Here we present our most recent interesting publications with abstracts and comments

• First observation of a second order quantum phase transition in algorithms of quantum computing:
  Phase diagram for the Grover algorithm with static imperfections
  A.A. Pomeransky, O.V. Zhirov and D.L. Shepelyansky
  Eur. Phys. J. D31, 131-135 (2004)

  Abstract.We study effects of static inter-qubit interactions on the stability of the Grover quantum search algorithm. Our numerical and analytical results show existence of regular and chaotic phases depending on the imperfection strength $\varepsilon$. The critical border $\varepsilon_c$ between two phases drops polynomially with the number of qubits $n_q$ as $\varepsilon_c \sim n_q^{-3/2}$. In the regular phase $(\varepsilon < \varepsilon_c)$ the algorithm remains robust against imperfections showing the efficiency gain $\varepsilon_c / \varepsilon$ for $\varepsilon \gtrsim 2^{-n_q/2}$. In the chaotic phase $(\varepsilon > \varepsilon_c)$ the algorithm is completely destroyed.

• First comprehensive study of Frenkel-Kontorova chain in quantum regime, with clear picture of elementary quantum excitations of the system and observation of the quantum phase transition:
  Quantum phase transition in the Frenkel-Kontorova chain: from pinned instanton glass to sliding phonon gas,
  O.V. Zhirov, G. Casati and D.L. Shepelyansky
  Phys. Rev. E67,056209(2003).

  Abstract: We study analytically and numerically the one-dimensional quantum Frenkel-Kontorova chain in the regime when the classical model is located in the pinned phase characterized by the gaped phonon excitations and devil's staircase. By extensive quantum Monte Carlo simulations we show that for the effective Plank constant $\hbar$ smaller than the critical value $\hbar_c$ the quantum chain is in the pinned instanton glass phase. In this phase the elementary excitations have two branches: phonons, separated from zero energy by a finite gap, and instantons which have an exponentially small excitation energy. At $\hbar=\hbar_c$ the quantum phase transition takes place and for $\hbar>\hbar_c$ the pinned instanton glass is transformed into the sliding phonon gas with gapless phonon excitations. This transition is accompanied by the divergence of the spatial correlation length and appearence of sliding modes at $\hbar>\hbar_c$.