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Generative quantum combinatorial optimization by means of a novel conditional generative quantum eigensolver

Shunya Minami, Kouhei Nakaji, Yohichi Suzuki, Alán Aspuru‐Guzik, Tadashi Kadowaki

Published Jan 1, 2025• 3 citations
Publisher

Abstract

We introduce conditional-GQE, a context-aware quantum circuit generator powered by an encoder–decoder transformer, and demonstrate its generative capability on combinatorial optimization tasks.
DOI
10.1039/d5dd00138b

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We introduce the generative quantum eigensolver (GQE), a new quantum computational framework that operates outside the variational quantum algorithm paradigm by applying classical generative models to quantum simulation. The GQE algorithm optimizes a classical generative model to produce quantum circuits with desired properties. Here, we develop a transformer-based implementation, which we name the generative pre-trained transformer-based (GPT) quantum eigensolver (GPT-QE). We show a proof-of-concept of training and pretraining of GPT-QE applied to electronic structure Hamiltonians, and demonstrate its ability illustrated by surpassing coupled cluster singles and doubles (CCSD) for the strong bond dissociation of the nitrogen molecule and approaching chemical accuracy. We also demonstrate the method on real quantum hardware.
Generative quantum eigensolver (GQE) is a hybrid quantum-classical algorithm that iteratively trains a classical generative machine learning model such that the model can generate quantum circuits with desired properties such as approximating molecular ground states. It offers as many potential applications and as much flexibility as variational quantum eigensolvers, while avoiding the problem of barren plateaus. Quantum circuit cutting (QCC) is a technique to perform quantum computations that require more qubits than available on single quantum devices. It comes with considerable sampling overhead depending on the structure of the circuit to be cut and how the circuit is cut. To make QCC practical, therefore, the circuits to be cut must be designed such that their execution is meaningful and QCC overhead is kept small. In this work, we extend GQE such that the generative model only produces circuits whose overhead by QCC is upper-bounded, while retaining the original purpose of GQE. Consequently, our proposal not only enhances the applicability of GQE through the use of QCC, but also provides a practical application for QCC. Using a transformer decoder implementation of GQE, we evaluate our method through simulated ground state search experiments on the BeH_2 molecule. A new loss function and a hybrid online/offline training strategy are also introduced and it is observed that these tools improve convergence and final energy values.
Combinatorial optimization is one of the key candidates in the race for practical quantum advantage. In this work, we introduce a new family of quantum-enhanced optimizers and demonstrate how quantum machine learning models known as quantum generative models can find lower minima than those found by means of stand-alone state-of-the-art classical solvers. We present two new quantum-enhanced optimization strategies. The first scheme leverages data points evaluated during the optimization search from any quantum or classical optimizer. In this scheme, we show how our quantum generative model boosts the performance of classical solvers in hard-to-solve instances where the classical solver is not capable of making progress as a stand-alone strategy. The second quantum optimization strategy works as a stand-alone solver. Here we show its superior performance when the goal is to find the best minimum within the least number of cost function evaluations. Under this setting, we benchmark our quantum-enhanced optimization strategies against several solvers, including Bayesian optimizers which are known to be one of the best competing solvers in such tasks. To illustrate our findings, these benchmarks are performed in the context of the portfolio optimization problem by constructing instances from the S&P 500 stock market index. We show that our quantum-inspired generative models based on tensor networks generalize to unseen candidates with lower cost function values than any of the candidates seen by the classical solvers. This is the first demonstration of the generalization capabilities of quantum generative models that brings real value in the context of an industrial-scale application.
Variational quantum eigensolver (VQE), a kind of quantum-classical hybrid method, has latent powers to leverage noisy quantum devices by effectively managing a limited number of qubits with finite coherent times. We implemented VQE with real quantum processors through IBM Quantum Experience and applied it to combinatorial optimizations. Notably, a short-depth quantum circuit consisting of a lower number of quantum gate than that in the quantum approximate optimization algorithm can search the ground states of diagonal Hamiltonians encoding the problems. Since the simplest case of the quantum circuit for VQE requires only single-qubit gates, it can be realized using current quantum computers.