Computing with many encoded logical qubits beyond break-even

High-rate quantum error correcting (QEC) codes encode many logical qubits in a given number of physical qubits, making them promising candidates for quantum computation. Implementing high-rate codes at a scale that both frustrates classical computing and improves performance by encoding requires both high fidelity gates and long-range qubit connectivity - both of which are offered by trapped-ion quantum computers. Here, we demonstrate computations that outperform their unencoded counterparts in the high-rate [[ k+2, k, 2 ]] iceberg quantum error detecting (QED) and \[[ \(k₂ + 2\)\(k₁ + 2\), k_2k1, 4\]] two-level concatenated iceberg QEC codes, using the 98-qubit Quantinuum Helios trapped-ion quantum processor. Utilizing new gadgets for encoded operations, we realize this "beyond break-even" performance with reasonable postselection rates across a range of fault-tolerant (FT) and partially-fault-tolerant (pFT) component and application benchmarks with between 48 and 94 logical qubits. These benchmarks include FT state preparation and measurement, QEC cycle benchmarking, logical gate benchmarking, GHZ state preparation, and a pFT quantum simulation of the three-dimensional XY model of quantum magnetism. Additionally, we illustrate that postselection rates can be suppressed by increasing the code distance via concatenation. Our results represent state-of-the-art logical component and state fidelities and provide evidence that high-rate QED/QEC codes are viable on contemporary quantum computers for near-term beyond-classical-scale computation.