Dissipative decoherence in the Grover algorithm

Using the methods of quantum trajectories we study effects of dissipative decoherence on the accuracy of the Grover quantum search algorithm. The dependence on the number of qubits and dissipation rate are determined and tested numerically with up to 16 qubits. As a result, our numerical and analytical studies give the universal law for decay of fidelity and probability of searched state which are induced by dissipative decoherence effects. This law is in agreement with the results obtained previously for quantum chaos algorithms. PACS. 03.67.Lx Quantum Computation – 03.65.Yz Decoherence; open systems; quantum statistical methods – 24.10.Cn Many-body theory Nowadays the quantum computing attracts a great interest of the scientific community [1]. The main reason of that is due to the fact that certain quantum algorithms allow to perform computations much faster than the usual classical algorithms. The famous example is the Shor algorithm which performs factorization of integers on a quantum computer exponentially faster than any known classical factorization algorithm [2]. However, at present there is no mathematical prove about efficiency of potentially possible classical algorithms that in this case gives certain restrictions on the comparative efficiency of classical and quantum algorithms. The situation is different in the case of the Grover quantum search algorithm [3]. Indeed, it has been proved that it is quadratically faster than any classical algorithm (see e.g. [1] and Refs. therein). In addition to the question of quantum algorithm efficiency it is also important to know what is the accuracy of a quantum algorithm in presence of realistic errors and imperfections. The accuracy can be characterized by the fidelity f [4] defined as a scalar product of the wave function of an ideal quantum algorithm and the wave function given by a realistic algorithm (see e.g. [1]). In general it is possible to distinguish three types (classes) of errors. The first class can be viewed as random unitary errors in rotational angles of quantum gates. This is the mostly studied case which has been also analyzed for various quantum algorithms with the help of numerical simulations with up to 28 qubits (see e.g. [5,6,7,8,9,10]). It has been shown that in such a case the fidelity decays exponentially with the number of quantum gates ng and with a rate γ which is proportional to a mean square of fluctuations in gate rotations. The second class of errors is related to static imperfections (static in time). They are produced by static residual couplings between qubits and static energy shifts of individual qubits which may generate many-body quantum chaos in a quantum computer hardware [11]. This second type of errors, e.g. static imperfections, gives a more rapid decay of fidelity as it has been shown in [12,10]. In the case of quantum algorithms for a complex dynamic these imperfections lead to the fidelity decrease described by a universal decay law given by the random matrix theory [10]. The two former classes are related to unitary errors. However, there is also the third class which corresponds to the case of nonunitary errors typical to the case of dissipative decoherence. This type of errors has been studied recently for the quantum baker map [13] and the quantum sawtooth map [14]. It has been shown that the exponential decay rate of fidelity is proportional to the number of qubits. The dissipative decoherence is treated in the frame of Lindblad equation for the density matrix [15]. A relatively large number of qubits can be reached by using the methods of quantum trajectories developed in [16,17,18,19,20,21,22]. The quantum algorithms studied in [7,8,9,10,12,13, 14] describe quantum and classical evolution of dynamical systems. However, it is also important to analyze the accuracy of realistic quantum computations for more standard algorithms, e.g. for the Grover algorithm. The effects of unitary errors of the first and second classes have been studied in [23] and [24] respectively. It has been shown that the accuracy of computation is qualitatively different in case of random errors in gates rotations [23] and in the case of static imperfections [24]. Thus it is important to analyze the effects of dissipative decoherence in the Grover algorithm to have a complete picture for this well known algorithm. For that aim we use the approach developed in [13,14]. 2 O.V.Zhirov and D.L.Shepelyansky: Dissipative decoherence in the Grover algorithm To study the effects of dissipative decoherence in the Grover algorithm we use the same notations as in [24]. Thus, the computational basis of a quantum register with N = 2q states ({|x〉}, x = 0, . . . , N − 1) is used for the algorithm itself. According to [1,3], the initial state |ψ0〉 = ∑N−1 x=0 |x〉/ √ N , is transfered to the state |ψ(t)〉 = Ĝ|ψ0〉 = sin ((t+ 1/2)ωG)|τ〉 + cos ((t+ 1/2)ωG)|η〉 , (1) after t applications of the Grover operator Ĝ. Here, ωG = 2 arcsin( √ 1/N) is the Grover frequency, |τ〉 is the search state and |η〉 = (0≤x<N) x 6=τ |x〉/ √ N − 1. Hence, the ideal algorithm gives a rotation in the 2D plane (|τ〉, |η〉). One iteration of the algorithm is given by the Grover operator Ĝ and can be implemented in ng = 12ntot − 42 elementary gates including one-qubit rotations, control-NOT and Toffolli gates as described in [24]. The implementation of all these gates requires an ancilla qubit so that the total number of qubits is ntot = nq + 1. To study the effects of dissipative decoherence on the accuracy of the Grover algorithm we follow the approach used in [13,14]. The evolution of the density operator ρ(t) of open system under weak Markovian noises is given by the master equation with Lindblad operators Lm (m = 1, · · · , ntot) [15]: ρ̇ = − i h̄ [Heffρ− ρH eff ] + ∑