Initialization is a Hamiltonian-control problem, not a measurement problem
Repeated surface-code error correction relies on stabilizer measurements, ancilla reset, and classical decoding. Before those cycles can protect anything, the processor must already sit in, or close to, the desired stabilizer-code state. The paper treats that initialization step as a separate problem: start from an easily prepared product state and evolve coherently toward a target Hamiltonian whose ground space defines the code state.
That reframing matters because the objective is not to read out Pauli stabilizers but to generate the multi-qubit correlations of their common eigenspace. Adiabatic evolution offers a measurement-free route, but its runtime is limited by the closing many-body gap and by finite coherence times. At experimentally accessible runtimes, diabatic excitations lower the final ground-state fidelity, which is exactly the gap counterdiabatic driving is meant to close.
Stabilizer geometry fixes which counterdiabatic operators are needed
The exact adiabatic gauge potential in an interacting many-body system is generally nonlocal and contains high-weight Pauli operators that are hard to realize directly. The paper therefore uses the Sels-Polkovnikov variational principle, which optimizes an approximate gauge potential inside a restricted operator manifold. The question becomes which operators belong in that manifold.
For the surface-code interpolation studied here, the answer follows from the stabilizer geometry. The initial Hamiltonian is a transverse Z-field and the target consists of X- and Z-type stabilizers. Because Z-type stabilizers commute with the driver, they do not contribute to the leading local gauge potential. The counterdiabatic terms come from the X-type stabilizers: each stabilizer generates operators obtained by replacing one X with a Y. A bulk plaquette XaXbXcXd produces four-body terms such as YaXbXcXd, while a boundary stabilizer XaXb produces two-body terms such as YaXb.
For the 3×3 patch this yields twelve candidate counterdiabatic operators: eight four-body terms from the two bulk plaquettes and four two-body terms from the boundary stabilizers. The paper also compares against a first-order nested-commutator ansatz, which contains the same bulk and boundary single-Y operators but fixes all relative amplitudes through a single coefficient, whereas the variational construction determines each coefficient independently.
Dressing pairwise exchange generates the required four-body operators
Near-term superconducting processors provide native one- and two-qubit controls, while the relevant counterdiabatic corrections are non-stoquastic higher-body Pauli terms. The paper bridges this gap with digital-analog synthesis: native analog exchange interactions plus short digital dressing layers.
The native analog resource is an XY exchange Hamiltonian on the hardware graph, with programmable coupling on each edge. A matching — a subset of pairwise disjoint edges — specifies which qubit pairs are dressed in parallel, and a local basis choice rotates the dressed interaction into different Pauli operators. Conjugating the native exchange block by these dressing circuits produces two-, three-, and four-body operators. Because each dressing layer consists of two-qubit gates on disjoint pairs, a single layer can raise operator weight from two to four, which is exactly the locality the plaquette counterdiabatic terms require.
The construction reverses the usual workflow. Instead of first building a variational gauge potential and then compiling its many-body Pauli operators into native gates, the paper first identifies the many-body operator manifold the hardware can generate, then uses that hardware-generated pool as the variational ansatz for the gauge potential. Solving the Sels-Polkovnikov equations assigns optimal coefficients to only a small subset of generated operators, determining both the relevant operators and their coefficients. Each retained operator is traced back to the dressing circuit and analog couplings that produced it.
At short times, counterdiabatic driving beats bare adiabatic evolution
The numerical benchmark uses a 3×3 patch, the rotated surface code with distance d = 3. Its bulk stabilizers are Z0Z1Z3Z4, X1X2X4X5, X3X4X6X7, and Z4Z5Z7Z8, and its boundary stabilizers are X0X3, X5X8, Z1Z2, and Z6Z7, giving eight stabilizers and a dimensionless ground-state energy of −8 for J = h.
The ground-state preparation dynamics are simulated over a total runtime T = 1, discretized into ten steps of size Δt = 0.1, with a linear schedule. Within that short runtime, bare adiabatic evolution does not reach the target ground space. Nested-commutator counterdiabatic driving improves preparation, digital variational counterdiabatic driving gives the highest final fidelity, and the hardware-native digital-analog implementation remains close to the fully digital variational protocol. That closeness is the evidence that the selected dressed exchange blocks capture the dominant counterdiabatic structure rather than an incidental subset.
The resource comparison is per Trotter step, not per protocol
For the 3×3 benchmark, the paper reports resources per Trotter step in native two-qubit exchange gates, single-qubit rotations, and analog blocks. In the hybrid digital-analog implementation, six of the twelve four-body operators selected by the variational gauge potential are synthesized using dressed analog blocks, while the two-body boundary operators are implemented digitally and included in the reported counts.
The digital-analog construction uses 25 two-qubit gates, 8 single-qubit rotations, and 4 analog blocks, for 28 explicit operations. The fully digital variational protocol uses 211 two-qubit gates and 92 iswap gates, for 305 operations, and the digital nested-commutator protocol uses 238 two-qubit gates and 108 iswap gates, for 340 operations. These are different decompositions of the same counterdiabatic task, so the comparison is about how the required operators are synthesized, not about different physical targets.
Matching transfer keeps four-body coverage while shrinking the search
For an n×n rotated patch with n² data qubits, the (n−1)² bulk checks and 2(n−1) boundary checks give n²−1 stabilizers, and the stabilizer-local variational counterdiabatic basis contains 2n(n−1) operators for odd n and 2n(n−1)−2 for even n. Both scale as O(n²).
The dominant classical cost is enumerating admissible matchings of the hardware graph. For matchings of fixed maximum size, the number of candidates is bounded by a polynomial in the number of edges, giving polynomial preprocessing complexity for fixed maximum matching size. For the 4×4 lattice, the paper reuses local matching patterns selected on a 3×4 reference system and generates their valid translated, rotated, and reflected copies. This transferred pool contains only six matchings, compared with 2593 candidates in a brute-force 4×4 search, yet it still covers all 16 required four-body counterdiabatic terms. Only the matching structure is transferred; the variational coefficients and corresponding couplings are recomputed for the 4×4 Hamiltonian.
The paper generalizes the construction to n×m stabilizer lattices and reports that digital-analog synthesis can reduce the entangling depth by an order of magnitude. The caveat stated with this result is that the dressing does not generate all terms for a given Hamiltonian, so enumerating the relevant matching pairs remains necessary.
Under noise, the coherent protocol approaches measurement-based preparation
For the 3×3 patch, the paper compares the coherent preparation protocols with a measurement-based stabilizer-preparation scheme under a hardware-motivated noise model. The noisy simulations are performed at the density-matrix level with a dimensionless parameter η that uniformly rescales baseline error probabilities, where η = 0 is ideal evolution and η = 1 defines the reference hardware setting. Stabilizer probabilities are evaluated at η = 0.5.
In that comparison, the measurement-based protocol achieves the highest fidelity. The digital-analog counterdiabatic implementation approaches its performance while remaining entirely coherent and measurement-free, and it substantially outperforms both fully digital counterdiabatic driving and bare adiabatic evolution over the investigated noise range. The reduced digital gate overhead is the stated reason the digital-analog version is more robust under this noise model.
This prepares code states, it does not run error correction
The protocol addresses state preparation of the surface code, not quantum error correction itself. Preparing a state that satisfies the stabilizer constraints is a prerequisite for logical-state initialization, and the paper frames its contribution as a coherent, measurement-free route to hardware-native preparation of stabilizer-code states.
The construction is not restricted to stabilizer Hamiltonians. Its essential ingredient is generating a higher-order counterdiabatic operator manifold from native two-body interactions before performing the variational gauge-potential optimization. In this implementation, basis transformations and matching-based dressing generate operators up to four-body locality from the native XY interaction. Extending the dressing structure, matching patterns, or sequence depth could give access to larger operator manifolds and higher-body interactions.
The most decision-relevant result for a reader weighing near-term superconducting initialization is the trade the paper quantifies: the digital-analog protocol gives up some fidelity relative to a measurement-based scheme, but it stays entirely coherent and measurement-free while cutting entangling depth by an order of magnitude relative to fully digital synthesis. The natural next step the authors identify is combining this preparation with repeated stabilizer measurements, decoding, and recovery, and quantifying how initial preparation fidelity propagates to logical observables and logical error rates.
Sources used in this answer
Measurement-free Preparation of Surface-Code States with Digital-Analog Counterdiabatic Drivings
A digital-analog counterdiabatic protocol prepares surface-code ground-state manifolds coherently without intermediate measurements, generating the dominant four-body counterdiabatic basis from dressed pairwise XY exchange and reducing entangling depth by an order of magnitude relative to fully digital synthesis.