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arXiv:1912.06959 · Physical Review Research

Efficient quantum algorithm for solving structured problems via multistep quantum computation

Hefeng Wang, Sixia Yu, Hua Xiang

Published Jan 17, 2023• 1 citation
arXivHTMLPDF

Abstract

In classical computation, a problem can be solved in multiple steps where the calculated results of each step can be copied and used repeatedly.However, in quantum computation, it is difficult to realize a similar multistep computation process because the no-cloning theorem forbids making copies of an unknown quantum state perfectly.We find a method based on a quantum resonant transition to protect and reuse an unknown quantum state that encodes the calculated results of an intermediate step without copying it, and present a quantum algorithm that solves a problem via multistep quantum computation.We demonstrate that this algorithm can solve a type of structured search problems efficiently.

DOI
10.1103/physrevresearch.5.l012004

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In classical computation, a problem can be solved in multiple steps where the calculated results of each step can be copied and used repeatedly.However, in quantum computation, it is difficult to realize a similar multistep computation process because the no-cloning theorem forbids making copies of an unknown quantum state perfectly.We find a method based on a quantum resonant transition to protect and reuse an unknown quantum state that encodes the calculated results of an intermediate step without copying it, and present a quantum algorithm that solves a problem via multistep quantum computation.We demonstrate that this algorithm can solve a type of structured search problems efficiently.
Abstract We present a quantum algorithm for finding the minimum of a function based on multistep quantum computation, and apply the algorithm for solving optimization problems with continuous variables. We construct the state space of the problem by discretizing the variables of the problem, and divide the state space according to the function values of the vectors of the state space. By comparing the function values of the vectors with a series of threshold values in decreasing order, we construct a sequence of Hamiltonians where the search space of a Hamiltonian is nested in that of the previous one. By applying a multistep quantum computation process for finding the ground state of the last Hamiltonian, the optimal vector of the state space of the problem is located in a small search space and can be determined efficiently. One of the most difficult problems in optimization algorithms is that a trial vector is trapped in a deep local minimum while the global minimum is missed, this problem can be alleviated in our algorithm and the run time is proportional to the number of the steps of the algorithm, provided that the reduction rate of the search spaces is polynomial large. We discuss the implementation of the algorithm and test the algorithm for some test functions.
We present a quantum algorithm for obtaining a quantum state that encodes the PageRank vector of the Google matrix through multistep quantum resonant transition (mQRT). In the algorithm, the PageRank vector is encoded in the ground state of a problem Hamiltonian associated with the Google matrix. By dividing the web graph corresponding to the Google matrix into a series of subgraphs with nested structure, we construct a sequence of Hamiltonians based on the subgraphs to form a Hamiltonian evolution path from a simple initial Hamiltonian to the problem Hamiltonian. The ground state of the problem Hamiltonian is obtained by going through ground states of the intermediate Hamiltonians via QRT step by step. This algorithm requires only one ancillary qubit, and the runtime of the algorithm is proportional to the number of steps. It provides a new way for efficiently obtaining the quantum state of the PageRank vector of large-scale networks.
A quantum computer is a device that processes information in a quantum-mechanically coherent fashion. In principle, it can exploit coherent quantum interference and entanglement to perform computations, such as factoring large numbers or searching an unsorted database, more rapidly than classical computers. Noise, decoherence, and manufacturing problems make constructing large scale quantum computers difficult. Ion traps and optical cavities offer promising experimental approaches, but no quantum algorithm has yet been implemented with those systems. On the other hand, because of their natural isolation from the environment, nuclear spins are particularly good 'quantum bits', and their use for quantum computation is possible by applying nuclear magnetic resonance (NMR) techniques in an unconventional manner. Here, we report on the experimental implementation of a quantum algorithm using NMR to solve a purely mathematical problem in fewer steps than is possible classically. Our simple quantum computer can determine the type of an unknown function using fewer function 'calls' than is possible using a classical computer.
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A quantum algorithm for a class of highly structured combinatorial search problems is introduced. This algorithm finds a solution in a single step, contrasting with the linear growth in the number of steps required by the best classical algorithms as the problem size increases, and the exponential growth required by classical and quantum methods that ignore the problem structure. In some cases, the algorithm can also guarantee that insoluble problems, in fact, have no solutions, unlike previously proposed quantum search algorithms.
Quantum theory fundamentally forbids the perfect copying of an arbitrary unknown quantum state, according to a principle known as the no-cloning theorem. Nevertheless, it is possible to construct a deterministic quantum map that produces multiple approximate copies of an unknown quantum state. This task is referred to as universal quantum cloning, further facilitating numerous quantum technologies such as quantum cryptography and quantum communication. In this work, we theoretically verify that the universal quantum cloning cannot be realized within a noncontextual theory, highlighting its intrinsically nonclassical nature. Our verification first {focuses on revealing that} 1→21\rightarrow21→2 cloning scenario {is fully contextual}, and {further covers general examples to observe the contextual behavior of} N→MN\rightarrow MN→M scenario. We believe that our results regarding quantum cloning serve a key role for understanding both quantum foundation and application.