Non-Adiabatic Holonomic QC

• Non-Adiabatic Holonomic Quantum Computing is a geometric quantum control method that employs cyclic evolution of a computational subspace to generate non-Abelian holonomies, eliminating the slow adiabatic requirement.
• It utilizes a canonical three-level Λ-system to implement universal single-qubit and two-qubit gates through bright/dark state decomposition that defines robust geometric operations.
• Advancements in NHQC include dynamically corrected, path-optimized, and decoherence-free designs applied across various platforms to enhance gate speed, fidelity, and error mitigation.

Non-Adiabatic Holonomic Quantum Computing (NHQC) is a geometric scheme for quantum information processing in which a computational subspace undergoes a cyclic evolution and acquires a non-Abelian holonomy, rather than an ordinary dynamical phase. In contrast to adiabatic holonomic quantum computation, NHQC removes the slow adiabatic requirement while retaining the geometric origin of the gate. In its standard form, the logical operation is determined by a loop of subspaces and can be written as $$U(C)= {\bf P}\exp\!\left(i\oint_C \boldsymbol{\mathcal A}\right)$$, with $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$ the matrix-valued connection one-form (Sjöqvist et al., 2011). Since the original non-adiabatic formulation, the field has developed into a broad control framework spanning single-loop and single-shot gates, unconventional and commutation-based holonomies, dynamically corrected and path-optimized constructions, decoherence-free-subspace encodings, qudit extensions, and hardware-specific realizations in trapped ions, superconducting circuits, nitrogen-vacancy centers, and Rydberg atoms (Liang et al., 2023).

1. Geometric foundations and defining conditions

The defining feature of NHQC is that the computational subspace, rather than a single state, is transported along a closed path in Hilbert space. The original non-adiabatic generalization formulated this as a time-dependent family of subspaces $M(0)\to M(t)\to M(\tau)=M(0)$, with the resulting gate identified as the holonomy associated with the closed loop $C$ in the Grassmann manifold of subspaces (Sjöqvist et al., 2011). The non-Abelian character is essential: universality requires at least two accessible loops based at the same subspace such that the corresponding holonomies do not commute.

In the standard formulation, two conditions define a purely holonomic gate. The first is cyclicity of the computational subspace, commonly written as

$$\sum_{k=1}^L|\varphi_k(\tau)\rangle\langle\varphi_k(\tau)| = \sum_{k=1}^L|\varphi_k(0)\rangle\langle\varphi_k(0)|.$$

The second is the parallel-transport condition,

$\langle\varphi_k(t)|H(t)|\varphi_l(t)\rangle=0,$

which removes the projected dynamical phase and leaves only the geometric contribution (Liang et al., 2023). Under these conditions, the evolution operator reduces to

$$U(\tau)=T{\rm exp}\left[i\int_0^\tau \mathcal A(t)\,dt\right],$$

with $$\mathcal A_{lm}(t)=i\langle \nu_l(t)|\dot{\nu}_m(t)\rangle$$.

A recurrent misconception is that holonomic computation is inherently adiabatic. NHQC was introduced precisely to remove that restriction. The geometric gate remains holonomic, but the path is traversed nonadiabatically, allowing fast control and making the scheme attractive for qubits with short coherence times (Sjöqvist et al., 2011). Another misconception is that “geometric” implies automatic robustness to all errors. The later NHQC literature repeatedly emphasizes that geometric dependence can mitigate certain control imperfections, but ancillary-state decoherence, amplitude miscalibration, detuning drift, and hardware-specific leakage remain central design constraints (Liang et al., 2023).

2. Canonical $\Lambda$-system formulation and universal gates

The canonical NHQC building block is a resonantly driven three-level $\Lambda$ system with computational states $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$0 and an auxiliary state. A standard interaction Hamiltonian is

$$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$1

where

$$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$2

$$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$3

and $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$4 (Li et al., 2020). The bright state $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$5 couples to the auxiliary level, whereas the dark state $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$6 remains decoupled. This bright/dark decomposition is the basic mechanism behind most NHQC gate constructions.

Canonical one-qubit holonomic gates are obtained by arranging a cyclic evolution of the bright component. In conventional single-loop NHQC, two resonant $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$7 segments with an appropriate phase jump produce

$$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$8

which, in the computational basis, becomes

$$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$9

Here the rotation axis $M(0)\to M(t)\to M(\tau)=M(0)$0 is set by $M(0)\to M(t)\to M(\tau)=M(0)$1 and $M(0)\to M(t)\to M(\tau)=M(0)$2, and the geometric phase $M(0)\to M(t)\to M(\tau)=M(0)$3 fixes the rotation angle, so arbitrary single-qubit operations are possible (Xu et al., 2024).

The original non-adiabatic proposal already included an entangling two-qubit gate. In the trapped-ion realization, the effective holonomic two-qubit unitary acts nontrivially on the $M(0)\to M(t)\to M(\tau)=M(0)$4 sector while leaving $M(0)\to M(t)\to M(\tau)=M(0)$5 and $M(0)\to M(t)\to M(\tau)=M(0)$6 unchanged; for $M(0)\to M(t)\to M(\tau)=M(0)$7, it reduces to a conditional phase shift, and together with arbitrary one-qubit holonomies it yields a universal gate set (Sjöqvist et al., 2011). This established the standard NHQC architecture: a universal family of single-qubit geometric rotations plus at least one entangling holonomic two-qubit gate.

The same $M(0)\to M(t)\to M(\tau)=M(0)$8-type logic underlies several later hardware realizations. In nitrogen-vacancy centers, for example, the ground-state triplet $M(0)\to M(t)\to M(\tau)=M(0)$9 provides the three-level structure, with microwave fields driving a bright state cyclically and leaving the orthogonal dark state untouched (Yan et al., 2017). The persistence of this model across platforms reflects its algebraic simplicity rather than any unique physical implementation.

3. Beyond strict parallel transport: generalized holonomic design principles

Although the conventional NHQC conditions are conceptually clean, they are restrictive. A major line of later work generalizes the design principle without abandoning the holonomic character. One influential development is NHQC$C$0, described as a “plug-and-play” approach. Instead of requiring the standard instantaneous NHQC constraints, NHQC$C$1 demands that the effective Hamiltonian in a moving auxiliary basis be diagonal and that the integrated dynamical phase vanish: $C$2 This relaxation makes it possible to incorporate optimal-control tools such as dynamical decoupling, composite pulses, shortcut-to-adiabaticity ideas, and dynamically corrected gates into single-loop geometric constructions (Liu et al., 2018).

A related generalization replaces the instantaneous parallel-transport condition by a commutation relation. In the commutation-relation-based scheme, one introduces geometric and dynamical generators,

$C$3

and imposes

$C$4

This allows the full evolution to factorize into geometric and dynamical parts,

$C$5

after which the dynamical factor is removed by a global parameter choice rather than by requiring $C$6 at every instant (Zhao et al., 2023). This is a broader admissible class of holonomic evolutions than the original NHQC condition.

Path design became a further degree of freedom in path-optimized NHQC. The central observation is that the same target holonomic gate can be produced by different loops, and these loops can have different sensitivity to systematic errors. In the path-optimized construction, the relevant path on the Bloch sphere is

$C$7

with $C$8 a tunable polar angle. A smaller $C$9 generally means less population in the auxiliary state and reduced sensitivity to leakage and decoherence, but it can also require a longer gate time. The scheme therefore frames robustness as a path-selection problem rather than only a pulse-compensation problem (Ji et al., 2022).

Another structural generalization extends NHQC beyond the standard three-level setting. By enlarging the building block to a bipartite-graph multilevel system, one preserves purely holonomic evolution while enabling fewer sequential steps for multiqubit controlled gates and native qudit operations. In the qubit case, four-level building blocks reduce controlled-phase-gate constructions from $$\sum_{k=1}^L|\varphi_k(\tau)\rangle\langle\varphi_k(\tau)| = \sum_{k=1}^L|\varphi_k(0)\rangle\langle\varphi_k(0)|.$$0 steps in the standard three-level $$\sum_{k=1}^L|\varphi_k(\tau)\rangle\langle\varphi_k(\tau)| = \sum_{k=1}^L|\varphi_k(0)\rangle\langle\varphi_k(0)|.$$1-based scheme to $$\sum_{k=1}^L|\varphi_k(\tau)\rangle\langle\varphi_k(\tau)| = \sum_{k=1}^L|\varphi_k(0)\rangle\langle\varphi_k(0)|.$$2 steps in the enlarged construction; in the qutrit case, five-level systems support arbitrary one-qutrit and two-qutrit holonomic gates (Xu et al., 2021).

4. Error channels and optimized control strategies

A central theme of the NHQC literature is that geometric origin alone does not remove the practical error budget. Conventional NHQC can be limited by fixed pulse-area constraints, ancillary-state decoherence, and sensitivity to amplitude or detuning errors. In brachistochrone NHQC, this limitation is addressed by time optimization. For the three-level $$\sum_{k=1}^L|\varphi_k(\tau)\rangle\langle\varphi_k(\tau)| = \sum_{k=1}^L|\varphi_k(0)\rangle\langle\varphi_k(0)|.$$3 system, the minimum-time solution yields

$$\sum_{k=1}^L|\varphi_k(\tau)\rangle\langle\varphi_k(\tau)| = \sum_{k=1}^L|\varphi_k(0)\rangle\langle\varphi_k(0)|.$$4

so the gate duration depends on the target angle $$\sum_{k=1}^L|\varphi_k(\tau)\rangle\langle\varphi_k(\tau)| = \sum_{k=1}^L|\varphi_k(0)\rangle\langle\varphi_k(0)|.$$5 rather than being fixed for all rotations. Under experimental conditions, this brachistochrone scheme was reported to reduce gate error by as much as $$\sum_{k=1}^L|\varphi_k(\tau)\rangle\langle\varphi_k(\tau)| = \sum_{k=1}^L|\varphi_k(0)\rangle\langle\varphi_k(0)|.$$6 compared with conventional NHQC (Liu et al., 2020). The point is not merely speed: shorter duration also reduces integrated exposure to decoherence.

Amplitude-miscalibration errors, often denoted $$\sum_{k=1}^L|\varphi_k(\tau)\rangle\langle\varphi_k(\tau)| = \sum_{k=1}^L|\varphi_k(0)\rangle\langle\varphi_k(0)|.$$7 errors, became a particularly important target for optimized NHQC control. In dynamically corrected NHQC, the error model

$$\sum_{k=1}^L|\varphi_k(\tau)\rangle\langle\varphi_k(\tau)| = \sum_{k=1}^L|\varphi_k(0)\rangle\langle\varphi_k(0)|.$$8

leads, for conventional NHQC, to a fidelity

$$\sum_{k=1}^L|\varphi_k(\tau)\rangle\langle\varphi_k(\tau)| = \sum_{k=1}^L|\varphi_k(0)\rangle\langle\varphi_k(0)|.$$9

whereas the dynamically corrected construction gives

$\langle\varphi_k(t)|H(t)|\varphi_l(t)\rangle=0,$0

The leading error is thus pushed from second to fourth order in $\langle\varphi_k(t)|H(t)|\varphi_l(t)\rangle=0,$1 (Li et al., 2020). Dynamically optimized NHQC refines this idea by engineering a composite single-loop path that preserves cyclicity and zero dynamical phase while further reducing the prefactor: $\langle\varphi_k(t)|H(t)|\varphi_l(t)\rangle=0,$2 The same work explicitly states that this is approximately half the infidelity of the earlier dynamically corrected NHQC formula, which scales as $\langle\varphi_k(t)|H(t)|\varphi_l(t)\rangle=0,$3 (Xu et al., 2024).

Another major error source is the auxiliary-state decay and dephasing intrinsic to $\langle\varphi_k(t)|H(t)|\varphi_l(t)\rangle=0,$4-system implementations. Decoherence-suppressed NHQC addresses this by reverse-engineering pulse shapes to minimize the integrated excited-state population

$\langle\varphi_k(t)|H(t)|\varphi_l(t)\rangle=0,$5

Using nitrogen-vacancy-center parameters, the resulting NOT- and Hadamard-gate fidelities were reported to improve from about $\langle\varphi_k(t)|H(t)|\varphi_l(t)\rangle=0,$6 in recent NHQC experiments to $\langle\varphi_k(t)|H(t)|\varphi_l(t)\rangle=0,$7 in the optimized construction, with the improvement attributed to reduced occupation of the lossy auxiliary state (Liu et al., 2022).

Additional variants target specific tradeoffs. The ultrafast detuned $\langle\varphi_k(t)|H(t)|\varphi_l(t)\rangle=0,$8-type scheme combines time-optimal control with time-independent detuning and reported $\langle\varphi_k(t)|H(t)|\varphi_l(t)\rangle=0,$9 and $$U(\tau)=T{\rm exp}\left[i\int_0^\tau \mathcal A(t)\,dt\right],$$0 fidelities of approximately $$U(\tau)=T{\rm exp}\left[i\int_0^\tau \mathcal A(t)\,dt\right],$$1 and $$U(\tau)=T{\rm exp}\left[i\int_0^\tau \mathcal A(t)\,dt\right],$$2, respectively, for $$U(\tau)=T{\rm exp}\left[i\int_0^\tau \mathcal A(t)\,dt\right],$$3 and $$U(\tau)=T{\rm exp}\left[i\int_0^\tau \mathcal A(t)\,dt\right],$$4 (Shen et al., 2021). Composite short-path NHQC uses inverse Hamiltonian engineering to realize the shortest circular loops compatible with holonomy, and reported average single-qubit fidelities of $$U(\tau)=T{\rm exp}\left[i\int_0^\tau \mathcal A(t)\,dt\right],$$5 for the $$U(\tau)=T{\rm exp}\left[i\int_0^\tau \mathcal A(t)\,dt\right],$$6 gate, $$U(\tau)=T{\rm exp}\left[i\int_0^\tau \mathcal A(t)\,dt\right],$$7 for the $$U(\tau)=T{\rm exp}\left[i\int_0^\tau \mathcal A(t)\,dt\right],$$8 gate, and $$U(\tau)=T{\rm exp}\left[i\int_0^\tau \mathcal A(t)\,dt\right],$$9 for the $$\mathcal A_{lm}(t)=i\langle \nu_l(t)|\dot{\nu}_m(t)\rangle$$0 gate under the stated decoherence model (Liang et al., 2021). A later qutrit-oriented framework combined inverse engineering with time-dependent perturbation theory to suppress second-order Rabi errors analytically and cancel second-order detuning errors through a compensation pulse (Lu et al., 7 Oct 2025).

5. Decoherence-free subspaces, scalability, and higher-dimensional encodings

Scalable NHQC increasingly relies on encoded logical subspaces rather than bare three-level systems. A standard strategy is the decoherence-free subspace (DFS), especially for collective dephasing noise. In dynamically corrected NHQC, a single logical qubit is encoded in

$$\mathcal A_{lm}(t)=i\langle \nu_l(t)|\dot{\nu}_m(t)\rangle$$1

with $$\mathcal A_{lm}(t)=i\langle \nu_l(t)|\dot{\nu}_m(t)\rangle$$2, $$\mathcal A_{lm}(t)=i\langle \nu_l(t)|\dot{\nu}_m(t)\rangle$$3, and $$\mathcal A_{lm}(t)=i\langle \nu_l(t)|\dot{\nu}_m(t)\rangle$$4, so that the logical Hamiltonian retains the same $$\mathcal A_{lm}(t)=i\langle \nu_l(t)|\dot{\nu}_m(t)\rangle$$5-type structure as the physical three-level model. In this encoded setting, the protocol becomes robust against both amplitude $$\mathcal A_{lm}(t)=i\langle \nu_l(t)|\dot{\nu}_m(t)\rangle$$6 errors and collective-dephasing $$\mathcal A_{lm}(t)=i\langle \nu_l(t)|\dot{\nu}_m(t)\rangle$$7 errors (Li et al., 2020). The dynamically optimized variant inherits the same logic and explicitly states that, when combined with DFS encoding, it is immune to both $$\mathcal A_{lm}(t)=i\langle \nu_l(t)|\dot{\nu}_m(t)\rangle$$8 and $$\mathcal A_{lm}(t)=i\langle \nu_l(t)|\dot{\nu}_m(t)\rangle$$9 errors (Xu et al., 2024).

Two-logical-qubit gates have been developed in larger DFSs. In one six-dimensional DFS, the logical basis

$\Lambda$0

is supplemented by two auxiliary states, yielding a Hamiltonian that splits into two NHQC-like three-level blocks. This supports a nontrivial entangling holonomic gate, and specific parameter choices produce a gate locally equivalent to CNOT (Li et al., 2020). In the optimized 2024 variant, choosing $\Lambda$1, $\Lambda$2, and $\Lambda$3 gives

$\Lambda$4

showing that the same optimization-plus-DFS philosophy extends to entangling operations (Xu et al., 2024).

An alternative route to protected NHQC against non-collective decoherence combines dynamical decoupling with DFS structure. For an $\Lambda$5-qubit register under linear local system-bath coupling, the decoupling group

$\Lambda$6

creates DFS sectors of dimension $\Lambda$7, giving an encoding rate $\Lambda$8. Within these sectors, the scheme realizes two noncommuting logical single-qubit gates and one nontrivial holonomic two-qubit gate using only two-qubit interactions (Sun et al., 2015).

Qudit extensions emphasize that NHQC is not intrinsically qubit-limited. The bipartite-graph multilevel framework realizes arbitrary one-qutrit and two-qutrit holonomic gates in five-level building blocks, while recent qutrit pulse-engineering work addresses second-order Rabi and detuning errors in the standard $\Lambda$9 structure (Xu et al., 2021, Lu et al., 7 Oct 2025). On superconducting hardware, super-robust NHQC embedded in DFS encoding has been proposed for a scalable two-dimensional square lattice of capacitively coupled transmons, with explicit single-logical-qubit and CNOT constructions and numerical superiority against global control errors and collective dephasing relative to conventional NHQC (Wang et al., 2024).

6. Physical implementations and experimental status

The physical realization of NHQC spans several major quantum-hardware platforms. Nitrogen-vacancy centers in diamond provide one of the clearest room-temperature solid-state realizations. In that setting, the electron-spin triplet ground state $\Lambda$0 supplies the levels $\Lambda$1, $\Lambda$2, and resonant microwave control drives a $\Lambda$3-type bright-state cycle with purely geometric action in the logical subspace (Yan et al., 2017). This platform is notable because initialization, control, and readout are optical or microwave, and the same physical system supports both one- and two-qubit NHQC proposals.

Superconducting circuits have become a principal NHQC platform because parametrically tunable couplings can emulate the required effective $\Lambda$4- or $\Lambda$5-type Hamiltonians. Path-optimized NHQC proposed a DFS-encoded transmon realization using experiment-friendly two-body exchange interaction and reported final optimized fidelities of about $\Lambda$6 for $\Lambda$7, about $\Lambda$8 for $\Lambda$9, and about $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$00 for a controlled-phase gate (Ji et al., 2022). Ultrafast NHQC on superconducting circuits used parametric modulation to realize a detuned $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$01-type logical three-level model in DFS encoding (Shen et al., 2021). Super-robust NHQC in coherence-protected superconducting circuits further proposed a square-lattice transmon architecture in which SR-NHQC in DFS outperforms conventional NHQC and ordinary dynamical gates under global control errors and collective dephasing (Wang et al., 2024).

Rydberg-atom implementations emphasize selective coupling, blockade, and heralding. In a three-atom blockade architecture with two computational atoms and one auxiliary atom, the no-decay branch is heralded by measuring the auxiliary atom in its Rydberg state, with success probability

$$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$02

For the reported single-qubit NOT gate, the full-Hamiltonian simulation used $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$03 and gave $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$04 for $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$05 and final average fidelity $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$06; for the two-qubit CNOT gate, the reported values were $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$07, $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$08, and $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$09 ideally (Kang et al., 2020). A distinct Förster-resonant NHQC$$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$10 proposal for two Rydberg atoms used invariant-based reverse engineering and zero-systematic-error-sensitivity optimization, reporting a CNOT average fidelity $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$11, fidelity about $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$12 at the experimental Förster defect $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$13 MHz, and $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$14 for $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$15 kHz spontaneous emission (Liu et al., 2021).

Trapped ions continue to serve both as a theoretical reference platform and as an experimental testbed. The 2026 demonstration of brachistochrone NHQC in a single trapped $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$16 ion compared conventional NHQC, BNHQC, and composite BNHQC on a $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$17 gate. The reported process fidelities were $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$18 for NHQC, $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$19 for BNHQC, and $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$20 for CBNHQC, while the state fidelities were $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$21, $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$22, and $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$23, respectively (Wang et al., 25 Mar 2026). The same work concluded that BNHQC offers a favorable balance between operation speed and robustness, and explicitly linked improved performance to decreased accumulated population of the auxiliary excited state.

Emerging directions include NHQC in non-Hermitian no-jump dynamics. A 2026 scheme for a driven three-level system in the conditional no-jump regime used a biorthogonal framework and complex pulse design to incorporate decay and dephasing of all bare eigenstates directly into the control, reporting $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$24 for representative Lindblad parameters and a $$\mathcal A_{kl}= i\langle \zeta_k(t)|d\zeta_l(t)\rangle$$25 ns gate time (Li et al., 25 Jun 2026). This does not replace unitary NHQC, but it extends holonomic design principles to monitored open-system settings.

Across these platforms, NHQC has evolved from a single geometric-gate proposal into a family of control strategies. The review literature states that, within state-of-the-art technology, implemented holonomic quantum gates can outperform conventional dynamical ones under certain conditions, but only when the chosen variant matches the dominant noise source, control constraints, and hardware architecture (Liang et al., 2023). That conclusion captures the present status of NHQC: not a single protocol, but a technically differentiated framework for geometric quantum control.