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arXiv:quant-ph/0307142 · Physical Review A

Effect of unitary noise on Grover’s quantum search algorithm

Daniel Shapira, Shay Mozes, Ofer Biham

Published Apr 1, 2003• 49 citations
arXivHTMLPDF

Abstract

The effect of unitary noise on the performance of Grover's quantum search algorithm is studied. This type of noise may result from tiny fluctuations and drift in the parameters of the (quantum) components performing the computation. The resulting operations are still unitary, but not precisely those assumed in the design of the algorithm. Here we focus on the effect of such noise in the Hadamard gate W, which is an essential component in each iteration of the quantum search process. To this end W is replaced by a noisy Hadamard gate U. The parameters of U at each iteration are taken from an arbitrary probability distribution (e.g., a Gaussian distribution) and are characterized by their statistical moments around the parameters of W. For simplicity, we assume that the noise is unbiased and isotropic, namely, all noise variables in the parametrization we use have zero average and the same standard deviation $\ensuremath{\epsilon}.$ The noise terms at different calls to U are assumed to be uncorrelated. For a search space of size ${N=2}^{n}$ (where n is the number of qubits used to span this space) it is found that as long as $\ensuremath{\epsilon}<{O(n}^{\ensuremath{-}1/2}{N}^{\ensuremath{-}1/4})$ the algorithm maintains significant efficiency, while above this noise level its operation is hampered completely. It is also found that below this noise threshold, when the search fails, it is likely to provide a state that differs from the marked state by only a few bits. This feature can be used to search for the marked state by a classical postprocessing, even if the quantum search has failed, thus improving the success rate of the search process.

DOI
10.1103/physreva.67.042301

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The effect of unitary noise on the performance of Grover's quantum search algorithm is studied. This type of noise may result from tiny fluctuations and drift in the parameters of the (quantum) components performing the computation. The resulting operations are still unitary, but not precisely those assumed in the design of the algorithm. Here we focus on the effect of such noise in the Hadamard gate W, which is an essential component in each iteration of the quantum search process. To this end W is replaced by a noisy Hadamard gate U. The parameters of U at each iteration are taken from an arbitrary probability distribution (e.g., a Gaussian distribution) and are characterized by their statistical moments around the parameters of W. For simplicity, we assume that the noise is unbiased and isotropic, namely, all noise variables in the parametrization we use have zero average and the same standard deviation \ensuremathϵ.\ensuremath{\epsilon}.\ensuremathϵ. The noise terms at different calls to U are assumed to be uncorrelated. For a search space of size N=2n{N=2}^{n}N=2n (where n is the number of qubits used to span this space) it is found that as long as \ensuremathϵ<O(n\ensuremath−1/2N\ensuremath−1/4)\ensuremath{\epsilon}<{O(n}^{\ensuremath{-}1/2}{N}^{\ensuremath{-}1/4})\ensuremathϵ<O(n\ensuremath−1/2N\ensuremath−1/4) the algorithm maintains significant efficiency, while above this noise level its operation is hampered completely. It is also found that below this noise threshold, when the search fails, it is likely to provide a state that differs from the marked state by only a few bits. This feature can be used to search for the marked state by a classical postprocessing, even if the quantum search has failed, thus improving the success rate of the search process.
We analyze the robustness of Grover's quantum search algorithm performed by a quantum register under a possibly time-correlated noise acting locally on the qubits. We model the noise as originating from an arbitrary but fixed unitary evolution UUU of some noisy qubits. The noise can occur with some probability in the interval between any pair of consecutive noiseless Grover evolutions. Although each run of the algorithm is a unitary process, the noise model leads to decoherence when all possible runs are considered. We derive a set of unitary UUU's, called good noises, for which the success probability of the algorithm at any given time remains unchanged with varying nontrivial total number mmm of noisy qubits in the register. The result holds irrespective of the presence of any time correlations in the noise. We show that only when UUU is either of the Pauli matrices ${\ensuremath{\sigma}}_{x}$ and ${\ensuremath{\sigma}}_{z}$ (which give rise to mmm-qubit bit-flip and phase-damping channels, respectively, in the time-correlation-less case), the algorithm's success probability stays unchanged when increasing or decreasing mmm. In contrast, when UUU is the Pauli matrix ${\ensuremath{\sigma}}_{y}$ (giving rise to mmm-qubit bit-phase flip channel in the time-correlation-less case), the success probability at all times stays unaltered as long as the parity (even or odd) of the total number mmm remains the same. This asymmetry between the Pauli operators stems from the inherent symmetry-breaking existing within the Grover circuit. We further show that the positions of the noisy sites are irrelevant in the case of any of the Pauli noises. The results are illustrated in the cases of time-correlated and time-correlation-less noise. We find that the former case leads to a better performance of the noisy algorithm. We also discuss physical scenarios where our chosen noise model is of relevance.
Abstract Grover’s quantum search algorithm promises a quadratic speedup for unstructured search over its classical counterpart. But this advantage is affected by noise acting on the search space. Here, we show that a quantum switch can act as a resource to mitigate the effects of noise. In this scenario, the noise is modeled by a depolarizing channel, which coherently acts on the entire quantum register. We show that a quantum switch can significantly reduce the error in Grover’s search algorithm. We consider the success probability of finding the marked item as the sole quantifier of diminishing the effect of noise in the search space in the presence of quantum switch. We propose two frameworks for the application of quantum switches. In the first framework, we apply the superposition of channel’s orders in the form of a switch and do a post-selection at every iteration of the applications of the Grover operator. In the second framework, we delay this measurement and post-selection until the very end. The number of post selections is minimal in the second scenario, and hence the noise reduction can be attributed more to the presence of quantum switch. We illustrate with an example of significant advantage in the success probability of Grover’s algorithm using quantum switch.
Grover's quantum algorithm improves any classical search algorithm. We show how random Gaussian noise at each step of the algorithm can be modeled easily because of the exact recursion formulas available for computing the quantum amplitude in Grover's algorithm. We study the algorithm's intrinsic robustess when no quantum correction codes are used, and evaluate how much noise the algorithm can bear with, in terms of the size of the phone book and a desired probability of finding the correct result. The algorithm loses efficiency when noise is added, but does not slow down. We also study the maximal noise under which the iterated quantum algorithm is just as slow as the classical algorithm. In all cases, the width of the allowed noise scales with the size of the phone book as ${N}^{\ensuremath{-}2/3}.$
External influences in the form of quantum noise on Grover’s search algorithm are investigated. The study shows that the algorithm can be robust under such external dissipation. The effect of noise is described by a completely positive trace preserving map, acting on an unsorted N‐dimensional database made of projective density matrices. To evaluate the efficiency of the algorithm two figures of merit are used, the radial fidelity giving the projection between the Bloch vectors of target and final density operators, and the cosine of angular fidelity between the same vectors. Explicitly we prove that the resulting search positive map depends on x, the strength of the coupling to the environment, and that there are infinitely many x values, named good x’s and bad x’s, for which search is successful after O(N) queries, or fails, respectively . Next we study the information theoretic aspects of the dissipative quantum search algorithm. To this end we view the initial state as a pure qubit signal entering through a cascade of quantum channels, each channel described by a completely positive map, and corresponds to the iterations of the algorithm. This map is identical to the dissipative search map of the algorithm for which the parameter x now characterizes the influence of the noisy channel exercised upon the incoming quantum signal. The process of searching is understood as transmission of the initial signal/item through the composite channel made by O(N) elementary channels. The quality of this transmission corresponds to the fidelity of targeting the wanted item in the quantum database, and is studied in terms of the majorization/quantum entropy of the input‐output density matrix, and by the entropy exchanged between the qubit signal/item and an external environment.
We briefly introduced some of our recent work related to the phase matching condition in quantum searching algorithms and the improved Grover algorithm. When one replaces the two phase inversions in the Grover algorithm with arbitrary phase rotations, the modified algorithm usually fails in searching the marked state unless a phase matching condition is satisfied between the two phases. The Grover algorithm is not 100% in success rate, an improved Grover algorithm with zero-failure rate is given by replacing the phase inversions with angles that depends on the size of the database. Other aspects of the Grover algorithm such as the SO(3) picture of quantum searching, the dominant gate imperfections in the Grover algorithm are also mentioned.
The recursion equation analysis of Grover's quantum search algorithm presented by Biham et al. [Phys. Rev. A 60, 2742 (1999)] is generalized. It is applied to the large class of Grover-type algorithms in which the Hadamard transform is replaced by any other unitary transformation and the phase inversion is replaced by a rotation by an arbitrary angle. The time evolution of the amplitudes of the marked and unmarked states, for any initial complex amplitude distribution, is expressed using first-order linear difference equations. These equations are solved exactly. The solution provides the number of iterations T after which the probability of finding a marked state upon measurement is the highest, as well as the value of this probability, Pmax.{P}_{\mathrm{max}}.Pmax​. Both T and Pmax{P}_{\mathrm{max}}Pmax​ are found to depend on the averages and variances of the initial amplitude distributions of the marked and unmarked states, but not on higher moments.