Pauli product rotations

--emit rotations rewrites the program in the form Litinski’s Game of Surface Codes starts from: a list of rotations about multi-qubit Pauli products followed by Pauli product measurements, with every Clifford gate gone. The program is lowered to Clifford gates, T gates and rotations as for a resource estimate, and qirc walks through it keeping the image of each qubit’s X and Z under the Clifford gates seen so far. A T gate on a qubit becomes a pi/8 rotation about the current image of its Z, any other rotation keeps its angle, and a measurement measures the image of Z. A new rotation is moved back past every rotation and measurement it commutes with, and when it meets one about the same Pauli product the two angles add: pi/8 and -pi/8 cancel, and two pi/8 rotations make a Clifford, which is folded into the images of the gates that follow.

$ qirc mod5_4.qasm --emit rotations
Pauli product rotations for main, 5 qubits

layer  step
    1  +pi/8     Z0 Z3 X4
    1  -pi/8     Z0 Z3
    1  +pi/8     Z2 Z3 X4
    1  -pi/8     Z2 Z3
    1  +pi/8     Z1 Z2 X4
    1  -pi/8     Z1 Z2
    1  +pi/8     Z0 Z1 X4
    1  -pi/8     Z0 Z1

8 pi/8 rotations in T depth 1, 0 other rotations, 0 measurements, 1 layer in all
10 rotations merged with an earlier one about the same Pauli product
a rotation by a about P applies exp(-i a P), and the Clifford gates are folded into the rotations and measurements

Each step goes in the layer after the last earlier step it does not commute with, so the steps within a layer commute and can run in any order, and the T depth counts the layers that hold pi/8 rotations. --emit resources reports it too. On the eleven Clifford+T benchmarks from the optimisation section, merging rotations this way reaches from -O0 the same 938 T gates that phase teleportation and PyZX reach, circuit by circuit, with T depths from 1 for mod5_4 and gf2^4_mult to 17 for tof_10, each in about a tenth of a second. The tests check on random five qubit circuits with arbitrary rotations that the rotations and measurements give the same outcome distribution as the circuit. A program with branches has no single list of rotations, and a reset after a qubit is used needs a correction that depends on a measurement, so both are refused. Qubits that are never measured are left with a Clifford gate at the end, which the list does not show.