Quantum computation with devices whose contents are never read

In classical computation, a "write-only memory" (WOM) is little more than an oxymoron, and the addition of WOM to a (deterministic or probabilistic) classical computer brings no advantage. We prove that quantum computers that are augmented with WOM can solve problems that neither a classical computer with WOM nor a quantum computer without WOM can solve, when all other resource bounds are equal. We focus on realtime quantum finite automata, and examine the increase in their power effected by the addition of WOMs with different access modes and capacities. Some problems that are unsolvable by two-way probabilistic Turing machines using sublogarithmic amounts of read/write memory are shown to be solvable by these enhanced automata.Comment: 32 pages, a preliminary version of this work was presented in the 9th International Conference on Unconventional Computation (UC2010

Undecidability of the Spectral Gap (full version)

We show that the spectral gap problem is undecidable. Specifically, we construct families of translationally-invariant, nearest-neighbour Hamiltonians on a 2D square lattice of d-level quantum systems (d constant), for which determining whether the system is gapped or gapless is an undecidable problem. This is true even with the promise that each Hamiltonian is either gapped or gapless in the strongest sense: it is promised to either have continuous spectrum above the ground state in the thermodynamic limit, or its spectral gap is lower-bounded by a constant in the thermodynamic limit. Moreover, this constant can be taken equal to the local interaction strength of the Hamiltonian.Comment: v1: 146 pages, 56 theorems etc., 15 figures. See shorter companion paper arXiv:1502.04135 (same title and authors) for a short version omitting technical details. v2: Small but important fix to wording of abstract. v3: Simplified and shortened some parts of the proof; minor fixes to other parts. Now only 127 pages, 55 theorems etc., 10 figures. v4: Minor updates to introductio

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 DMRG - 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 neighbour 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 also predicts the existence of qualitatively new types of complex physics in 1D spin chains. In particular, it implies there are 1D systems with constant spectral gap and non-degenerate classical ground state for all systems sizes up to an uncomputably large size, whereupon they switch to a gapless behaviour with dense spectrum.Comment: 7 figure

Computational power of quantum and probabilistic automata

Elektroniskā versija nesatur pielikumusAnot¹acija ņSis darbs apvieno p¹et¹³jumus par diviem autom¹atu veidiem: varb¹utiskajiem apgrieņzamajiem autom¹atiem (PRA), kas ir saist¹³ti ar kvantu gal¹³gajiem auto- m¹atiem (QFA), un vienvirziena kvantu autom¹atiem ar skait¹³t¹aju (Q1CA), kas ir l»oti ierobeņzots kvantu autom¹atu modelis, kam atbilstoņsa kvantu sist¹ema nav gal¹³ga. Darba m¹erk»is ir aprakst¹³t valodu klases, ko paz¹³st ņsie autom¹ati, un sal¹³dzin¹at kvantu un varb¹utiskos autom¹atus. M¹es pied¹av¹ajam varb¹utisk¹a apgrieņzama autom¹ata modeli. M¹es p¹et¹am vienvirziena PRA gan ar klasisko (C-PRA) v¹ardu akcept¹eņsanu, gan ar ap- st¹adin¹aņsanu (DH-PRA). M¹es par¹ad¹am valodu klases a¤1 a¤2 : : : a¤n paz¹³ņsanu ar PRA. M¹es par¹ad¹am vai valodu klase, ko paz¹³st PRA, ir sl¹egta pret B¹ula oper¹acij¹am. M¹es par¹ad¹am visp¹ar¹³gas valodu klases, ko C-PRA un DH-PRA nepaz¹³st. M¹es apskat¹am v¹ajas apgrieņzam¹³bas de¯n¹³ciju un par¹ad¹am atņsk»ir¹³bu no apgrieņzam¹³bas. M¹es pied¹av¹ajam vispar¹³gu kvantu vienvirziena autom¹ata modeli ar skai- t¹³t¹aju (Q1CA). M¹es pier¹ad¹am ka ņsis modelis apmierina transform¹acijas uni- tarit¹ates principu. Tiek pied¹av¹ats speci¹als Q1CA veids - vienk¹arņsais Q1CA, kas l»auj konstru¹et autom¹atu piem¹erus konkr¹et¹am valod¹am. M¹es par¹ad¹am vair¹aku kontekstatkar¹³go valodu paz¹³ņsanu ar Q1CA. M¹es pier¹ad¹am ka past¹av valodas, ko paz¹³st Q1CA, bet ko nepaz¹³st varb¹utiskais autom¹ats ar skait¹³t¹aju.The thesis assembles research on two models of automata - probabilistic reversible (PRA) that appear very similar to 1-way quantum ¯nite automata (1-QFA) and quantum one-way one counter automata (Q1CA), that is the most restricted model of non-¯nite space quantum automata. The objective of the research is to describe classes of languages recognizable by these models and compare related quantum and probabilistic automata. We propose the model of probabilistic reversible automata. We study both one-way PRA with classical (1-C-PRA) and decide and halt (1-DH- PRA) acceptance. We show recognition of general class of languages Ln = a¤1a¤2 : : : a¤n with probability 1 ¡ ". We show whether the classes of languages they recognize are closed under boolean operations and describe general class of languages not recognizable by these automata in terms of \forbidden con- structions" for the minimal deterministic automaton of the language. We also consider \weak" reversibility as equivalent de¯nition for 1-way automata and show the di®erence from ordinary reversibility in 1.5-way case. We propose the general notion of quantum one-way one counter au- tomata(Q1CA). We describe well-formedness conditions for the Q1CA that ensure unitarity of its evolution. A special kind of Q1CA, called simple, that satis¯es the well-formedness conditions is introduced. We show recognition of several non context free languages by Q1CA. We show that there is a lan- guage that can be recognized by quantum one-way one counter automaton, but not by the probabilistic one counter automaton

MFCS\u2798 Satellite Workshop on Cellular Automata

For the 1998 conference on Mathematical Foundations of Computer Science (MFCS\u2798) four papers on Cellular Automata were accepted as regular MFCS\u2798 contributions. Furthermore an MFCS\u2798 satellite workshop on Cellular Automata was organized with ten additional talks. The embedding of the workshop into the conference with its participants coming from a broad spectrum of fields of work lead to interesting discussions and a fruitful exchange of ideas. The contributions which had been accepted for MFCS\u2798 itself may be found in the conference proceedings, edited by L. Brim, J. Gruska and J. Zlatuska, Springer LNCS 1450. All other (invited and regular) papers of the workshop are contained in this technical report. (One paper, for which no postscript file of the full paper is available, is only included in the printed version of the report). Contents: F. Blanchard, E. Formenti, P. Kurka: Cellular automata in the Cantor, Besicovitch and Weyl Spaces K. Kobayashi: On Time Optimal Solutions of the Two-Dimensional Firing Squad Synchronization Problem L. Margara: Topological Mixing and Denseness of Periodic Orbits for Linear Cellular Automata over Z_m B. Martin: A Geometrical Hierarchy of Graph via Cellular Automata K. Morita, K. Imai: Number-Conserving Reversible Cellular Automata and Their Computation-Universality C. Nichitiu, E. Remila: Simulations of graph automata K. Svozil: Is the world a machine? H. Umeo: Cellular Algorithms with 1-bit Inter-Cell Communications F. Reischle, Th. Worsch: Simulations between alternating CA, alternating TM and circuit families K. Sutner: Computation Theory of Cellular Automat

Universal computation and other capabilities of hybrid and continuous dynamical systems

Caption title.Includes bibliographical references (p. 25-27).Supported by the Army Research Office and the Center for Intelligent Control Systems. DAAL03-92-G-0164 DAAL03-92-G-0115Michael S. Branicky