Simulating imperfect quantum optical circuits using unsymmetrized bases

Fault-tolerant photonic quantum computing requires the generation of large entangled resource states. The required size of these states makes it challenging to simulate the effects of errors such as loss and partial distinguishability. For an interferometer with $N$ partially distinguishable input photons and $M$ spatial modes, the Fock basis can have up to ${N+NM-1\choose N}$ elements. We show that it is possible to use a much smaller unsymmetrized basis with size $M^N$ without discarding any information. This enables simulations of the joint effect of loss and partial distinguishability on larger states than is otherwise possible. We demonstrate the technique by providing the first-ever simulations of the generation of imperfect qubits encoded using quantum parity codes, including an example where the Hilbert space is over $60$ orders of magnitude smaller than the $N$-photon Fock space. As part of the analysis, we derive the loss mechanism for partially distinguishable photons.