On the success probability of quantum order finding
Martin Ekerå
备注：
This revision resolves a minor issue in Alg. 4, and addresses a number of other minor issues. It furthermore adds a number of extensions and clarifications, in particular with respect to potential optimizations. No key results or findings in the original version of the manuscript are affected by this revision
We prove a lower bound on the probability of Shor's order-finding algorithm successfully recovering the order $r$ in a single run. The bound implies that by performing two limited searches in the classical post-processing part of the algorithm, a high success probability can be guaranteed, for any $r$, without re-running the quantum part or increasing the exponent length compared to Shor. Asymptotically, in the limit as $r$ tends to infinity, the probability of successfully recovering $r$ in a single run tends to one. Already for moderate $r$, a high success probability exceeding e.g. $1 - 10^{-4}$ can be guaranteed. As corollaries, we prove analogous results for the probability of completely factoring any integer $N$ in a single run of the order-finding algorithm.
We compared the noise correlation and the success probability of coherent Ising machines (CIMs) with optical delay-line, measurement feedback, and mean-field couplings. We theoretically studied three metrics for the noise correlations in these CIMs: quantum entanglement, quantum discord, and normalized correlation of canonical coordinates. The success probability was obtained through numerical simulations of truncated stochastic differential equations based on the Wigner distribution function. The results indicate that the success probability is more directly related to the normalized correlat…
Upper and Lower Bounds on Optimal Success Probability of Quantum State Discrimination with and without Inconclusive Results
给出了判别给定量子态的最大成功概率的上、下界。建议的上界是从相应的最优状态判别问题的对偶问题的次优解得到的。我们还给出了一个充分必要条件的上限，以实现最大的成功概率，建议的下限可以从这个条件得到。推导出所提出的上界的略微修改版本比Qiu等人提出的上界更紧[Phys. Rev. A 81，042329（2010）]。此外，我们提出了最大成功概率的上限和下限与一个固定的利率不确定的结果。所提出的界限的性能进行了评估，通过数值实验。
A probabilistic algorithm for preparing Bethe eigenstates of the spin-1/2 Heisenberg spin chain on a quantum computer has recently been found. We derive an exact formula for the success probability of this algorithm in terms of the Gaudin determinant, and we study its large-length limit. We demonstrate the feasibility of computing antiferromagnetic ground-state spin-spin correlation functions for short chains. However, the success probability decreases exponentially with the chain length, which precludes the computation of these correlation functions for chains of moderate length. Some conject…
We present a universal mechanism that, acting on any target qubit, propagates it to the state it had T time units before the experiment started. This protocol works by setting the target on a superposition of flight paths, where it is acted on by uncharacterized, but repeatable, quantum operations. Independently of the effect of each of these individual operations on the target, the successful interference of the paths causes it to leap to its past state. We prove that, for generic interaction effects, the system will reach the desired state with probability 1 after some finite number of steps…
Coherence plays a very important role in Grover search algorithm (GSA). In this paper, we define the normalization coherence N(C), where C is a coherence measurement. In virtue of the constraint of large N and Shannon's maximum entropy principle, a surprising complementary relationship between the coherence and the success probability of GSA is obtained. Namely, P_s(t)+N(C(t))\simeq 1, where C is in terms of the relative entropy of coherence and l_1 norm of coherence, t is the number of the search iterations in GSA. Moreover, the equation holds no matter in ideal or noisy environments. Conside…
The textbook adversary bound for function evaluation states that to evaluate a function $f\colon D\to C$ with success probability $\frac{1}{2}+δ$ in the quantum query model, one needs at least $\left( 2δ-\sqrt{1-4δ^2} \right) Adv(f)$ queries, where $Adv(f)$ is the optimal value of a certain optimization problem. For $δ\ll 1$, this only allows for a bound of $Θ\left(δ^2 Adv(f)\right)$ even after a repetition-and-majority-voting argument. In contrast, the polynomial method can sometimes prove a bound that doesn't converge to $0$ as $δ\to 0$. We improve the $δ$-dependent prefactor and achieve a b…
We consider the problem of discriminating finite-dimensional quantum processes, also called quantum supermaps, that can consist of multiple time steps. Obtaining the ultimate performance for discriminating quantum processes is of fundamental importance, but is challenging mainly due to the necessity of considering all discrimination strategies allowed by quantum mechanics, including entanglement-assisted strategies and adaptive strategies. In the case in which the processes to be discriminated have internal memories, the ultimate performance would generally be more difficult to analyze. In thi…
In the emerging domain of quantum algorithms, the Grover's quantum search is certainly one of the most significant. It is relatively simple, performs a useful task and more importantly, does it in an optimal way. However, due to the success of quantum walks in the field, it is logical to study quantum search variants over several kind of walks. In this paper, we propose an in-depth study of the quantum search over a hypercube layout. First, through the analysis of elementary walk operators restricted to suitable eigenspaces, we show that the acting component of the search algorithm takes place…
In recent years, quantum annealing has gained the status of being a promising candidate for solving various optimization problems. Using a set of hard 2-satisfiabilty (2-SAT) problems, consisting of upto 18-variables problems, we analyze the scaling complexity of the quantum annealing algorithm and study the distributions of the minimum energy gap and the success probability. We extend the analysis of the standard quantum annealing Hamiltonian by introducing an additional term, the trigger Hamiltonian, which can be of two types : ferromagnetic and antiferromagnetic. We use these trigger Hamilt…
Comment on Nahimovs et al. `On the probability of finding marked connected components using quantum walks'
在这篇评论文章中，我们提出了在Nahimovs等人的论文中发现的两个误解。\emph{On the probability of finding marked connected components using quantum walks}。首先，我们证明了定理2（一个状态是平稳的充分必要条件）是不完整的-它只在未标记的顶点形成一个连通分量时才有效。其次，我们纠正了定理3（概率的下限）中\emph{a}系数的推导，并给出了如何求出\emph{a}的上界值。
Present-day, noisy, small or intermediate-scale quantum processors---although far from fault-tolerant---support the execution of heuristic quantum algorithms, which might enable a quantum advantage, for example, when applied to combinatorial optimization problems. On small-scale quantum processors, validations of such algorithms serve as important technology demonstrators. We implement the quantum approximate optimization algorithm (QAOA) on our hardware platform, consisting of two superconducting transmon qubits and one parametrically modulated coupler. We solve small instances of the NP-comp…
One-sided confidence intervals are presented for the average of non-identical Bernoulli parameters. These confidence intervals are expressed as analytical functions of the total number of Bernoulli games won, the number of rounds and the confidence level. Tightness of these bounds in the sense of Buehler, i.e. as the strictest possible monotonic intervals, is demonstrated for all confidence levels. A simple interval valid for all confidence levels is also provided with a tightness guarantee. Finally, an application of the proposed confidence intervals to sequential sampling is discussed.
Asymptotic Quantum Search and a Quantum Algorithm for Calculation of a Lower Bound of the Probability of Finding a Diophantine Equation That Accepts Integer Solutions
We analyze the robustness of Grover's quantum search algorithm performed by a quantum register under a possibly time-correlated noise acting locally on the qubits. We model the noise as originating from an arbitrary but fixed unitary evolution, $U$, of some noisy qubits. The noise can occur with some probability in the interval between any pair of consecutive noiseless Grover evolutions. Although each run of the algorithm is a unitary process, the noise model leads to decoherence when all possible runs are considered. We derive a set of unitary $U$'s, called the 'good noises,' for which the su…
Quantum walks are powerful tools for building quantum search algorithms or quantum sampling algorithms named the construction of quantum stationary state. However, the success probability of those algorithms are all far away from 1. Amplitude amplification is usually used to amplify success probability, but the soufflé problems follow. Only stop at the right step can we achieve a maximum success probability. Otherwise, as the number of steps increases, the success probability may decrease, which will cause troubles in practical application of the algorithm when the optimal number of steps is n…
Quantum walk search may exhibit phenomena beyond the intuition from a conventional random walk theory. One of such examples is exceptional configuration phenomenon -- it appears that it may be much harder to find any of two or more marked vertices, that if only one of them is marked. In this paper, we analyze the probability of finding any of marked vertices in such scenarios and prove upper bounds for various sets of marked vertices. We apply the upper bounds to large collection of graphs and show that the quantum search may be slow even when taking real-world networks.
