Measurement-free Preparation of Surface-Code States with Digital-Analog Counterdiabatic Drivings

AI summary

Overview

Research area: Quantum error correction and quantum control — specifically, measurement-free preparation of surface-code states on superconducting hardware using counterdiabatic (CD) driving and digital-analog quantum computing (DAQC).

Technical level: Advanced. The paper assumes familiarity with stabilizer codes, the adiabatic theorem, the adiabatic gauge potential, Trotterization, and superconducting gate primitives. The conceptual narrative is accessible, but the derivations (Sels-Polkovnikov variational principle, matching-based dressing circuits) require graduate-level quantum information background.

Scope (1 sentence): The paper introduces and benchmarks a hardware-native, measurement-free protocol that synthesizes plaquette-local counterdiabatic terms from dressed analog XY exchange blocks to prepare rotated surface-code ground-state manifolds on grids up to 4×4 qubits.

What This Paper Is About

Before a superconducting processor can run repeated error-correction cycles, it must first be initialized in (or near) the target stabilizer-code state. Doing this with measurements requires ancillas, resets, and classical decoding; doing it coherently requires generating four-body stabilizer correlations from hardware that natively offers only one- and two-qubit controls. The paper's goal is to close that gap: to show that the counterdiabatic terms needed to speed up adiabatic state preparation have a structure dictated by the stabilizer geometry itself, and that this structure can be realized with fixed-angle two-qubit gates plus analog exchange-interaction evolutions on a nearest-neighbor superconducting lattice.

Key Contributions

1. A stabilizer-local CD basis derived from geometry. The authors show that, for an interpolation from a Z-polarized product state to the rotated surface-code Hamiltonian, only the X-type stabilizers contribute to the leading local adiabatic gauge potential, because Z-type stabilizers commute with the driver. Each X-type stabilizer generates operators obtained by replacing one X with a Y (Eqs. 15–17), giving a compact, plaquette-local CD basis.

2. A digital-analog synthesis that maps CD terms onto hardware-native blocks. Dressed analog XY blocks — native exchange evolution conjugated by matching-based dressing circuits built from fixed-angle two-qubit XY gates and single-qubit rotations — are shown to generate two-, three-, and four-body Pauli strings (Eqs. 22–23). The same matching pattern, read out in different local Pauli frames, can generate the plaquette-local CD terms.

3. A reversed workflow that uses the hardware pool as the variational ansatz. Instead of choosing an operator basis first and then compiling it, the protocol first generates a hardware-native Pauli pool and then solves the Sels-Polkovnikov equations on that pool (Eq. 24), retaining only operators with non-negligible weight and tracing each back to its source dressing circuit and exchange couplings.

4. Benchmarks up to 4×4 and an entangling-depth analysis. The method is benchmarked on 2×3, 3×3, 3×4, and 4×4 grids, with a noise-model comparison against measurement-based stabilizer preparation on the 3×3 patch, and an analytic generalization to n×m lattices showing an order-of-magnitude reduction in entangling depth.

Main Findings

• The relevant gauge potential occupies a tiny subspace of the hardware pool. For the 3×3 patch, the dressed analog blocks generate a pool of 188 Pauli strings, but solving the Sels-Polkovnikov equations assigns nonzero coefficients to only six of them.

• CD consistently beats bare adiabatic evolution at short runtime. At T = 1, adiabatic evolution does not reach the target ground space; NC-CD improves preparation, and digital VarCD gives the highest final fidelity among the coherent digital protocols on the 3×3 patch. (Exact fidelity values appear in the figures but are not quoted numerically in the text.)

• DAQC closely tracks the fully digital VarCD result. The hardware-native implementation remains close to digital VarCD on the 3×3 patch and follows it as lattice size increases, even though the synthesis cannot cover all operator support present in the digital protocol.

• Depth, not Pauli count, governs the DAQC cost. In the digital approach each many-body Pauli evolution is transpiled into the iSWAP entangling basis, so depth grows with both the number and the weight of Pauli terms. In DAQC, multiple CD operators are generated collectively within one dressed analog block, and disjoint dressing operations on a matching run in parallel.

• Digital-analog synthesis reduces entangling depth by an order of magnitude relative to the fully digital construction, as stated in the abstract and visualized in the depth-scaling figure.

• The DAQC protocol is far less sensitive to Trotter step size. Digital VarCD shows the strongest dependence on Δt because its Hamiltonian contains non-commuting CD Pauli strings subject to first-order product-formula error. The DAQC (CD-only) result is comparatively insensitive, because each dressed native interaction is implemented as a complete analog block and the CD-only evolution is naturally discretized in the interpolation parameter s (Eqs. 29–31).

• Measurement-based preparation still wins under noise, but the gap is narrow. On the 3×3 patch at T = 1 with ten first-order Trotter steps (Δt = 0.1), the single-shot stabilizer projection protocol achieves the highest fidelity; the digital-analog CD implementation approaches it while remaining entirely coherent and measurement-free, and substantially outperforms fully digital CD and bare adiabatic evolution.

• Resource scaling is polynomial. For an n×n patch the number of stabilizers is N_stab = n² − 1 and the stabilizer-local CD basis contains N_CD = 2n(n−1) for odd n and 2n(n−1) − 2 for even n — both O(n²). Matching enumeration is bounded by a sum of binomial coefficients over hardware edges, giving polynomial preprocessing for fixed maximum matching size.

• Pattern reuse replaces brute-force search at 4×4. The brute-force matching search becomes rapidly more expensive from 2×3 to 3×4; for 4×4, the authors instead reuse local matching patterns found on the 3×4 reference system and generate their valid translated, rotated, and reflected copies, recomputing only the VarCD coefficients.

• Reference noise parameters are stated explicitly. At η = 1: p_1q = 2.0×10⁻⁴ and p_2q = 2.0×10⁻³ (gate fidelities 99.98% and 99.8%), analog block error p_an = 2.0×10⁻³, T₁ = 150 μs, T₂* = 100 μs, t_1q = 25 ns, t_2q = 70 ns. The parameter η uniformly rescales error probabilities while gate durations and coherence times stay fixed.

Methodology in Plain English

The authors frame code-state initialization as a control problem rather than a measurement problem. Start from an easy state — all qubits in |0⟩, which is a Z-polarized product state — and slowly morph the Hamiltonian into the surface-code Hamiltonian whose ground space is the code. Slow morphing means a long runtime, and long runtimes lose to decoherence, so they add a counterdiabatic correction term that cancels the unwanted diabatic transitions.

The obstacle is that the exact correction is highly nonlocal. Their workaround has three layers:

1. Find which corrections matter, using geometry. Because the starting Hamiltonian is a Z field, only X-type stabilizers produce leading-order corrections. Each X-plaquette yields four operators (one X swapped for a Y on each site); each two-body boundary check yields two. This gives twelve candidate operators on the 3×3 patch.

2. Build those operators from what the hardware already does. The native resource is an analog XY exchange interaction on the coupler graph. By sandwiching that analog evolution between layers of fixed-angle two-qubit XY gates applied on disjoint edges (a "matching") plus single-qubit rotations, the effective interaction is rotated into different Pauli frames and grows into three- and four-body strings. Different dressings of the same matching produce different operators.

3. Let the variational principle pick winners. The full set of generated Pauli strings becomes the ansatz pool for the Sels-Polkovnikov variational gauge potential. Solving the variational equations assigns coefficients; almost all turn out negligible (six out of 188 survive on 3×3). Each survivor is traced back to the dressing circuit and exchange couplings that produced it, and those couplings become the amplitudes of the analog blocks in the actual time evolution.

In the fast-driving impulse regime (JT ≪ 1), the adiabatic part of the evolution becomes negligible while the integrated CD contribution stays finite, so the implementation keeps only the CD term. The four-body terms run as analog blocks; the leftover two-body boundary terms are done digitally.

Why This Matters

Impact on research. The paper forges an explicit link between three things usually treated separately: stabilizer geometry, the operator structure of the adiabatic gauge potential, and the primitive set of a superconducting processor. It shows that variational CD need not be a purely classical optimization exercise — the hardware's native interaction can define the ansatz. It also demonstrates that coherent, measurement-free initialization can approach the fidelity of measurement-based preparation while avoiding mid-circuit measurement entirely, which is attractive for architectures where measurement and reset are costly or error-prone.

Real-world applications (downstream implications):

• Fault-tolerant quantum computing: faster, shallower logical-state initialization as a front-end primitive before repeated syndrome-extraction cycles begin.
• Superconducting processor control stacks: pulse-level compilation that targets tunable-coupler XY exchange and fixed-angle two-qubit gates rather than generic iSWAP decompositions.
• Hardware-aware ansatz design for near-term devices: using generated operator pools as variational bases for other state-preparation and simulation tasks.
• Benchmarking digital-analog paradigms: quantitative depth and noise-robustness comparisons that inform when analog blocks beat purely digital circuits.

Industry relevance. The author list spans Kipu Quantum (a quantum software/hardware-co-design company) and the Universitat de València, and the entire construction is expressed in the vocabulary of commercial superconducting hardware: tunable couplers, exchange couplings, 25 ns single-qubit and 70 ns two-qubit gate times, T₁ = 150 μs. The order-of-magnitude entangling-depth reduction is directly relevant to vendors whose bottleneck is two-qubit gate error and circuit depth.

Future Directions

• Extend beyond the pattern-reuse shortcut at 4×4. The 4×4 results rely on matching patterns transferred from the 3×4 reference geometry; whether this transfer remains adequate at larger lattices, and how the classical preprocessing cost scales without it, is left open.
• Close the residual synthesis gap. The DAQC protocol does not cover all operator support available to the fully digital VarCD protocol, which leaves a measurable fidelity gap on the 3×3 benchmark; expanding the dressing repertoire to capture the remaining weight is a natural next step.
• Validate on real hardware. All results are numerical, using a hardware-motivated density-matrix noise model with representative (not device-specific) parameters. Experimental demonstration is not reported.
• Push toward larger and rectangular lattices. The paper derives the n×m generalization analytically and benchmarks up to 4×4 (16 qubits); demonstrating the protocol at sizes where the many-body gap closes more severely is an open question.

Target Audience

Quantum error-correction researchers working on logical-state initialization; superconducting-qubit experimentalists interested in pulse-level, hardware-native compilation; quantum-control theorists working on counterdiabatic driving and the adiabatic gauge potential in interacting many-body systems; and digital-analog quantum computing practitioners evaluating when analog blocks outperform decomposed digital circuits. Readers without background in stabilizer formalism or adiabatic gauge potentials will find the introduction readable but the methods section demanding.

Authors’ abstract

Preparing stabilizer-code states with shallow circuits is an important primitive for near-term quantum error correction in superconducting circuits, where logical-state initialization requires four-body stabilizer correlations from native one- and two-body controls. We introduce a digital-analog method for the preparation of surface-code ground-state manifolds. The method exploits a structural property of the adiabatic gauge potential, mapping stabilizer-local counterdiabatic terms to digital-analog blocks on the superconducting layout using fixed-angle two-qubit XY gates, single-qubit rotations, and analog exchange-interaction evolutions. The dressed blocks generate higher-order operators used as the variational ansatz to determine the counterdiabatic terms within the Sels-Polkovnikov adiabatic-gauge-potential framework. We benchmark checkerboard plaquette-stabilizer grids up to 4 x 4; the square-grid instances correspond to rotated surface codes, while the rectangular cases probe the scaling of the method. For all grids, the proposed method generates the dominant four-body counterdiabatic basis and, at short evolution times, substantially improves ground-state preparation compared with bare adiabatic evolution. We further generalize the construction to n x m stabilizer lattices and show that digital-analog synthesis can reduce the entangling depth by an order of magnitude. Together, these results establish a direct connection between stabilizer geometry, the structure of the counterdiabatic gauge potential, and digital-analog control for measurement-free preparation of stabilizer-code states on near-term superconducting architectures.