Quantum Branching Programs and Space-Bounded Nonuniform Quantum Complexity

In this paper, the space complexity of nonuniform quantum computations is investigated. The model chosen for this are quantum branching programs, which provide a graphic description of sequential quantum algorithms. In the first part of the paper, simulations between quantum branching programs and nonuniform quantum Turing machines are presented which allow to transfer lower and upper bound results between the two models. In the second part of the paper, different variants of quantum OBDDs are compared with their deterministic and randomized counterparts. In the third part, quantum branching programs are considered where the performed unitary operation may depend on the result of a previous measurement. For this model a simulation of randomized OBDDs and exponential lower bounds are presented.Comment: 45 pages, 3 Postscript figures. Proofs rearranged, typos correcte

Reversible Simulation of Irreversible Computation by Pebble Games

Reversible simulation of irreversible algorithms is analyzed in the stylized form of a `reversible' pebble game. While such simulations incur little overhead in additional computation time, they use a large amount of additional memory space during the computation. The reacheable reversible simulation instantaneous descriptions (pebble configurations) are characterized completely. As a corollary we obtain the reversible simulation by Bennett and that among all simulations that can be modelled by the pebble game, Bennett's simulation is optimal in that it uses the least auxiliary space for the greatest number of simulated steps. One can reduce the auxiliary storage overhead incurred by the reversible simulation at the cost of allowing limited erasing leading to an irreversibility-space tradeoff. We show that in this resource-bounded setting the limited erasing needs to be performed at precise instants during the simulation. We show that the reversible simulation can be modified so that it is applicable also when the simulated computation time is unknown.Comment: 11 pages, Latex, Submitted to Physica

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

Reversibility and Adiabatic Computation: Trading Time and Space for Energy

Future miniaturization and mobilization of computing devices requires energy parsimonious `adiabatic' computation. This is contingent on logical reversibility of computation. An example is the idea of quantum computations which are reversible except for the irreversible observation steps. We propose to study quantitatively the exchange of computational resources like time and space for irreversibility in computations. Reversible simulations of irreversible computations are memory intensive. Such (polynomial time) simulations are analysed here in terms of `reversible' pebble games. We show that Bennett's pebbling strategy uses least additional space for the greatest number of simulated steps. We derive a trade-off for storage space versus irreversible erasure. Next we consider reversible computation itself. An alternative proof is provided for the precise expression of the ultimate irreversibility cost of an otherwise reversible computation without restrictions on time and space use. A time-irreversibility trade-off hierarchy in the exponential time region is exhibited. Finally, extreme time-irreversibility trade-offs for reversible computations in the thoroughly unrealistic range of computable versus noncomputable time-bounds are given.Comment: 30 pages, Latex. Lemma 2.3 should be replaced by the slightly better ``There is a winning strategy with $n+2$ pebbles and $m-1$ erasures for pebble games $G$ with $T_G= m2^n$, for all $m \geq 1$'' with appropriate further changes (as pointed out by Wim van Dam). This and further work on reversible simulations as in Section 2 appears in quant-ph/970300

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