The quantum approximate optimisation algorithm (QAOA) and its variants are at the core of many scenarios that aim at combining the power of quantum computers (QC) and classical high-performance computing (HPC) appliances for combinatorial optimisation. Several obstacles challenge concrete benefits now and in the foreseeable future: Imperfections quickly degrade algorithmic performance below practical utility; over-heads arising from alternating between classical and quantum primitives can counter any advantage; and the choice of parameters or algorithmic variant can substantially influence runtime and result quality. Selecting the appropriate combination is a non-trivial issue, as it not only depends on end-user requirements, but also on details of the hardware and software stack. Appropriate automation can alleviate the burden of choosing optimal combinations for end-users: They should not be required to understand technicalities like detail differences between QAOA variants, required number of QAOA layers, or necessary measurement samples. Yet, they should receive the best possible satisfaction of their non-functional requirements, be it performance or other. We determine factors that influence approximation quality and temporal behaviour of four QAOA variants using comprehensive density-matrix-based numerical simulations targeting three widely studied optimisation problems. Our simulations consider ideal quantum computation, and a continuum of scenarios troubled by realistic imperfections. Our quantitative results, accompanied by a comprehensive reproduction package, show strong performance differences between QAOA variants that can be pinpointed to narrow and specific effects. We identify influential co-variables and relevant non-functional quality goals that, as we argue, mark the rele-vant ingredients for designing appropriate software engineering abstraction mechanisms and automated tool-chains for devising quantum solutions from higher-level problem specifications.
The quantum approximate optimisation algorithm (QAOA) and its variants are at the core of many scenarios that aim at combining the power of quantum computers (QC) and classical high-performance computing (HPC) appliances for combinatorial optimisation. Several obstacles challenge concrete benefits now and in the foreseeable future: Imperfections quickly degrade algorithmic performance below practical utility; over-heads arising from alternating between classical and quantum primitives can counter any advantage; and the choice of parameters or algorithmic variant can substantially influence runtime and result quality. Selecting the appropriate combination is a non-trivial issue, as it not only depends on end-user requirements, but also on details of the hardware and software stack. Appropriate automation can alleviate the burden of choosing optimal combinations for end-users: They should not be required to understand technicalities like detail differences between QAOA variants, required number of QAOA layers, or necessary measurement samples. Yet, they should receive the best possible satisfaction of their non-functional requirements, be it performance or other. We determine factors that influence approximation quality and temporal behaviour of four QAOA variants using comprehensive density-matrix-based numerical simulations targeting three widely studied optimisation problems. Our simulations consider ideal quantum computation, and a continuum of scenarios troubled by realistic imperfections. Our quantitative results, accompanied by a comprehensive reproduction package, show strong performance differences between QAOA variants that can be pinpointed to narrow and specific effects. We identify influential co-variables and relevant non-functional quality goals that, as we argue, mark the rele-vant ingredients for designing appropriate software engineering abstraction mechanisms and automated tool-chains for devising quantum solutions from higher-level problem specifications.
Combining quantum computers with classical compute power has become a standard means for developing algorithms and heuristics that are, eventually, supposed to beat any purely classical alternatives. While in-principle advantages for solution quality or runtime are expected for increasingly many approaches, substantial challenges remain: Non-functional properties like runtime or solution quality of many suggested approaches are not yet fully understood, and need to be explored empirically. This, in turn, makes it unclear which approach is best suited for a given problem. Accurately predicting behaviour and properties of quantum-classical algorithms opens possibilities for software abstraction layers, which in turn can automate decisionmaking for algorithm selection and parametrisation. While such techniques find frequent use in classical high-performance computing, they are still mostly absent from quantum software toolchains. In this paper, we present a methodology (accompanied by a reproducible reference implementation) to perform algorithm selection based on desirable non-functional requirements. This greatly simplifies decision-making processes for end users. Based on meta-information annotations at the source code level, our framework traces key characteristics of quantum-classical heuristics and algorithms, and uses this information to predict the most suitable approach and its parameters for given computational challenges and their non-functional requirements. As combinatorial optimisation is a very extensively studied aspect of quantumclassical systems, we perform a comprehensive case study based on numerical simulations of algorithmic approaches to implement and validate our ideas. We develop statistical models to quantify the influence of various factors on non-functional properties, and establish predictions for optimal algorithmic choices without manual user effort. We argue that our methodology generalises to problem classes beyond combinatorial optimisation, such as Hamiltonian optimisation, and lays a foundation for integrated software layers for quantum design automation.
Quantum computing promises solutions to classically difficult and new-found problems through controlling the subtleties of quantum computing. The Quantum Approximate Optimisation Algorithm (QAOA) is a recently proposed quantum algorithm designed to tackle difficult combinatorial optimisation problems utilising both quantum and classical computation. The hybrid nature, generality and typically low gate-depth make it a strong candidate for near-term implementation in quantum computing. Finding the practical limits of the algorithm is currently an open problem. Until now, no tools to facilitate the design and validation of probabilistic quantum optimisation algorithms such as the QAOA on a non-trivial scale exist. Graph similarity is a long standing classically difficult problem withstanding decades of research from academia and industry. Determining the maximal edge overlap between all possible node label permutations is an NP-Complete task and provides an apt measure of graph similarity. We introduce a novel quantum optimisation simulation package facilitating investigation of all constituent components of the QAOA from desktop to cluster scale using graph similarity as an example. Our simulation provides flexibility and performance. We investigate eight classical optimisation methods each at six levels of decomposition. Moreover an encoding for permutation based problems such as graph similarity through edge overlap to the QAOA allows for significant quantum memory savings at the cost of additional operations. This compromise extends into the classical portion of the algorithm as the inclusion of infeasible solutions creates a challenging cost-function landscape. We present performance analysis of our simulation and of the QAOA setting a precedent for investigating and validating numerous other difficult problems to the QAOA as we move towards realising practical quantum computation.
The quantum approximate optimization algorithm (QAOA) is a promising quantum-classical hybrid technique to solve combinatorial optimization problems in near-term gate-based noisy quantum devices. In QAOA, the objective is a function of the quantum state, which itself is a function of the gate parameters of a multi-level parameterized quantum circuit (PQC). A classical optimizer varies the continuous gate parameters to generate distributions (quantum state) with significant support to the optimal solution. Even at the lowest circuit depth, QAOA offers non-trivial provable performance guarantee which is expected to increase with the circuit depth. However, the existing analysis fails to consider non-idealities in the qubit quality i.e., short lifetime and imperfect gate operations in realistic quantum hardware. In this article, we investigate the impact of various noise sources on the performance of QAOA both in simulation and on a real quantum computer from IBM. Our analyses indicate that the optimal number of stages (p-value) for any QAOA instance is limited by the noise characteristics (gate error, coherence time, etc.) of the target hardware as opposed to the current perception that higher-depth QAOA will provide monotonically better performance for a given problem compared to the low-depth implementations.
Structured variational quantum algorithms such as the Quantum Approximate Optimisation Algorithm (QAOA) have emerged as leading candidates for exploiting advantages of near-term quantum hardware. They interlace classical computation, in particular optimisation of variational parameters, with quantum-specific routines, and combine problem-specific advantages -- sometimes even provable -- with adaptability to the constraints of noisy, intermediate-scale quantum (NISQ) devices. While circuit depth can be parametrically increased and is known to improve performance in an ideal (noiseless) setting, on realistic hardware greater depth exacerbates noise: The overall quality of results depends critically on both, variational parameters and circuit depth. Although identifying optimal parameters is NP-hard, prior work has suggested that they may exhibit regular, predictable patterns for increasingly deep circuits and depending on the studied class of problems. In this work, we systematically investigate the role of classical parameters in QAOA performance through extensive numerical simulations and suggest a simple, yet effective heuristic scheme to find good parameters for low-depth circuits. Our results demonstrate that: (i) high-quality parameters often deviate substantially from expected patterns; (ii) QAOA performance becomes progressively less sensitive to specific parameter choices as depth increases; and (iii) iterative component-wise fixing performs on par with, and at shallow depth may even outperform, several established parameter-selection strategies. We identify conditions under which structured parameter patterns emerge, and when deviations from the patterns warrant further consideration. These insights for low-depth circuits may inform more robust pathways to harnessing QAOA in realistic quantum computing scenarios.
The Quantum Approximate Optimisation Algorithm (QAOA) is a widely studied quantum-classical iterative heuristic for combinatorial optimisation. While QAOA targets problems in complexity class NP, the classical optimisation procedure required in every iteration is itself known to be NP-hard. Still, advantage over classical approaches is suspected for certain scenarios, but nature and origin of its computational power are not yet satisfactorily understood. By introducing means of efficiently and accurately approximating the QAOA optimisation landscape from solution space structures, we derive a new algorithmic variant of unit-depth QAOA for two-level Hamiltonians (including all problems in NP): Instead of performing an iterative quantum-classical computation for each input instance, our non-iterative method is based on a quantum circuit that is instance-independent, but problem-specific. It matches or outperforms unit-depth QAOA for key combinatorial problems, despite reduced computational effort. Our approach is based on proving a long-standing conjecture regarding instance-independent structures in QAOA. By ensuring generality, we link existing empirical observations on QAOA parameter clustering to established approaches in theoretical computer science, and provide a sound foundation for understanding the link between structural properties of solution spaces and quantum optimisation.
Quantum computers promise considerable speedups over classical approaches, which has raised interest from many disciplines. Since any currently available implementations suffer from noise and imperfections, achieving concrete speedups for meaningful problem sizes remains a major challenge. Yet, imperfections and noise may remain present in quantum computing for a long while. Such limitations play no role in classical software computing, and software engineers are typically not well accustomed to considering such imperfections, albeit they substantially influence core properties of software and systems.In this paper, we show how to model imperfections with an approach tailored to (quantum) software engineers. We intuitively illustrate, using numerical simulations, how imperfections influence core properties of quantum algorithms on NISQ systems, and show possible options for tailoring future NISQ machines to improve system performance in a co-design approach.Our results are obtained from a software framework that we provide in form of an easy-to-use reproduction package. It does not require computer scientists to acquire deep physical knowledge on noise, yet provide tangible and intuitively accessible means of interpreting the influence of noise on common software quality and performance indicators.