  Quantum integrated photonics requires large-scale linear optical circuitry,
and for many applications it is desirable to have a universally programmable
circuit, able to implement an arbitrary unitary transformation on a number of
modes. This has been achieved using the Reck scheme, consisting of a network of
Mach Zehnder interferometers containing a variable phase shifter in one path,
as well as an external phase shifter after each Mach Zehnder. It subsequently
became apparent that with symmetric Mach Zehnders containing a phase shift in
both paths, the external phase shifts are redundant, resulting in a more
compact circuit. The rectangular Clements scheme improves on the Reck scheme in
terms of circuit depth, but it has been thought that an external phase-shifter
was necessary after each Mach Zehnder. Here, we show that the Clements scheme
can be realised using symmetric Mach Zehnders, requiring only a small number of
external phase-shifters that do not contribute to the depth of the circuit.
This will result in a significant saving in the length of these devices,
allowing more complex circuits to fit onto a photonic chip, and reducing the
propagation losses associated with these circuits. We also discuss how similar
savings can be made to alternative schemes which have robustness to imbalanced
beam-splitters.
