Optimal compilation of parametrised quantum circuits
1University of Amsterdam
2University of Oxford
3Quantinuum
| Published: | 2025-08-27, volume 9, page 1828 |
| Editor: | Jordi Tura |
| Eprint: | arXiv:2401.12877v5 |
| Doi: | https://doi.org/10.22331/q-2025-08-27-1828 |
| Citation: | Quantum 9, 1828 (2025). |
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Abstract
Parametrised quantum circuits contain phase gates whose phase is determined by a classical algorithm prior to running the circuit on a quantum device. Such circuits are used in variational algorithms like QAOA and VQE. In order for these algorithms to be as efficient as possible it is important that we use the fewest number of parameters. We show that, while the general problem of minimising the number of parameters is NP-hard, when we restrict to circuits that are Clifford apart from parametrised phase gates and where each parameter is used just once, we $can$ efficiently find the optimal parameter count. We show that when parameter transformations are required to be sufficiently well-behaved, the only rewrites that reduce parameters correspond to simple `fusions'. Using this we find that a previous circuit optimisation strategy by some of the authors [Kissinger, van de Wetering. PRA (2019)] finds the optimal number of parameters. Our proof uses the ZX-calculus. We also prove that the standard rewrite rules of the ZX-calculus suffice to prove any equality between parametrised Clifford circuits.
Featured image: Abstract representation of the type of ZX-diagrams that get produced during the optimisation step of the algorithm that PyZX uses.
Popular summary
In both of these examples, removing redundant parametrised gates from the parametrised quantum circuit not only reduces the size of the quantum circuit, it also reduces the number of iterations the quantum circuit needs to be executed on the quantum computer. Therefore, reducing parameter count allows us to get a high quality output from the quantum computer in less time.
In this work we show that an earlier efficient circuit optimisation algorithm by some of the authors that is implemented in the software library PyZX is optimal for compiling parametrised quantum circuits. Specifically, given a parametrised quantum circuit as input, the algorithm always produces an equivalent circuit with optimal parameter count. Importantly, this optimality guarantee requires two assumptions: 1) the parameters of the parametrised gates are all used exactly one in the circuit, 2) the parameters of the new circuit are built from linear combinations of the parameters of the original circuit. While this second assumption might seem restrictive, we show that it actually encapsulates any smooth continuous transformation on the unit circle and so is rather general. We further show in this work that relaxing the first assumption makes the problem NP-hard, and hence that we don't expect there to exist an efficient optimisation algorithm that guarantees optimality.
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► References
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Cited by
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