We describe a general method for turning quantum circuits into sparse quantum subsystem codes. The idea is to turn each circuit element into a set of low-weight gauge generators that enforce the input-output relations of that circuit element. Using this prescription, we can map an arbitrary stabilizer code into a new subsystem code with the same distance and number of encoded qubits but where all the generators have constant weight, at the cost of adding some ancilla qubits. With an additional overhead of ancilla qubits, the new code can also be made spatially local. Applying our construction to certain concatenated stabilizer codes yields families of subsystem codes with constant-weight generators and with minimum distance $d = n^{1-ε}$, where $ε= O(1/\sqrt{\log n})$. For spatially local codes in $D$ dimensions we nearly saturate a bound due to Bravyi and Terhal and achieve $d = n^{1-ε-1/D}$. Previously the best code distance achievable with constant-weight generators in any dimension, due to Freedman, Meyer and Luo, was $O(\sqrt{n\log n})$ for a stabilizer code.