Gaussian Stabilizer Group in Quantum Theory

• Gaussian Stabilizer Group is a symmetry group defined by symplectic phase-space dynamics that preserves stabilizer states in both discrete-variable and bosonic GKP settings.
• It plays dual roles by acting as the Clifford group in DV systems—normalizing Pauli/Weyl operators—and as a subgroup of Gaussian unitaries with trivial logical action in grid codes.
• Its framework supports practical insights including quantum convolution, state and channel entropy extremality, central limit theorems, and noise-optimized compilation strategies.

The expression Gaussian Stabilizer Group appears in two closely related but technically distinct settings. In discrete-variable quantum information, the phrase denotes the role played by the Clifford group as the discrete analogue of the continuous-variable Gaussian unitary group: Clifford unitaries normalize the Pauli/Weyl group, implement symplectic linear maps on finite phase space, preserve stabilizer states, and underlie a discrete-variable convolution theory in which stabilizer states behave as “discrete quantum Gaussians” (Bu et al., 2023). In the square-GKP setting, the same expression refers to the subgroup of Gaussian unitaries that act as encoded identities on the logical code space; this group is characterized by affine symplectic automorphisms of the dual lattice and is used to search over logically equivalent physical implementations of Clifford circuits under noise (Pelletier et al., 15 Sep 2025). In both usages, the core structure is symplectic phase-space dynamics constrained by stabilizer preservation, but the ambient Hilbert spaces, group definitions, and operational roles differ.

1. Terminological scope and conceptual setting

In the discrete-variable formulation, the guiding claim is that stabilizer states play a role in DV quantum systems similar to the role Gaussian states play in continuous-variable systems, which motivates the designation “discrete quantum Gaussians” for stabilizer states (Bu et al., 2023). From this viewpoint, the natural group of “Gaussian operations” is the Clifford group, because it preserves stabilizer states and acts symplectically on finite phase space.

In the GKP formulation, the starting point is different. For grid codes, the unitary stabilizer group consists of those unitaries that act as encoded identities on the logical subspace, and the Gaussian stabilizer group is obtained by restricting this unitary stabilizer group to Gaussian unitaries (Pelletier et al., 15 Sep 2025). Here the question is not whether Gaussian states are replaced by stabilizer states, but which Gaussian transformations leave the logical action trivial while possibly changing the physical implementation.

These usages are structurally aligned because both are formulated in terms of Heisenberg–Weyl covariance and symplectic transformations. They are nevertheless not identical. In the DV setting, “Gaussian stabilizer group” is effectively a group-theoretic interpretation of the Clifford group as the discrete Gaussian group. In the GKP setting, it is a subgroup of Gaussian unitaries defined by trivial encoded action on a bosonic stabilizer code.

2. Finite-dimensional formulation: Weyl systems, stabilizers, and the Clifford group

For prime local dimension $d$ and $n$ qudits, the finite phase space is

$V^n = \mathbb{Z}_d^n \times \mathbb{Z}_d^n.$

The single-qudit Pauli operators satisfy

$$X|k\rangle = |k+1\rangle,\qquad Z|k\rangle = \chi(k)|k\rangle,\qquad \chi(k)=\exp(2\pi i k/d).$$

The Weyl operators are

$$w(p,q)=\chi(-2^{-1}pq)Z^pX^q \quad \text{for odd } d,$$

and

$w(p,q)=i^{-pq}Z^pX^q \quad \text{for } d=2,$

with symplectic inner product

$\{(p,q),(p',q')\}_s = pq' - qp'.$

For $n$ qudits,

$w(\vec p,\vec q)=\prod_k w(p_k,q_k).$

A pure stabilizer state is a common eigenvector of an abelian subgroup $S$ of the Weyl group of size $n$0, with $n$1 commuting generators. General stabilizer states are convex mixtures of pure stabilizers. Equivalently, the stabilizer algebra $n$2 is abelian and has minimal projections; normalized minimal stabilizer projections (MSPS) are the elementary stabilizer states (Bu et al., 2023).

An $n$3-qudit unitary $n$4 is Clifford if it normalizes the Weyl operators: $n$5 for some symplectic linear transformation $n$6 on the discrete phase space. Clifford unitaries preserve stabilizer states. This normalizer property is the decisive group-theoretic reason the Clifford group functions as the DV analogue of the Gaussian unitary group. The relevant phase-space action is linear and symplectic, and the preserved family of states is the stabilizer family, which the paper identifies as the discrete analogue of Gaussian states (Bu et al., 2023).

3. Characteristic functions, convolution, and discrete Gaussian behavior

The discrete-variable characteristic function of a state $n$7 is

$n$8

and the Weyl expansion is

$n$9

For MSPS, and more generally stabilizer states, $V^n = \mathbb{Z}_d^n \times \mathbb{Z}_d^n.$0 with symplectic-abelian support. This support structure is central to the discrete Gaussian analogy.

The paper introduces a DV quantum convolution using a Clifford unitary $V^n = \mathbb{Z}_d^n \times \mathbb{Z}_d^n.$1 associated with an invertible $V^n = \mathbb{Z}_d^n \times \mathbb{Z}_d^n.$2 matrix $V^n = \mathbb{Z}_d^n \times \mathbb{Z}_d^n.$3 over $V^n = \mathbb{Z}_d^n \times \mathbb{Z}_d^n.$4: $V^n = \mathbb{Z}_d^n \times \mathbb{Z}_d^n.$5 Its action on characteristic functions is multiplicative: $V^n = \mathbb{Z}_d^n \times \mathbb{Z}_d^n.$6 where $V^n = \mathbb{Z}_d^n \times \mathbb{Z}_d^n.$7 in $V^n = \mathbb{Z}_d^n \times \mathbb{Z}_d^n.$8. Special cases include a discrete beam splitter, discrete amplifier, and, for odd $V^n = \mathbb{Z}_d^n \times \mathbb{Z}_d^n.$9, a Hadamard convolution (Bu et al., 2023).

This formalism yields a closure theorem: if $$X|k\rangle = |k+1\rangle,\qquad Z|k\rangle = \chi(k)|k\rangle,\qquad \chi(k)=\exp(2\pi i k/d).$$0 and $$X|k\rangle = |k+1\rangle,\qquad Z|k\rangle = \chi(k)|k\rangle,\qquad \chi(k)=\exp(2\pi i k/d).$$1 are stabilizer states, then $$X|k\rangle = |k+1\rangle,\qquad Z|k\rangle = \chi(k)|k\rangle,\qquad \chi(k)=\exp(2\pi i k/d).$$2 is again a stabilizer state. For MSPS, the proof uses the fact that $$X|k\rangle = |k+1\rangle,\qquad Z|k\rangle = \chi(k)|k\rangle,\qquad \chi(k)=\exp(2\pi i k/d).$$3 and $$X|k\rangle = |k+1\rangle,\qquad Z|k\rangle = \chi(k)|k\rangle,\qquad \chi(k)=\exp(2\pi i k/d).$$4 take only the values $$X|k\rangle = |k+1\rangle,\qquad Z|k\rangle = \chi(k)|k\rangle,\qquad \chi(k)=\exp(2\pi i k/d).$$5 and $$X|k\rangle = |k+1\rangle,\qquad Z|k\rangle = \chi(k)|k\rangle,\qquad \chi(k)=\exp(2\pi i k/d).$$6, and that the support of the product remains abelian under the symplectic commutation criterion. Convexity extends the result to general stabilizer states.

The same framework supports several extremality results. The mean state $$X|k\rangle = |k+1\rangle,\qquad Z|k\rangle = \chi(k)|k\rangle,\qquad \chi(k)=\exp(2\pi i k/d).$$7 is defined by

$$X|k\rangle = |k+1\rangle,\qquad Z|k\rangle = \chi(k)|k\rangle,\qquad \chi(k)=\exp(2\pi i k/d).$$8

The paper proves that $$X|k\rangle = |k+1\rangle,\qquad Z|k\rangle = \chi(k)|k\rangle,\qquad \chi(k)=\exp(2\pi i k/d).$$9 is an MSPS determined by the abelian subgroup

$$w(p,q)=\chi(-2^{-1}pq)Z^pX^q \quad \text{for odd } d,$$0

and that it obeys Clifford covariance: $$w(p,q)=\chi(-2^{-1}pq)Z^pX^q \quad \text{for odd } d,$$1 For Rényi entropies,

$$w(p,q)=\chi(-2^{-1}pq)Z^pX^q \quad \text{for odd } d,$$2

with equality only at $$w(p,q)=\chi(-2^{-1}pq)Z^pX^q \quad \text{for odd } d,$$3. Thus MSPS maximize all Schur-concave spectral functionals among states with the same mean state up to Clifford conjugation (Bu et al., 2023).

For “positive” $$w(p,q)=\chi(-2^{-1}pq)Z^pX^q \quad \text{for odd } d,$$4, the convolution obeys majorization: $$w(p,q)=\chi(-2^{-1}pq)Z^pX^q \quad \text{for odd } d,$$5 hence

$$w(p,q)=\chi(-2^{-1}pq)Z^pX^q \quad \text{for odd } d,$$6

Iterated convolution gives a discrete-variable “second law for quantum convolution”: $$w(p,q)=\chi(-2^{-1}pq)Z^pX^q \quad \text{for odd } d,$$7

The paper also proves a Fisher-information inequality for positive $$w(p,q)=\chi(-2^{-1}pq)Z^pX^q \quad \text{for odd } d,$$8: $$w(p,q)=\chi(-2^{-1}pq)Z^pX^q \quad \text{for odd } d,$$9 where

$w(p,q)=i^{-pq}Z^pX^q \quad \text{for } d=2,$0

A DV de Bruijn identity is established: $w(p,q)=i^{-pq}Z^pX^q \quad \text{for } d=2,$1

The culmination is a DV quantum central limit theorem. For a zero-mean $w(p,q)=i^{-pq}Z^pX^q \quad \text{for } d=2,$2-qudit state $w(p,q)=i^{-pq}Z^pX^q \quad \text{for } d=2,$3, repeated beam-splitter convolution converges exponentially in Hilbert–Schmidt norm to the mean state: $w(p,q)=i^{-pq}Z^pX^q \quad \text{for } d=2,$4 where the magic gap is

$w(p,q)=i^{-pq}Z^pX^q \quad \text{for } d=2,$5

The limit state is an MSPS and therefore a stabilizer state. In this precise sense, stabilizer states are attractors under DV convolution in the same way Gaussian states are attractors in CV central limit theory (Bu et al., 2023).

4. Channel-level extension and the “Gaussian” interpretation of the Clifford group

The discrete Gaussian analogy extends from states to channels. A channel $w(p,q)=i^{-pq}Z^pX^q \quad \text{for } d=2,$6 is represented by its Choi state

$w(p,q)=i^{-pq}Z^pX^q \quad \text{for } d=2,$7

Channel convolution is defined by convolving Choi states: $w(p,q)=i^{-pq}Z^pX^q \quad \text{for } d=2,$8 Equivalently,

$w(p,q)=i^{-pq}Z^pX^q \quad \text{for } d=2,$9

where $\{(p,q),(p',q')\}_s = pq' - qp'.$0 is the DV convolutional channel induced by $\{(p,q),(p',q')\}_s = pq' - qp'.$1 (Bu et al., 2023).

A stabilizer channel maps stabilizer states to stabilizer states. Because $\{(p,q),(p',q')\}_s = pq' - qp'.$2 is Clifford and convolution preserves stabilizer structure, the convolution of stabilizer channels is stabilizer, and the mean channel $\{(p,q),(p',q')\}_s = pq' - qp'.$3, defined through the Choi state, is likewise a stabilizer channel. This is the channel-level counterpart of the statement that stabilizer states are the discrete Gaussians.

The paper proves channel entropy extremality using the Gour–Wilde channel entropy

$\{(p,q),(p',q')\}_s = pq' - qp'.$4

For $\{(p,q),(p',q')\}_s = pq' - qp'.$5,

$\{(p,q),(p',q')\}_s = pq' - qp'.$6

with equality only for $\{(p,q),(p',q')\}_s = pq' - qp'.$7 in the abelian algebra determined by the stabilizer support of $\{(p,q),(p',q')\}_s = pq' - qp'.$8. For positive $\{(p,q),(p',q')\}_s = pq' - qp'.$9,

$n$0

and iterated convolution yields the channel version of the second law: $n$1

The convolutional channel $n$2 achieves minimal output entropy exactly on pure stabilizer inputs and achieves maximal Holevo capacity iff the state is stabilizer. These results are presented as direct analogues of Gaussian extremality for continuous-variable channels. The resulting interpretation is explicit: the Clifford group is the DV “Gaussian stabilizer group” because it normalizes the Weyl group, implements the convolutional mixing unitaries, preserves stabilizer states and stabilizer channels, and organizes the entropy, Fisher-information, and central-limit structure of the theory (Bu et al., 2023).

5. Square-GKP Gaussian stabilizer group

For square-GKP codes, the setting is bosonic rather than finite-dimensional. Single-mode phase space is coordinated by

$n$3

and the displacement operator is

$n$4

In the notation of the paper, translations are

$n$5

The standard square-GKP code is defined by a square stabilizer lattice of spacing $n$6 in the $n$7 plane, with stabilizer generators

$n$8

Equivalently,

$n$9

Logical Pauli operators are half-stabilizer displacements: $w(\vec p,\vec q)=\prod_k w(p_k,q_k).$0 with displacement vectors

$w(\vec p,\vec q)=\prod_k w(p_k,q_k).$1

The stabilizer lattice is

$w(\vec p,\vec q)=\prod_k w(p_k,q_k).$2

and logical displacements sit at the midpoints of the stabilizer grid (Pelletier et al., 15 Sep 2025).

A Gaussian unitary is parameterized by a symplectic matrix $w(\vec p,\vec q)=\prod_k w(p_k,q_k).$3 and a displacement $w(\vec p,\vec q)=\prod_k w(p_k,q_k).$4, acting as

$w(\vec p,\vec q)=\prod_k w(p_k,q_k).$5

For grid codes, the Gaussian stabilizer group is defined as

$w(\vec p,\vec q)=\prod_k w(p_k,q_k).$6

The paper shows that Gaussian stabilizers act as affine automorphisms of the dual lattice: $w(\vec p,\vec q)=\prod_k w(p_k,q_k).$7 For the single-mode square code, the fixed-point symplectic maps are integral matrices with $w(\vec p,\vec q)=\prod_k w(p_k,q_k).$8, so

$w(\vec p,\vec q)=\prod_k w(p_k,q_k).$9

The necessary and sufficient conditions for trivial logical action on the square-GKP code are: $S$0 Equivalently, $S$1 maps every Pauli coset grid to itself and the displacement is by a stabilizer vector. In the multimode formulation, with Pauli subgrids $S$2, the condition is

$S$3

The symplectic part admits an algebraic characterization: $S$4 subject to

$S$5

In a canonical basis $S$6, the solutions form a group under

$S$7

Generators are built from the symmetric basis $S$8 and skew-symmetric basis $S$9, with constraints determined by the nonzero entries of $n$00. The paper presents this as a complete generating framework for the fixed-point symplectic part and, after adding appropriate translations, for $n$01 itself (Pelletier et al., 15 Sep 2025).

6. Explicit generators, compilation over $n$02, and noise-protection use

In the single-mode square-GKP case, $n$03 and $n$04 are already canonical. The paper derives three fundamental integer solutions $n$05 and corresponding symplectic matrices: $n$06

$n$07

$n$08

These implement the squares of encoded Clifford generators: $n$09 After accounting for the displacement part needed to return Pauli cosets to the stabilizer grid, the single-mode Gaussian stabilizer group is generated by

$n$10

For two modes, the paper identifies four inter-mode symplectic generators

$n$11

$n$12

implementing squares of encoded controlled-Pauli operations: $n$13

$n$14

together with the analogous $n$15. For $n$16 modes, the full Gaussian stabilizer group is generated by these inter-mode elements together with the single-mode generators and lattice displacements by stabilizer vectors (Pelletier et al., 15 Sep 2025).

This explicit generating set is used for noisy Clifford compilation. Every logical Clifford $n$17 admits multiple Gaussian implementations $n$18 differing by multiplication by Gaussian stabilizers $n$19. The compiler represents an implementation as

$n$20

where $n$21 is a fixed physical representative “closest to identity” and $n$22 is a short word in the generators of $n$23. The search is therefore over the coset $n$24.

The physical criterion is expressed in terms of the envelope of a finite-energy GKP state. Under a Gaussian circuit,

$n$25

The compiler minimizes, at each gate, the envelope displacement magnitude and squeezing through

$n$26

and

$n$27

The search is restricted to short walks, typically $n$28, because long words increase squeezing and displacement. Selection is lexicographic: first minimize $n$29, then $n$30.

The noise model combines bosonic loss and dephasing. Loss with parameter $n$31 has Kraus operators

$n$32

and updates Gaussian peaks as

$n$33

Dephasing with rate $n$34 is

$n$35

and induces angular smearing. Because the channels commute, the average magnitude of displacement experienced by a phase-space feature at radius $n$36 per application is

$n$37

When $n$38 approaches the logical spacing $n$39, logical shifts proliferate, so the compiler suppresses envelope radius and squeezing.

The numerical demonstration uses logical randomized benchmarking. The paper reports that for $n$40 and $n$41 logical qubits at $n$42 and $n$43, the GS compiler consistently yields higher survival probabilities at all lengths than both a constant compiler and a random-walk compiler. Lifetime is defined as the number of gates $n$44 needed to reach a $n$45 drop from the initial baseline under the Gaussian fit

$n$46

The lifetime improvement factor $n$47 increases as loss weakens; at very high loss, all compilers perform poorly and gains vanish. The same advantage appears in two-qubit-only randomized benchmarking over the group $n$48, which the paper identifies as relevant for magic-state-based computations where single-qubit Cliffords are tracked in a frame and only two-qubit gates are physically executed. Diagnostic metrics show that the maximum $n$49 and $n$50 attained along random sequences remain significantly lower under the GS compiler than under the other strategies (Pelletier et al., 15 Sep 2025).

A plausible implication is that, in the GKP setting, the Gaussian stabilizer group is not merely a structural symmetry group but also a compilation resource: it parameterizes physically distinct Gaussian realizations of the same logical Clifford operation and thereby exposes an optimization space adapted to Gaussian noise. In the DV setting, by contrast, the emphasis is foundational: the Clifford group is “Gaussian” because it preserves the discrete stabilizer sector and supports convolutional, entropic, Fisher-information, and central-limit phenomena directly analogous to continuous-variable Gaussian theory.