The Iceberg code for error detection on trapped-ion computers

Short answer: The Iceberg code is a quantum error detection code that encodes k logical qubits into k + 2 physical qubits and is designed for existing trapped-ion computers. Its authors, in a 2022 arXiv preprint later published as a 2023 journal article, report fault-tolerant initialisation and syndrome measurement circuits that detect any single-qubit error, and an experiment protecting eight logical qubits. It detects errors and does not correct them.

The code

The authors start from the point that fully fault-tolerant quantum error correction is too resource intensive for existing quantum computers, so they develop an error detection code for current trapped-ion machines. Encoding k logical qubits into k + 2 physical qubits, the code has fault-tolerant state-initialisation and syndrome-measurement circuits that can detect any single-qubit error. They nickname it the Iceberg code because of the particular structure of its logical operators. The supplementary material identifies it as a [[k + 2, k, 2]] code.

It provides a universal set of local and global logical rotations that have physical support on only two qubits. The authors say a high-fidelity, though non fault-tolerant, compilation of this gate set is possible because trapped-ion computers have two-qubit physical rotations with all-to-all connectivity.

What fault tolerant means here

The supplementary material says the initialisation and syndrome-measurement circuits need two additional ancilla qubits, which can be reset and reused, so the physical circuits need k + 4 physical qubits, a constant overhead. The circuits are fault-tolerant in the sense that no single fault gives rise to an undetectable logical error, where single faults are local Pauli errors from the failure of a circuit element. The authors verify initialisation, syndrome measurement and measurement circuits by exhaustively checking all code sizes up to k = 16: for each possible local error they insert the Pauli operator, propagate it to the end of the circuit and assess the result.

At the end of a circuit all physical qubits are measured in the computational basis, which lets them reconstruct the logical Z outcomes and the Z stabiliser. The X stabiliser cannot be reconstructed from that, so it is extracted before the final readout with a flagged fault-tolerant circuit.

The experiment

The authors report protecting circuits of eight logical qubits with up to 256 layers, saturating the logical quantum volume of 2 to the power 8, and showing a positive effect from increasing the frequency of syndrome measurements. They say these results illustrate the practical usefulness of the code for protecting expressive circuits on existing trapped-ion computers. The paper's acknowledgements state that the experiments used the Quantinuum system model H1-2, powered by Honeywell ion traps. These are the authors' reported results from one hardware platform.

Limits

The code detects errors. It does not correct them, and the gate set compilation is not fault-tolerant. The fault-tolerance check described covers single local Pauli faults for code sizes up to k = 16.

FAQ

Does the Iceberg code correct errors?

No. The authors describe it as an error detection code that can detect any single-qubit error.

How many physical qubits does it need?

The code uses k + 2 physical qubits for k logical qubits, and the supplementary material says the physical circuits need k + 4 including two reusable ancilla qubits.

Is the whole gate set fault-tolerant?

No. The authors describe the compilation of the universal gate set as high-fidelity but not fault-tolerant.

What hardware ran the experiment?

The paper states the experiments were done on the Quantinuum system model H1-2, powered by Honeywell ion traps.

How was fault tolerance checked?

By exhaustively checking all code sizes up to k = 16, inserting each local Pauli error and propagating it to the end of the circuit.