We consider unstructured search using multimode bosonic quantum systems under energy constraints. The task is to locate an unknown marked mode among $N$ possibilities using arbitrary control, subject to a time-averaged mean-boson-number budget $n_{\text{b}}\gg 1$ . We analyze a continuous-time phase-shift oracle of strength $B_{0}$ , with cost given by the total oracle time, and a discrete oracle with fixed phase $φ_{0}$ , with cost given by the number of queries. We give algorithms that identify the target with query complexity $O(\sqrt{N}/{n_{\text{b}}}B_{0})$ for continuous-time oracles or $O(\sqrt{N}/n_{\text{b}} |{\sin(φ_{0}/{2})}|)$ for discrete oracles with $n_{\text{b}} |{\sin(φ_{0}/{2})}|=O(1)$ . In contrast, for discrete oracles with $n_{\text{b}} |{\sin(φ_{0}/{2})}|=Ω(1)$ , we provide an algorithm using $O(\max\{1,\sqrt{N/n_{\text{b}}|{\sin(φ_{0}/{2})}|}\})$ queries. We also prove matching lower bounds on the query complexity, showing that these algorithms are asymptotically optimal. Furthermore, we show that, in the absence of loss, non-Gaussianity is necessary to achieve the optimal joint query scaling in $N$ and $n_{\text{b}}$ , but entanglement is not. Under uniform boson loss at rate $γ$ , we prove a query lower bound of $Ω( γN / n _{\text{b}} B_{0}^{2} )$ , attained for $γ= Ω(B_{0} \sqrt{n _{\text{b}}/N})$ using coherent-state probes. Finally, we propose experimental implementations using multipass optical microscopy or superconducting circuits as potential routes to speeding up imaging or device calibration with a low boson-number budget.We consider unstructured search using multimode bosonic quantum systems under energy constraints. The task is to locate an unknown marked mode among $N$ possibilities using arbitrary control, subject to a time-averaged mean-boson-number budget $n_{\text{b}}\gg 1$. We analyze a continuous-time phase-shift oracle of strength $B_{0}$, with cost given by the total oracle time, and a discrete oracle with fixed phase $φ_{0}$, with cost given by the number of queries. We give algorithms that identify the target with query complexity $O(\sqrt{N}/{n_{\text{b}}}B_{0})$ for continuous-time oracles or $O(\sqrt{N}/n_{\text{b}} |{\sin(φ_{0}/{2})}|)$ for discrete oracles with $n_{\text{b}} |{\sin(φ_{0}/{2})}|=O(1)$. In contrast, for discrete oracles with $n_{\text{b}} |{\sin(φ_{0}/{2})}|=Ω(1)$, we provide an algorithm using $O(\max\{1,\sqrt{N/n_{\text{b}}|{\sin(φ_{0}/{2})}|}\})$ queries. We also prove matching lower bounds on the query complexity, showing that these algorithms are asymptotically optimal. Furthermore, we show that, in the absence of loss, non-Gaussianity is necessary to achieve the optimal joint query scaling in $N$ and $n_{\text{b}}$, but entanglement is not. Under uniform boson loss at rate $γ$, we prove a query lower bound of $Ω( γN / n _{\text{b}} B_{0}^{2} )$, attained for $γ= Ω(B_{0} \sqrt{n _{\text{b}}/N})$ using coherent-state probes. Finally, we propose experimental implementations using multipass optical microscopy or superconducting circuits as potential routes to speeding up imaging or device calibration with a low boson-number budget.

quant-pharXiv:2610.17163Oct 13, 20260 citations

Parameterized Quantum Query Algorithms for Graph Problems

Tatsuya Terao, Ryuhei Mori

In this paper, we consider the parameterized quantum query complexity for graph problems. We design parameterized quantum query algorithms for k-vertex cover and k-matching problems, and present lower bounds on the parameterized quantum query complexity. Then, we show that our quantum query algorithms are optimal up to a constant factor when the parameters are small. Our main results are as follows. Parameterized quantum query complexity of vertex cover. In the k-vertex cover problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a vertex cover of size at most k. We show that the quantum query complexity of the k-vertex cover problem is O(√kn + k^{3/2}√n) in the adjacency matrix model. For the design of the quantum query algorithm, we use the method of kernelization, a well-known tool for the design of parameterized classical algorithms, combined with Grover’s search. Parameterized quantum query complexity of matching. In the k-matching problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a matching of size at least k. We show that the quantum query complexity of the k-matching problem is O(√kn + k²) in the adjacency matrix model. We obtain this upper bound by using Grover’s search carefully and analyzing the number of Grover’s searches by making use of potential functions. We also show that the quantum query complexity of the maximum matching problem is O(√pn + p²) where p is the size of the maximum matching. For small p, it improves known bounds Õ(n^{3/2}) for bipartite graphs [Blikstad-v.d.Brand-Efron-Mukhopadhyay-Nanongkai, FOCS 2022] and O(n^{7/4}) for general graphs [Kimmel-Witter, WADS 2021]. Lower bounds on parameterized quantum query complexity. We also present lower bounds on the quantum query complexities of the k-vertex cover and k-matching problems. The lower bounds prove the optimality of the above parameterized quantum query algorithms up to a constant factor when k is small. Indeed, the quantum query complexities of the k-vertex cover and k-matching problems are both Θ(√k n) when k = O(√n) and k = O(n^{2/3}), respectively.In this paper, we consider the parameterized quantum query complexity for graph problems. We design parameterized quantum query algorithms for k-vertex cover and k-matching problems, and present lower bounds on the parameterized quantum query complexity. Then, we show that our quantum query algorithms are optimal up to a constant factor when the parameters are small. Our main results are as follows. Parameterized quantum query complexity of vertex cover. In the k-vertex cover problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a vertex cover of size at most k. We show that the quantum query complexity of the k-vertex cover problem is O(√kn + k^{3/2}√n) in the adjacency matrix model. For the design of the quantum query algorithm, we use the method of kernelization, a well-known tool for the design of parameterized classical algorithms, combined with Grover’s search. Parameterized quantum query complexity of matching. In the k-matching problem, we are given an undirected graph G with n vertices and an integer k, and the objective is to determine whether G has a matching of size at least k. We show that the quantum query complexity of the k-matching problem is O(√kn + k²) in the adjacency matrix model. We obtain this upper bound by using Grover’s search carefully and analyzing the number of Grover’s searches by making use of potential functions. We also show that the quantum query complexity of the maximum matching problem is O(√pn + p²) where p is the size of the maximum matching. For small p, it improves known bounds Õ(n^{3/2}) for bipartite graphs [Blikstad-v.d.Brand-Efron-Mukhopadhyay-Nanongkai, FOCS 2022] and O(n^{7/4}) for general graphs [Kimmel-Witter, WADS 2021]. Lower bounds on parameterized quantum query complexity. We also present lower bounds on the quantum query complexities of the k-vertex cover and k-matching problems. The lower bounds prove the optimality of the above parameterized quantum query algorithms up to a constant factor when k is small. Indeed, the quantum query complexities of the k-vertex cover and k-matching problems are both Θ(√k n) when k = O(√n) and k = O(n^{2/3}), respectively.

arXiv:2408.03864Leibniz international proceedings in informaticsJan 1, 20240 citations

Optimal Parallel Quantum Query Algorithms

Stacey Jeffery, Frédéric Magniez, Ronald de Wolf

Lecture notes in computer scienceJan 1, 20142 citations

Research on Quantum Searching Algorithms Based on Phase Shifts

钟普查, 鲍皖苏

Chinese Physics LettersJan 1, 20080 citations

Quantum search for unknown number of target items by hybridizing fixed-point method with trail-and-error method*

Tan Li, Shuo Zhang, Xiang-Qun Fu +4 more

For the unsorted database quantum search with the unknown fraction λ of target items, there are mainly two kinds of methods, i.e., fixed-point and trail-and-error. (i) In terms of the fixed-point method, Yoder et al. [Phys. Rev. Lett. 113 210501 (2014)] claimed that the quadratic speedup over classical algorithms has been achieved. However, in this paper, we point out that this is not the case, because the query complexity of Yoder’s algorithm is actually in O ( 1 / λ 0 ) rather than O ( 1 / λ ) , where λ 0 is a known lower bound of λ. (ii) In terms of the trail-and-error method, currently the algorithm without randomness has to take more than 1 times queries or iterations than the algorithm with randomly selected parameters. For the above problems, we provide the first hybrid quantum search algorithm based on the fixed-point and trail-and-error methods, where the matched multiphase Grover operations are trialed multiple times and the number of iterations increases exponentially along with the number of trials. The upper bound of expected queries as well as the optimal parameters are derived. Compared with Yoder’s algorithm, the query complexity of our algorithm indeed achieves the optimal scaling in λ for quantum search, which reconfirms the practicality of the fixed-point method. In addition, our algorithm also does not contain randomness, and compared with the existing deterministic algorithm, the query complexity can be reduced by about 1/3. Our work provides a new idea for the research on fixed-point and trial-and-error quantum search.For the unsorted database quantum search with the unknown fraction λ of target items, there are mainly two kinds of methods, i.e., fixed-point and trail-and-error. (i) In terms of the fixed-point method, Yoder et al. [Phys. Rev. Lett. 113 210501 (2014)] claimed that the quadratic speedup over classical algorithms has been achieved. However, in this paper, we point out that this is not the case, because the query complexity of Yoder’s algorithm is actually in O ( 1 / λ 0 ) rather than O ( 1 / λ ) , where λ 0 is a known lower bound of λ. (ii) In terms of the trail-and-error method, currently the algorithm without randomness has to take more than 1 times queries or iterations than the algorithm with randomly selected parameters. For the above problems, we provide the first hybrid quantum search algorithm based on the fixed-point and trail-and-error methods, where the matched multiphase Grover operations are trialed multiple times and the number of iterations increases exponentially along with the number of trials. The upper bound of expected queries as well as the optimal parameters are derived. Compared with Yoder’s algorithm, the query complexity of our algorithm indeed achieves the optimal scaling in λ for quantum search, which reconfirms the practicality of the fixed-point method. In addition, our algorithm also does not contain randomness, and compared with the existing deterministic algorithm, the query complexity can be reduced by about 1/3. Our work provides a new idea for the research on fixed-point and trial-and-error quantum search.

arXiv:1912.01773Chinese Physics BNov 1, 20198 citations

We consider unstructured search using multimode bosonic quantum systems under energy constraints. The task is to locate an unknown marked mode among

N

possibilities using arbitrary control, subject to a time-averaged mean-boson-number budget

n_{\text{b}}\gg 1

. We analyze a continuous-time phase-shift oracle of strength

B_{0}

, with cost given by the total oracle time, and a discrete oracle with fixed phase

φ_{0}

, with cost given by the number of queries. We give algorithms that identify the target with query complexity

O(\sqrt{N}/{n_{\text{b}}}B_{0})

for continuous-time oracles or

O(\sqrt{N}/n_{\text{b}} |{\sin(φ_{0}/{2})}|)

for discrete oracles with

n_{\text{b}} |{\sin(φ_{0}/{2})}|=O(1)

. In contrast, for discrete oracles with

n_{\text{b}} |{\sin(φ_{0}/{2})}|=Ω(1)

, we provide an algorithm using

O(\max\{1,\sqrt{N/n_{\text{b}}|{\sin(φ_{0}/{2})}|}\})

queries. We also prove matching lower bounds on the query complexity, showing that these algorithms are asymptotically optimal. Furthermore, we show that, in the absence of loss, non-Gaussianity is necessary to achieve the optimal joint query scaling in

N

and

n_{\text{b}}

, but entanglement is not. Under uniform boson loss at rate

γ

, we prove a query lower bound of

Ω( γN / n _{\text{b}} B_{0}^{2} )

, attained for

γ= Ω(B_{0} \sqrt{n _{\text{b}}/N})

using coherent-state probes. Finally, we propose experimental implementations using multipass optical microscopy or superconducting circuits as potential routes to speeding up imaging or device calibration with a low boson-number budget.