Jenga-Krotov: Efficient Exchange Gate Compilation

• The paper introduces JK, a two-stage protocol that compresses multi-qubit gate sequences (e.g., Toffoli gate from 216 to 92 exchange unitaries) via over-parameterized Krotov optimization followed by iterative pulse deletion.
• JK is designed for exchange-only qubits in semiconductor quantum-dot systems, addressing the challenge of synthesizing high-fidelity multi-qubit gates with hardware-native constraints.
• Benchmark results show that JK not only reduces pulse counts and time steps but also enhances noise robustness against charge fluctuations and crosstalk compared to standard decomposition methods.

Searching arXiv for the JK paper and core Krotov background. Attempting to retrieve relevant arXiv records for Jenga-Krotov and Krotov-method background. Jenga-Krotov (JK) is a two-stage quantum-control and gate-compilation algorithm introduced for exchange-only (EO) qubits in semiconductor quantum-dot architectures. It is designed to discover compact, high-fidelity pulse sequences for multi-qubit gates by combining over-parameterized Krotov optimization with iterative pulse deletion and re-optimization. In the benchmark application to the Toffoli gate on three linear EO qubits, JK reduces the required sequence from 216 exchange unitaries and 162 time steps in a standard direct decomposition to 92 exchange unitaries and 50 time steps, while maintaining the target fidelity and improving robustness under charge noise and crosstalk (Wu et al., 16 Jul 2025).

1. Definition and problem setting

JK addresses a specific bottleneck in EO quantum computing: the synthesis of nontrivial multi-qubit gates under the restriction to exchange-only control. In EO architectures, each logical qubit is encoded into three physical spins and manipulated using only Heisenberg exchange interactions. This control model is experimentally attractive because it avoids ESR or magnetic-field-gradient control, but it makes gate synthesis highly constrained. The paper frames Toffoli as the representative benchmark because high-fidelity single- and two-qubit EO gates are already known, whereas efficient multi-qubit compilation remains difficult (Wu et al., 16 Jul 2025).

The algorithm’s central motivation is that conventional approaches are inefficient in this setting. The paper states that direct decomposition into one- and two-qubit gates produces prohibitively long exchange sequences, and that standard optimal-control methods such as GRAPE and vanilla Krotov either converge poorly under realistic sequence-length constraints or get stuck in local minima or barren plateaus. JK is proposed as a remedy through a deliberately over-complete optimization stage followed by systematic structural compression (Wu et al., 16 Jul 2025).

The name “Jenga-Krotov” is literal in the sense used by the authors: after a dense Krotov-optimized pulse sequence is found, exchange-pulse “bricks” are removed one at a time, with each candidate removal accepted only if re-optimization preserves the target infidelity threshold. The method is therefore not a replacement for Krotov’s method, but a meta-algorithm built around it (Wu et al., 16 Jul 2025).

2. Exchange-only qubit architecture and control model

The single-EO-qubit setting is the standard triple-quantum-dot architecture with one electron spin per dot and nearest-neighbor exchange coupling. The Hamiltonian is written as

$$\mathcal{H}=\sum_{\langle i,j\rangle}\mathcal{H}_{i,j},\qquad \mathcal{H}_{i,j}=J_{i,j}(t)\boldsymbol{S}_i\cdot\boldsymbol{S}_j,\qquad (i,j=1,2,3),$$

with $\boldsymbol{S}_i=\frac{1}{2}\boldsymbol{\sigma}_i$. The logical qubit is encoded in a decoherence-free subspace, and the paper chooses the $S_{z,\mathrm{tot}^{(3)}}=1/2$ DFS with

$$|\mathbb{0}\rangle=|1\rangle,\qquad |\mathbb{1}\rangle=|3\rangle.$$

Leakage states can arise under inhomogeneous magnetic field, but the analysis assumes a homogeneous external magnetic field and neglects them (Wu et al., 16 Jul 2025).

Control is parameterized by piecewise-constant exchange pulses. For a segment of duration $\tau$ with fixed coupling $J$, the paper defines

$$\mathcal{H}_{i,j}^{\mathrm{ex}}=\frac{1}{4}\boldsymbol{\sigma}_i\cdot\boldsymbol{\sigma}_j-\frac{1}{4},\qquad p=\frac{J\tau}{\pi},$$

and the corresponding exchange unitary

$$U_{i,j}^{\mathrm{ex}}(p)=e^{-ip\pi \mathcal{H}_{i,j}^{\mathrm{ex}}}.$$

The optimization variables are these pulse parameters $p$ (Wu et al., 16 Jul 2025).

For the three-logical-qubit Toffoli problem, the physical system is a chain of nine spins. Although the full Hilbert space has dimension $2^9=512$, conservation of total spin and $\boldsymbol{S}_i=\frac{1}{2}\boldsymbol{\sigma}_i$0 reduces the relevant problem to a 90-dimensional basis. The computational target is the logical three-qubit action in the replicated $\boldsymbol{S}_i=\frac{1}{2}\boldsymbol{\sigma}_i$1 computational blocks embedded in this reduced representation (Wu et al., 16 Jul 2025).

The available controls are alternating odd and even layers of commuting nearest-neighbor exchanges. The paper writes

$\boldsymbol{S}_i=\frac{1}{2}\boldsymbol{\sigma}_i$2

and

$\boldsymbol{S}_i=\frac{1}{2}\boldsymbol{\sigma}_i$3

using the commutativity relation $\boldsymbol{S}_i=\frac{1}{2}\boldsymbol{\sigma}_i$4 when all indices are distinct (Wu et al., 16 Jul 2025).

3. Two-stage JK procedure

JK begins with parameter-space expansion. A long initial sequence of length $\boldsymbol{S}_i=\frac{1}{2}\boldsymbol{\sigma}_i$5 is chosen, and standard Krotov optimization is used to drive the gate infidelity below a preset threshold $\boldsymbol{S}_i=\frac{1}{2}\boldsymbol{\sigma}_i$6. The paper repeatedly uses

$\boldsymbol{S}_i=\frac{1}{2}\boldsymbol{\sigma}_i$7

If random initializations fail to reach infidelity below $\boldsymbol{S}_i=\frac{1}{2}\boldsymbol{\sigma}_i$8, the starting sequence length $\boldsymbol{S}_i=\frac{1}{2}\boldsymbol{\sigma}_i$9 is increased. This stage is explicitly motivated as a way to escape the failure of short-sequence optimization (Wu et al., 16 Jul 2025).

The second stage is the “Jenga” step. The paper introduces a $S_{z,\mathrm{tot}^{(3)}}=1/2$0 matrix $S_{z,\mathrm{tot}^{(3)}}=1/2$1, initialized to $S_{z,\mathrm{tot}^{(3)}}=1/2$2 to denote unexplored entries. An unexplored entry $S_{z,\mathrm{tot}^{(3)}}=1/2$3 is selected at random, the corresponding exchange amplitude is set to zero,

$S_{z,\mathrm{tot}^{(3)}}=1/2$4

and the remaining nonzero pulses are re-optimized with Krotov while their locations are kept fixed. If the re-optimized sequence still satisfies

$S_{z,\mathrm{tot}^{(3)}}=1/2$5

the deletion is accepted and the matrix entry is updated to $S_{z,\mathrm{tot}^{(3)}}=1/2$6; otherwise the deletion is rejected and marked with $S_{z,\mathrm{tot}^{(3)}}=1/2$7. After all entries have been tested, any time step whose four control slots are all removed is deleted, shrinking the sequence in both pulse count and time-step count (Wu et al., 16 Jul 2025).

The algorithm is therefore a hard-threshold pruning scheme, not a soft sparsity penalty. The paper explicitly distinguishes JK from plain Krotov on this basis: standard Krotov optimizes pulse amplitudes for a fixed control structure and fixed sequence length, whereas JK first enlarges the parameterization and then optimizes the support pattern of the pulse sequence through deletion-and-reoptimization (Wu et al., 16 Jul 2025).

This structure is closely aligned with broader interpretations of Krotov methods as sequential update algorithms over time, rather than merely fixed continuous-time formulas. Schirmer and de Fouquieres describe the “Krotov” family as sequential-in-time update methods and emphasize that finite-grid optimization should be treated as fundamental, not as an afterthought to continuum derivations (Schirmer et al., 2011). A plausible implication is that JK should be viewed less as a new variational principle than as a structured control-design workflow layered on top of an existing Krotov solver.

4. Objective functionals and Krotov layer

The paper defines a GRAPE baseline in terms of the unitary fidelity

$S_{z,\mathrm{tot}^{(3)}}=1/2$8

with $S_{z,\mathrm{tot}^{(3)}}=1/2$9, and the update

$$|\mathbb{0}\rangle=|1\rangle,\qquad |\mathbb{1}\rangle=|3\rangle.$$0

using learning rate $$|\mathbb{0}\rangle=|1\rangle,\qquad |\mathbb{1}\rangle=|3\rangle.$$1 in the reported comparison (Wu et al., 16 Jul 2025).

The Krotov layer of JK is formulated through the objective

$$|\mathbb{0}\rangle=|1\rangle,\qquad |\mathbb{1}\rangle=|3\rangle.$$2

For three qubits, the optimization uses all eight logical computational basis states. The final-time objective is the real-part state infidelity

$$|\mathbb{0}\rangle=|1\rangle,\qquad |\mathbb{1}\rangle=|3\rangle.$$3

with

$$|\mathbb{0}\rangle=|1\rangle,\qquad |\mathbb{1}\rangle=|3\rangle.$$4

The running cost is

$$|\mathbb{0}\rangle=|1\rangle,\qquad |\mathbb{1}\rangle=|3\rangle.$$5

in the intended notation given in the paper summary (Wu et al., 16 Jul 2025).

The control update is written as

$$|\mathbb{0}\rangle=|1\rangle,\qquad |\mathbb{1}\rangle=|3\rangle.$$6

with

$$|\mathbb{0}\rangle=|1\rangle,\qquad |\mathbb{1}\rangle=|3\rangle.$$7

Forward states and co-states are propagated according to the standard Krotov forward-backward structure, with the co-state boundary condition determined by the derivative of $$|\mathbb{0}\rangle=|1\rangle,\qquad |\mathbb{1}\rangle=|3\rangle.$$8 (Wu et al., 16 Jul 2025).

Within the broader quantum-control literature, this places JK squarely in the Krotov lineage. In Schirmer and de Fouquieres, sequential Krotov-type methods are analyzed as local time-slice updates with search-length control, and fixed penalty terms are treated as unnecessary or even counterproductive at finite time resolution (Schirmer et al., 2011). In a separate comparison for Bose-Einstein-condensate control, GRAPE and Krotov are both derived from a variational principle but differ in how the control is updated, how additional costs are handled, and whether second-order information is included; that study reports that GRAPE tends to yield smoother controls, whereas Krotov can converge faster with a suitable step-size parameter but may generate sharp features (Jäger et al., 2014). JK does not alter those foundational distinctions. It uses Krotov as the optimization engine and adds a structural pruning layer above it (Wu et al., 16 Jul 2025).

5. Toffoli compilation and quantitative performance

The direct-decomposition benchmark, denoted DIR, builds Toffoli from standard logical components. In the decomposition used by the paper, Toffoli requires 2 Hadamard gates, 7 $$|\mathbb{0}\rangle=|1\rangle,\qquad |\mathbb{1}\rangle=|3\rangle.$$9 or $\tau$0 gates, 6 CNOT gates, and 4 SWAP gates, because two of the CNOTs are between non-nearest-neighbor logical qubits. This gives a sequence of 216 exchange unitaries and 162 time steps (Wu et al., 16 Jul 2025).

JK instead starts from a dense optimized sequence and compresses it. The paper reports a successful uncompressed run with 55 time steps and a final compressed Toffoli sequence with 92 exchange unitaries and 50 time steps. It divides the final sequence into five sections,

$\tau$1

with

$\tau$2

$\tau$3

Odd and even time-step unitaries apply the commuting exchange layers described above (Wu et al., 16 Jul 2025).

Sequence	Exchange unitaries	Time steps
DIR	216	162
JK	92	50

The paper also reports a constrained comparison at $\tau$4 time steps. Because each time step carries four control channels, this already corresponds to 100 pulses. After 100 iterations, GRAPE plateaus at infidelity $\tau$5 and standard Krotov at infidelity $\tau$6, far above the target threshold. This is the explicit motivation for the over-parameterize-then-prune logic of JK (Wu et al., 16 Jul 2025).

For the optimized JK sequence, the paper states that the intrinsic noiseless infidelity saturates around $\tau$7, which it interprets as the optimization precision limit of the 92-pulse sequence. The final pulse table is given explicitly in the paper, so the 50-step compiled sequence is fully specified (Wu et al., 16 Jul 2025).

Noise robustness is evaluated under two channels. For charge-noise or exchange-strength fluctuation,

$\tau$8

and for crosstalk,

$\tau$9

The parameters are sampled as

$J$0

with means varied from $J$1 to $J$2 and infidelity averaged over 100 Monte Carlo samples. The gate fidelity used for this evaluation is

$J$3

with $J$4 (Wu et al., 16 Jul 2025).

Under realistic noise, the abstract states that the accumulated gate error of the JK sequence is an order of magnitude lower than that of the conventional direct-decomposition approach. In the body text, Fig. 11 is summarized as showing that once noise strength exceeds $J$5, JK consistently yields lower infidelity than DIR in both charge-noise and crosstalk simulations (Wu et al., 16 Jul 2025).

6. Interpretation, scope, and common misconceptions

JK is presented as a general and scalable EO-gate-synthesis strategy, but the concrete demonstration is limited to one case: Toffoli compilation for three linear EO qubits. The paper argues conceptually that the same approach should apply to arbitrary multi-qubit gates and other EO geometries, including triangular configurations, yet it does not empirically validate those broader claims across multiple gates or system sizes (Wu et al., 16 Jul 2025).

Several implementation caveats remain important. The paper does not provide wall-clock runtime, hardware specifications, full convergence curves for the pruning stage, or scaling-law fits. It names shape functions $J$6 and inverse step widths $J$7 for Krotov, but does not specify their numerical values. It also contains a minor internal inconsistency regarding the size of the initial dense sequence: the Results section describes the 55-step starting point as 192 exchange unitaries, whereas elsewhere the same setup implies 220. The stable reported result is the compressed 92-unitary, 50-step sequence (Wu et al., 16 Jul 2025).

A common misconception arises from the name. JK is not related to the recreational Jenga game in the sense of the mathematical topology literature. An unrelated paper, “Maximum genus of the Jenga like configurations,” studies generalized Jenga towers as polyhedral closed surfaces and derives a maximum-genus formula, but it does not mention “Jenga-Krotov” or “JK” (Akiyama et al., 2017). In JK, “Jenga” refers only to the deletion analogy used for pulse pruning (Wu et al., 16 Jul 2025).

Another possible misconception is to treat JK as a new Krotov update law. The paper does not make that claim. Its defining move is architectural rather than variational: first use standard Krotov to find a high-fidelity dense solution, then iteratively delete exchange unitaries and re-optimize the remaining amplitudes. In that sense, JK is best characterized as Krotov-based structure optimization for hardware-native gate compilation in EO systems (Wu et al., 16 Jul 2025).

Its significance lies precisely there. EO hardware admits only exchange interactions, and direct modular decompositions become long and noise-sensitive. JK exploits the empirical asymmetry between optimization and implementation: dense parameterizations can be easier to optimize globally, while sparse pulse schedules are easier to execute experimentally. The Toffoli result—216 to 92 exchange unitaries and 162 to 50 time steps—makes that asymmetry operational in a semiconductor-qubit setting (Wu et al., 16 Jul 2025).