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arXiv:1712.06558 · Physical Review A

Grover search under localized dephasing

Daniel Reitzner, Mark S. Hillery

Published Jan 23, 2019• 20 citations
arXivHTMLPDF

Abstract

Decoherence in quantum searches, and in the Grover search, in particular, has already been extensively studied, leading very quickly to the loss of the quadratic speedup over the classical case, when searching for some target (marked) element within a set of size $N$. The noise models used were, however, almost always global. In this paper, we study Grover search under the influence of localized partially dephasing noise of rate $p$. We find that, in the case when the size $k$ of the affected subspace is much smaller than $N$ and the target is unaffected by the noise, namely when $kp\ensuremath{\ll}\sqrt{N}$, the quadratic speedup is retained. Once these restrictions are not met, the quadratic speedup is lost. If the target is affected by the noise, the noise rate needs to scale as $1/\sqrt{N}$ to keep the speedup. We also observe an intermediate region, where if $k\ensuremath{\sim}{N}^{\ensuremath{\mu}}$ and the target is unaffected, the speedup seems to obey ${N}^{\ensuremath{\mu}}$, which for $\ensuremath{\mu}>0.5$ is worse than the quantum, but better than the classical case. We also put obtained results for quantum searches into perspective of quantum walks and searches on graphs.

DOI
10.1103/physreva.99.012339

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Decoherence in quantum searches, and in the Grover search, in particular, has already been extensively studied, leading very quickly to the loss of the quadratic speedup over the classical case, when searching for some target (marked) element within a set of size NNN. The noise models used were, however, almost always global. In this paper, we study Grover search under the influence of localized partially dephasing noise of rate ppp. We find that, in the case when the size kkk of the affected subspace is much smaller than NNN and the target is unaffected by the noise, namely when kp\ensuremath≪Nkp\ensuremath{\ll}\sqrt{N}kp\ensuremath≪N​, the quadratic speedup is retained. Once these restrictions are not met, the quadratic speedup is lost. If the target is affected by the noise, the noise rate needs to scale as 1/N1/\sqrt{N}1/N​ to keep the speedup. We also observe an intermediate region, where if k\ensuremath∼N\ensuremathμk\ensuremath{\sim}{N}^{\ensuremath{\mu}}k\ensuremath∼N\ensuremathμ and the target is unaffected, the speedup seems to obey N\ensuremathμ{N}^{\ensuremath{\mu}}N\ensuremathμ, which for \ensuremathμ>0.5\ensuremath{\mu}>0.5\ensuremathμ>0.5 is worse than the quantum, but better than the classical case. We also put obtained results for quantum searches into perspective of quantum walks and searches on graphs.
Abstract Grover’s quantum search algorithm promises a quadratic speedup for unstructured search over its classical counterpart. But this advantage is affected by noise acting on the search space. Here, we show that a quantum switch can act as a resource to mitigate the effects of noise. In this scenario, the noise is modeled by a depolarizing channel, which coherently acts on the entire quantum register. We show that a quantum switch can significantly reduce the error in Grover’s search algorithm. We consider the success probability of finding the marked item as the sole quantifier of diminishing the effect of noise in the search space in the presence of quantum switch. We propose two frameworks for the application of quantum switches. In the first framework, we apply the superposition of channel’s orders in the form of a switch and do a post-selection at every iteration of the applications of the Grover operator. In the second framework, we delay this measurement and post-selection until the very end. The number of post selections is minimal in the second scenario, and hence the noise reduction can be attributed more to the presence of quantum switch. We illustrate with an example of significant advantage in the success probability of Grover’s algorithm using quantum switch.
Adiabatic quantum algorithms must evolve slowly enough to suppress non-adiabatic transitions while remaining fast enough to be practical. In open systems, this trade-off is reshaped by decoherence. For Hamiltonians subject to dephasing Lindbladians, Avron et al. [Phys. Rev. A 82, 040304(R) (2010)] showed that a unique timetable exists that maximizes the fidelity with a target state. This optimal schedule is characterized by a constant tunneling rate along the adiabatic path. In this work, we revisit their analysis and apply it to the adiabatic Grover search framework, obtaining closed-form expressions for the optimal evolution schedule, the minimum runtime, and the resulting achievable fidelity. Moreover, by invoking an energy-time uncertainty argument, we identify a critical dephasing threshold beyond which further noise-assisted acceleration is prohibited, showing that the apparent advantage of strong dephasing is a Zeno-type stabilization mechanism rather than a conventional algorithmic speedup over unitary Grover search.
Quantum search is among the most important algorithms in quantum computing. At its core is quantum amplitude amplification, a technique that achieves a quadratic speedup over classical search by combining two global reflections: the oracle, which marks the target, and the diffusion operator, which reflects about the initial state. We show that this speedup can be preserved when the oracle is the only global operator, with all other operations acting locally on non-overlapping partitions of the search register. We present a recursive construction that, when the initial and target states both decompose as tensor products over these chosen partitions, admits an exact closed-form solution for the algorithm's dynamics. This is enabled by an intriguing degeneracy in the principal angles between successive reflections, which collapse to just two distinct values governed by a single recursively defined angle. Applied to unstructured search, a problem that naturally satisfies the tensor decomposition, the approach retains the O(N)O(\sqrt{N})O(N​) oracle complexity of Grover search when each partition contains at least log⁡2(log⁡2N)\log_2(\log_2 N)log2​(log2​N) qubits. On an 18-qubit search problem, partitioning into two stages reduces the non-oracle circuit depth by as much as 51%-96% relative to Grover, requiring up to 9% additional oracle calls. For larger problem sizes this oracle overhead rapidly diminishes, and valuable depth reductions persist when the oracle circuit is substantially deeper than the diffusion operator. More broadly, these results show that a global diffusion operator is not necessary to achieve the quadratic speedup in quantum search, offering a new perspective on this foundational algorithm. Moreover, the scalar reduction at the heart of our analysis inspires and motivates new directions and innovations in quantum algorithm design and evaluation.
The theory of decoherent histories is an attempt to derive classical physics from positing only quantum laws at the fundamental level without notions of a classical apparatus or collapse of the wave-function. Searching for a marked target in a list of N items requires Ω(N) oracle queries when using a classical computer, while a quantum computer can accomplish the same task in O{\sqrt{N}} queries using Grover's quantum algorithm. We study a closed quantum system executing Grover algorithm in the framework of decoherent histories and find it to be an exactly solvable model, thus yielding an alternate derivation of Grover's famous result. We also subject the Grover-executing computer to a generic external influence without needing to know the specifics of the Hamiltonian insofar as the histories decohere. Depending on the amount of decoherence, which is captured in our model by a single parameter related to the amount of information obtained by the environment, the search time can range from quantum to classical. Thus, we identify a key effect induced by the environment that can adversely affect a quantum computer's performance and demonstrate exactly how classical computing can emerge from quantum laws.
Fixed-point quantum search algorithms succeed at finding one of MMM target items among NNN total items even when the run time of the algorithm is longer than necessary. While the famous Grover's algorithm can search quadratically faster than a classical computer, it lacks the fixed-point property---the fraction of target items must be known precisely to know when to terminate the algorithm. Recently, Yoder, Low, and Chuang [Phys. Rev. Lett. 113, 210501 (2014)] gave an optimal gate-model search algorithm with the fixed-point property. Previously, it had been discovered by Roland and Cerf [Phys. Rev. A 65, 042308 (2002)] that an adiabatic quantum algorithm, operating by continuously varying a Hamiltonian, can reproduce the quadratic speedup of gate-model Grover search. We ask, can an adiabatic algorithm also reproduce the fixed-point property? We show that the answer depends on what interpolation schedule is used, so as in the gate model, there are both fixed-point and non-fixed-point versions of adiabatic search, only some of which attain the quadratic quantum speedup. Guided by geometric intuition on the Bloch sphere, we rigorously justify our claims with an explicit upper bound on the error in the adiabatic approximation. We also show that the fixed-point adiabatic search algorithm can be simulated in the gate model with neither loss of the quadratic Grover speedup nor of the fixed-point property. Finally, we discuss natural uses of fixed-point algorithms such as preparation of a relatively prime state and oblivious amplitude amplification.
Grover's quantum search algorithm provides a proven quadratic speedup over classical linear search for unstructured databases. While its theoretical advantage is well established, experimental validation on real noisy intermediate-scale quantum (NISQ) hardware remains limited by noise, decoherence, and hardware constraints. In this work, we present an experimental comparison between classical linear search and Grover's quantum search algorithm executed on IBM Quantum real superconducting hardware using Qiskit Runtime. A four-qubit implementation corresponding to a 16-element search space is evaluated across multiple experimental runs. The results demonstrate consistent amplitude amplification, achieving an average success probability of 54.25% and an amplification factor of 8.68× over random guessing. Although classical execution remains faster in wall-clock time due to quantum hardware overhead, the experiment empirically validates the algorithmic advantage of Grover's search and quantifies the performance gap between ideal and real quantum processors in the NISQ era.