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Quantum Information and Computation

Corrected quantum walk for optimal Hamiltonian simulation

BerryDominic, NovoLeonardo

Published Nov 1, 2016• 2 citations
Publisher

Abstract

We describe a method to simulate Hamiltonian evolution on a quantum computer by repeatedly using a superposition of steps of a quantum walk, then applying a correction to the weightings for the num...
DOI
10.5555/3179439.3179442

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We describe a method to simulate Hamiltonian evolution on a quantum computer by repeatedly using a superposition of steps of a quantum walk, then applying a correction to the weightings for the numbers of steps of the quantum walk. This correction enables us to obtain complexity which is the same as the lower bound up to doublelogarithmic factors for all parameter regimes. The scaling of the query complexity is given. This technique should also be useful for improving the scaling of the Taylor series approach to simulation, which is relevant to applications such as quantum chemistry.
We describe an improved version of the quantum algorithm for Hamiltonian simulation based on the implementation of a truncated Taylor series of the evolution operator. The idea is to add an extra step to the previously known algorithm which implements an operator that corrects the weightings of the Taylor series. This way, the desired accuracy is achieved with an improvement in the overall complexity of the algorithm. This quantum simulation method is applicable to a wide range of Hamiltonians of interest, including to quantum chemistry problems.
We describe an improved version of the quantum simulation method based on the implementation of a truncated Taylor series of the evolution operator. The idea is to add an extra step to the previously known algorithm which implements an operator that corrects the weightings of the Taylor series. This way, the desired accuracy is achieved with an improvement in the overall complexity of the algorithm. This quantum simulation method is applicable to a wide range of Hamiltonians of interest, including to quantum chemistry problems.
This paper investigates the potential of quantum walks for optimizing the weights within neural networks. Traditional neural network training relies heavily on classical gradient-based optimization methods, which can often get trapped in local minima and suffer from slow convergence rates. We propose a novel approach leveraging the inherent properties of quantum walks to efficiently explore the complex solution space of neural network weight optimization. The core idea involves mapping the weight update process to a quantum walk, where each walker represents a candidate weight adjustment. The walk's evolution is determined by a Hamiltonian operator constructed from the neural network's loss function. We aim to demonstrate that, under specific conditions, quantum walks can outperform classical gradient descent in terms of convergence speed and potentially escape local minima more effectively. This research represents a novel combination of quantum walk methodology with neural network optimization, offering a fundamentally different paradigm for training complex artificial intelligence models.
The physics of quantum mechanics is the inspiration for, and underlies, quantum computation. As such, one expects physical intuition to be highly influential in the understanding and design of many quantum algorithms, particularly simulation of physical systems. Surprisingly, this has been challenging, with current Hamiltonian simulation algorithms remaining abstract and often the result of sophisticated but unintuitive constructions. We contend that physical intuition can lead to optimal simulation methods by showing that a focus on simple single-qubit rotations elegantly furnishes an optimal algorithm for Hamiltonian simulation, a universal problem that encapsulates all the power of quantum computation. Specifically, we show that the query complexity of implementing time evolution by a d-sparse Hamiltonian H[over ^] for time-interval t with error ε is O[td∥H[over ^]∥_{max}+log(1/ε)/loglog(1/ε)], which matches lower bounds in all parameters. This connection is made through general three-step "quantum signal processing" methodology, comprised of (i) transducing eigenvalues of H[over ^] into a single ancilla qubit, (ii) transforming these eigenvalues through an optimal-length sequence of single-qubit rotations, and (iii) projecting this ancilla with near unity success probability.