NOISE EFFECTS IN THE QUANTUM SEARCH ALGORITHM FROM ...

More documents

Recommendations

Info

$Positive stable realizations of fractional continuous-time linear systems$

496 P. Gawron et al. where s = 1 ∑ α i . (15) N A k-fold application of Grover’s iteration G to the initial state |s〉 leads to (Bouwmeester et al., 2000; Bugajski, 2001) G k |s〉 = α k ∑ x≠x 0 |x〉 + β k |x 0 〉, (16) with real coefficients 1 α k = √ cos (2k +1)θ, β k =sin(2k +1)θ, N − 1 (17) where θ is an angle that fulfils the relation sin(θ) = √ 1 . (18) N Therefore the coefficients α k ,β k are periodic functions of k. After a series of iterations, β k rises. The influence of the marked state |x 0 〉 on the state of the register results in the evolution of the initial state |s〉 towards the marked state. β k attains its maximum after approximately π 4 √ N steps. The number of steps needed to transfer the initial state towards the marked state is of order O( √ N). Inthe classical case the number of steps is of order O(N). 4.2.6. Measurement. The last step of Grover’s algorithm is a von Neumann measurement. The probability of obtaining the proper result is |β k | 2 . 5. Noise model The above discussion of the quantum search algorithm has been conducted using the state vector formalism. In order to incorporate noise into the quantum computation model, we have to make use of density operators which define the quantum state in the most general way. 5.1. Quantum noise. Microscopic systems that are governed by the laws of quantum mechanics are hard to control and, at the same time, to separate from the environment. The interaction with the environment introduces noise into the quantum system. Therefore any future quantum computer will also be prone to noise. One-qubit noise. There are several one-parameter families of one-qubit noisy channels that are typically discussed in the literature (Nielsen and Chuang, 1999). We present them briefly below. Depolarising channel. This is a bi-stochastic channel that transforms any state into a maximally mixed state with i a given probability α. The family of channels can be defined using a four-element set of Kraus operators { √ √ √ √1 α α α − αI, 3 σ x, 3 σ y, z} 3 σ , where [ ] [ ] 1 0 0 1 I = , σ 0 1 x = , 1 0 [ ] [ ] 0 −i 1 0 σ y = , σ i 0 z = 0 −1 are Pauli matrices. Amplitude damping. The amplitude damping channel transforms |1〉 into |0〉 with a given probability α. The state |0〉 remains unchanged. The set of Kraus operators is {[ 1 0 0 √ 1 − α ] , [ 0 √ α 0 0 ]} . Phase damping. Phase damping in a quantum phenomenon describes the loss of quantum information without the loss of energy. It is described by the following set of Kraus operators: {[ ] [ ]} 1 √ 0 0 0 , 0 1 − α 0 √ . α Bit flip. The bit flip family of channels is the quantum version of the classical binary symmetric channel. The action of the channel might be interpreted in the following way: it flips the state of a qubit from |0〉 to |1〉 and from |1〉 to |0〉 with probability α. Kraus operators for this family of channels consist of a matrix proportional to the identity and a matrix proportional to the negation gate, {√ √ } 1 − αI, ασx . Phase flip. The phase flip channel acts similarly to the bit flip channel with the distinction that a σ z gate is applied randomly to the qubit {√ √ } 1 − αI, ασz . Bit-phase flip. The bit-phase flip channel may be considered a joint application of bit and phase flip gates to a qubit. Its Kraus operators are as follows: {√ √ } 1 − αI, ασy . In all of the above families of channels, the real parameter α ∈ [0, 1] can be interpreted as the amount of noise introduced by the channel.
Noise effects in the quantum search algorithm from the viewpoint of computational complexity 497 |0〉 |0〉 . |0〉 H ⊗n Iterate √ N π 4 times Diffusion Oracle |x〉→(−1) f(x) |x〉 H ⊗n |0〉→|0〉 |x〉→−|x〉 H ⊗n Noise ρ t+1=Φ(ρ t) forx>0 . Fig. 1. Circuit for Grover’s algorithm extended with a non-unitary noisy channel. Multiqubit local channels. Our goal is to extend the noise acting on distinct qubits to the entire registers. We assume that the appearance of an error on a given qubit is independent of an error appearing on any other qubits. In order to apply noise operators to multiple qubits, we form a new set of Kraus operators acting on a larger Hilbert space. We assume that we have the set of n one-qubit Kraus operators {e k } n k=1 . We construct the new set of nN operators {E k } nN k=1 that act on a Hilbert space of dimension 2 N by applying the following formula: where {E k } = ⋃ {e i1 ⊗ e i2 ⊗ ...⊗ e iN }, (19) I I = {i 1 } n i 1=1 ×{i 2 } n i 2=1 × ...×{i N } n i N =1. One should note that the extended channel Φ(ρ) = ∑ k E kρE † k is by definition local (Bengtsson and Życzkowski, 2006). By applying Eqn. (19) to the sets of operators listed above, we obtain one-parameter families of local noisy channels, which we use in further investigations. 5.2. Application of noise to the algorithm. In order to simulate noisy behaviour of the system implementing the algorithm, we apply a noisy channel after every Grover iteration. The evolution of the system is described by the following procedure, which is graphically depicted in Fig. 1: 1. Prepare the system in state ρ 0 := |0 ⊗n 〉〈0 ⊗n |. 2. ρ := H ⊗n ρ 0 H ⊗n† 3. ⌊ π 4 √ N⌋ times do: (a) apply Grover’s iteration ρ := GρG † , (b) apply noise ρ := Φ(ρ). 4. Perform an orthogonal measurement in the computational basis. The probability of finding the sought element ξ is p = 〈ξ|ρ|ξ〉. This approach simplifies the physical reality, but it is sufficient to study the robustness of the algorithm in the presence of noise. In order to study the discussed problem, we make use of numerical simulations. Therefore some simplification is necessary as the size of the problem grows exponentially fast with the number of qubits. The tool we use is quantum-octave (Gawron et al., 2010), a library that contains functions for simulation and analysis of quantum processes. In our model we assume that it is easy to verify the correctness of the quantum search algorithm. It is an assumption usually made in the complexity analysis of search algorithms. 6. Analysis of the influence of noise on the efficiency of the algorithm An interesting question arises: “What is the maximal amount of noise for which Grover’s algorithm is more efficient than any classical search algorithm” Grover’s algorithm is probabilistic, therefore we cannot expect to obtain a valid outcome with certainty. We assume that if the algorithm fails in a given run we will rerun it. There is a certain number of reruns for which the quantum algorithm is worse than the classical. We are interested only in the statistical behaviour of algorithm and calculate the mean value of repetitions. Let k = ⌊ N 2 / √ π 4 N⌋ be the maximal number of single runs of Grover’s algorithm for which quantum search is faster than the classical one. We compute the minimal value of success probability p min of a single run of Grover’s algorithm for which we obtain a valid result with confidence C, { p min =min 1 − (1 − p) k ≥ C } . (20) p Numerically obtained values of p min for the confidence level C =0.95 for Grover’s algorithm are listed in Table 1.
Page 1 and 2: Int. J. Appl. Math. Comput. Sci., 2
Page 3: Noise effects in the quantum search
Page 7: Noise effects in the quantum search

NOISE EFFECTS IN THE QUANTUM SEARCH ALGORITHM FROM ...

Create successful ePaper yourself

Delete template?

Save as template?