Shallow circuits, denser codes, surface loss (09/21/26)

TL;DR Single-crystal bulk ${\mathrm{LiNbO}}_3$ and $\mathrm{AlN}$ show two-level-system (TLS) microwave loss—parasitic saturating defects that absorb at single-photon power—from millikelvin temperatures to above $1\mathrm{K}$, but the dissipation sits on the surface rather than in the bulk and piezoelectric channels are ruled out. For cQED hybrid cavities that host electro-optic crystals for microwave–optical transduction, that surface localization means microwave quality factors compatible with high-efficiency conversion are reachable with surface engineering rather than swapping the crystal itself.

• Main result: Bulk single-crystal ${\mathrm{LiNbO}}_3$ and $\mathrm{AlN}$ both display TLS-like microwave dissipation down to single-photon drive, while piezoelectric-related loss channels are excluded.
• Surface vs bulk: TLS-induced loss is predominantly surface-localized rather than an intrinsic bulk property of either crystal.
• Materials corroboration: Room-temperature 3D X-ray photoelectron spectroscopy (3D XPS) and time-of-flight secondary-ion mass spectrometry (ToF-SIMS) support a surface-dominated defect picture.
• cQED implication: Integrating these bulk electro-optic crystals into superconducting cavities can still reach microwave quality factors compatible with high-efficiency microwave–optical transduction if surface loss is controlled.

TL;DR A two-round one-dimensional CHSH problem—a relation task built from the Clauser–Horne–Shimony–Holt nonlocal game, with CHSH questions issued only after Pauli-frame data (the byproduct Paulis tracking prior measurements) are fixed—admits a nearest-neighbor quantum circuit on $2N$ qubits of operational depth at most eight that hits the Tsirelson success probability $(2+\sqrt{2})/4<1$. Matching that rate up to any fixed $\delta <(\sqrt{2}-1)/4$ forces any bounded-fan-in classical circuit of fixed wiring to use depth $\Omega (\log N)$ after the second-round questions, giving an unconditional shallow-circuit advantage without needing perfect quantum success or classical geometric locality.

• Main result: Constant-depth 1D quantum circuits achieve the optimal CHSH win rate $(2+\sqrt{2})/4$, while classical circuits need logarithmic depth to approach that rate.
• Problem design: The two-round schedule withholds CHSH questions until Pauli-frame bits are fixed, blocking the trivial classical strategy that solves the simultaneous-input version perfectly.
• Quantum protocol: On $2N$ qubits with only adjacent two-qubit gates and operational depth $\le 8$, the circuit realizes the Tsirelson CHSH correlations (strictly less than unit success).
• Classical lower bound: For every fixed $0\le \delta <(\sqrt{2}-1)/4$, any randomized classical circuit with fixed wiring and bounded fan-in that averages success $\ge (2+\sqrt{2})/4-\delta$ needs depth $\Omega (\log N)$ after the second-round questions.
• Significance: Unconditional quantum advantage for shallow circuits holds even when the quantum side is geometrically 1D and classical wiring is unrestricted, and without requiring perfect quantum success.

TL;DR A nearest-neighbor brickwork layout of the 4.8.8 color code (faces are squares and octagons) plus periodic superdense syndrome-extraction circuits—twelve CNOT layers that pack checks tightly while sharing ancillas—appear to keep the full code distance $d$, using only $q=(2d^2+5d-5)/2$ physical qubits including ancillas. That is asymptotically half the qubit count of a rotated surface code at the same $d$, and circuit-level simulations aimed at superconducting hardware show favorable scaling for both planar patches and toroidal cousins.

• Main result: Periodic twelve-layer superdense CNOT schedules on the brickwork 4.8.8 color code appear to preserve the full distance $d$ under nearest-neighbor gates.
• Overhead: Including ancillas, the family uses $q=(2d^2+5d-5)/2$ physical qubits—asymptotically $2/3$ the qubits of the triangular 6.6.6 color code and $1/2$ those of the rotated surface code at equal $d$.
• Layout: The construction sits on the square lattice with only nearest-neighbor couplings, matching planar superconducting connectivity.
• Evidence: Circuit-level noise simulations targeting superconducting architectures show favorable logical-error scaling for planar codes and for toroidal cousins under periodic boundary conditions.