Abstract
Parametric quantum circuits play a crucial role in the performance of many variational quantum algorithms. To successfully implement such algorithms, one must design efficient quantum circuits that sufficiently approximate the solution space while maintaining a low parameter count and circuit depth. In this paper, develop a method to analyze the dimensional expressivity of parametric quantum circuits. Our technique allows for identifying superfluous parameters in the circuit layout and for obtaining a maximally expressive ansatz with a minimum number of parameters. Using a hybrid quantum-classical approach, we show how to efficiently implement the expressivity analysis using quantum hardware, and we provide a proof of principle demonstration of this procedure on IBM's quantum hardware. We also discuss the effect of symmetries and demonstrate how to incorporate or remove symmetries from the parametrized ansatz.
Highlights
Current noisy intermediate-scale quantum (NISQ) computers [1] open up a new route to tackle a variety of computational problems that cannot be addressed efficiently with classical computers
We demonstrate how to modify a given quantum circuit to remove unwanted symmetries from the states generated by the quantum circuit, and how to approach the construction of a maximally expressive quantum circuit respecting the physical symmetries
We proposed a dimensional expressivity analysis for the study and design of parametric quantum circuits
Summary
Current noisy intermediate-scale quantum (NISQ) computers [1] open up a new route to tackle a variety of computational problems that cannot be addressed efficiently with classical computers. For VQSs using a local optimizer in parameter space, this effectively translates into the question whether or not the physical state space is path connected and whether or not a path from the initial parameter set to a parameter set for the solution can be found using only information available in the physical state space, that is, information that can be obtained from quantum device calls These path connectedness questions are addressed in App. A, since they are highly technical and explore a direction that is not necessary to delve into deeply for the examples provided in this paper. As a simple introductory example, let us analyze QISKIT’s [33] hardware efficient SU(2) 2local circuit EfficientSU2(3, reps=N), which the QISKIT documentation [34] proposes as “a heuristic pattern that can be used to prepare trial wave functions for variational quantum algorithms or classification circuit for machine learning.” This circuit consists of N + 1 blocks of RY and RZ gates applied to every qubit.
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