Time-optimal multi-qubit gates: Complexity, efficient heuristic and gate-time bounds

TL;DR

This paper investigates the complexity of synthesizing time-optimal multi-qubit gates, provides explicit constructions, develops a heuristic algorithm, and establishes bounds and conjectures on gate times, aiming to enhance quantum algorithm efficiency.

Contribution

It proves NP-hardness of time-optimal multi-qubit gate synthesis, offers explicit constructions, and introduces a polynomial-time heuristic algorithm with bounds on gate times.

Findings

Time-optimal multi-qubit gate synthesis is NP-hard.

Explicit constructions of special time-optimal gates with constant times.

Heuristic algorithm achieves near-optimal gate times with polynomial runtime.

Abstract

Multi-qubit entangling interactions arise naturally in several quantum computing platforms and promise advantages over traditional two-qubit gates. In particular, a fixed multi-qubit Ising-type interaction together with single-qubit X-gates can be used to synthesize global ZZ-gates (GZZ gates). In this work, we first show that the synthesis of such quantum gates that are time-optimal is NP-hard. Second, we provide explicit constructions of special time-optimal multi-qubit gates. They have constant gate times and can be implemented with linearly many X-gate layers. Third, we develop a heuristic algorithm with polynomial runtime for synthesizing fast multi-qubit gates. Fourth, we derive lower and upper bounds on the optimal GZZ gate-time. Based on explicit constructions of GZZ gates and numerical studies, we conjecture that any GZZ gate can be executed in a time O(n) for n qubits. Our heuristic synthesis algorithm leads to GZZ gate-times with a similar scaling, which is optimal in this sense. We expect that our efficient synthesis of fast multi-qubit gates allows for faster and, hence, also more error-robust execution of quantum algorithms.