GHZ-Preserving Gates and Optimized Distillation Circuits

TL;DR

The paper addresses the challenge of efficiently designing and simulating GHZ-state distillation circuits under realistic noise. It introduces a GHZ-preserving gate framework decomposed into a bilocal B group and a homogeneous H group, plus Pauli corrections, enabling constant-time ($O(1)$) simulation via permutation of GHZ-basis states. This permits large-scale circuit optimization with a genetic algorithm, yielding distillation circuits that outperform prior baselines under practical noise and resource constraints, and extending naturally to graph states through local Clifford equivalence. The approach supports scalable quantum networks and MBQC applications, and an open-source implementation is provided. The work thus offers a principled, scalable path to high-fidelity multipartite entanglement in realistic hardware.

Abstract

Greenberger-Horne-Zeilinger (GHZ) states play a central role in quantum computing and communication protocols, as a typical multipartite entanglement resource. This work introduces an efficient enumeration and simulation method for circuits that preserve and distill noisy GHZ states, significantly reducing the simulation complexity of a gate on $n$ qubits, from exponential $O(2^n)$ for standard state-vector methods or $O(n)$ for Clifford circuits, to a constant $O(1)$ for the method presented here. This method has profound implications for the design of quantum networks, where preservation and purification of entanglement with minimal resource overhead is critical. In particular, we demonstrate the use of the new method in an optimization procedure enabled by the fast simulation, that discovers GHZ distillation circuits far outperforming the state of the art. Fine-tuning to arbitrary noise models is possible as well. We also show that the method naturally extends to graph states that are local Clifford equivalent to GHZ states.

Figures (13)

• Figure 1: General GHZ-preserving circuits. Example with two 3-qubit GHZ states, every node holding one qubit from each state (lines 1, 3 and 5 correspond to the first GHZ state, lines 2, 4 and 6 correspond to the second GHZ state). The red blocks (left) together make up a unitary from the H group, where all nodes apply the same gate chosen from $\{I, SWAP, CNOT_{12}, CNOT_{21},DCX_{12},DCX_{21}\}$. The blue (second from left) and green (third from left) blocks make up different unitaries from the B group, where $n-1$ pairs of nodes each apply the same gate chosen from $\left\{ (g\cdot(f_1 \otimes f_2))\, |\, g \in \{I \otimes I, CZ\}, (f_1, f_2) \in \{I, S\}^2\right\}$ (see Tables \ref{['tab:Fgates']} and \ref{['tab:Hgates']}). Lastly, we need $2n-2$ Pauli operations, $P_1,P_2,P_3,P_4$.
• Figure 2: Example of ket notation, stabilizer tableau representation, and bitstring representation for the same state.
• Figure 3: Comparison between our optimized circuits and the recurrent pumping method. Input fidelity vs output fidelity (top) and input fidelity vs success probability (bottom). N denotes the number of raw GHZ states used by our circuits (encoded by marker shape); R denotes the register size (how many qubits a node can store at a time, encoded by color). Each point corresponds to a circuit obtained by our optimization under the given parameters. Solid, dashed, and dotted lines are the standard pumping baselines (see Appendix \ref{['entanglement pumping']}); the numbers listed under "Pumping" in the legend indicate the number of raw GHZ states used by the pumping protocol (solid: $N{=}4$, dashed: $N{=}5$, dotted: $N{=}6$). All points are optimized to maximize the output fidelity given the lower bound of the pumping method’s success probability (gate error rate $p{=}0.01$, measurement error rate $\eta{=}0.01$). Our circuits achieve higher output fidelity for any given success probability.
• Figure 4: Comparison between our optimized circuits and the nested distillation method. Input fidelity vs output fidelity (left) and input fidelity vs success probability (right), comparing against 2-round and 3-round nested protocols using 4 and 8 raw GHZ states, respectively. Standard nested protocols typically assume twirling after each distillation round, where random Pauli gates are applied to symmetrize errors. This removes the strong $Z$-bias of the intermediate measurement outcomes and replaces it with an isotropic noise model (see Appendix \ref{['asymmetry']}). In contrast, our circuits directly model and detect both $X$ and $Z$ errors without relying on twirling, and are optimized under realistic noise (gate error rate $p=0.01$, measurement error rate $\eta=0.01$). As a result, we achieve higher output fidelities even with fewer raw GHZ states at moderate input fidelities. Moreover, unlike nested protocols, which require exponentially growing and fixed numbers of raw states, our method supports arbitrary input counts, offering greater flexibility for circuit-level optimization. Both plots use the same data set as in Fig. \ref{['fig:twocolumn']}, but are shown separately for clarity and to highlight different comparison baselines.
• Figure 5: Example of generated circuit. This circuit is generated based on 5-to-1 3-qubit GHZ states, with constraint of register number $R=3$, gate error $p=0.01$, measurement error $\eta=0.01$, and input fidelity $f_{in}=0.9$. The small dot in the circuit means adding a new raw state after one is measured (register reuse). Notably, the figure only shows Alice's circuit, while Bob and Charlie apply the same operations on their respective registers. We deliberately selected this circuit for illustration because it only contains H-group gates, which ensures that all operations are applied homogeneously across all nodes. This homogeneity makes the circuit easier to visualize, as opposed to other generated circuits that may contain B-group gates, introducing bilocal operations between two nodes and making the diagram more complex to represent.

Theorems & Definitions (28)

• Definition 1: GHZ basis
• Definition 2: GHZ preserving
• Definition
• Lemma 1: Single-qubit gates in the GHZ-preserving group
• proof
• Lemma 2: Bilocal requirement
• proof
• Lemma 3: Constraints on two-qubit gates in the $B$ group
• proof
• Theorem 1