A quantum system will stay near its instantaneous ground state if the Hamiltonian that governs its evolution varies slowly enough. This quantum adiabatic behavior is the basis of a new class of algorithms for quantum computing. We test one such algorithm by applying it to randomly generated, hard, instances of an NP-complete problem. For the small examples that we can simulate, the quantum adiabatic algorithm works well, and provides evidence that quantum computers (if large ones can be built) may be able to outperform ordinary computers on hard sets of instances of NP-complete problems.
When a composite Lindblad system consists of weakly coupled sub-systems with fast and slow timescales, the description of slow dynamics can be simplified by discarding fast degrees of freedom. This model reduction technique is called adiabatic elimination. While second-order perturbative expansion with respect to the timescale separation has revealed that the evolution of a reduced state is completely positive, this paper presents an example exhibiting complete positivity violation in the fourth-order expansion. Despite the non-uniqueness of slow dynamics parametrization, we prove that complet…
Prime factorization is a difficult problem with classical computing, whose exponential hardness is the foundation of Rivest-Shamir-Adleman (RSA) cryptography. With programmable quantum devices, adiabatic quantum computing has been proposed as a plausible approach to solve prime factorization, having promising advantage over classical computing. Here, we find there are certain hard instances that are consistently intractable for both classical simulated annealing and un-configured adiabatic quantum computing (AQC). Aiming at an automated architecture for optimal configuration of quantum adiabat…
On the relevance of avoided crossings away from quantum critical point to the complexity of quantum adiabatic algorithm
两个最近的预印本[B。阿尔茨赫姆湾Krovi和J. Roland，“Quantum adiabatic optimization fails for random instances of NP-complete problems”，arXiv：0908.2782和“安德森局域化casts clouds over adiabatic quantum optimization”，arXiv：0912.0746]认为，对于作为实例大小N的逆幂缩放的非常小的横向场，对NP-complete问题的随机实例的局部最小值的能量的随机4阶微扰校正导致避免的交叉，所述交叉导致接近末端的量子绝热算法的失败（由于指数小间隙）。本文的理论部分没有考虑零场下基态和激发态的指数简并。修正后的分析表明，不像那些在中间的频谱，避免交叉的边缘将需要高[O（1）]的横向领域，在这一点上的微扰理论可能会成为发散由于量子相变。这种效应只在大的情况下才表现出来[exp（0.02 N）>> 1]，这可能是作者的数值工作中没有观察到这种效应的原因。虽然我们对量子绝热算法失败的机制提出了质疑，但我们不能对它的最终复杂性得出任何结论。
Mechanism of Efficacy in QAOA for Random k-SAT: From Adiabatic Manifold to Sublinear Parameter Optimization
School of Computer Science and Engineering, Southeast University（东南大学计算机科学与工程学院）；Key Laboratory of Computer Network and Information Integration (Southeast University), Ministry of Education（教育部计算机网络与信息集成重点实验室（东南大学））
A novel and readily understandable derivation of the Golden Rule of time dependent perturbation theory is presented. The derivation is based on adiabatic turning on of the perturbation as used, for instance, in some formal developments of scattering theory. Energy conservation is expressed in terms of an intuitively and physically appealing Lorentzian line shape rather than the artificial, oscillatory sin(x)=x type line shape that appears in conventional derivations. The conditions for the derivation's validity are compactly and conveniently expressed in the frequency/energy domain rather than…
Scaling QAOA: transferring optimal adiabatic schedules from small-scale to large-scale variational circuits
Aix-Marseille Université, Université de Toulon（艾克斯-马赛大学、土伦大学）；ENSTA, Institut Polytechnique de Paris（巴黎理工学院）；Conservatoire National des Arts et Métiers（国立工艺博物馆）；Université Paris Cité（巴黎西岱大学）
Adiabatic quantum computation (AQC), which is particularly useful for combinatorial optimization, becomes more powerful by using excited states, instead of ground states. However, the excited-state AQC is prone to errors due to dissipation. Here we propose the excited-state AQC started with the most stable state, i.e., the vacuum state. This counterintuitive approach becomes possible by using a driven quantum system, or more precisely, a network of Kerr-nonlinear parametric oscillators (KPOs). By numerical simulations, we show that some hard instances, where standard ground-state AQC with KPOs…
Adiabatic quantum algorithms as quantum phase transitions: first versus second order
在连续极限（大量的量子比特），绝热量子算法显示出一个显着的相似性，通过量子相变扫描。我们发现，第二或更高的顺序相比，那些一阶的过渡是有利的。有了这个洞察力，我们提出了一种新的绝热量子算法来解决3-SAT问题（精确覆盖），根据数值模拟（高达20个量子比特），它比以前的建议要快得多。这些发现表明，绝热量子算法可以比Grover搜索程序（产生二次增强）更快地解决NP完全问题，例如3-SAT，甚至可能具有指数加速。PACS：03.67。- a，03.67.Lx，73.43.Nq，64.70.- p.
