Leakage Elimination Operator (LEO)

• LEO is a control operation that eliminates leakage by enforcing block-diagonal dynamics between protected and leakage subspaces.
• Various pulse constructions, including bang-bang control, projector-based adiabatic control, and Floquet engineering, enable precise leakage suppression.
• Experimental implementations on superconducting processors and ion traps demonstrate significant leakage reduction and enhanced gate fidelity.

Leakage Elimination Operator (LEO) denotes a control operation designed to suppress transitions between a designated computational subspace and its orthogonal complement, while preserving dynamics inside the computational subspace. In the standard formulation, the Hilbert space is decomposed as $\mathcal{H}=\mathcal{H}_P\oplus\mathcal{H}_Q$, with $P$ the projector onto the protected subspace and $Q=I-P$; an ideal LEO acts as identity on $\mathcal{H}_P$ and as a sign flip on $\mathcal{H}_Q$, so that the effective Hamiltonian becomes block diagonal with respect to protected and leakage sectors (Fan et al., 31 May 2026). Across later developments, this objective has been realized through bang-bang parity kicks, projector-based adiabatic control, exact composite propagators in finite-dimensional systems, dissipative pumping, and hardware-level Hamiltonian block-diagonalization on superconducting processors (Wang et al., 19 Jul 2025).

1. Definition and operator structure

In the canonical description, leakage is population transfer between a computational or code subspace and its complement. With $\mathcal{H}=\mathcal{H}_P\oplus\mathcal{H}_Q$, an ideal LEO may be written as

$R_L P=P,\qquad R_L Q=-Q,$

or equivalently

$R_L=P-Q.$

In block form relative to $P\oplus Q$, this is the same as a diagonal operator that is $+I$ on one sector and $P$0 on the other sector. The defining algebraic property is that the LEO commutes with operators acting purely inside the protected and leakage sectors, and anti-commutes with operators that couple them. In the notation used in the leakage-control literature,

$P$1

where $P$2 acts inside the code, $P$3 acts inside the complement, and $P$4 contains the off-diagonal leakage terms (Markaida et al., 2020).

This definition isolates the specific role of LEOs within subspace protection. Standard decoherence inside $P$5 leaves the state in the encoded space, whereas leakage transfers population out of the space in which logical operations and error correction are defined. The LEO target is therefore not generic noise suppression, but elimination of the $P$6 coupling in the effective dynamics. In average-Hamiltonian language, the objective is

$P$7

so that the effective unitary is block diagonal with respect to computational and leakage subspaces (Wang et al., 19 Jul 2025).

2. Exact and nonperturbative formulations

A central formal development is the Feshbach $P$8–$P$9 partitioning of a general linear equation of motion. In a basis whose first vector is a target time-dependent state $Q=I-P$0, the generator is written as

$Q=I-P$1

where $Q=I-P$2 is the scalar generator on the target amplitude, $Q=I-P$3 acts on the orthogonal sector, and $Q=I-P$4 couple the two sectors. Eliminating the orthogonal amplitudes yields an exact one-component integro-differential equation for the target amplitude. After a reparametrization $Q=I-P$5, the dynamics become

$Q=I-P$6

with the control entering through the phase factor $Q=I-P$7, where

$Q=I-P$8

For quantum systems, $Q=I-P$9 is purely imaginary, so $\mathcal{H}_P$0 is real and the suppression mechanism is phase engineering of $\mathcal{H}_P$1 (Jing et al., 2021).

In this formulation, a rotating-frame LEO is introduced as

$\mathcal{H}_P$2

with $\mathcal{H}_P$3 a scalar control function. The control modifies $\mathcal{H}_P$4, and thus only changes the one-component dynamics through the accumulated phase. Leakage suppression occurs when the oscillation frequency of $\mathcal{H}_P$5 exceeds the characteristic frequency content of the smooth kernel $\mathcal{H}_P$6, so that the kernel integral is canceled by destructive interference. This nonperturbative viewpoint reframes LEOs as a mechanism for creating dynamical leakage-free paths rather than merely as a bang-bang decoupling sequence (Jing et al., 2021).

The same logic appears in adiabatic settings. In an instantaneous eigenbasis, leakage means nonadiabatic transitions out of a chosen eigenstate, and a projector-type control

$\mathcal{H}_P$7

acts inside the adiabatic subspace. The Feshbach $\mathcal{H}_P$8 partition then yields an integro-differential equation whose kernel oscillates with the average control frequency, so sufficiently rapid modulation suppresses nonadiabatic leakage even when the bare evolution is not adiabatically slow (Wang et al., 2016).

3. Pulse constructions and generalized control conditions

The original LEO picture is closely associated with parity-kick or bang-bang control. For a Hamiltonian decomposed as $\mathcal{H}_P$9, repeated application of an operator $\mathcal{H}_Q$0 satisfying $\mathcal{H}_Q$1 and $\mathcal{H}_Q$2 averages out the leakage part. In the ideal limit,

$\mathcal{H}_Q$3

so leakage is removed while the sector-preserving dynamics remain (Jing et al., 2014).

Projector-based pulse constructions make this idea explicit. In adiabatic-speedup experiments on a superconducting qubit, the LEO Hamiltonian is

$\mathcal{H}_Q$4

with a zero-area rectangular modulation

$\mathcal{H}_Q$5

This produces a unitary that phase-shifts the target instantaneous eigenstate while leaving the orthogonal state unchanged, and thus averages away the nonadiabatic coupling in the adiabatic frame (Fan et al., 31 May 2026).

A later Floquet–Magnus reformulation generalized this high-frequency picture. For a periodic LEO drive, the earlier $\mathcal{H}_Q$6-partition control condition is exactly the zeroth-order Magnus condition on the Floquet Hamiltonian,

$\mathcal{H}_Q$7

In this language, the usual high-frequency LEO formulas are the zero-order approximation, while low-frequency control requires first- and higher-order Magnus terms. For sinusoidal pulses $\mathcal{H}_Q$8, the zeroth-order condition becomes

$\mathcal{H}_Q$9

whereas low-frequency corrections shift the optimal control parameters away from the simple Bessel-zero solutions (Yu et al., 29 Jun 2026).

This establishes an important distinction. High-frequency bang-bang control is one realization of LEO theory, but not its only mathematically controlled regime. A plausible implication is that modern LEO design is better viewed as a hierarchy of effective-Hamiltonian constraints, from average-Hamiltonian cancellation at zeroth order to Floquet-engineered dynamical localization at higher orders (Yu et al., 29 Jun 2026).

4. Exact, experimental, and dissipative realizations

LEOs have been realized in several experimentally distinct ways, ranging from exact finite-dimensional composite circuits to dissipative pumping.

Platform	LEO mechanism	Reported outcome
Charge quadrupole qubit (Sun et al., 2017)	Exact composite unitaries in a three-level $\mathcal{H}=\mathcal{H}_P\oplus\mathcal{H}_Q$0 subalgebra	Ideal $\mathcal{H}=\mathcal{H}_P\oplus\mathcal{H}_Q$1 and $\mathcal{H}=\mathcal{H}_P\oplus\mathcal{H}_Q$2 rotations generated by two or three propagators
IBM superconducting processor (Markaida et al., 2020)	Standard-gate LEOs on encoded 2- and 3-qubit subspaces	LEOs significantly suppress leakage
$\mathcal{H}=\mathcal{H}_P\oplus\mathcal{H}_Q$3 hyperfine qubit (Hayes et al., 2019)	Optical pumping from leak states back to the qubit subspace	Each cycle reduces leakage by a factor of $\mathcal{H}=\mathcal{H}_P\oplus\mathcal{H}_Q$4
IBM ibm_marrakesh (Fan et al., 31 May 2026)	Projector-based zero-area LEO pulses during adiabatic evolution	Fidelity improved from $\mathcal{H}=\mathcal{H}_P\oplus\mathcal{H}_Q$5 to over $\mathcal{H}=\mathcal{H}_P\oplus\mathcal{H}_Q$6–$\mathcal{H}=\mathcal{H}_P\oplus\mathcal{H}_Q$7 at $\mathcal{H}=\mathcal{H}_P\oplus\mathcal{H}_Q$8

The charge-quadrupole example is conceptually notable because it is exact rather than perturbative. In a three-level system $\mathcal{H}=\mathcal{H}_P\oplus\mathcal{H}_Q$9, the logical control and leakage operators form an embedded $R_L P=P,\qquad R_L Q=-Q,$0 algebra. Using this structure, noisy $R_L P=P,\qquad R_L Q=-Q,$1- and $R_L P=P,\qquad R_L Q=-Q,$2-rotations can be decomposed into finite sequences of experimentally available Hamiltonians, yielding exact leakage-free $R_L P=P,\qquad R_L Q=-Q,$3 and $R_L P=P,\qquad R_L Q=-Q,$4 gates without additional control axes (Sun et al., 2017).

The trapped-ion hyperfine realization is instead dissipative. A quadrupole transition selectively pumps the leakage states $R_L P=P,\qquad R_L Q=-Q,$5 of $R_L P=P,\qquad R_L Q=-Q,$6 out of the leakage manifold, and subsequent optical pumping returns population to the qubit subspace. Each pumping cycle reduces the leakage population by a factor of $R_L P=P,\qquad R_L Q=-Q,$7, while interleaved randomized benchmarking bounds the induced qubit memory error by $R_L P=P,\qquad R_L Q=-Q,$8 per cycle and the induced qubit population decay by $R_L P=P,\qquad R_L Q=-Q,$9 per cycle (Hayes et al., 2019).

The experimental realization on IBM’s hardware established that LEOs can be compiled into standard gate sets and applied to encoded subspaces in two- and three-qubit Hilbert spaces. The authors describe this as the first experimental implementation of LEOs and report that the operators significantly suppress leakage on the superconducting processor (Markaida et al., 2020). In a different superconducting setting, projector-based LEO control was used to realize adiabatic speedup on ibm_marrakesh. There, ideal zero-area pulses increased adiabatic fidelity from $R_L=P-Q.$0 to over $R_L=P-Q.$1–$R_L=P-Q.$2 at short evolution time, while Bayesian-optimized pulses produced an additional $R_L=P-Q.$3 gain in simulation but not on hardware under present experimental conditions (Fan et al., 31 May 2026).

LEOs also appear in exact open-system settings. For a logical qubit encoded in a mechanical oscillator, a nonperturbative LEO generated by $R_L=P-Q.$4 suppresses motional decoherence, and its effectiveness depends on the integral of the pulse sequence in the time domain rather than the detailed pulse shape when the time period is chosen appropriately (Zheng et al., 2020). In a Jaynes–Cummings atom coupled to a structured environment, a diagonal control $R_L=P-Q.$5 suppresses the relevant dissipative coefficients, and fidelity remains above $R_L=P-Q.$6 on average even with $R_L=P-Q.$7 random errors in amplitude and timing (Luo et al., 2024).

5. Hardware-level block diagonalization in superconducting processors

A recent superconducting realization recasts the LEO objective as dynamic Hamiltonian engineering during a two-qubit gate. In fast flux-tuned CZ gates on fixed-frequency transmons coupled through tunable couplers, spectator-induced frequency collisions enlarge the intended $R_L=P-Q.$8 avoided crossing into a three-level problem $R_L=P-Q.$9. In the effective model,

$P\oplus Q$0

the spectator state $P\oplus Q$1 mediates leakage out of the gate dynamics. A bright–dark rotation,

$P\oplus Q$2

with

$P\oplus Q$3

is then chosen so that the dark state decouples from the active gate subspace. The key condition

$P\oplus Q$4

makes the effective Hamiltonian block diagonal, with the desired dynamics confined to

$P\oplus Q$5

and the leakage pathway isolated in

$P\oplus Q$6

In LEO language, the control enforces $P\oplus Q$7 throughout the gate, so the gate unitary is effectively $P\oplus Q$8 up to decoherence and higher-order errors (Wang et al., 19 Jul 2025).

This Hamiltonian-engineered LEO is implemented by tuning a spectator coupler during the CZ so that the block-diagonalization condition holds at the operating point. Experimentally, the method suppresses leakage rates to the order of $P\oplus Q$9 across a wide near-resonant detuning range. In the single-spectator case, the per-gate leakage rate drops from $+I$0 to $+I$1, while the total CZ gate error changes from $+I$2 to $+I$3. With three simultaneous spectators, the total leakage is reduced from $+I$4 to $+I$5, remaining below the threshold relevant for surface code error correction (Wang et al., 19 Jul 2025).

A related superconducting development is a passive leakage removal unit based on a disordered transmon array. The proposal is explicitly described as not a LEO in the original dynamical-decoupling sense, but as realizing a very similar map: the qubit subspace is protected by localization through energy-level mismatch, whereas leakage excitations are resonant and mobile, propagate to the end of the array, and are removed by feedback measurement or dissipation. For $+I$6, $+I$7, and $+I$8, the reported values are $+I$9, $P$00, $P$01, and $P$02 under realistic conditions (Martín-Vázquez et al., 20 Feb 2025). This suggests a useful distinction between Hamiltonian-engineered LEOs that prevent leakage during a gate and passive leakage-removal units that export leakage after it has occurred.

6. Scope, limitations, and current directions

LEOs are sometimes treated as synonymous with one specific control technology, but the literature shows a broader category. They include unitary parity kicks, projector Hamiltonians in adiabatic frames, nonperturbative phase engineering in exact open-system equations, exact composite circuits in finite-dimensional systems, dissipative optical pumping, and hardware-level Hamiltonian block diagonalization. They are therefore closer to a control principle—decoupling protected and leakage sectors—than to a unique pulse family or gate pattern (Jing et al., 2021).

Several common distinctions are explicit in the current literature. LEOs are conceptually close to dynamical decoupling and subspace protection, but they are not the same as counterdiabatic driving, which adds a term that exactly cancels nonadiabatic couplings for the full adiabatic path, and they are not the same as DRAG, which is designed to reduce leakage to higher transmon levels by compensating microwave-drive-induced couplings (Fan et al., 31 May 2026). They are also distinct from Petz-recovery-based protocols, which reverse a specific dissipative trajectory rather than suppress leakage in situ (Luo et al., 2024).

Another misconception is that LEOs inherently require ideal high-frequency bang-bang control. Earlier adiabatic and open-system analyses already showed that effectiveness can depend on the average frequency of the control function or on the integral of the pulse sequence rather than on the detailed pulse shape (Wang et al., 2016). Later nonperturbative work on mechanical oscillators made the same point in exact open-system dynamics at arbitrary temperature and arbitrary system-bath coupling strength (Zheng et al., 2020). More recently, the Floquet–Magnus reformulation proved that the high-frequency $P$03-partition conditions are only the zero-order approximation, and that higher-order Magnus terms must be included in the low-frequency regime (Yu et al., 29 Jun 2026).

The limitations are correspondingly platform dependent. Accurate block diagonalization can require precise calibration of coupler biases, and leakage pathways that are dominated by fixed stray couplings or interactions not tunable through the available controls may not be fully suppressed (Wang et al., 19 Jul 2025). In projector-based adiabatic implementations, optimized pulses derived from incomplete noise models may not transfer faithfully from simulation to hardware, as illustrated by the modest simulation-only advantage of Bayesian-refined LEO pulses on ibm_marrakesh (Fan et al., 31 May 2026). In non-Markovian open-system settings, control becomes less effective as the environment approaches the Markovian limit (Jing et al., 2014). These constraints do not invalidate the LEO program, but they delimit the regimes in which phase engineering can outperform simpler controls.

Current directions are correspondingly technical rather than conceptual. They include multi-qubit extensions of projector-based adiabatic speedup, more realistic noise-aware pulse optimization, generalization from three-level block-diagonalization to larger invariant sectors, and combinations of hardware-level LEOs with active error correction, dynamical decoupling, and decoherence-free subspaces (Wang et al., 19 Jul 2025). The cumulative picture is that LEOs have evolved from an abstract subspace-protection operator into a family of experimentally grounded leakage-control strategies, unified by the requirement that the effective dynamics preserve a chosen subspace while canceling or exporting the couplings that lead out of it (Yu et al., 29 Jun 2026).