GHZ-Preserving Operations

• GHZ-Preserving Operations are unitary gates or channels on n-qubit systems that permute the GHZ basis states while preserving local Pauli equivalence.
• They feature a complete algebraic classification into Pauli, bilocal (B group), and homogeneous (H group) operations, allowing for constant-time simulation through precomputed state permutations.
• Circuit-level optimization using these operations enhances entanglement distillation by improving output fidelity and reducing raw-state requirements in realistic quantum network scenarios.

A Greenberger–Horne–Zeilinger (GHZ)-preserving operation is a unitary or channel on $n$-qubit systems that permutes the GHZ basis states (up to a global phase), ensuring that evolution never leaves the subspace of states locally Pauli-equivalent to the canonical GHZ state. This property enables efficient simulation and optimization of entanglement distillation protocols for quantum networks. Recent advances provide a complete algebraic classification of such operations, a constant-time simulation method, and circuit optimization frameworks delivering superior performance under realistic noise, with extensions to graph states local-Clifford-equivalent to GHZ states (Wang et al., 29 Oct 2025).

1. Definition: GHZ States and GHZ-Preserving Operations

The $n$-qubit GHZ state is

$$|\mathrm{GHZ}_n\rangle = \frac{1}{\sqrt{2}}(|0\rangle^{\otimes n} + |1\rangle^{\otimes n})\,.$$

The GHZ basis comprises all states of the form

$$\{ P_1\otimes P_2\otimes\cdots\otimes P_n\,|\,P_i\in\{I,X,Y,Z\}\}\,|\mathrm{GHZ}_n\rangle\,,$$

i.e., local Pauli operations on each qubit applied to $|\mathrm{GHZ}_n\rangle$.

A unitary $U$ (or quantum channel $\mathcal{E}$) is GHZ-preserving if, for any basis vector $|\mathrm{GHZ}_i\rangle$,

$$U\,|\mathrm{GHZ}_i\rangle = e^{i\phi}\,|\mathrm{GHZ}_j\rangle,$$

for some $j$ and phase $n$0. In the density-matrix formalism, this reads

$n$1

Intuitively, GHZ-preserving gates act exclusively within the GHZ stabilizer structure, only relabeling basis states.

2. Algebraic Structure and Enumeration of GHZ-Preserving Gates

GHZ-preserving unitaries, up to phase and local Paulis, can be decomposed into three distinct subgroups [(Wang et al., 29 Oct 2025), Thm. II.3]:

• Local Pauli gates: Arbitrary single-qubit $n$2 on each of $n$3 qubits (for multiple GHZ copies), absorbing global phases.
• B group (bilocal): Eight elements for each pair of nodes, generated by taking the same two-qubit gate $n$4 on corresponding pairs, with $n$5 and $n$6.
• H group (homogeneous): Six elements generated by applying the identical two-qubit gate at every node, drawn from $n$7.

Any GHZ-preserving unitary may be written as

$n$8

where $n$9 are local Paulis, $$|\mathrm{GHZ}_n\rangle = \frac{1}{\sqrt{2}}(|0\rangle^{\otimes n} + |1\rangle^{\otimes n})\,.$$0 are B-group bilocals, and $$|\mathrm{GHZ}_n\rangle = \frac{1}{\sqrt{2}}(|0\rangle^{\otimes n} + |1\rangle^{\otimes n})\,.$$1 is an H-group homogeneous gate.

The total number of distinct (phaseless) GHZ-preserving unitaries is $$|\mathrm{GHZ}_n\rangle = \frac{1}{\sqrt{2}}(|0\rangle^{\otimes n} + |1\rangle^{\otimes n})\,.$$2.

Subgroup	Generator Elements	Algebraic Structure
B group	$$|\mathrm{GHZ}_n\rangle = \frac{1}{\sqrt{2}}(|0\rangle^{\otimes n} + |1\rangle^{\otimes n})\,.$$3, $$|\mathrm{GHZ}_n\rangle = \frac{1}{\sqrt{2}}(|0\rangle^{\otimes n} + |1\rangle^{\otimes n})\,.$$4, $$|\mathrm{GHZ}_n\rangle = \frac{1}{\sqrt{2}}(|0\rangle^{\otimes n} + |1\rangle^{\otimes n})\,.$$5, $$|\mathrm{GHZ}_n\rangle = \frac{1}{\sqrt{2}}(|0\rangle^{\otimes n} + |1\rangle^{\otimes n})\,.$$6, etc.$$|\mathrm{GHZ}_n\rangle = \frac{1}{\sqrt{2}}(|0\rangle^{\otimes n} + |1\rangle^{\otimes n})\,.$$7	$$|\mathrm{GHZ}_n\rangle = \frac{1}{\sqrt{2}}(|0\rangle^{\otimes n} + |1\rangle^{\otimes n})\,.$$8
H group	$$|\mathrm{GHZ}_n\rangle = \frac{1}{\sqrt{2}}(|0\rangle^{\otimes n} + |1\rangle^{\otimes n})\,.$$9, SWAP, $$\{ P_1\otimes P_2\otimes\cdots\otimes P_n\,|\,P_i\in\{I,X,Y,Z\}\}\,|\mathrm{GHZ}_n\rangle\,,$$0, $$\{ P_1\otimes P_2\otimes\cdots\otimes P_n\,|\,P_i\in\{I,X,Y,Z\}\}\,|\mathrm{GHZ}_n\rangle\,,$$1, $$\{ P_1\otimes P_2\otimes\cdots\otimes P_n\,|\,P_i\in\{I,X,Y,Z\}\}\,|\mathrm{GHZ}_n\rangle\,,$$2, ...$$\{ P_1\otimes P_2\otimes\cdots\otimes P_n\,|\,P_i\in\{I,X,Y,Z\}\}\,|\mathrm{GHZ}_n\rangle\,,$$3	$$\{ P_1\otimes P_2\otimes\cdots\otimes P_n\,|\,P_i\in\{I,X,Y,Z\}\}\,|\mathrm{GHZ}_n\rangle\,,$$4 (dihedral), order $$\{ P_1\otimes P_2\otimes\cdots\otimes P_n\,|\,P_i\in\{I,X,Y,Z\}\}\,|\mathrm{GHZ}_n\rangle\,,$$5

B gates enable bilocal error correlation; H gates homogeneously shuffle errors; Paulis absorb phases.

3. Simulation Framework: Constant-Time GHZ Stabilizer Updating

Traditional simulation methods require $$\{ P_1\otimes P_2\otimes\cdots\otimes P_n\,|\,P_i\in\{I,X,Y,Z\}\}\,|\mathrm{GHZ}_n\rangle\,,$$6 (full state vector) or $$\{ P_1\otimes P_2\otimes\cdots\otimes P_n\,|\,P_i\in\{I,X,Y,Z\}\}\,|\mathrm{GHZ}_n\rangle\,,$$7 (tableau) time per gate. For GHZ-preserving operations, all action is limited to permutations of the "phase bits" in a block-diagonal stabilizer tableau; there is no mixing of basis elements.

The entire configuration of $$\{ P_1\otimes P_2\otimes\cdots\otimes P_n\,|\,P_i\in\{I,X,Y,Z\}\}\,|\mathrm{GHZ}_n\rangle\,,$$8 GHZ basis states is encoded in an $$\{ P_1\otimes P_2\otimes\cdots\otimes P_n\,|\,P_i\in\{I,X,Y,Z\}\}\,|\mathrm{GHZ}_n\rangle\,,$$9 bitstring, tracking the $|\mathrm{GHZ}_n\rangle$0 sign of each generator. For two GHZ states ($|\mathrm{GHZ}_n\rangle$1), there are $|\mathrm{GHZ}_n\rangle$2 possible basis elements.

For each elementary B or H gate $|\mathrm{GHZ}_n\rangle$3, precompute a permutation $|\mathrm{GHZ}_n\rangle$4. Application of any GHZ-preserving gate $|\mathrm{GHZ}_n\rangle$5 to a state with index $|\mathrm{GHZ}_n\rangle$6 then reduces to: $|\mathrm{GHZ}_n\rangle$7 which is $|\mathrm{GHZ}_n\rangle$8 time.

All permutations are computed once at initialization and reused for subsequent simulations, eliminating further bookkeeping. Since gate application never exits the GHZ basis, this method maintains high efficiency irrespective of $|\mathrm{GHZ}_n\rangle$9 (Wang et al., 29 Oct 2025).

4. Circuit-Level Optimization for GHZ Distillation

Leveraging the $U$0 simulation of GHZ-preserving gates and the drastically shrunken gate set ($U$1 per layer), a genetic-algorithm optimizer with simulated annealing can efficiently search for short distillation circuits maximizing output fidelity under realistic noise (independent depolarizing noise $U$2 per gate, measurement error $U$3, input fidelity $U$4). Hardware constraints include register size $U$5 and total raw states $U$6.

In a demonstrative scenario (three nodes, $U$7 distillation, $U$8, $U$9, $\mathcal{E}$0, $\mathcal{E}$1), circuits constructed from H-group gates achieved:

• Higher output fidelity at fixed success probability than standard recurrence methods (e.g., BBPSSW-style).
• Fewer raw states to reach a desired fidelity versus nested protocols.
• Uniform gains across $\mathcal{E}$2 up to $\mathcal{E}$3.

For $\mathcal{E}$4 and $\mathcal{E}$5, output fidelity improved by several percent absolute over previous protocols, saving on average one raw input copy (Wang et al., 29 Oct 2025).

5. GHZ-Preserving Operations for Local-Clifford-Equivalent Graph States

GHZ states correspond to special two-colorable graph states (star graphs with an additional Hadamard on the central node). Any graph state that is local-Clifford (LC)-equivalent to a GHZ state inherits a preserving-gate algebra via conjugation: the B and H generators are mapped through the LC that interconverts the graph and GHZ states.

For example, the 3-qubit complete graph (triangle) with standard generators $\mathcal{E}$6 admits a GHZ-matching generator set under local basis change ($\mathcal{E}$7). The same enumeration of preserving operations and $\mathcal{E}$8 simulation extend, with minor relabeling of state indexing.

Full extension to arbitrary graph state topologies is ongoing work (Wang et al., 29 Oct 2025).

6. Impact on Quantum Networking and Open Resources

By restricting attention to the GHZ basis and its stabilizer group, the search space for circuit synthesis shrinks from superexponential (over the full Clifford group) to $\mathcal{E}$9 elements per layer. Simulation complexity drops from $|\mathrm{GHZ}_i\rangle$0 or $|\mathrm{GHZ}_i\rangle$1 per gate to a constant $|\mathrm{GHZ}_i\rangle$2. These advances enable rapid circuit-level optimization, discovering practical distillation circuits outperforming all known protocols under realistic noise regimes.

These methods are directly relevant for near-term quantum networks conducting multipartite entanglement distillation under resource and noise constraints. An open-source Julia implementation is publicly available at https://github.com/umass-qc/GHZpreserving.jl (Wang et al., 29 Oct 2025).