Universal Nonadiabatic Passages in Quantum Control

• Universal nonadiabatic passages are engineered finite-time channels that enable exact state, mode, or operator transport without relying on slow adiabatic evolution.
• They leverage ancillary basis constructions and path engineering across platforms such as discrete systems, bosonic networks, and Majorana modes to achieve precise control.
• Applications include robust quantum state transfer, nonadiabatic geometric gate implementation, and scalable designs with quantifiable fidelity bounds.

Universal nonadiabatic passages are explicitly engineered finite-time evolution channels in which a quantum system follows a chosen time-dependent basis state, mode, or subspace exactly, rather than approximately tracking instantaneous eigenstates in an adiabatic limit. In recent quantum-control literature, the term is used most directly for ancillary-basis constructions in general discrete systems, bosonic networks, and Majorana platforms, where the propagator transports selected states or operators along prescribed nonadiabatic routes with controlled phases (Jin et al., 2024, Jin et al., 8 Sep 2025, Jin et al., 22 Aug 2025). Closely related work in nonadiabatic geometric quantum computation recasts the same design logic as path engineering in Hilbert space, where cyclic evolution and vanishing dynamical phase produce purely geometric gates (Mousolou, 2017, Li et al., 2020, Xue et al., 10 Nov 2025).

1. Definition and conceptual scope

In the ancillary-basis formulation, the defining object is a family of projectors

$\Pi_k(t)=|\mu_k(t)\rangle\langle \mu_k(t)|$

that satisfy the von Neumann equation

$\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$

When this condition holds, the corresponding basis vector is an exact transitionless channel in the chosen moving frame. The resulting propagator can be written as

$$U(t,0)=\sum_{k=1}^{K}e^{if_k(t)}|\mu_k(t)\rangle\langle\mu_k(0)|,$$

so the passage is literally the transport of an initial basis state into a target basis state up to a phase (Jin et al., 2024).

This notion is broader than standard adiabatic passage. In the general $M+N$-dimensional framework, the construction is not tied to a specific $\Lambda$-system, dark state, or slow-driving assumption; it is an exact state-engineering method that can connect arbitrary initial and target states through parametric ancillary bases (Jin et al., 2024). In bosonic networks, the same idea appears in the Heisenberg picture: a time-dependent bosonic mode $\mu_k(t)$ is engineered so that the initial operator $\mu_k(0)$ evolves into $\mu_k(t)$ with only a phase factor, thereby transporting any state encoded in that mode (Jin et al., 8 Sep 2025). In Majorana control, the passage states are time-dependent superpositions of two Majorana zero-mode states, designed so that their evolution reproduces the exchange unitary without slow braiding (Jin et al., 22 Aug 2025).

A common source of confusion is terminological. In these control papers, “universal” refers to generality of target selection, system size, or pair selection. In nonadiabatic geometric quantum computation, by contrast, “universal” refers to a universal gate set generated by arbitrary one-qubit gates plus at least one entangling two-qubit gate (Mousolou, 2017, Li et al., 2020, Xue et al., 10 Nov 2025).

2. Ancillary-basis construction in general discrete systems

The most explicit general theory is formulated for an $M+N$-dimensional Hilbert space decomposed into an assistant subspace $\{|e_m\rangle\}$ of dimension $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$0 and a working subspace $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$1 of dimension $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$2. The Hamiltonian is written as

$\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$3

and the ancillary basis is built recursively from bright states and dark or nonadiabatic states (Jin et al., 2024).

Within this recursion, most ancillary states remain internal to either the assistant or working sector, while the final coupled pair,

$\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$4

forms the actual passage between the two sectors. Their structure is controlled by angles $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$5, $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$6, $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$7 and phases $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$8, $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$9, $$U(t,0)=\sum_{k=1}^{K}e^{if_k(t)}|\mu_k(t)\rangle\langle\mu_k(0)|,$$0. Substituting the ancillary basis into the von Neumann equation yields closed-form control conditions for the drive phases, couplings, and detuning: $$U(t,0)=\sum_{k=1}^{K}e^{if_k(t)}|\mu_k(t)\rangle\langle\mu_k(0)|,$$1

$$U(t,0)=\sum_{k=1}^{K}e^{if_k(t)}|\mu_k(t)\rangle\langle\mu_k(0)|,$$2

$$U(t,0)=\sum_{k=1}^{K}e^{if_k(t)}|\mu_k(t)\rangle\langle\mu_k(0)|,$$3

These relations are the operative core of the construction: once the ancillary angles and phases are chosen, the Hamiltonian is fixed analytically (Jin et al., 2024).

The formalism is explicitly transitionless in the ancillary basis. If

$$U(t,0)=\sum_{k=1}^{K}e^{if_k(t)}|\mu_k(t)\rangle\langle\mu_k(0)|,$$4

then the coefficients satisfy

$$U(t,0)=\sum_{k=1}^{K}e^{if_k(t)}|\mu_k(t)\rangle\langle\mu_k(0)|,$$5

so each selected passage channel evolves independently with phase

$$U(t,0)=\sum_{k=1}^{K}e^{if_k(t)}|\mu_k(t)\rangle\langle\mu_k(0)|,$$6

This separates the problem of target selection from the problem of exact solvability (Jin et al., 2024).

The same work also notes that many ancillary states become static dark states when the parameters are time-independent, satisfying

$$U(t,0)=\sum_{k=1}^{K}e^{if_k(t)}|\mu_k(t)\rangle\langle\mu_k(0)|,$$7

Such states can be activated into useful passages by adding auxiliary interactions inside the assistant or working subspaces, for example through an added Hamiltonian $$U(t,0)=\sum_{k=1}^{K}e^{if_k(t)}|\mu_k(t)\rangle\langle\mu_k(0)|,$$8 that promotes a previously dark basis state into an active transport channel (Jin et al., 2024).

3. Relation to nonadiabatic geometric control

Universal nonadiabatic passages intersect directly with nonadiabatic geometric quantum computation, although the two notions are not identical. In the geometric setting, a gate is nonadiabatic and geometric when the chosen states undergo cyclic evolution and the dynamical phase vanishes. One representative formulation imposes

$$U(t,0)=\sum_{k=1}^{K}e^{if_k(t)}|\mu_k(t)\rangle\langle\mu_k(0)|,$$9

or equivalently the parallel-transport condition

$M+N$0

An auxiliary periodic basis $M+N$1 then generates a Hamiltonian

$M+N$2

which realizes any prescribed closed path and yields

$M+N$3

For a single qubit, the geometric phase is

$M+N$4

that is, half the solid angle enclosed by the path $M+N$5 on the Bloch sphere (Li et al., 2020).

This path-design viewpoint is also the organizing principle of recent reviews of nonadiabatic geometric quantum computation. In that literature, one specifies a state trajectory $M+N$6 on the Bloch sphere and reconstructs the Hamiltonian from

$M+N$7

The same framework encompasses multi-loop cyclic schemes, orange-slice loops, short-path variants, noncyclic geometric gates, and optimal-control-assisted protocols (Xue et al., 10 Nov 2025).

A more structured realization of nonadiabatic geometric control appears in decoherence-free holonomic computation. There, a logical qubit is encoded in the decoherence-free subspace

$M+N$8

temporarily embedded in

$M+N$9

and controlled through anisotropic $\Lambda$0 interactions plus local fields. Cyclic finite-time evolution with vanishing dynamical contribution on the logical subspace yields a nonadiabatic quantum holonomy in $\Lambda$1 for one logical qubit and $\Lambda$2 for two logical qubits, producing arbitrary single-logical-qubit rotations and an entangling two-logical-qubit gate (Mousolou, 2017).

Path length itself is an independent design variable. In a three-level $\Lambda$3-system used for nonadiabatic holonomic gates, a multi-segment construction extends the admissible class of nonadiabatic paths beyond the original $\Lambda$4-rotation bright-state route. The matching condition

$\Lambda$5

allows a gate to be assembled from split segments, and the paper gives an example with total effective angle about $\Lambda$6, rather than $\Lambda$7. That reduction lowers exposure to decoherence and also supports a dynamical-decoupling-based decoherence-elimination strategy without redundant encoding qubits (Xu et al., 2018).

4. Bosonic networks and Majorana zero modes

In bosonic networks, universal nonadiabatic passages are formulated directly at the level of mode operators. For an $\Lambda$8-node network with

$\Lambda$9

one introduces time-dependent ancillary modes $\mu_k(t)$0 through a unitary transformation

$\mu_k(t)$1

After a further rotation to a time-independent ancillary basis $\mu_k(t)$2, the rotating-frame Hamiltonian becomes

$\mu_k(t)$3

The central solvability condition is

$\mu_k(t)$4

with $\mu_k(t)$5 the projector onto the $\mu_k(t)$6-th ancillary mode. This is presented as the necessary and sufficient condition for determining the time evolution exactly: if it holds, the relevant passage mode is decoupled from the others and simply accumulates a phase (Jin et al., 8 Sep 2025).

The scope claimed for the bosonic framework is broad. It is described as applicable to arbitrary system size $\mu_k(t)$7, arbitrary network connectivity, arbitrary target pair or subset of nodes, and arbitrary bosonic states, including Fock, coherent, cat, NOON, and thermal states. The paper illustrates this by arbitrary state exchange between two nodes, chiral NOON-state transfer among three bosonic nodes, and chiral Fock-state transfer among three of four bosonic nodes (Jin et al., 8 Sep 2025).

A topological version appears in universal quantum control over Majorana zero modes. There, a local defect mediates an effective interaction between any chosen pair of Majorana end modes from three Kitaev chains. The two-state passage basis is

$\mu_k(t)$8

$\mu_k(t)$9

and the effective Hamiltonian in the reduced subspace is

$\mu_k(0)$0

with

$\mu_k(0)$1

Under

$\mu_k(0)$2

the resulting unitary reproduces the braid operator

$\mu_k(0)$3

for the selected pair of Majorana modes (Jin et al., 22 Aug 2025).

The Majorana work also introduces robustness enhancement through rapid modulation of a passage-dependent global phase. For local-defect frequency errors, the fidelity drops significantly at $\mu_k(0)$4, whereas for $\mu_k(0)$5 it stays above $\mu_k(0)$6 across $\mu_k(0)$7. For Rabi-frequency errors, the fidelity exceeds $\mu_k(0)$8 when $\mu_k(0)$9. The same formalism also yields clockwise and counterclockwise chiral population transfer among three Majorana modes (Jin et al., 22 Aug 2025).

5. State transfer, Raman passages, and entanglement generation

An analytically transparent precursor to later universal constructions is the explicit nonadiabatic passage for stimulated Raman transitions in a three-level $\mu_k(t)$0 system. On exact one-photon resonance, the interaction-picture Hamiltonian is

$\mu_k(t)$1

and the passage is obtained by choosing

$\mu_k(t)$2

With

$\mu_k(t)$3

the system follows exactly the invariant eigenstate

$\mu_k(t)$4

which evolves from $\mu_k(t)$5 to $\mu_k(t)$6 as $\mu_k(t)$7 goes from $\mu_k(t)$8 to $\mu_k(t)$9. The protocol uses only pump and Stokes pulses; no additional coupling field is required (Cao et al., 2019).

That work explicitly distinguishes the scheme from standard delayed-pulse STIRAP and then reinterprets it within shortcut-to-adiabaticity. By scaling the reference Hamiltonian as $M+N$0, the intermediate-state population can be reduced from

$M+N$1

to

$M+N$2

with reduction factor

$M+N$3

For constant $M+N$4, the maximum suppression occurs at $M+N$5, where

$M+N$6

This establishes a direct link between explicit nonadiabatic passage design and dressed-state shortcut methods (Cao et al., 2019).

In the more general $M+N$7-dimensional passage theory, these ideas are applied to distant superconducting qubits. For two Andreev spin qubits with tunable longitudinal coupling

$M+N$8

the first step prepares the single-excitation Bell state

$M+N$9

from the ground state. The reported fidelity is $\{|e_m\rangle\}$0 under dissipation $\{|e_m\rangle\}$1, with $\{|e_m\rangle\}$2 for $\{|e_m\rangle\}$3 and $\{|e_m\rangle\}$4 for $\{|e_m\rangle\}$5. A second step converts that Bell state into the double-excitation Bell state

$\{|e_m\rangle\}$6

with reported fidelity $\{|e_m\rangle\}$7 in the relevant dissipative setting (Jin et al., 2024).

The same framework extends to multipartite entanglement. For three qubits, the protocol proceeds through

$\{|e_m\rangle\}$8

$\{|e_m\rangle\}$9

$\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$00

and the general $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$01-qubit construction yields

$\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$02

in $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$03 steps. The reported fidelity for the three-qubit GHZ state is $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$04 at $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$05, decreasing to $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$06 at $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$07 (Jin et al., 2024).

6. Bounds, critical passages, and the meaning of universality

The control-oriented literature on universal nonadiabatic passages is complemented by a theoretical literature on nonadiabatic transition bounds. For transitions from the $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$08-th level to the $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$09-th level, one rigorous bound is

$\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$10

or equivalently

$\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$11

For nondegenerate levels, the integrand is the norm of the component of $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$12 orthogonal to $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$13, i.e. the standard quantum-geometric-tensor quantity governing adiabaticity. The same paper also derives a universal bound valid for any nonadiabatic transition,

$\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$14

where

$\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$15

Here “universal” means independent of the particular target level $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$16 (Hatomura et al., 2020).

Near many-body criticality, a different use of “universal” concerns scaling rather than constructive controllability. For slow passages that approach but do not cross a critical point, excitation probabilities retain exponential suppression but acquire nontrivial algebraic prefactors: $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$17 In the nonlinear Landau-Zener model, the paper identifies $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$18 away from touching and $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$19 at touching, with

$\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$20

near the critical point. In an exactly solvable Stark-ladder model driven by

$\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$21

the exact excitation number is

$\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$22

whose slow-passage asymptotics yield

$\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$23

while at touching $\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$24 the behavior becomes a pure power law,

$\frac{d}{dt}\Pi_k(t)=-i[H(t),\Pi_k(t)].$25

The paper presents these features as universal aspects of near-critical nonadiabatic passages, while emphasizing that the exponents themselves remain model-dependent (Sinitsyn et al., 2023).

Taken together, these strands show that “universal nonadiabatic passages” is not a single formalism but a family of closely related ideas. In exact control theory, it denotes analytically designed passage channels in ancillary bases that connect arbitrary initial and target states or modes (Jin et al., 2024, Jin et al., 8 Sep 2025, Jin et al., 22 Aug 2025). In nonadiabatic geometric quantum computation, it denotes path-engineered state or subspace evolutions that realize a universal gate set through cyclicity and dynamical-phase cancellation (Mousolou, 2017, Li et al., 2020, Xue et al., 10 Nov 2025). In transition theory, it denotes bounds or scaling structures that apply across broad classes of nonadiabatic processes (Hatomura et al., 2020, Sinitsyn et al., 2023). This suggests that the unifying content of the term is not any one Hamiltonian architecture, but the replacement of slow eigenstate following by exact or rigorously characterized finite-time passage engineering.