Quantum annealing: The fastest route to quantum computation? (original) (raw)
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Adiabatic quantum computation and quantum phase transitions
Physical Review A, 2004
We analyze the ground state entanglement in a quantum adiabatic evolution algorithm designed to solve the NP-complete Exact Cover problem. The entropy of entanglement seems to obey linear and universal scaling at the point where the energy gap becomes small, suggesting that the system passes near a quantum phase transition. Such a large scaling of entanglement suggests that the effective connectivity of the system diverges as the number of qubits goes to infinity and that this algorithm cannot be efficiently simulated by classical means. On the other hand, entanglement in Grover's algorithm is bounded by a constant. PACS numbers: 03.67.-a, 03.65.Ud, 03.67.Hk
Computing Research Repository, 2010
Two recent preprints [B. Altshuler, H. Krovi, and J. Roland, "Quantum adiabatic optimization fails for random instances of NP-complete problems", arXiv:0908.2782 and "Anderson localization casts clouds over adiabatic quantum optimization", arXiv:0912.0746] argue that random 4th order perturbative corrections to the energies of local minima of random instances of NP-complete problem lead to avoided crossings that cause the failure of quantum adiabatic algorithm (due to exponentially small gap) close to the end, for very small transverse field that scales as an inverse power of instance size N . The theoretical portion of this work does not to take into account the exponential degeneracy of the ground and excited states at zero field. A corrected analysis shows that unlike those in the middle of the spectrum, avoided crossings at the edge would require high [O(1)] transverse fields, at which point the perturbation theory may become divergent due to quantum phase transition. This effect manifests itself only in large instances [exp(0.02N ) ≫ 1], which might be the reason it had not been observed in the authors' numerical work. While we dispute the proposed mechanism of failure of quantum adiabatic algorithm, we cannot draw any conclusions on its ultimate complexity.
First-order quantum phase transition in adiabatic quantum computation
Physical Review A, 2009
We investigate the connection between local minima in the problem Hamiltonian and first order quantum phase transitions during an adiabatic quantum computation. We demonstrate how some properties of the local minima can lead to an extremely small gap that is exponentially sensitive to the Hamiltonian parameters. Using perturbation expansion, we derive an analytical formula that can not only predict the behavior of the gap, but also provide insight on how to controllably vary the gap size by changing the parameters. We show agreement with numerical calculations for a weighted maximum independent set problem instance.
A study of heuristic guesses for adiabatic quantum computation
2008
Adiabatic quantum computation (AQC) is a universal model for quantum computation which seeks to transform the initial ground state of a quantum system into a final ground state encoding the answer to a computational problem. AQC initial Hamiltonians conventionally have a uniform superposition as ground state. We diverge from this practice by introducing a simple form of heuristics: the ability to start the quantum evolution with a state which is a guess to the solution of the problem. With this goal in mind, we explain the viability of this approach and the needed modifications to the conventional AQC (CAQC) algorithm. By performing a numerical study on hard-to-satisfy 6 and 7 bit random instances of the satisfiability problem (3-SAT), we show how this heuristic approach is possible and we identify that the performance of the particular algorithm proposed is largely determined by the Hamming distance of the chosen initial guess state with respect to the solution. Besides the possibility of introducing educated guesses as initial states, the new strategy allows for the possibility of restarting a failed adiabatic process from the measured excited state as opposed to restarting from the full superposition of states as in CAQC. The outcome of the measurement can be used as a more refined guess state to restart the adiabatic evolution. This concatenated restart process is another heuristic that the CAQC strategy cannot capture.
Adiabatic quantum computation: Enthusiast and Sceptic's perspectives
Enthusiast's perspective: We analyze the effectiveness of AQC for a small rank problem Hamiltonian HF with the arbitrary initial Hamiltonian HI . We prove that for the generic HI the running time cannot be smaller than O( √ N ), where N is a dimension of the Hilbert space. We also construct an explicit HI for which the running time is indeed O( √ N ). Our algorithm can be used to solve the unstructured search problem with the unknown number of marked items.
A Quantum Adiabatic Evolution Algorithm Applied to Random Instances of an NP-Complete Problem
Science, 2001
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 tested one such algorithm by applying it to randomly generated hard instances of an NP-complete problem. For the small examples that we could simulate, the quantum adiabatic algorithm worked well, providing 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.
Adiabatic quantum optimization fails for random instances of NP-complete problems
2009
Adiabatic quantum optimization has attracted a lot of attention because small scale simulations gave hope that it would allow to solve NP-complete problems efficiently. Later, negative results proved the existence of specifically designed hard instances where adiabatic optimization requires exponential time. In spite of this, there was still hope that this would not happen for random instances of NP-complete problems. This is an important issue since random instances are a good model for hard instances that can not be solved by current classical solvers, for which an efficient quantum algorithm would therefore be desirable. Here, we will show that because of a phenomenon similar to Anderson localization, an exponentially small eigenvalue gap appears in the spectrum of the adiabatic Hamiltonian for large random instances, very close to the end of the algorithm.
Theory versus practice in annealing-based quantum computing
Theoretical Computer Science, 2020
This paper introduces basic concepts of annealing-based quantum computing, also known as adiabatic quantum computing (AQC) and quantum annealing (QA), and surveys what is known about this novel computing paradigm. Extensive empirical research on physical quantum annealing processers built by D-Wave Systems has exposed many interesting features and properties. However, because of longstanding differences between abstract and empirical approaches to the study of computational performance, much of this work may not be considered relevant to questions of interest to complexity theory; by the same token, several theoretical results in quantum computing may be considered irrelevant to practical experience. To address this communication gap, this paper proposes models of computation and of algorithms that lie on a scale of instantiation between pencil-and-paper abstraction and physical device. Models at intermediate points on these scales can provide a common language, allowing researchers from both ends to communicate and share their results. The paper also gives several examples of common terms that have different technical meanings in different regions of this highly multidisciplinary field, which can create barriers to effective communication across disciplines.
Quantum search by local adiabatic evolution
Physical Review A, 2002
The adiabatic theorem has been recently used to design quantum algorithms of a new kind, where the quantum computer evolves slowly enough so that it remains near its instantaneous ground state which tends to the solution [1]. We apply this time-dependent Hamiltonian approach to the Grover's problem, i. e., searching a marked item in an unstructured database. We find that, by adjusting the evolution rate of the Hamiltonian so as to keep the evolution adiabatic on each infinitesimal time interval, the total running time is of order √ N , where N is the number of items in the database. We thus recover the advantage of Grover's standard algorithm as compared to a classical search, scaling as N . This is in contrast with the constant-rate adiabatic approach developed in [1], where the requirement of adiabaticity is expressed only globally, resulting in a time of order N .
From quantum circuits to adiabatic algorithms
Physical Review A, 2005
This paper explores several aspects of the adiabatic quantum computation model. We first show a way that directly maps any arbitrary circuit in the standard quantum computing model to an adiabatic algorithm of the same depth. Specifically, we look for a smooth time-dependent Hamiltonian whose unique ground state slowly changes from the initial state of the circuit to its final state. Since this construction requires in general an n-local Hamiltonian, we will study whether approximation is possible using previous results on ground state entanglement and perturbation theory. Finally we will point out how the adiabatic model can be relaxed in various ways to allow for 2-local partially adiabatic algorithms as well as 2-local holonomic quantum algorithms.