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arXiv:1806.03295 · Physical Review A

Experimental realization of quantum algorithms for a linear system inspired by adiabatic quantum computing

Jingwei Wen, Xiangyu Kong, Shijie Wei, Bi-Xue Wang, Tao Xin, Guilu Long

Published Jan 14, 2019• 53 citations
arXivHTMLPDF

Abstract

Adiabatic quantum computation is of fundamental importance in the field of quantum computation as it offers an alternative approach to the gate-based model for the manipulation of quantum systems. Recently, an interesting work [arXiv:1805.10549] indicated that we can solve a linear equation system via an algorithm inspired by adiabatic quantum computing. Here we demonstrate the algorithm in a four-qubit nuclear magnetic resonance system by determining the solution of an eight-dimensional linear equation $A\mathbf{x}=\mathbf{b}$. The result is by far the maximum-dimensional linear equation solution with a limited number of qubits in experiments, which include some ingenious simplifications. Our experiment provides the possibility of solving so many practical problems related to linear equations systems and has the potential applications in designing the future quantum algorithms.

DOI
10.1103/physreva.99.012320

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Adiabatic quantum computation is of fundamental importance in the field of quantum computation as it offers an alternative approach to the gate-based model for the manipulation of quantum systems. Recently, an interesting work [arXiv:1805.10549] indicated that we can solve a linear equation system via an algorithm inspired by adiabatic quantum computing. Here we demonstrate the algorithm in a four-qubit nuclear magnetic resonance system by determining the solution of an eight-dimensional linear equation Ax=bA\mathbf{x}=\mathbf{b}Ax=b. The result is by far the maximum-dimensional linear equation solution with a limited number of qubits in experiments, which include some ingenious simplifications. Our experiment provides the possibility of solving so many practical problems related to linear equations systems and has the potential applications in designing the future quantum algorithms.
Many important problems in science and engineering can be reduced to the problem of solving linear equations. The quantum algorithm discovered recently indicates that one can solve an NNN-dimensional linear equation in O(logN)O(logN)O(logN) time, which provides an exponential speedup over the classical counterpart. Here we report an experimental demonstration of the quantum algorithm when the scale of the linear equation is $2\ifmmode\times\else\texttimes\fi{}2$ using a nuclear magnetic resonance quantum information processor. For all sets of experiments, the fidelities of the final four-qubit states are all above 96%. This experiment gives the possibility of solving a series of practical problems related to linear systems of equations and can serve as the basis to realize many potential quantum algorithms.
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We present two quantum algorithms based on evolution randomization, a simple variant of adiabatic quantum computing, to prepare a quantum state ∣x⟩|x⟩∣x⟩ that is proportional to the solution of the system of linear equations $A\stackrel{\ensuremath{\rightarrow}}{x}=\stackrel{\ensuremath{\rightarrow}}{b}$. The time complexities of our algorithms are $O\mathbf{(}{\ensuremath{\kappa}}^{2}\mathrm{log}(\ensuremath{\kappa})/\ensuremath{\epsilon}\mathbf{)}$ and $O\mathbf{(}\ensuremath{\kappa}\mathrm{log}(\ensuremath{\kappa})/\ensuremath{\epsilon}\mathbf{)}$, where $\ensuremath{\kappa}$ is the condition number of AAA and $\ensuremath{\epsilon}$ is the precision. Both algorithms are constructed using families of Hamiltonians that are linear combinations of products of AAA, the projector onto the initial state ∣b⟩|b⟩∣b⟩, and single-qubit Pauli operators. The algorithms are conceptually simple and easy to implement. They are not obtained from equivalences between the gate model and adiabatic quantum computing. They do not use phase estimation or variable-time amplitude amplification, and do not require large ancillary systems. We discuss a gate-based implementation via Hamiltonian simulation and prove that our second algorithm is almost optimal in terms of $\ensuremath{\kappa}$. Like previous methods, our techniques yield an exponential quantum speed-up under some assumptions. Our results emphasize the role of Hamiltonian-based models of quantum computing for the discovery of important algorithms.
We report the realization of a nuclear magnetic resonance computer with three quantum bits that simulates an adiabatic quantum optimization algorithm. Adiabatic quantum algorithms offer new insight into how quantum resources can be used to solve hard problems. This experiment uses a particularly well-suited three quantum bit molecule and was made possible by introducing a technique that encodes general instances of the given optimization problem into an easily applicable Hamiltonian. Our results indicate an optimal run time of the adiabatic algorithm that agrees well with the prediction of a simple decoherence model.
We present two quantum algorithms based on evolution randomization, a simple variant of adiabatic quantum computing, to prepare a quantum state ∣x⟩\vert x \rangle∣x⟩ that is proportional to the solution of the system of linear equations Ax⃗=b⃗A \vec{x}=\vec{b}Ax=b. The time complexities of our algorithms are O(κ2log⁡(κ)/ϵ)O(\kappa^2 \log(\kappa)/\epsilon)O(κ2log(κ)/ϵ) and O(κlog⁡(κ)/ϵ)O(\kappa \log(\kappa)/\epsilon)O(κlog(κ)/ϵ), where κ\kappaκ is the condition number of AAA and ϵ\epsilonϵ is the precision. Both algorithms are constructed using families of Hamiltonians that are linear combinations of products of AAA, the projector onto the initial state ∣b⟩\vert b \rangle∣b⟩, and single-qubit Pauli operators. The algorithms are conceptually simple and easy to implement. They are not obtained from equivalences between the gate model and adiabatic quantum computing. They do not use phase estimation or variable-time amplitude amplification, and do not require large ancillary systems. We discuss a gate-based implementation via Hamiltonian simulation and prove that our second algorithm is almost optimal in terms of κ\kappaκ. Like previous methods, our techniques yield an exponential quantum speedup under some assumptions. Our results emphasize the role of Hamiltonian-based models of quantum computing for the discovery of important algorithms.
Controllable adiabatic evolution of a multi-qubit system can be used for adiabatic quantum computation (AQC). This evolution ends at a configuration where the Hamiltonian of the system encodes the solution of the problem to be solved. As a first steps towards realization of AQC we have investigated two, three and four flux qubit systems. These systems were characterized by making use of a radio-frequency method. We designed two-qubit systems with coupling energies up to several kelvins. For the three-flux-qubit systems we determined the complete ground-state flux diagram in the three dimensional flux space around the qubits common degeneracy point. We show that the system's Hamiltonian can be completely reconstructed from our measurements. Our concept for the implementation of AQC, by making use of flux qubits, is discussed.