A polynomial size model with implicit SWAP gate counting for exact qubit reordering

Abstract

Due to the physics behind quantum computing, quantum circuit designers must adhere to the constraints posed by the limited interaction distance of qubits. Existing circuits need therefore to be modified via the insertion of SWAP gates, which alter the qubit order by interchanging the location of two qubits' quantum states. We consider the Nearest Neighbor Compliance problem on a linear array, where the number of required SWAP gates is to be minimized. We introduce an Integer Linear Programming model of the problem of which the size scales polynomially in the number of qubits and gates. Furthermore, we solve 131 benchmark instances to optimality using the commercial solver CPLEX. The benchmark instances are substantially larger in comparison to those evaluated with exact methods before. The largest circuits contain up to 18 qubits or over 100 quantum gates. This formulation also seems to be suitable for developing heuristic methods since (near) optimal solutions are discovered quickly in the search process.

Optimal qubit assignment and routing via integer programming

“…Results are reported in Figures 13,14 and 15 in the Appendix. Each figure contains 6 subfigures corresponding to the six possible orders of the three objective functions.…”

“…The most well-studied aspect of this problem is the initial qubit assignment: this is discussed, e.g., in [13,16,17,18], see also the numerous references therein. The aspect of qubit routing is comparatively less studied, perhaps due to the inherent difficulty of the problem; some notable works in this area are [12,14,16,17]. Most of the literature proposes heuristic approaches, and exact formulations are limited to certain topologies (e.g., [14]) or to some aspects of the problem (e.g., [5]).…”

“…The aspect of qubit routing is comparatively less studied, perhaps due to the inherent difficulty of the problem; some notable works in this area are [12,14,16,17]. Most of the literature proposes heuristic approaches, and exact formulations are limited to certain topologies (e.g., [14]) or to some aspects of the problem (e.g., [5]). The most significant differences between our work and the existing literature on exact solution approaches are therefore: (i) we simultaneously consider the problem of allocating and routing qubits, using a novel network flow formulation with design variables that is unlike any of the previously proposed approaches; (ii) we take into account different costs for inserted SWAPs depending on whether or not they can be merged with adjacent two-qubit gates; (iii) we allow arbitrary hardware topologies.…”

A Constraint Programming Approach to Electric Vehicle Routing with Time Windows

“…, |V |} which represents the node location of logical qubit q ∈ Q at layer ℓ ∈ L ′ . Since ILP is less flexible than CP (e.g., due to linearity restrictions) the ILP models from the literature [10,3,9] must introduce additional variables to properly model the problem. In our models, it is sufficient to constrain the values of x qℓ from one layer to the next.…”

“…A variety of techniques have been proposed for solving this problem in the literature, including both exact and heuristic approaches. Exact methods typically involve the use of search-based solvers leveraging smart inference techniques that, given enough time, will find and prove the optimal solution [4,3,10,9,14]. Alternatively, heuristic methods (which have, until recently, been the focus of previous work) sacrifice completeness in favor of rapidly producing high quality circuits [8,15,6]; these methods tend to scale more effectively to larger problem instances as well.…”