53 research outputs found
Nonuniversal entanglement level statistics in projection-driven quantum circuits and glassy dynamics in classical computation circuits
In this thesis, I describe research results on three topics : (i) a phase transition in the area-law regime of quantum circuits driven by projection measurements; (ii) ultra slow dynamics in two dimensional spin circuits; and (iii) tensor network methods applied to boolean satisfiability problems.
(i) Nonuniversal entanglement level statistics in projection-driven quantum circuits; Non-thermalized closed quantum many-body systems have drawn considerable attention, due to their relevance to experimentally controllable quantum systems. In the first part of the thesis, we study the level-spacing statistics in the entanglement spectrum of output states of random universal quantum circuits where, at each time step, qubits are subject to a finite probability of projection onto states of the computational basis. We encounter two phase transitions with increasing projection rate: The first is the volume-to-area law transition observed in quantum circuits with projective measurements; The second separates the pure Poisson level statistics phase at large projective measurement rates from a regime of residual level repulsion in the entanglement spectrum within the area-law phase, characterized by non-universal level spacing statistics that interpolates between the Wigner-Dyson and Poisson distributions. The same behavior is observed in both circuits of random two-qubit unitaries and circuits of universal gates, including the set implemented by Google in its Sycamore circuits.
(ii) Ultra-slow dynamics in a translationally invariant spin model for multiplication and factorization; Slow relaxation of glassy systems in the absence of disorder remains one of the most intriguing problems in condensed matter physics. In the second part of the thesis we investigate slow relaxation in a classical model of short-range interacting Ising spins on a translationally invariant two-dimensional lattice that mimics a reversible circuit that, depending on the choice of boundary conditions, either multiplies or factorizes integers. We prove that, for open boundary conditions, the model exhibits no finite-temperature phase transition. Yet we find that it displays glassy dynamics with astronomically slow relaxation times, numerically consistent with a double exponential dependence on the inverse temperature. The slowness of the dynamics arises due to errors that occur during thermal annealing that cost little energy but flip an extensive number of spins. We argue that the energy barrier that needs to be overcome in order to heal such defects scales linearly with the correlation length, which diverges exponentially with inverse temperature, thus yielding the double exponential behavior of the relaxation time.
(iii) Reversible circuit embedding on tensor networks for Boolean satisfiability; Finally, in the third part of the thesis we present an embedding of Boolean satisfiability (SAT) problems on a two-dimensional tensor network. The embedding uses reversible circuits encoded into the tensor network whose trace counts the number of solutions of the satisfiability problem. We specifically present the formulation of #2SAT, #3SAT, and #3XORSAT formulas into planar tensor networks. We use a compression-decimation algorithm introduced by us to propagate constraints in the network before coarse-graining the boundary tensors. Iterations of these two steps gradually collapse the network while slowing down the growth of bond dimensions. For the case of #3XORSAT, we show numerically that this procedure recognizes, at least partially, the simplicity of XOR constraints for which it achieves subexponential time to solution. For a #P-complete subset of #2SAT we find that our algorithm scales with size in the same way as state-of-the-art #SAT counters, albeit with a larger prefactor. We find that the compression step performs less efficiently for #3SAT than for #2SAT
Tensor Network States: Optimizations and Applications in Quantum Many-Body Physics and Machine Learning
Tensor network states are ubiquitous in the investigation of quantum many-body (QMB) physics. Their advantage over other state representations is evident from their reduction in the computational complexity required to obtain various quantities of interest, namely observables. Additionally, they provide a natural platform for investigating entanglement properties within a system. In this dissertation, we develop various novel algorithms and optimizations to tensor networks for the investigation of QMB systems, including classical and quantum circuits. Specifically, we study optimizations for the two-dimensional Ising model in a transverse field, we create an algorithm for the -SAT problem, and we study the entanglement properties of random unitary circuits. In addition to these applications, we reinterpret renormalization group principles from QMB physics in the context of machine learning to develop a novel algorithm for the tasks of classification and regression, and then utilize machine learning architectures for the time evolution of operators in QMB systems
Arboreal Bound Entanglement
In this paper, we discuss the entanglement properties of graph-diagonal
states, with particular emphasis on calculating the threshold for the
transition between the presence and absence of entanglement (i.e. the
separability point). Special consideration is made of the thermal states of
trees, including the linear cluster state. We characterise the type of
entanglement present, and describe the optimal entanglement witnesses and their
implementation on a quantum computer, up to an additive approximation. In the
case of general graphs, we invoke a relation with the partition function of the
classical Ising model, thereby intimating a connection to computational
complexity theoretic tasks. Finally, we show that the entanglement is robust to
some classes of local perturbations.Comment: 9 pages + appendices, 3 figure
Quasi-elementary Landscapes and Superpositions of Elementary Landscapes
Whitley, D., & Chicano F. (2012). Quasi-elementary Landscapes and Superpositions of Elementary Landscapes. (Hamadi, Y., & Schoenauer M., Ed.).Learning and Intelligent Optimization - 6th International Conference, LION 6, Paris, France, January 16-20, 2012, Revised Selected Papers. 277â291.There exist local search landscapes where the evaluation function is an eigenfunction of the graph Laplacian that corresponds to the neighborhood structure of the search space. Problems that display this structure are called âElementary Landscapesâ and they have a number of special mathematical properties. The term âQuasi-elementary landscapesâ is introduced to describe landscapes that are âalmostâ elementary; in quasi-elementary landscapes there exists some efficiently computed âcorrectionâ that captures those parts of the neighborhood structure that deviate from the normal structure found in elementary landscapes. The âshiftâ operator, as well as the â3-optâ operator for the Traveling Salesman Problem landscapes induce quasi-elementary landscapes. A local search neighborhood for the Maximal Clique problem is also quasi-elementary. Finally, we show that landscapes which are a superposition of elementary landscapes can be quasi-elementary in structure.Universidad de MĂĄlaga. Campus de Excelencia Internacional AndalucĂa Tech. Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant number FA9550-11-1-0088. Spanish Ministry of Science and Innovation and FEDER under contract TIN2008-06491-C04-01 (the Mâ project). Andalusian Government under contract P07-TIC-03044 (DIRICOM project)
On Efficiently Solvable Cases of Quantum k-SAT
The constraint satisfaction problems k-SAT and Quantum k-SAT (k-QSAT) are canonical NP-complete and QMA_1-complete problems (for k >= 3), respectively, where QMA_1 is a quantum generalization of NP with one-sided error. Whereas k-SAT has been well-studied for special tractable cases, as well as from a parameterized complexity perspective, much less is known in similar settings for k-QSAT. Here, we study the open problem of computing satisfying assignments to k-QSAT instances which have a "matching" or "dimer covering"; this is an NP problem whose decision variant is trivial, but whose search complexity remains open.
Our results fall into three directions, all of which relate to the "matching" setting: (1) We give a polynomial-time classical algorithm for k-QSAT when all qubits occur in at most two clauses. (2) We give a parameterized algorithm for k-QSAT instances from a certain non-trivial class, which allows us to obtain exponential speedups over brute force methods in some cases by reducing the problem to solving for a single root of a single univariate polynomial. (3) We conduct a structural graph theoretic study of 3-QSAT interaction graphs which have a "matching". We remark that the results of (2), in particular, introduce a number of new tools to the study of Quantum SAT, including graph theoretic concepts such as transfer filtrations and blow-ups from algebraic geometry; we hope these prove useful elsewhere
Towards the First Practical Applications of Quantum Computers
Noisy intermediate-scale quantum (NISQ) computers are coming online. The lack of error-correction in these devices prevents them from realizing the full potential of fault-tolerant quantum computation, a technology that is known to have significant practical applications, but which is years, if not decades, away. A major open question is whether NISQ devices will have practical applications.
In this thesis, we explore and implement proposals for using NISQ devices to achieve practical applications. In particular, we develop and execute variational quantum algorithms for solving problems in combinatorial optimization and quantum chemistry. We also execute a prototype of a protocol for generating certified random numbers. We perform our experiments on a superconducting qubit processor developed at Google. While we do not perform any quantum computations that are beyond the capabilities of classical computers, we address many implementation challenges that must be overcome to succeed in such an endeavor, including optimization, efficient compilation, and error mitigation. In addressing these challenges, we push the limits of what can currently be done with NISQ technology, going beyond previous quantum computing demonstrations in terms of the scale of our experiments and the types of problems we tackle. While our experiments demonstrate progress in the utilization of quantum computers, the limits that we reached underscore the fundamental challenges in scaling up towards the classically intractable regime. Nevertheless, our results are a promising indication that NISQ devices may indeed deliver practical applications.PHDComputer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163016/1/kevjsung_1.pd
Estimation et rĂ©duction du coĂ»t dâalgorithmes quantiques
Plusieurs algorithmes ont Ă©tĂ© marquants dans le dĂ©veloppement de lâinformatique quantique : on peut penser notamment Ă lâalgorithme de Shor ou Ă lâalgorithme de Grover. Bien que pour ce dernier, par exemple, nous ayons une accĂ©lĂ©ration quadratique prouvĂ©e, il nâest pas clair que ce sera suffisant pour offrir un avantage pratique par rapport aux algorithmes classiques. Dâabord, celui-ci est formulĂ© en termes dâoracle, une boĂźte noire qui cache une sous-routine non comprise dans le calcul du coĂ»t. Ensuite, lorsque lâon prend en considĂ©ration le surcoĂ»t engendrĂ© notamment par la correction dâerreur quantique, il est possible de perdre lâaccĂ©lĂ©ration promise. Mais Ă©galement, la durĂ©e dâune porte logique quantique est considĂ©rablement plus long que son homologue classique. Quel est le coĂ»t rĂ©el dâun algorithme quantique lorsque lâon prend en considĂ©ration toutes les sous-routines ? Ă quand un ordinateur quantique utile qui surpassera les performances dâun superordinateur ?
Dans cette thĂšse, nous prĂ©senterons trois projets visant tous Ă estimer et rĂ©duire le coĂ»t dâalgorithmes ou sous-routines quantiques. Les algorithmes abordĂ©s sont issus dâune discrĂ©tisation de lâĂ©volution adiabatique. Le premier se concentre sur un algorithme de prĂ©paration dâĂ©tat dâun systĂšme Ă N corps par une Ă©volution adiabatique via lâeffet ZĂ©non. Le second porte sur une version quantique des algorithmes de marches alĂ©atoires et de recuit simulĂ© pouvant, par exemple, prĂ©parer un Ă©tat stationnaire. Le dernier dĂ©crit un nouvel algorithme : une Ă©volution adiabatique basĂ©e sur la rĂ©flexion. Celui-ci permet, entre autres, de rĂ©soudre des problĂšmes MAX-kSAT, une classe de problĂšmes NP-difficile. Avec ces projets, nous voulons, dâune part, proposer des algorithmes efficaces ainsi que leur implĂ©mentation de A Ă Z et, dâautre part, estimer les caractĂ©ristiques nĂ©cessaires Ă un ordinateur quantique utile (p. ex. taille, rĂ©sistance au bruit, vitesse dâopĂ©ration).
Les rĂ©sultats prĂ©sentĂ©s dĂ©montrent le coĂ»t Ă©levĂ© associĂ© aux algorithmes tolĂ©rants aux fautes. Bien quâon sâattende Ă avoir une accĂ©lĂ©ration par rapport au classique, lorsque lâon prend en considĂ©ration le nombre de qubits physiques, le nombre dâopĂ©rations physiques et la durĂ©e de chacune de ces opĂ©rations, en incluant la correction dâerreur notamment, la taille des instances offrant un avantage rĂ©el est loin dâĂȘtre atteignable pour les processeurs quantiques Ă court terme. Toutefois, en combinant des mĂ©thodes astucieuses et au moyen de diffĂ©rents procĂ©dĂ©s dâoptimisation, il est possible de rĂ©duire considĂ©rablement le coĂ»t des algorithmes quantiques, et donc de rĂ©duire le dĂ©lai pour atteindre la suprĂ©matie quantique
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