This paper studies the computational power of quantum computers to explore as to whether they can recognize properties which are in nondeterministic polynomial-time class (NP) and beyond. To study the computational power, we use the Feynman's path integral (FPI) formulation of quantum mechanics. From a computational point of view the Feynman's path integral computes a quantum dynamical analogue of the k-ary relation computed by an Alternating Turing machine (ATM) using AND-OR Parallelism. Hence, if we can find a suitable mapping function between an instance of a mathematical problem and the corresponding interference problem, using suitable potential functions for which FPI can be integrated exactly, the computational power of a quantum computer can be bounded to that of an alternating Turing machine that can solve problems in NP (e.g, factorization problem) and in polynomial space. Unfortunately, FPI is exactly integrable only for a few problems (e.g., the harmonic oscillator) involving quadratic potentials; otherwise, they may be only approximately computable or noncomputable. This means we cannot in general solve all quantum dynamical problems exactly except for those special cases of quadratic potentials, e.g., harmonic oscillator. Since there is a one to one correspondence between the quantum mechanical problems that can be analytically solved and the path integrals that can be exactly evaluated, we can say that the noncomputability of FPI implies quantum unsolvability. This is the analogue of classical unsolvability. The Feynman's path graph can be considered as a semantic parse graph for the quantum mechanical sentence. It provides a semantic valuation function of the terminal sentence based on probability amplitudes to disambiguate a given quantum description and obtain an interpretation in a linear time. In Feynman's path integral, the kernels are partially ordered over time (different alternate paths acting concurrently at the same time) and multiplied. The semantic valuation is computable only if the FPI is computable. Thus both the expressive power and complexity aspects quantum computing are mirrored by the exact and efficient integrability of FPI.This paper studies the computational power of quantum computers to explore as to whether they can recognize properties which are in nondeterministic polynomial-time class (NP) and beyond. To study the computational power, we use the Feynman's path integral (FPI) formulation of quantum mechanics. From a computational point of view the Feynman's path integral computes a quantum dynamical analogue of the k-ary relation computed by an Alternating Turing machine (ATM) using AND-OR Parallelism. Hence, if we can find a suitable mapping function between an instance of a mathematical problem and the corresponding interference problem, using suitable potential functions for which FPI can be integrated exactly, the computational power of a quantum computer can be bounded to that of an alternating Turing machine that can solve problems in NP (e.g, factorization problem) and in polynomial space. Unfortunately, FPI is exactly integrable only for a few problems (e.g., the harmonic oscillator) involving quadratic potentials; otherwise, they may be only approximately computable or noncomputable. This means we cannot in general solve all quantum dynamical problems exactly except for those special cases of quadratic potentials, e.g., harmonic oscillator. Since there is a one to one correspondence between the quantum mechanical problems that can be analytically solved and the path integrals that can be exactly evaluated, we can say that the noncomputability of FPI implies quantum unsolvability. This is the analogue of classical unsolvability. The Feynman's path graph can be considered as a semantic parse graph for the quantum mechanical sentence. It provides a semantic valuation function of the terminal sentence based on probability amplitudes to disambiguate a given quantum description and obtain an interpretation in a linear time. In Feynman's path integral, the kernels are partially ordered over time (different alternate paths acting concurrently at the same time) and multiplied. The semantic valuation is computable only if the FPI is computable. Thus both the expressive power and complexity aspects quantum computing are mirrored by the exact and efficient integrability of FPI.

International Journal of Modern Physics CMar 1, 19987 citations

Quantum Integrability

L. E. Reichl

Institute for Nonlinear ScienceJan 1, 19921 citation

On classical and quantum integrability

Hans Jürgen Korsch

Physics Letters AJun 1, 198215 citations

On the quantum complexity of integration of a function with unknown singularity

Maciej Goćwin

In this paper we study the quantum complexity of the integration of a function with an unknown singularity. We assume that the function has $r$ continuous derivatives, with the derivative of order $r$ being H\"older continuous with the exponent $\rho$ on the whole integration interval except the one singular point. We show that the $\ve$-complexity of this problem is of order $\ve^{-1/(r+\rho+1)}$. Since the classical deterministic complexity of this problem is $\ve^{-1/(r+\rho)}$, quantum computers give a speed-up for this problem for all values of parameters $r$ and $\rho$ .In this paper we study the quantum complexity of the integration of a function with an unknown singularity. We assume that the function has $r$ continuous derivatives, with the derivative of order $r$ being H\"older continuous with the exponent $\rho$ on the whole integration interval except the one singular point. We show that the $\ve$-complexity of this problem is of order $\ve^{-1/(r+\rho+1)}$. Since the classical deterministic complexity of this problem is $\ve^{-1/(r+\rho)}$, quantum computers give a speed-up for this problem for all values of parameters $r$ and $\rho$.

Quantum Information and ComputationMay 1, 20230 citations

Dynamical deviation from quantum integrability

Peter Barmettler, Vladimir Gritsev

Jan 21, 20120 citations

In this paper we study the quantum complexity of the integration of a function with an unknown singularity. We assume that the function has

r

continuous derivatives, with the derivative of order

r

being H\"older continuous with the exponent

\rho

on the whole integration interval except the one singular point. We show that the $\ve$-complexity of this problem is of order $\ve^{-1/(r+\rho+1)}$. Since the classical deterministic complexity of this problem is $\ve^{-1/(r+\rho)}$, quantum computers give a speed-up for this problem for all values of parameters

r

and

\rho