To appear in the Proceedings of the 2021 IEEE International Conference on Web Services (ICWS), ©2021 IEEE Best-approximation error for parametric quantum circuits Lena Funcke Perimeter Institute for Theoretical Physics Waterloo, ON, Canada lfuncke@perimeterinstitute.ca Tobias Hartung Department of Mathematical Sciences University of Bath Bath, United Kingdom th2040@bath.ac.uk Karl Jansen NIC, DESY Zeuthen Zeuthen, Germany Karl.Jansen@desy.de Stefan K¨ uhn Computation-Based Science and Technology Research Center The Cyprus Institute Nicosia, Cyprus s.kuehn@cyi.ac.cy Manuel Schneider NIC, DESY Zeuthen Zeuthen, Germany manuel.schneider@desy.de Paolo Stornati NIC, DESY Zeuthen Zeuthen, Germany paolo.stornati@desy.de Abstract—In Variational Quantum Simulations, the construc- tion of a suitable parametric quantum circuit is subject to two counteracting effects. The number of parameters should be small for the device noise to be manageable, but also large enough for the circuit to be able to represent the solution. Dimensional expressivity analysis can optimize a candidate circuit considering both aspects. In this article, we will first discuss an inductive construction for such candidate circuits. Furthermore, it is sometimes necessary to choose a circuit with fewer parameters than necessary to represent all relevant states. To characterize such circuits, we estimate the best-approximation error using Voronoi diagrams. Moreover, we discuss a hybrid quantum- classical algorithm to estimate the worst-case best-approximation error, its complexity, and its scaling in state space dimensionality. This allows us to identify some obstacles for variational quantum simulations with local optimizers and underparametrized circuits, and we discuss possible remedies. Index Terms—parametric quantum circuits, dimensional ex- pressivity analysis, best-approximation error, variational quan- tum simulations, Voronoi diagrams I. I NTRODUCTION Noisy intermediate-scale quantum (NISQ) computers [1] are opening up a new avenue to address a large class of computa- tional problems that cannot be solved efficiently with classical computers. Applications of quantum computing range from machine learning [2] to finance [3] to various optimization problems [4], [5]. In physics, quantum computers intrinsically circumvent the sign problem that prevents Monte Carlo simu- lations of strongly correlated quantum-many body problems in certain parameter regimes [6]. Although current hardware is of limited size and suffers from a considerable level of noise, the Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Industry Canada and by the Province of Ontario through the Ministry of Colleges and Uni- versities. S.K. acknowledges financial support from the Cyprus Research and Innovation Foundation under project ”Future-proofing Scientific Applications for the Supercomputers of Tomorrow (FAST)”, contract no. COMPLEMEN- TARY/0916/0048. M.S. and P.S. thank the Helmholtz Einstein International Berlin Research School in Data Science (HEIBRiDS) for funding. ability for NISQ devices to outperform classical computers has already been demonstrated successfully [7], [8] and techniques for mitigating the effects of noise are rapidly developing [9]– [13]. Many algorithms designed for NISQ devices make use of parametric quantum circuits. Variational quantum simula- tions (VQSs) [14], [15], a class of hybrid quantum-classical algorithms for solving optimization problems, are a particu- larly important example. Using a parametric quantum circuit, i.e., a quantum circuit composed of parameter dependent gates, a cost function is evaluated efficiently for a given set of parameters using the quantum coprocessor. The cost function is then minimized on a classical computer in a feedback loop based on the measurement outcome obtained from the quantum device. Using cost functions related to the energy of the quantum state prepared on the quantum device, VQSs have been successfully applied to quantum many-body systems in quantum chemistry [10], [14], [16], [17] and even quantum mechanics and quantum field theory [18]–[22]. Since VQSs depend on the choice of parametric quantum circuits, there are many open questions related to finding a good or optimal quantum circuit. For example, in order to be able to find the solution – or at least find a good approximation – a parametric quantum circuit needs to have many parameters. However, many parameters means many gates and thus large noise. One measure for a “good” quantum circuit is therefore to have as many parameters as necessary while being able to parametrize the entire state space of the simulated model. An optimal circuit taking this point of view would be minimal (there are no redundant parameters) and maximally expressive (the circuit can generate all relevant states). Parametric quantum circuits can be analyzed from this point of view using dimensional expressivity analysis (DEA) [23] which we will review in section II. However, DEA can only tell us whether or not a given circuit is minimal and maximally expressive. The construction of a maximally 1 arXiv:2107.07378v1 [quant-ph] 15 Jul 2021