Hybrid simplification rules for boundaries of quantum circuits

Abstract

We describe rules to simplify quantum circuits at their boundaries, i.e. at state preparation and measurement. There, any strictly incoherent operation may be pushed into a preor post-processing of classical data. The rules can greatly simplify the implementation of quantum circuits and are particularly useful for hybrid algorithms on noisy intermediatescale quantum hardware, e.g. in the context of quantum simulation. Finally we illustrate how circuit cutting can enable the rules to be applied to more locations. Near-term quantum computers do not have the capabilities for full-fledged error correction. Thus they suffer from noise and can only run circuits of limited size. Nevertheless, useful computations might be performed on such devices. In this so called noisy intermediate-scale quantum (NISQ) regime, it is crucial to optimize the quantum circuits to get meaningful results. The literature on this important building-block of a quantum software stack is steadily increasing. It includes work on compilation techniques [1, 2, 3, 4], which can also be quite hardware specific [5]. There are also several approaches to optimizing quantum circuits, e.g. w.r.t. the circuit depth or the controlled-not or T-gate count [6]. For short circuits on few qubits optimal solutions can be found by meet-in-the-middle algorithms [7]. However, larger circuits can only be optimized using heuristics. Circuits can be simplified via a detour over ZX diagrams [8, 9, 10]. But often rewrite rules are used to simplify the circuit step by step [11, 12]. The present work can be viewed to follow the latter approach as well. The focus is slightly different, though, because we are interested in hybrid simplification rules. With “hybrid” we mean rules which affect both the quantum and the classical part of a computation. For NISQ applications it is quite natural to think about hybrid Michael Epping: Michael.Epping@dlr.de simplification rules, because the most promising applications for NISQ devices are hybrid anyways. That is they use the quantum computer only for those parts of the algorithm, where it is actually beneficial. An important class of such algorithms are variational quantum algorithms [13].