# Mapping Quantum Circuits to IBM QX Architectures Using the Minimal Number of SWAP and H Operations

## TL;DR

This paper presents a method to optimally map quantum circuits onto IBM QX architectures by minimizing SWAP and H operations using Boolean satisfiability solvers, outperforming existing heuristics.

## Contribution

It formulates the circuit mapping as a symbolic optimization problem and provides a practical implementation that achieves minimal operation counts.

## Key findings

- IBM's heuristic adds over 100% more operations than the lower bound on average.
- The proposed method effectively minimizes SWAP and H operations.
- Implementation is publicly available for use and further research.

## Abstract

The recent progress in the physical realization of quantum computers (the first publicly available ones--IBM's QX architectures--have been launched in 2017) has motivated research on automatic methods that aid users in running quantum circuits on them. Here, certain physical constraints given by the architectures which restrict the allowed interactions of the involved qubits have to be satisfied. Thus far, this has been addressed by inserting SWAP and H operations. However, it remains unknown whether existing methods add a minimum number of SWAP and H operations or, if not, how far they are away from that minimum--an NP-complete problem. In this work, we address this by formulating the mapping task as a symbolic optimization problem that is solved using reasoning engines like Boolean satisfiability solvers. By this, we do not only provide a method that maps quantum circuits to IBM's QX architectures with a minimal number of SWAP and H operations, but also show by experimental evaluation that the number of operations added by IBM's heuristic solution exceeds the lower bound by more than 100% on average. An implementation of the proposed methodology is publicly available at http://iic.jku.at/eda/research/ibm_qx_mapping.