Measurement-based quantum computation beyond the one-way model

We introduce novel schemes for quantum computing based on local measurements on entangled resource states. This work elaborates on the framework established in [Phys. Rev. Lett. 98, 220503 (2007), quant-ph/0609149]. Our method makes use of tools from many-body physics - matrix product states, finitely correlated states or projected entangled pairs states - to show how measurements on entangled states can be viewed as processing quantum information. This work hence constitutes an instance where a quantum information problem - how to realize quantum computation - was approached using tools from many-body theory and not vice versa. We give a more detailed description of the setting, and present a large number of new examples. We find novel computational schemes, which differ from the original one-way computer for example in the way the randomness of measurement outcomes is handled. Also, schemes are presented where the logical qubits are no longer strictly localized on the resource state. Notably, we find a great flexibility in the properties of the universal resource states: They may for example exhibit non-vanishing long-range correlation functions or be locally arbitrarily close to a pure state. We discuss variants of Kitaev's toric code states as universal resources, and contrast this with situations where they can be efficiently classically simulated. This framework opens up a way of thinking of tailoring resource states to specific physical systems, such as cold atoms in optical lattices or linear optical systems.Comment: 21 pages, 7 figure

Ala- ja ylärajoja merkkijonon etsinnälle verkosta

String Matching in Labelled Graphs (SMLG) is a generalisation of the classic problem of finding a match for a string into a text. In SMLG, we are given a pattern string and a graph with node labels, and we want to find a path whose node labels match the pattern string. This problem has been studied since 1992, and it was initially intended to model the problem of finding a link in a hypertext. Recently, the problem received attention due to its applications in bioinformatics, but all of the solutions, old and new, failed to run in truly sub-quadratic time. In this work, based on four published papers, we study SMLG from different angles, first proving conditional lower bounds, and then proposing efficient algorithms for special classes of graphs. In the first paper, we unveil the reason behind the hardness of SMLG, showing a quadratic conditional lower bound based on the Orthogonal Vectors Hypothesis and the Strong Exponential Time Hypothesis. The techniques that we employ come from the fine-grained complexity, and involve finding linear-time reductions from the Orthogonal Vectors problem to different variations of SMLG. In the second paper, we strengthen our findings by showing that an indexing data structure built in polynomial time is not enough to provide subquadratic time queries for SMLG. We devise a general framework for obtaining indexing lower bounds out of regular lower bounds, and we prove the indexing lower bound for SMLG as an application of this technique. In the third paper, we surpass the limitations of our lower bounds by identifying a class of graphs, called founder block graphs, which support linear time queries after subquadratic indexing. This class of graph effectively represents collections of strings called multiple sequence alignments, if gap characters are not present. In the fourth paper, we significantly improve our previous results on efficiently indexable graphs. We propose elastic founder graphs, a superset of founder block graphs, that are able to represent multiple sequence alignments with gaps. Moreover, we propose algorithms for constructing elastic founder graph, indexing them, and perform queries in linear time.Merkkijonon etsintä verkosta (engl. String Matching in Labelled Graphs, SMLG) on yleistys klassiselle ongelmalle etsiä merkkijonohahmon osumaa tekstistä. SMLG ongelmassa syötteenä ovat merkkijonohahmo ja verkko, jonka solmuilla on merkkijonotunnisteet. Tavoitteena on löytää polku, jonka solmujen tunnisteet muodostavat tekstin, joka sisältää annetun merkkijonohahmon. Ongelmaa on tutkittu vuodesta 1992 alun alkaen mallintamaan linkkien etsintää hypertekstistä. Viime aikoina ongelma on tullut uudestaan esille bioinformatiikan saralla. Sekä vanhat että uudet ratkaisut eivät ole onnistuneet oleellisesti murtamaan neliöllistä aikavaativuutta ongelman ratkaisussa. Tässä työssä SMLG ongelmaa tarkastellaan eri näkökulmista perustuen neljään julkaisuun. Ensin todistetaan ehdollinen alaraja ongelman vaativuudelle. Sitten esitetään tehokkaita ratkaisuja erilaisille verkkojen aliluokille. Ensimmäisessä julkaisussa paljastamme syyn SMLG ongelman vaikeudelle johtamalla ehdollisen alarajan perustuen kohtisuorien vektorien hypoteesiin (engl. Orthogonal Vectors Hypothesis) ja vahvaan eksponentiaalisen aikavaativuuden hypoteesiin (engl. Strong Exponential Time Hypothesis). Tähän tulokseen käytämme hienorakenteisen vaativuusteorian (engl. fine-grained complexity) tekniikoita, kuten lineaariaikaista reduktiota kohtisuorien vektoreiden ongelmasta kohdeongelmaan, tässä tapauksessa eri variaatioille SMLG ongelmasta. Toisessa julkaisussa vahvistamme edellistä tulosta osoittamalla, että polynomiaikainen verkon indeksointi ei riitä tukemaan alle neliöaikaista merkkijonohahmon etsintää. Kehitämme yleisen kehikon tämän kaltaisten indeksointialarajojen johtamiseen tavallisista alarajoista, ja todistamme SMLG ongelman alarajan sovellutuksena tästä tekniikasta. Kolmannessa julkaisussa ohitamme alarajat identifioimalla verkkojen aliluokan, kantasegmentteihin perustuvat verkot (engl. founder block graphs), joilla indeksointi onnistuu alle neliöllisessä ajassa, jonka jälkeen merkkijonohahmon etsintää voidaan suorittaa lineaarisessa ajassa. Kantasegmentteihin perustuvilla verkoilla voidaan esittää merkkijonokokoelmien monilinjaukset, mikäli linjauksessa ei tarvita poistoja ja lisäyksiä. Neljännessä julkaisussa parannamme merkittävästi aiempia tuloksiamme indeksoitavista verkoista. Laajennamme kantasegmentteihin perustuvat verkot elastisuuden käsitteellä, jolloin ne voivat esittää mielivaltaisia monilinjauksia, joissa linjauksessa sallitaan poistot ja lisäykset. Tämän lisäksi johdamme algoritmeja näiden elastisten kantasegmentteihin perustuvien verkkojen muodostamiseen, indeksointiin, sekä merkkijonohahmojen etsintään

Translationally Invariant Constraint Optimization Problems

We study the complexity of classical constraint satisfaction problems on a 2D grid. Specifically, we consider the complexity of function versions of such problems, with the additional restriction that the constraints are translationally invariant, namely, the variables are located at the vertices of a 2D grid and the constraint between every pair of adjacent variables is the same in each dimension. The only input to the problem is thus the size of the grid. This problem is equivalent to one of the most interesting problems in classical physics, namely, computing the lowest energy of a classical system of particles on the grid. We provide a tight characterization of the complexity of this problem, and show that it is complete for the class $FP^{NEXP}$. Gottesman and Irani (FOCS 2009) also studied classical translationally-invariant constraint satisfaction problems; they show that the problem of deciding whether the cost of the optimal solution is below a given threshold is NEXP-complete. Our result is thus a strengthening of their result from the decision version to the function version of the problem. Our result can also be viewed as a generalization to the translationally invariant setting, of Krentel's famous result from 1988, showing that the function version of SAT is complete for the class $FP^{NP}$. An essential ingredient in the proof is a study of the complexity of a gapped variant of the problem. We show that it is NEXP-hard to approximate the cost of the optimal assignment to within an additive error of $\Omega(N^{1/4})$, for an $N \times N$ grid. To the best of our knowledge, no gapped result is known for CSPs on the grid, even in the non-translationally invariant case. As a byproduct of our results, we also show that a decision version of the optimization problem which asks whether the cost of the optimal assignment is odd or even is also complete for $P^{NEXP}$.Comment: 75 pages, 13 figure

Hamiltonian Complexity in Many-Body Quantum Physics

The development of quantum computers has promised to greatly improve our understanding of quantum many-body physics. However, many physical systems display complex and unpredictable behaviour which is not amenable to analytic or even computational solutions. This thesis aims to further our understanding of what properties of physical systems a quantum computer is capable of determining, and simultaneously explore the behaviour of exotic quantum many-body systems. First, we analyse the task of determining the phase diagram of a quantum material, and thereby charting its properties as a function of some externally controlled parameter. In the general case we find that determining the phase diagram to be uncomputable, and in special cases show it is P^{QMA_{EXP}}-complete. Beyond this, we examine how a common method for determining quantum phase transitions --- the Renormalisation Group (RG) --- fails when applied to a set of Hamiltonians with uncomputable properties. We show that for such Hamiltonians (a) there is a well-defined RG procedure, but this procedure must fail to predict the uncomputable properties (b) this failure of the RG procedure demonstrates previously unseen and novel behaviour. We also formalise in terms of a promise problem, the question of computing the ground state energy per particle of a model in the limit of an infinitely large system, and show that approximating this quantity is likely intractable. In doing this we develop a new kind of complexity question concerned with determining the precision to which a single number can be determined. Finally we consider the problem of measuring local observables in the low energy subspace of systems --- an important problem for experimentalists and theorists alike. We prove that if a certain kind of construction exists for a class of Hamiltonians, , the results about hardness of determining the ground state energy directly implies hardness results for measuring observables at low energies

The Complexity of Translationally Invariant Spin Chains with Low Local Dimension

We prove that estimating the ground state energy of a translationally-invariant, nearest-neighbour Hamiltonian on a 1D spin chain is QMAEXP-complete, even for systems of low local dimension (roughly 40). This is an improvement over the best previously-known result by several orders of magnitude, and it shows that spin-glass-like frustration can occur in translationally-invariant quantum systems with a local dimension comparable to the smallest-known non-translationally-invariant systems with similar behaviour. While previous constructions of such systems rely on standard models of quantum computation, we construct a new model that is particularly well-suited for encoding quantum computation into the ground state of a translationally-invariant system. This allows us to shift the proof burden from optimizing the Hamiltonian encoding a standard computational model to proving universality of a simple model. Previous techniques for encoding quantum computation into the ground state of a local Hamiltonian allow only a linear sequence of gates, hence only a linear (or nearly linear) path in the graph of all computational states. We extend these techniques by allowing significantly more general paths, including branching and cycles, thus enabling a highly efficient encoding of our computational model. However, this requires more sophisticated techniques for analysing the spectrum of the resulting Hamiltonian. To address this, we introduce a framework of graphs with unitary edge labels. After relating our Hamiltonian to the Laplacian of such a unitary labelled graph, we analyse its spectrum by combining matrix analysis and spectral graph theory techniques

Combinatorial Algorithms for Subsequence Matching: A Survey

In this paper we provide an overview of a series of recent results regarding algorithms for searching for subsequences in words or for the analysis of the sets of subsequences occurring in a word.Comment: This is a revised version of the paper with the same title which appeared in the Proceedings of NCMA 2022, EPTCS 367, 2022, pp. 11-27 (DOI: 10.4204/EPTCS.367.2). The revision consists in citing a series of relevant references which were not covered in the initial version, and commenting on how they relate to the results we survey. arXiv admin note: text overlap with arXiv:2206.1389

Undecidability of the Spectral Gap in One Dimension

The spectral gap problem—determining whether the energy spectrum of a system has an energy gap above ground state, or if there is a continuous range of low-energy excitations—pervades quantum many-body physics. Recently, this important problem was shown to be undecidable for quantum-spin systems in two (or more) spatial dimensions: There exists no algorithm that determines in general whether a system is gapped or gapless, a result which has many unexpected consequences for the physics of such systems. However, there are many indications that one-dimensional spin systems are simpler than their higher-dimensional counterparts: For example, they cannot have thermal phase transitions or topological order, and there exist highly effective numerical algorithms such as the density matrix renormalization group—and even provably polynomial-time ones—for gapped 1D systems, exploiting the fact that such systems obey an entropy area law. Furthermore, the spectral gap undecidability construction crucially relied on aperiodic tilings, which are not possible in 1D. So does the spectral gap problem become decidable in 1D? In this paper, we prove this is not the case by constructing a family of 1D spin chains with translationally invariant nearest-neighbor interactions for which no algorithm can determine the presence of a spectral gap. This not only proves that the spectral gap of 1D systems is just as intractable as in higher dimensions, but it also predicts the existence of qualitatively new types of complex physics in 1D spin chains. In particular, it implies there are 1D systems with a constant spectral gap and nondegenerate classical ground state for all systems sizes up to an uncomputably large size, whereupon they switch to a gapless behavior with dense spectrum