Reducing the Amount of Single‐Qubit Rotations in VQE and Related Algorithms

Abstract

With the advent of hybrid quantum classical algorithms using parameterized quantum circuits, the question of how to optimize these algorithms and circuits emerges. In this paper, it is shown that the number of single-qubit rotations in parameterized quantum circuits can be decreased without compromising the relative expressibility or entangling capability of the circuit. It is also shown that the performance of a variational quantum eigensolver (VQE) is unaffected by a similar decrease in single-qubit rotations. Relative expressibility and entangling capability are compared across different number of qubits in parameterized quantum circuits. High-dimensional qudits as a platform for hybrid quantum classical algorithms is a rarity in the literature. Therefore, quantum frequency comb photonics is considered as a platform for such algorithms and it is shown that a relative expressibility and entangling capability comparable to the best regular parameterized quantum circuits can be obtained.

Dimensional Expressivity Analysis of Parametric Quantum Circuits

“…This question of circuit expressivity is an active area of research, see Refs. [22][23][24][25][26][27][28][29][30][31][32] and references therein. In particular, it has been proposed to assess the expressivity of a parametric quantum circuit by quantifying the circuit's ability to uniformly reach the full Hilbert space [23], which was accomplished by computing statistical properties based on randomly sampling states from a given circuit template.…”

“…In particular, it has been proposed to assess the expressivity of a parametric quantum circuit by quantifying the circuit's ability to uniformly reach the full Hilbert space [23], which was accomplished by computing statistical properties based on randomly sampling states from a given circuit template. Using this ansatz, it has been shown that the number of single-qubit rotations in certain parameterized quantum circuits can be decreased without compromising the expressivity of the circuit [26]. On the other hand, it has been shown that the inclusion of redundant parametrized gates can make the quantum circuits more resilient to noise [29,32].…”

Quantum Energy Landscape and VQA Optimization

“…Unlike the initial circuit energy L(θ 0 ) that cannot differ much from the ensemble average (7), the energy L(θ τ ) at an intermediate time τ should significantly deviate almost by definition (27) of the steepest descent method. It indicates how distinctive the intermediate states |ψ(θ τ ) are from initial states, thus requiring independent exploration of their geometric properties.…”

“…The curves in Figures 3a and 3b are respectively the energy gap ∆E and Renyi-2 entanglement entropy R (2) evaluated under an equal partitioning of n = 12 qubits. The blue/orange colors indicate whether the displayed values are before/after applying the gradient descent (27) to circuit parameters τ = 5000 times. When the average entanglement entropy of the pre-optimization states saturates to the maximum possible value, as the cases for p ≤ 0.1, Figure 3a exhibits the formulation of orange dot clusters around ∆E ∼ 9.…”

“…When the average entanglement entropy of the pre-optimization states saturates to the maximum possible value, as the cases for p ≤ 0.1, Figure 3a exhibits the formulation of orange dot clusters around ∆E ∼ 9. It means the failure of many circuit instances in reducing ∆E via the local gradient search (27). The gradient descent fails to make a trajectory towards the Ising ground state, while stopping at a suboptimal extremum in the quantum energy landscape.…”