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arXiv:1404.1977 · Physical Review A

Runtime of unstructured search with a faulty Hamiltonian oracle

Kristan Temme

Published Aug 11, 2014• 13 citations
arXivHTMLPDF

Abstract

We show that it is impossible to obtain a quantum speedup for a faulty Hamiltonian oracle. The effect of dephasing noise to this continuous-time oracle model has first been investigated by Shenvi, Brown, and Whaley [Phys. Rev. A 68, 052313 (2003).]. The authors consider a faulty oracle described by a continuous-time master equation that acts as dephasing noise in the basis determined by the marked item. The analysis focuses on the implementation with a particular driving Hamiltonian. A universal lower bound for this oracle model, which rules out a better performance with a different driving Hamiltonian, has so far been lacking. Here, we derive an adversary-type lower bound which shows that the evolution time $T$ has to be at least in the order of $N$, i.e., the size of the search space, when the error rate of the oracle is constant. This means that quadratic quantum speedup vanishes and the runtime assumes again the classical scaling. For the standard quantum oracle model this result was first proven by Regev and Schiff [in Automata, Languages and Programming, Lecture Notes in Computer Science Vol. 5125 (Springer, Berlin, 2008), pp. 773--781]. Here, we extend this result to the continuous-time setting.

DOI
10.1103/physreva.90.022310

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We show that it is impossible to obtain a quantum speedup for a faulty Hamiltonian oracle. The effect of dephasing noise to this continuous-time oracle model has first been investigated by Shenvi, Brown, and Whaley [Phys. Rev. A 68, 052313 (2003).]. The authors consider a faulty oracle described by a continuous-time master equation that acts as dephasing noise in the basis determined by the marked item. The analysis focuses on the implementation with a particular driving Hamiltonian. A universal lower bound for this oracle model, which rules out a better performance with a different driving Hamiltonian, has so far been lacking. Here, we derive an adversary-type lower bound which shows that the evolution time TTT has to be at least in the order of NNN, i.e., the size of the search space, when the error rate of the oracle is constant. This means that quadratic quantum speedup vanishes and the runtime assumes again the classical scaling. For the standard quantum oracle model this result was first proven by Regev and Schiff [in Automata, Languages and Programming, Lecture Notes in Computer Science Vol. 5125 (Springer, Berlin, 2008), pp. 773--781]. Here, we extend this result to the continuous-time setting.
This paper investigates the impact of noise in the quantum query model, a fundamental framework for quantum algorithms. We focus on the scenario where the oracle is subject to non-unitary (or irreversible) noise, specifically under the \textit{faulty oracle} model, where the oracle fails with a constant probability and acts as identity. Regev and Schiff (ICALP'08) showed that quantum advantage is lost for the search problem under this noise model. Our main result shows that every quantum query algorithm can be made robust in this noise model with a roughly quadratic blow-up in query complexity, thereby preserving quantum speedup for all problems where the quantum advantage is super-cubic. This is the first non-trivial robustification of quantum query algorithms against an oracle that is noisy.
We study quantum query complexity under several models of imperfect oracle access, and develop lower bounds through a common framework based on fractional block sensitivity (fbs⁡\operatorname{fbs}fbs). For a negligent oracle that applies the correct query with probability 1−p1-p1−p, we prove a lower bound in terms of fbs⁡(f,0n)\operatorname{fbs}(f,0^n)fbs(f,0n). We also show, perhaps surprisingly, that negligence need not destroy quantum speedups: any function fff can be transformed into a partial function f′f'f′ whose negligent query complexity essentially preserves the quantum query complexity of fff. This gives partial functions with exponential quantum speedups even under negligent queries, and in particular rules out a general lower bound in terms of fbs⁡(f)\operatorname{fbs}(f)fbs(f) for partial functions in this model. For two other models, we obtain general lower bounds in terms of fbs⁡(f)\operatorname{fbs}(f)fbs(f) for all Boolean functions. For hybrid algorithms using QQQ coherent and CCC classical queries, we prove the tradeoff C+Q2=Ω(fbs⁡(f))C+Q^2=Ω(\operatorname{fbs}(f))C+Q2=Ω(fbs(f)). For an IID dephasing noisy model where each query dephases the query-index register at rate ppp independently, we prove Ω ⁣(p fbs⁡(f))Ω\!\left(p\,\operatorname{fbs}(f)\right)Ω(pfbs(f)) queries are necessary. Finally, we introduce a broader family of time-varying dephasing models and identify a variational resource that is always lower bounded by fbs⁡(f)\operatorname{fbs}(f)fbs(f). Computing this resource reduces to a convex optimization problem, providing a simple way to derive lower bounds for new noise schedules. As applications, we recover the hybrid and IID dephasing bounds and determine the query complexity of unstructured search when the dephasing rate grows over time.
We consider quantum search algorithms that have access to a noisy oracle that, for every oracle call, with probability p>0p>0p>0 completely depolarizes the query registers, while otherwise working properly. Previous results had not ruled out quantum O(n)\mathrm{O}(\sqrt{n})O(n​)-query algorithms in this setting, even for constant ppp. We show that, for all p≤0.99p\le 0.99p≤0.99, the quantum noisy-query complexity of the unstructured search is Θ~(max⁡{np,n})\tildeΘ(\max\{np,\sqrt{n}\})Θ~(max{np,n​}). The lower bound Ω(max⁡{np,n})Ω(\max\{np,\sqrt n\})Ω(max{np,n​}) holds also for the dephasing noise and even when, for every oracle call, the algorithm is provided with a flag indicating whether the error has occurred.
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.
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 $kp\ensuremath{\ll}\sqrt{N}$, 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/N1/\sqrt{N}1/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.
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.