# Fault-tolerant quantum computation with few qubits

## TL;DR

This paper presents a fault-tolerant quantum computation scheme using only 17 qubits with the 15-qubit Hamming code, enabling universal operations with minimal overhead suitable for small-scale quantum devices.

## Contribution

It introduces fault-tolerant procedures for Clifford operations on encoded qubits using only two additional qubits, significantly reducing qubit overhead for fault-tolerance.

## Key findings

- Fault-tolerant Clifford operations achieved with 17 qubits.
- Universal fault-tolerant computation possible with 19 qubits.
- Techniques include gadgets, code symmetries, and pieceable fault tolerance.

## Abstract

Reliable qubits are difficult to engineer, but standard fault-tolerance schemes use seven or more physical qubits to encode each logical qubit, with still more qubits required for error correction. The large overhead makes it hard to experiment with fault-tolerance schemes with multiple encoded qubits. The 15-qubit Hamming code protects seven encoded qubits to distance three. We give fault-tolerant procedures for applying arbitrary Clifford operations on these encoded qubits, using only two extra qubits, 17 total. In particular, individual encoded qubits within the code block can be targeted. Fault-tolerant universal computation is possible with four extra qubits, 19 total. The procedures could enable testing more sophisticated protected circuits in small-scale quantum devices. Our main technique is to use gadgets to protect gates against correlated faults. We also take advantage of special code symmetries, and use pieceable fault tolerance.