Quantum Computing of Quantum Chaos and Imperfection Effects

Abstract

We study numerically the imperfection effects in the quantum computing of the kicked rotator model in the regime of quantum chaos. It is shown that there are two types of physical characteristics: for one of them the quantum computation errors grow exponentially with the number of qubits in the computer while for the other the growth is polynomial. Certain similarity between classical and quantum computing errors is also discussed.PACS numbers: 05.45. Mt, 03.67.Lx, 24.10.Cn A great interest to quantum computers has been generated recently by prominent theoretical results and impressive experimental progress which allowed to realise operations with a few qubits (see [1] for a review). The most striking theoretical advantage is an enormous parallelism of quantum computing. Using the Shor algorithm [2] the factorization of large numbers can be done exponentially faster on a quantum computer than by any known algorithm on a classical computer. Also a search of an item in a long list is much faster on a quantum computer as shown by Grover [3]. Experimentally a variety of physical systems is considered for realisation of one qubit, viewed as a two level system, and controlled coupling between a few qubits that forms the basis for realisation of a quantum computer. These systems include ion traps [4,5], nuclear magnetic resonance systems [6], nuclear spins with interaction controlled electronically [7,8] or by laser pulses [9], electrons in quantum dots [10], Cooper pair boxes [11], optical lattices [12] and electrons floating on liquid helium [13]. As a result, a two-qubit gate has been experimentally realized with cold ions [14], and the Grover algorithm has been performed for three qubits made from nuclear spins in a molecule [15].It is clear that in any realistic realisation of a quantum computer a special attention should be paid to the imperfection effects. Indeed, the imperfections are always present and they in principle may seriously modify the computation results comparing to the algorithms based on ideal qubit operations. At present the imperfection effects are tested in the numerical simulations of the quantum Fourier transform (QFT) [4] and the Shor algorithm factorization of 15 [16,17]. The obtained results look to be promising for the quantum computing indicating that a small amount of noise does not change strongly the computations [4] even if in some cases only rather low level of noise is tolerable [17]. However, for different reasons these studies do not allow to obtain analytical estimates of a tolerable imperfection level for a large number of qubits n. Indeed, the Shor algorithm is rather complicated and the capability of nowadays computers become too restrictive [16,17]. Recently the effects of static imperfections on the the stability of quantum computer hardware have been determined for a broad regime of parameters and it has been shown that the quantum hardware is sufficiently robust [18]. However, these results cannot be directly generalised for a specific quantum algorithm operating in ti...

Quantitative model for the effective decoherence of a quantum computer with imperfect unitary operations

“…Grover's algorithm [12] was studied in [13] with the help of a phenomenological model of probability diffusion, but the dependence of the model parameters on the number of qubits and ǫ was not determined. Conclusions similar to those in [7] were reached also in [14], where again the test algorithm was Grover's one 1 , and in a number of articles about the simulation of quantum chaotic maps [9,15,16,17,18]. These results show that the timescale for the degradation of the fidelity of a q.c.…”

“…Although there are other quantities which are exponentially sensitive to the number of qubits [15,18,19]. 3 Satisfying this condition corresponds to tuning the gate implementation in order to eliminate the systematic part of the error.…”

Imperfection Effects for Multiple Applications of the Quantum Wavelet Transform

“…Here n q can be viewed as the number of qubits (two-level quantum systems) of which a quantum computer is built. Apart from Shor's algorithm, the QFT finds a number of various applications in quantum computation, including the simulation of quantum chaos models showing rich and complex dynamics [5,6,7]. The sensitivity of the QFT to imperfections was tested in numerical simulations and the time-scales for reliable computation of the algorithm were established [6,7,8,9].…”

“…Apart from Shor's algorithm, the QFT finds a number of various applications in quantum computation, including the simulation of quantum chaos models showing rich and complex dynamics [5,6,7]. The sensitivity of the QFT to imperfections was tested in numerical simulations and the time-scales for reliable computation of the algorithm were established [6,7,8,9].A few years after the discovery of the QFT algorithm, it has been shown that certain WT can also be implemented on a quantum computer in a polynomial number of quantum gates [10,11,12]. In fact, explicit quantum circuits were developed for the most popular discrete WT, namely the 4-coefficient Daubechies WT (D (4) ) and the Haar WT, both for pyramidal and packet algorithms [10,11,12].…”

Quantum Computing and Information Extraction for Dynamical Quantum Systems

“…Furthermore, we emphasize that the quantum computation of quantities like dynamical localization or fidelity is a demanding testing ground for quantum computers. In the first case, we want to simulate dynamical localization, a purely quantum phenomena which is quite fragile in the presence of noise [30,31]; in the latter case, fidelity is computed as a result of a sophisticated many-qubit Ramsey-type interference experiment. Therefore the computation of these quantities appears to be a relevant test for quantum processors operating in the presence of decoherence and imperfection effects.…”