Pauli Product Measurement

Overview

The Pauli product measurement (PPM) framework is used and introduced in A Game of Surface Codes [1] to facilitate the mapping of quantum programs to surface codes. Quantum circuits are represented by a small set of building blocks, namely Pauli product measurements (PPMs) as well as Pauli product rotations (PPRs). By distinguishing between operations which can be implemented classically (through clever book keeping) from those requiring real hardware operations, a program is ultimately converted into a sequence consisting of only Pauli product measurements.

Often, we may assume that the input circuit consists of only Clifford and T gates, e.g. \{H, S, T, \text{CNOT}\}, in order to simplify the mapping. Any circuit can always be approximated to arbitrary precision with the (Clifford + T) gate set.

Inputs

• (Clifford + T) circuit

Intermediate representation

• Pauli product measurements (PPMs) \langle P \rangle and Pauli product rotations (PPRs) e^{-i \phi P}, where \phi \in \mathbb{R} and P is a Pauli word. The angle \phi can have three different values for:
  • Pauli operators: multiples of \frac{\pi}{2}, in that case e^{-i \phi P} = \pm P
  • Clifford operators: multiples of \frac{\pi}{4}; all such operators can be commuted to the right of the circuit and merged with PPMs
  • T-gates: multiples of \frac{\pi}{8}; all such operators can be realized with a magic state injection, leaving only classically controlled Clifford and Pauli operators that can be merged into PPMs again.

Outputs

• Circuit consisting of only PPMs

Example

Using the rewrite rules for (Clifford +T) circuits in [1], we can perform the following three steps on the circuit below:

1. Decomposition of (Clifford + T) into Pauli product rotations (PPRs) (catalyst.passes.to_ppr)
2. Commutation of Clifford PPRs past other PPRs (catalyst.passes.commute_ppr)
3. Absorption of Clifford PPRs into terminal measurements (catalyst.passes.merge_ppr_ppm)

Reducing a (Clifford + T) circuit to non-Clifford Pauli product rotations and Pauli product measurements. Orange boxes are Clifford gates with angles \pm \frac{\pi}{4}, neon-yellow boxes are non-Clifford gates with angles \pm \frac{\pi}{8}, and blue boxes are terminal measurements. Note that the final circuit uses a short-hand notation to write two independent terminal measurements on overlapping wires. In particular, this is not a mid-circuit measurement XI followed by a terminal measurement XZ.

In a final fourth step, the non-Clifford PPRs can be replaced by a magic state injection together with classically controlled Clifford PPRs that can be absorbed by the PPMs again.

Typical usage

The Pauli product measurement decomposition was introduced in [1] and used for the ideal operation on surface codes for fault tolerant quantum computing. To transform the circuit entirely to PPMs, we require the injection of magic states from auxiliary qubits. Otherwise, we obtain a circuit containing PPMs and non-Clifford PPRs.

References

[1] "A Game of Surface Codes: Large-Scale Quantum Computing with Lattice Surgery", Daniel Litinski, 1808.02892, 2018

@misc{PennyLane-pauli-product-measurement,
  title={Pauli Product Measurement},
  howpublished={\url{https://pennylane.ai/compilation/pauli-product-measurement}},
  year={2025}
}