Bosonic quantum codes for amplitude damping

Traditional quantum error correction involves the redundant encoding of $k$ quantum bits using $n$ quantum bits to allow the detection and correction of any $t$ bit error. The smallest general $t=1\mathrm{}$ code requires $n=5\mathrm{}$ for $k=1\mathrm{}$. However, the dominant error process in a physical system is often well known, thus inviting the following question: Given a specific error model, can more efficient codes be devised? We demonstrate alternative codes that correct just amplitude damping errors that allow, for example, a $t=1\mathrm{}$, $k=1\mathrm{}$ code using effectively $n=4.6\mathrm{}$. Our scheme is based on using bosonic states of photons in a finite number of optical modes. We present necessary and sufficient conditions for the codes and describe construction algorithms, physical implementation, and performance bounds.