Hamiltonian simulation with optimal sample complexity

Kimmel, Shelby; Lin, Cedric Yen-Yu; Low, Guang Hao; Ozols, Māris; Yoder, Theodore J.

doi:10.1038/s41534-017-0013-7

Cited by 51 publications

(53 citation statements)

References 48 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The resource requirements for this result are given from lemma 2 for the sample-based Hamiltonian simulation. The extension to the controlled simulation as required by phase estimation is done in an overhead that factors in as a constant into the resource requirements [34]. Replacing…”

Section: Simulating the Gradientmentioning

confidence: 99%

Quantum gradient descent and Newton’s method for constrained polynomial optimization

et al. 2019

View full text Add to dashboard Cite

Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton's method takes into account curvature information and thereby often improves convergence. Here, we develop quantum versions of these iterative optimization algorithms and apply them to polynomial optimization with a unit norm constraint. In each step, multiple copies of the current candidate are used to improve the candidate using quantum phase estimation, an adapted quantum state exponentiation scheme, as well as quantum matrix multiplications and inversions. The required operations perform polylogarithmically in the dimension of the solution vector and exponentially in the number of iterations. Therefore, the quantum algorithm can be useful for highdimensional problems where a small number of iterations is sufficient.

show abstract

Section: Simulating the Gradientmentioning

confidence: 99%

Quantum gradient descent and Newton’s method for constrained polynomial optimization

et al. 2019

View full text Add to dashboard Cite

show abstract

“…A throughout description of this density matrix-based Hamiltonian simulation procedure is presented in Ref. [42]. Here we will first give an overall description of the quantum method, while the detailed analysis is presented later in the paper.…”

Section: B Multi-layer Casementioning

confidence: 99%

Bayesian deep learning on a quantum computer

Zhao

Pozas-Kerstjens

Rebentrost

et al. 2019

Quantum Mach. Intell.

View full text Add to dashboard Cite

Bayesian methods in machine learning, such as Gaussian processes, have great advantages compared to other techniques. In particular, they provide estimates of the uncertainty associated with a prediction. Extending the Bayesian approach to deep architectures has remained a major challenge. Recent results connected deep feedforward neural networks with Gaussian processes, allowing training without backpropagation. This connection enables us to leverage a quantum algorithm designed for Gaussian processes and develop a new algorithm for Bayesian deep learning on quantum computers. The properties of the kernel matrix in the Gaussian process ensure the efficient execution of the core component of the protocol, quantum matrix inversion, providing an at least polynomial speedup over classical algorithms. Furthermore, we demonstrate the execution of the algorithm on contemporary quantum computers and analyze its robustness with respect to realistic noise models.

show abstract

“…steps in the algorithm [24], where    F max denotes the maximal absolute element of  F . The phase estimation is performed as discussed in [42] to obtain the e 1 2 scaling compared to the e 1 3 scaling of the original work [20,24]. Note that in our setting = Q  F 1 ) .…”

Section: Simulating the Hankel Matricesmentioning

confidence: 99%

An efficient quantum algorithm for spectral estimation

et al. 2017

View full text Add to dashboard Cite

We develop an efficient quantum implementation of an important signal processing algorithm for line spectral estimation: the matrix pencil method, which determines the frequencies and damping factors of signals consisting of finite sums of exponentially damped sinusoids. Our algorithm provides a quantum speedup in a natural regime where the sampling rate is much higher than the number of sinusoid components. Along the way, we develop techniques that are expected to be useful for other quantum algorithms as well-consecutive phase estimations to efficiently make products of asymmetric low rank matrices classically accessible and an alternative method to efficiently exponentiate non-Hermitian matrices. Our algorithm features an efficient quantum-classical division of labor: the time-critical steps are implemented in quantum superposition, while an interjacent step, requiring much fewer parameters, can operate classically. We show that frequencies and damping factors can be obtained in time logarithmic in the number of sampling points, exponentially faster than known classical algorithms.

show abstract

Hamiltonian simulation with optimal sample complexity

Cited by 51 publications

References 48 publications

Quantum gradient descent and Newton’s method for constrained polynomial optimization

Quantum gradient descent and Newton’s method for constrained polynomial optimization

Bayesian deep learning on a quantum computer

An efficient quantum algorithm for spectral estimation

Contact Info

Product

Resources

About