36,703 research outputs found

    Power of Uninitialized Qubits in Shallow Quantum Circuits

    Get PDF
    We study the computational power of shallow quantum circuits with O(log n) initialized and n^{O(1)} uninitialized ancillary qubits, where n is the input length and the initial state of the uninitialized ancillary qubits is arbitrary. First, we show that such a circuit can compute any symmetric function on n bits that is classically computable in polynomial time. Then, we regard such a circuit as an oracle and show that a polynomial-time classical algorithm with the oracle can estimate the elements of any unitary matrix corresponding to a constant-depth quantum circuit on n qubits. Since it seems unlikely that these tasks can be done with only O(log n) initialized ancillary qubits, our results give evidences that adding uninitialized ancillary qubits increases the computational power of shallow quantum circuits with only O(log n) initialized ancillary qubits. Lastly, to understand the limitations of uninitialized ancillary qubits, we focus on near-logarithmic-depth quantum circuits with them and show the impossibility of computing the parity function on n bits

    Two-Level Rectilinear Steiner Trees

    Get PDF
    Given a set PP of terminals in the plane and a partition of PP into kk subsets P1,...,PkP_1, ..., P_k, a two-level rectilinear Steiner tree consists of a rectilinear Steiner tree TiT_i connecting the terminals in each set PiP_i (i=1,...,ki=1,...,k) and a top-level tree TtopT_{top} connecting the trees T1,...,TkT_1, ..., T_k. The goal is to minimize the total length of all trees. This problem arises naturally in the design of low-power physical implementations of parity functions on a computer chip. For bounded kk we present a polynomial time approximation scheme (PTAS) that is based on Arora's PTAS for rectilinear Steiner trees after lifting each partition into an extra dimension. For the general case we propose an algorithm that predetermines a connection point for each TiT_i and TtopT_{top} (i=1,...,ki=1,...,k). Then, we apply any approximation algorithm for minimum rectilinear Steiner trees in the plane to compute each TiT_i and TtopT_{top} independently. This gives us a 2.372.37-factor approximation with a running time of O(PlogP)\mathcal{O}(|P|\log|P|) suitable for fast practical computations. The approximation factor reduces to 1.631.63 by applying Arora's approximation scheme in the plane

    An Ordered Approach to Solving Parity Games in Quasi Polynomial Time and Quasi Linear Space

    Get PDF
    Parity games play an important role in model checking and synthesis. In their paper, Calude et al. have shown that these games can be solved in quasi-polynomial time. We show that their algorithm can be implemented efficiently: we use their data structure as a progress measure, allowing for a backward implementation instead of a complete unravelling of the game. To achieve this, a number of changes have to be made to their techniques, where the main one is to add power to the antagonistic player that allows for determining her rational move without changing the outcome of the game. We provide a first implementation for a quasi-polynomial algorithm, test it on small examples, and provide a number of side results, including minor algorithmic improvements, a quasi bi-linear complexity in the number of states and edges for a fixed number of colours, and matching lower bounds for the algorithm of Calude et al

    On the time complexity of 2-tag systems and small universal Turing machines

    Get PDF
    We show that 2-tag systems efficiently simulate Turing machines. As a corollary we find that the small universal Turing machines of Rogozhin, Minsky and others simulate Turing machines in polynomial time. This is an exponential improvement on the previously known simulation time overhead and improves a forty year old result in the area of small universal Turing machines.Comment: Slightly expanded and updated from conference versio

    A Discrete Logarithm-based Approach to Compute Low-Weight Multiples of Binary Polynomials

    Full text link
    Being able to compute efficiently a low-weight multiple of a given binary polynomial is often a key ingredient of correlation attacks to LFSR-based stream ciphers. The best known general purpose algorithm is based on the generalized birthday problem. We describe an alternative approach which is based on discrete logarithms and has much lower memory complexity requirements with a comparable time complexity.Comment: 12 page
    corecore