Abstract
Imperfections and noise in realistic quantum computers may seriously affect the accuracy of quantum algorithms. In this article we explore the impact of static imperfections on quantum entanglement as well as non-entangled quantum correlations in Grover’s search algorithm. Using the metrics of concurrence and geometric quantum discord, we show that both the evolution of entanglement and quantum discord in Grover algorithm can be restrained with the increasing strength of static imperfections. For very weak imperfections, the quantum entanglement and discord exhibit periodic behavior, while the periodicity will most certainly be destroyed with stronger imperfections. Moreover, entanglement sudden death may occur when the strength of static imperfections is greater than a certain threshold.
1 Introduction
Quantum algorithms, both in certainty and probability, have great advantages over classical algorithm in the potential of computing. It is believed that quantum algorithm provides speedup depending largely on quantum entanglement and other quantum phenomenon. Entanglement - a special kind of quantum state, is used to describe a compound system which contains two or more members in the quantum system. However, recent studies have demonstrated that there are other quantum correlations which cannot be captured by entanglement [1, 2]. In order to quantify and describe these non-entangled quantum correlations, quantum discord was proposed by Zurek et al [3–5].
In realistic situations, a quantum computing system is not an ideally closed system and thus external environments will have an impact on it. Even if a quantum computing system is isolated, there will be unavoidable interqubit couplings in the hardware of quantum computers - which is called static imperfections [6, 7]. These static imperfections may lead to many-body quantum chaos, which strongly modifies the hardware properties of realistic quantum computers. Therefore, studying the behavior of quantum entanglement and quantum discord under static imperfections has far-reaching consequences.
Grover’s quantum search algorithm plays an important role in the advance of quantum computation and quantum information because it provides a proof that quantum computers are faster than classical ones in database searching. The algorithm uses
As quantum discord is a novel and not yet fully understood measure of quantum correlations, we will mainly focus on the evolution of it as well as quantum entanglement in the disturbed Grover search algorithm. Specifically, we will examine the behaviors of entanglement and quantum discord in the presence of static interqubit couplings. The paper is organized as follows. In Sec.2, we briefly review the Grover search algorithm. In Sec.3, we introduce quantum entanglement, quantum correlations and their measurements. The effects of static imperfections on quantum entanglement and quantum discord in Grover search algorithm are discussed in Sec.4. We conclude the paper with some discussions in Sec.5.
2 The Grover Search Algorithm
The search problem for an unstructured database is a fundamental and practical problem which appears in many different fields. It was first shown by Grover that searching a database using quantum mechanics can be substantially faster than any classical search of unsorted data.
Grover’s search algorithm begins with a quantum register of n qubits, where n is the number of qubits necessary to represent the search space of size N = 2n, all initialized to |0〉. By applying the Hadamard operation H⊗n, the qubits are put into an equal superposition state |φ0〉:
We then repeat the Grover transformation a number of times in order to enlarge the probability amplitude of the searched state |τ〉 while restraining the probability amplitudes of other states. The Grover transformation operator G is composed of two operators: G=DO. Here the quantum oracle O is a quantum black-box that will rotate the phase of the state by radians if the system is indeed in the searched state while do nothing otherwise. The diffusion operator D is independent of the searched state: Dii = –1 + 2/N and Dij = 2/N(i ≠ j).
For the equal superposition state |φ0〉, t applications of the Grover transformation operator G give
where the Grover frequency
3 Quantum Entanglement and Quantum Discord
3.1 Quantum Entanglement and Its Measurement
Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. Entangled states display strong correlations that are impossible in classical mechanics. It is believed to be an essential resource for quantum computation, quantum communication etc. To quantify entanglement, a number of measures have been proposed, such as entanglement of formation, entanglement of distillation, entanglement cost, etc [15, 16]. Nevertheless, most proposed measures of entanglement involve extremizations which are difficult to handle analytically. For that reason we choose concurrence to investigate the behavior of entanglement because it is well justified and mathematically tractable.
Suppose ρAB to be the density matrix of a quantum system that is composed of two qubits, qA and qB. The concurrence of the two-qubit density matrix C(ρAB) is computed as:
where the λi’s are the square root of the eigenvalues of
3.2 Quantum Discord
If measurement of a single quantum subsystem alters global correlations, then the subsystems are quantum correlated. Entanglement is not the only kind of genuine quantum correlation. Compared with entanglement in composite quantum systems, quantum correlations are more general and fundamental. Recent studies have demonstrated that quantum entanglement is just one form of quantum correlation and there are other forms of correlations even in separable states [17, 18].
The use of quantum discord as a measure of quantum correlations has attracted increasing interest. Here geometric quantum discord is used to describe and quantify non-entangled quantum correlation. This measure is significant in capturing quantum correlations from a geometric perspective. The geometric measure of quantum discord is defined as [19, 20]
where the minimum is over the set of zero-discord states and the geometric quantity ‖ρAB – χAB‖ = Tr(ρAB – χAB)2 is the squared Hilbert-Schmidt distance between the Hermitian operators ρAB and χAB.
Specifically, the density matrix ρAB for a two-qubit state can be written as
where I is the identity matrix, xi = Trρ(σi ⊗ I) and yi = Trρ(I ⊗σi) are the components of the local Bloch vectors, σi(i = 1, 2, 3) are the Pauli spin matrices {σx, σy, σz}, wij = Trρ(σi ⊗ σj). As a consequence, the geometric quantum discord of ρAB is evaluated as [21]
where x = (x1, x2, x3)TW is a matrix consisted by wij and λmax is the largest eigenvalue of the matrix K = xxT + WWT (the superscript T denotes the transpose). The geometric measure of quantum discord for an arbitrary state has been evaluated in Ref. [20] and an explicit and tight lower bound has been obtained.
4 The Effect of Static Imperfections
To study the effects of static imperfections on quantum algorithms, we use the standard generic quantum computer model proposed in Refs. [6, 7, 9]. The quantum computer model is defined as a two-dimensional lattice of qubits with static imperfections in the individual qubit energies and residual short-range interqubit couplings. The Hamiltonian of this model is
where
In our numerical experiments, we first defined a 3 × 3 square lattice which contains 9 qubits. As the implementation of Grover algorithm with the elementary quantum gates requires an ancillary qubit, the size of the search space becomes N = 28. As a result, we can find the searched state after
According to Eq. (3) and Eq. (5), we have numerically calculated the concurrence and the geometric quantum discord with iterations of Grover search algorithm in the presence of imperfections. The experimental results of the concurrence C and the geometric quantum discord DG for the 3 × 3 lattice model are shown in Fig. 1. For the 4 × 3 lattice model they are shown in Fig. 2. In Fig. 1 and 2, the concurrence C exhibits periodical behavior when the strength of the static imperfections become very weak. Furthermore, the period is almost equal to the number of iterations that we will find the searched state |τ〉 with highest probability (it is approximately 12 in Fig. 1 and around 35 in Fig. 2). As the strength of the imperfections increase, the amplitude decreases and the periodical behavior of entanglement is destroyed gradually. Hence the quantum search algorithm is unable to function properly.
Although the geometric quantum discord DG exhibit similar dynamical behaviors with concurrence C for small or moderate imperfections, they show different behaviors for large imperfections. When the strength of static imperfections exceeds a certain value, we see the sudden death of entanglement (as depicted in the left bottom panels of Fig. 1 and Fig. 2), while this phenomenon cannot be observed in dynamics of geometric quantum discord. Our numerical results suggests that quantum discord in quantum algorithms is more robust to static imperfections than quantum entanglement.
5 Conclusions
Quantum entanglement and non-entangled quantum correlations are valuable resources in quantum information processing, but at the same time, they are very fragile. In this paper we have numerically studied the dynamics of quantum entanglement and quantum discord in quantum computing systems with the static interqubit couplings. Static imperfections in quantum computing systems can break quantum correlations and seriously destroy operability, and thus lead to unpredictable computation results. Although the evolutions of both entanglement and quantum discord will be suppressed by static imperfections in the same way, quantum discord is slightly more robust against imperfections in comparison with quantum entanglement.
Acknowledgement
The authors wish to express their gratitude to the Advanced Analysis and Computation Center of CUMT for the award of CPU hours to accomplish this work. This work is supported by the Fundamental Research Funds for the Central Universities (Grant No. 2015QNA67).
References
[1] Horodecki R., Horodecki P., Horodecki M., Horodecki K., Quantum entanglement, Rev. Mod. Phys., 2009, 81, 865–942.10.1103/RevModPhys.81.865Search in Google Scholar
[2] Datta A., Shaji A., Quantum discord and quantum computing - an appraisal, Int. J. Quantum Inf., 2011, 9, 1787–1805.10.1142/S0219749911008416Search in Google Scholar
[3] Zurek W.H., Einselection and decoherence from an information theory perspective, Ann. Phys., 2000, 9, 855–864.10.1007/0-306-47114-0_17Search in Google Scholar
[4] Ollivier H., Zurek W.H., Quantum discord: a measure of the quantumness of correlations, Phys. Rev. Lett., 2001, 88, 017901.10.1103/PhysRevLett.88.017901Search in Google Scholar PubMed
[5] Henderson L., Vedral V., Classical quantum and total correlations, J. Phys. A, 2001, 34, 6899–6905.10.1088/0305-4470/34/35/315Search in Google Scholar
[6] Georgeot B., Shepelyansky D.L., Emergence of quantum chaos in the quantum computer core and how to manage it, Phys. Rev. E, 2000, 62, 6366–6375.10.1103/PhysRevE.62.6366Search in Google Scholar PubMed
[7] Georgeot B., Shepelyansky D.L., Quantum chaos border for quantum computing, Phys. Rev. E, 2000, 62, 3504–3507.10.1103/PhysRevE.62.3504Search in Google Scholar PubMed
[8] Grover L.K., From Schrödinger’s equation to quantum search algorithm, Am. J. Phys., 2001, 69, 769–777.10.1119/1.1359518Search in Google Scholar
[9] Pomeransky A.A., Zhirov O.V., Shepelyansky D.L., Phase diagram for the Grover algorithm with static imperfections, Eur. Phys. J. D, 2004, 31, 131–135.10.1140/epjd/e2004-00113-4Search in Google Scholar
[10] Zhirov O.V., Shepelyansky D.L., Dissipative decoherence in the Grover algorithm, Eur. Phys. J. D, 2006, 38, 405–408.10.1140/epjd/e2006-00046-xSearch in Google Scholar
[11] Azuma H., Decoherence in Grover’s quantum algorithm: Perturbative approach, Phys. Rev. A, 2002, 65, 042311.10.1103/PhysRevA.65.042311Search in Google Scholar
[12] Song P.H., Kim I., Computational leakage: Grover’s algorithm with imperfections, Eur. Phys. J. D, 2003, 23, 299-303.10.1140/epjd/e2003-00030-0Search in Google Scholar
[13] Shapira D., Mozes S., Biham O., Effect of unitary noise on Grover’s quantum search algorithm, Phys. Rev. A, 2003, 67, 042301.10.1103/PhysRevA.67.042301Search in Google Scholar
[14] Roosi M., Bruß D., Macchiavello C., Scale invariance of entanglement dynamics in Grover’s quantum search algorithm, Phys. Rev. A, 2013, 87, 022331.10.1103/PhysRevA.87.022331Search in Google Scholar
[15] Bennett C.H., DiVincenzo D.P., Smolin J.A., Wootters W.K., Mixed-state entanglement and quantum error correction, Phys. Rev. A, 1996, 54, 3824.10.1103/PhysRevA.54.3824Search in Google Scholar PubMed
[16] Bennett C.H., Bernstein H.J., Popescu S., Schumacher B., Concentrating partial entanglement by local operations, Phys. Rev. A, 1996, 53, 2046.10.1103/PhysRevA.53.2046Search in Google Scholar
[17] Dorner R., Vedral V., Correlations in quantum physics, Int. J. Mod. Phys. B, 2013, 27, 1345017.10.1142/S0217979213450173Search in Google Scholar
[18] Sadhukhan D., et al., Quantum discord length is enhanced while entanglement length is not by introducing disorder in a spin chain, Phys. Rev. E, 2016, 93, 012131.10.1103/PhysRevE.93.012131Search in Google Scholar PubMed
[19] Dakić B., Vedral V., Brukner C., Necessary and sufficient condition for nonzero quantum discord, Phys. Rev. Lett., 2010, 105, 190502.10.1103/PhysRevLett.105.190502Search in Google Scholar PubMed
[20] Luo S.L., Fu S.S., Geometric measure of quantum discord, Phys. Rev. A, 2010, 82, 034302.10.1103/PhysRevA.82.034302Search in Google Scholar
[21] Chen H., Fu Y.Q., Fang J.X., Geometric discord of non-X-structured state under decoherence channels, Int. J. Theor. Phys., 2014, 53, 2967–2979.10.1007/s10773-014-2094-9Search in Google Scholar
[22] Ye B., Qiu L., 1/f noise in Ising quantum computers, Fluct. Noise Lett., 2014, 13, 1450006.10.1142/S0219477514500060Search in Google Scholar
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