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Resonant Quantum Search with Monitor Qubits

Frank A. Wilczek, Wilczek, F, Biao Wu, Wu, B, Wu, B, Wu, B

Published Jan 1, 2020• 6 citations
OpenAlex

Abstract

© 2020 Chinese Physical Society and IOP Publishing Ltd. We present an algorithm for the generalized search problem (searching k marked items among N items) based on a continuous Hamiltonian and exploiting resonance. This resonant algorithm has the same time complexity as the Grover algorithm. A natural extension of the algorithm, incorporating auxiliary "monitor"qubits, can determine k precisely, if it is unknown. The time complexity of our counting algorithm is O(√N), similar to the best quantum approximate counting algorithm, or better, given appropriate physical resources.

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Resonant Quantum Search with Monitor Qubits*

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Searching a database is a central task in computer science and is paradigmatic of transport and optimization problems in physics. For an unstructured search, Grover's algorithm predicts a quadratic speedup, with the search time $τ(N)=Θ(\sqrt{N})$ and $N$ the database size. Numerical studies suggest that the time complexity can change in the presence of feedback, injecting information during the search. Here, we determine the time complexity of the quantum analog of a randomized algorithm, which implements feedback in a simple form. The search is a continuous-time quantum walk on a complete graph, where the target is continuously monitored by a detector. Additionally, the quantum state is reset if the detector does not click within a specified time interval. This yields a non-unitary, non-Markovian dynamics. We optimize the search time as a function of the hopping amplitude, detection rate, and resetting rate, and identify the conditions under which time complexity could outperform Grover's scaling. The overall search time does not violate Grover's optimality bound when including the time budget of the physical implementation of the measurement. For databases of finite sizes monitoring can warrant rapid convergence and provides a promising avenue for fault-tolerant quantum searches.

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In this paper, Grover's quantum search algorithm is analyzed using a classical computer by calculating the amplitudes and the probabilities of finding a single marked state for n=5, 10, 15, 20, 25, and 27 qubit states.The calculations show that the marked state can be found in 𝒪(√𝑁) iterations, where 𝑁 = 2 ( is the number of items.The possibility of improving Grover's search algorithm to find a single item in N search elements is discussed by calculating the amplitudes and hence the probabilities of finding a single marked state for n=5, 10, 15, 20, 25, 30, 35, 40, 45, and 50 qubit states.The calculations showed that the marked state could be found with sufficiently high probability in 𝒪(ln (𝑁)) iterations.This is quite a remarkable speed-up that can be achieved to find a single marked element in an unsorted N search element.

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We present an extension of Adiabatic Quantum Computing (AQC) algorithm for the unstructured search to the case when the number of marked items is unknown. The algorithm maintains the optimal Grover speedup and includes a small counting subroutine. Our other results include a lower bound on the amount of time needed to perform a general Hamiltonian-based quantum search, a lower bound on the evolution time needed to perform a search that is valid in the presence of control error and a generic upper bound on the minimum eigenvalue gap for evolutions. In particular, we demonstrate that quantum speedup for the unstructured search using AQC type algorithms may only be achieved under very rigid control precision requirements.

•Jan 1, 2012•6 citations
© 2020 Chinese Physical Society and IOP Publishing Ltd. We present an algorithm for the generalized search problem (searching k marked items among N items) based on a continuous Hamiltonian and exploiting resonance. This resonant algorithm has the same time complexity as the Grover algorithm. A natural extension of the algorithm, incorporating auxiliary "monitor"qubits, can determine k precisely, if it is unknown. The time complexity of our counting algorithm is O(√N), similar to the best quantum approximate counting algorithm, or better, given appropriate physical resources.
We present an algorithm for the generalized search problem (searching k marked items among N items) based on a continuous Hamiltonian and exploiting resonance. This resonant algorithm has the same time complexity O ( N / k ) as the Grover algorithm. A natural extension of the algorithm, incorporating auxiliary “monitor” qubits, can determine k precisely, if it is unknown. The time complexity of our counting algorithm is O ( N ) , similar to the best quantum approximate counting algorithm, or better, given appropriate physical resources.
We propose in this paper a rough counting algorithm and modified version of Grover's search algorithm. The proposed counting algorithm uses Deutsch-Jozsa's algorithm to roughly estimate the number of items satisfying some search conditions by reconstructing the output distribution. The modified Grover's search combines the counting into one part of the algorithm and uses the counting result to determine the number of Grover's iterations taken to get such items. We provide an analysis of expected probability of the proposed algorithm taking into account the problem size, the number of satisfying items and the query complexity to show that the proposed modified Grover's search is optimal.
Searching a database is a central task in computer science and is paradigmatic of transport and optimization problems in physics. For an unstructured search, Grover's algorithm predicts a quadratic speedup, with the search time τ(N)=Θ(N)τ(N)=Θ(\sqrt{N})τ(N)=Θ(N​) and NNN the database size. Numerical studies suggest that the time complexity can change in the presence of feedback, injecting information during the search. Here, we determine the time complexity of the quantum analog of a randomized algorithm, which implements feedback in a simple form. The search is a continuous-time quantum walk on a complete graph, where the target is continuously monitored by a detector. Additionally, the quantum state is reset if the detector does not click within a specified time interval. This yields a non-unitary, non-Markovian dynamics. We optimize the search time as a function of the hopping amplitude, detection rate, and resetting rate, and identify the conditions under which time complexity could outperform Grover's scaling. The overall search time does not violate Grover's optimality bound when including the time budget of the physical implementation of the measurement. For databases of finite sizes monitoring can warrant rapid convergence and provides a promising avenue for fault-tolerant quantum searches.
We present two continuous-time quantum search algorithms similar to the adiabatic search algorithm, but now without the requirement of adiabatic evolution. Both algorithms can find the marked state in a time proportional to N\sqrt{N}N​. The behavior of the first algorithm is, essentially, similar to Grover's algorithm, but the second model possesses the important property that one does not need to single out a given time in order to find the searched state. After a well-defined transition time, this second algorithm will converge towards the marked state with a high probability, provided the parameters of the Hamiltonian are chosen appropriately. This convergence shows a resemblance to quantum search algorithms with a fixed point [L. K. Grover, Phys. Rev. Lett. 95, 150501 (2005)].
In this paper, Grover's quantum search algorithm is analyzed using a classical computer by calculating the amplitudes and the probabilities of finding a single marked state for n=5, 10, 15, 20, 25, and 27 qubit states.The calculations show that the marked state can be found in 𝒪(√𝑁) iterations, where 𝑁 = 2 ( is the number of items.The possibility of improving Grover's search algorithm to find a single item in N search elements is discussed by calculating the amplitudes and hence the probabilities of finding a single marked state for n=5, 10, 15, 20, 25, 30, 35, 40, 45, and 50 qubit states.The calculations showed that the marked state could be found with sufficiently high probability in 𝒪(ln (𝑁)) iterations.This is quite a remarkable speed-up that can be achieved to find a single marked element in an unsorted N search element.
We present an extension of Adiabatic Quantum Computing (AQC) algorithm for the unstructured search to the case when the number of marked items is unknown. The algorithm maintains the optimal Grover speedup and includes a small counting subroutine. Our other results include a lower bound on the amount of time needed to perform a general Hamiltonian-based quantum search, a lower bound on the evolution time needed to perform a search that is valid in the presence of control error and a generic upper bound on the minimum eigenvalue gap for evolutions. In particular, we demonstrate that quantum speedup for the unstructured search using AQC type algorithms may only be achieved under very rigid control precision requirements.