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

Abstract

e recent progress in the physical realization of quantum computers (the first publicly available ones-IBM's QX architectureshave 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.us 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 N P-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 h p://iic.jku.at/eda/research/ibm qx mapping.

Realizing Quantum Algorithms on Real Quantum Computing Devices

“…Hence, the main objective of every designer is to keep this overhead as small as possible -an N P-hard problem [56]. In fact, in order to map a quantum [57]. Accordingly, IBM itself but also researchers started to investigate more e cient solutions to tackle this problem.…”

“…This resulted in exact approaches such as introduced in [57], [58] as well as heuristic solutions such as introduced in [26], [40], [49], [54], [58], [59]. While the former can guarantee minimal or, at least, close-to-minimal solutions, they are often not that scalable even though the use of reasoning engines such as SAT solvers yield impressive improvements [57]. Nevertheless, these solutions are still important, e.g., to determine minimal building blocks and to evaluate the quality of heuristic approaches.…”

Optimal layout synthesis for quantum computing

“…This problem is then passed to Z3 SMT solver [9] which guarantees optimal solutions. Compared to Wille et al [25], a leading optimal approach which also formulates LSQC to SMT, we provide exponential reduction on the number of variables, resulting in orders-of-magnitude reductions in runtime and the memory usage. The key to this improvement is to represent the solution space with mapping variables, not with mapping transformation variables as in [21,25].…”

“…The task of deriving such information is called layout synthesis for quantum computing (LSQC) [23]. Other names have also been used to refer to this task, e.g., quantum circuit compilation [1,8], transpilation (in Qiskit 1 ), placement [14,19], mapping [6,13,15,25,28], qubit allocation [21], and routing [22]. We choose to use the term 'layout synthesis' as it involves both placement and scheduling.…”

“…Therefore, despite the NP-completeness of the problem [14,21,23], optimal approaches to LSQC are still worthy of research because they provide the baseline of measuring the available LSQC solutions. There have been a few previous works on exact or optimal LSQC [6,19,21,25,27]. However, they either just focus on specific coupling graphs, or has some implicit and unnecessary constraints.…”