Discretized Quantum Exhaustive Search for Variational Quantum Algorithms

TL;DR

Variational Quantum Algorithms on NISQ devices suffer from trainability challenges due to the exponential Hilbert-space size $2^n$ and barren plateaus. The paper introduces Discretized Quantum Exhaustive Search (DQES) based on Mutually Unbiased Bases (MUBs) to sample the cost landscape, along with Efficient Partial DQES for larger problems, and provides circuit representations for MUB states and shifted-MUB initializations. It formalizes DQSS and DQES, develops full and partial sampling schemes, proposes two MUB-based initialization strategies, and demonstrates them on molecular electronic-structure, Transverse-Field Ising, and Max-Cut problems to reveal energy landscapes and convergence benefits. These results offer ansatz-agnostic insights into the cost landscape and practical initializations that can improve convergence and mitigate barren plateaus in VQAs on NISQ hardware.

Abstract

Quantum computers promise a great computational advantage over classical computers, yet currently available quantum devices have only a limited amount of qubits and a high level of noise, limiting the size of problems that can be solved accurately with those devices. Variational Quantum Algorithms (VQAs) have emerged as a leading strategy to address these limitations by optimizing cost functions based on measurement results of shallow-depth circuits. However, the optimization process usually suffers from severe trainability issues as a result of the exponentially large search space, mainly local minima and barren plateaus. Here we propose a novel method that can improve variational quantum algorithms -- ``discretized quantum exhaustive search''. On classical computers, exhaustive search, also named brute force, solves small-size NP complete and NP hard problems. Exhaustive search and efficient partial exhaustive search help designing heuristics and exact algorithms for solving larger-size problems by finding easy subcases or good approximations. We adopt this method to the quantum domain, by relying on mutually unbiased bases for the $2^n$-dimensional Hilbert space. We define a discretized quantum exhaustive search that works well for small size problems. We provide an example of an efficient partial discretized quantum exhaustive search for larger-size problems, in order to extend classical tools to the quantum computing domain, for near future and far future goals. Our method enables obtaining intuition on NP-complete and NP-hard problems as well as on Quantum Merlin Arthur (QMA)-complete and QMA-hard problems. We demonstrate our ideas in many simple cases, providing the energy landscape for various problems and presenting two types of energy curves via VQAs.

Figures (10)

• Figure 1: Circuits transforming states from the computational basis to each basis in (a) a 2 qubit MUB set. (b) 3 qubit MUB set. (c) 2 qubit MUB set and the relevant parameters of a hardware efficient ansatz to create these circuits.
• Figure 2: Optimization paths (energy curves) for 1 qubit search space example. (a) Optimization path obtained by COBYLA for a single-qubit example of the use of MUB states for VQA: the full path, starting from state $|-\rangle$. (b) Optimization path obtained by COBYLA for a single-qubit example of the use of MUB states for VQA: the full path, starting from the state $|-_i\rangle$. (c) A closeup on reaching the solution for the optimization path presented in (a). (d) A closeup on reaching the solution for the optimization path presented in (b).
• Figure 3: Exhaustive search of all the 2 qubit MUB states and their energy with respect to the $H_2$ molecule in $0.75 [\text{\AA}]$ atomic distance.
• Figure 4: Convergence results of the optimization process of a VQE algorithm on the $H_2$ molecule in $0.75 [\text{\AA}]$ atomic distance using a noiseless simulator, the COBYLA optimizer, and different initial states. The initial states are the 3 MUB states that had the lowest energy in the landscape calculation of the molecule. The right figure is a zoom to the final iterations of the optimization.
• Figure 5: Exhaustive search of all the 2 qubit MUB states and their energy with respect to the $HeH^+$ molecule in $1 [\text{\AA}]$ atomic distance.

Theorems & Definitions (4)

• Definition 1
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• Definition 3
• Definition 4