rangle=|\psi_S\rangle$3 to fix phases and complete the full generator set $g_i|\psi_S\rangle=|\psi_S\rangle$4 (Mao et al., 6 Mar 2026).

For large systems, the search is recast as a combinatorial optimization over an $g_i|\psi_S\rangle=|\psi_S\rangle$5 binary symplectic tableau with a phase vector. A commutation graph is built on the Hamiltonian-supported Pauli set, and a genetic algorithm optimizes the fitness

$g_i|\psi_S\rangle=|\psi_S\rangle$6

with an additional fidelity term in benchmarks where the true ground state is known. The resulting classical preprocessing scales as

$g_i|\psi_S\rangle=|\psi_S\rangle$7

for population size $g_i|\psi_S\rangle=|\psi_S\rangle$8, generations $g_i|\psi_S\rangle=|\psi_S\rangle$9, and $n$00 nonzero Pauli terms. Once the generator set is fixed, standard tableau synthesis yields a Clifford circuit $n$01 with worst-case $n$02 gates and $n$03 classical synthesis time (Mao et al., 6 Mar 2026).

This overlap-sensitive perspective aligns naturally with older stabilizer-manipulation results. Efficient inner-product algorithms compute overlaps between stabilizer states in $n$04 time in general and show quadratic behavior for many practical instances, which is directly relevant whenever one wishes to compare multiple stabilizer minimizers against a target state or against each other (Garcia et al., 2012).

5. Applications in quantum simulation and many-body physics

The stabilizer-ground-state framework was proposed not only as a variational endpoint but also as a building block for broader simulation workflows. In 1D many-body problems, stabilizer ground states can track phases and topological features. A generalized 1D cluster model,

$n$05

shows cluster, ferromagnetic, and polarized phases consistent with DMRG, with simple stabilizer generators in each phase. For the 2D toric code in fields,

$n$06

one can scan single-qubit rotations $n$07 and compute the stabilizer ground state of $n$08, producing an extended stabilizer ground state phase diagram consistent with Monte Carlo and capturing transitions along the $n$09 line qualitatively (Sun et al., 2024).

These states also function as algorithmic reference states. Stabilizer initializations improve VQE energy optimization versus zero-state or product-state starts; stabilizer CMCS can guide a splitting $n$10 in which $n$11 aligns with stabilizers and $n$12 is treated perturbatively; and the framework has stated synergies with stabilizer tensor networks, Clifford-augmented DMRG, and stabilizer-based Monte Carlo (Sun et al., 2024).

In measurement-based deterministic imaginary time evolution, the OSGS is used as the anchor state. Imaginary-time evolution

$n$13

is implemented through weak measurements and deterministic feedback. The threshold is set to the stabilizer-ground-state energy $n$14, and when the running energy estimate exceeds $n$15, a Clifford reset $n$16 reprepares $n$17. Because $n$18 maximizes the initial ground-state overlap inside the SGS manifold, fewer measurements are needed to make the $n$19 component dominant (Mao et al., 6 Mar 2026).

A closely related stabilizer-first strategy appears in stabilizer-accelerated many-body estimation. There the Hamiltonian is split as

$n$20

with $n$21 determined by an optimal stabilizer group and $n$22 carrying the non-stabilizer residue. In the Lipkin–Meshkov–Glick model,

$n$23

the optimal stabilizer ground state is $n$24 for $n$25 and an equal-weight parity superposition stabilized by $n$26 for $n$27. In this setting, stabilizer ground states accelerate imaginary-time convergence because the initial overlap with the exact ground state is larger, and a single non-unitary reweighting $n$28 captures much of the remaining correlation (Robin, 5 May 2025).

Molecular applications further broaden the scope. Stabilizer configuration interaction finds the best stabilizer approximations to molecular ground states up to 36 qubits and constructs generalized stabilizer states that improve the approximation further, while adaptive SCI scales as $n$29 (Anand et al., 2024). Independent stabilizer-approximation studies report that water in STO-3G requires 14 qubits and benzene 72 qubits, and that the best stabilizer states approximate the true ground states very well, especially when the molecules are strongly distorted (Wang et al., 2023).

6. Preparation, verification, limitations, and related directions

Once an optimal stabilizer ground state has been identified, it remains to prepare and certify it. For 1D $n$30-local Hamiltonians with an $n$31 generator output, the stabilizer ground states can be prepared with

$n$32

single- and two-qubit Clifford gates, where $n$33; in the non-degenerate case this becomes $n$34 (Sun et al., 2024). More generally, any stabilizer state is locally Clifford equivalent to a graph state, and graph-decimation methods synthesize preparation circuits by reducing graph edges with two-qubit Clifford gates. The AI-guided method QuSynth combines reinforcement learning and Monte Carlo tree search and reports reductions in two-qubit gate count by up to a factor of $n$35 compared to previous approaches while retaining low circuit depth (Doherty et al., 18 Mar 2026).

Experimental certification is also unusually structured. For a target stabilizer state $n$36, a Pauli-measurement verification scheme has acceptance operator

$n$37

spectral gap $n$38, and exact sample complexity

$n$39

For any entangled stabilizer state, any separable-measurement protocol satisfies the universal bound $n$40, yielding

$n$41

Optimal Pauli-measurement protocols saturate this bound for all connected graph states up to seven qubits, and X/Z-only protocols satisfy $n$42 (Dangniam et al., 2020).

The main conceptual limitation is explicit. The method is exact within the stabilizer ansatz, not for arbitrary quantum states. In noncommuting or frustrated Pauli Hamiltonians, or whenever the true ground state is strongly non-stabilizer, the optimal stabilizer ground state need not coincide with the physical ground state. The formalism still returns the best stabilizer state and its energy, but not a claim of exactness beyond that ansatz (Sun et al., 2024, Mao et al., 6 Mar 2026).

A second limitation is computational. For 1D local sparse Hamiltonians the state-machine algorithm is exact and efficient, but in higher dimensions exact computation is generally NP-hard, so approximate or heuristic extensions may be needed (Sun et al., 2024). This suggests a practical division of labor: exact CMCS-based methods where locality and dimensionality permit them, and overlap-aware heuristic or hybrid procedures when the search space or non-stabilizer residue becomes dominant.

In that sense, the optimal stabilizer ground state is best understood as a certified Clifford backbone for ground-state estimation: exact for commuting-stabilizer instances, variationally controlled for general Pauli Hamiltonians, and frequently useful as the reference state from which more expressive classical or quantum refinements proceed.