この論文は国立情報学研究所の電子図書館事業により電子化されました。We study the effects of imperfections in a spin model of a quantum computer. We identify different regimes, ranging from low-frequency fluctuations, where the imperfections can be considered static, to the high-frequency case, where the imperfections are purely dynamical and their effects are shown to be completely wiped out

Facchi, Paolo

Fazio, Rosario

Montangero, Simone

Pascazio, Saverio

Kyoto University Research Information Repository

TitleDynamical versus static imperfections in quantumcomputers(6) Approaches from mathematical science andquantum information, Chaos and Nonlinear Dynamics inQuantum-Mechanical and Macroscopic Systems)Author(s)Facchi, Paolo; Fazio, Rosario; Montangero, Simone; Pascazio,SaverioCitation物性研究 (2005), 84(3): 519-522Issue Date2005-06-20URL http://hdl.handle.net/2433/110193RightType Departmental Bulletin PaperTextversionpublisherKyoto University「量子系およびマクロ系におけるカオスと非線形動力学jDynamical versus static imperfections in quantum computers Pao16 Facchi，l Rosario Fazio，2 Simone Montangero，2 and Saverio Pascazio1 1 Dipαrtimento di Fisicα， Universita di Bαri， 1・70126Bαri， Itαly αηd 1NFN， Sezione di Bαri， 1-70126 Bαri， Itαly 2NEST-1NFM & Scuola No門η~ale Superiore， Piazzαdei Cavαl'ieri 7， 56126 Pisα， 1tαly 切ww.qti.sns. it Abstract We study the e百ectsof imperfections in a spin model of a quantum computer. We identify different regimes， ranging from low-frcquency fluctuations， where the imperfections can be considered static， tothe high-frequency case， where the imperfections are purely dynamical and their effects are shown to be completely wiped out. Dynamical errors， yielding decoherence， can be extremely detrimental in quantum computation [1]. On the other hand， the role of static imperfections， such as small inaccuracies in the coupling constants， isoften considered on a different footing， as these do not induce， strictly speaking， any decoherence， but rather errors that can be tolerated up to a certain threshold [2]. Also， the role of static imperfections is regime dependent， and can be utilized as an indicator of an underlying chaotic dynamics [2]. However， strictu senso， a discrimination between “static" imperfections and “dy-namical" noise is given by the physics and depends on the speed of the quantum computer: dynamical noise plays the role of static imperfectionsうifits timescale is much larger than the computational time. We intend to explore this problem in more details and discuss the suggestion [3] that static imperfections can be more disruptive than noise for quantum computation. We model a quantum computer as a lattice of interacting spins (qubits). Due to the imperfections， the couplings between the qubits and with an external field are both random and fluctuate in time. We consider n qubits on a ιdimensionallatticeうdescribed by the Hamiltonian Hr(t) =玄[ムo+ oj(t)]aY) +乞Jij(t)ali)σ;j)? (1) j=l (i，j) where the σど)ぅs(α=ιy， z)are the Pauli matrices for q山 iti and the second sum runs over nearest-neighbor pairs. The number nc of terms in the second sum depends both on the arrangement and dimensionality and is proportional to nd. The energy spacing between the up and down states of a q山 itisムo+ Oi(t)うwherethe Oi (t) 's are uniformly distributed in the interval [-0/2ぅ0/2]and the Jij(t)'s in the interval [ーよJ](zero means and variances 02σ2 and 4J2σ2， respectively， with σ2 = 1/12). ? ????10-1 Elt) 10-2 研究会報告FIG. 1: Error E as a function of T for t = 25， n= 10， J= 5 X10-3， inthe ergodic regime d = 5 X10-3 = J， wiもhnU = 8，nj↑= 5 (squares)， nU = 13， n↑↑= 0 (triangles)， and in七heFGR regime d = 3 X10-1 < J， with nU = 8， n↑j = 5 (circles). We set 0'2 = 1/12， nc = 13， ムo= 1. The fits are given by Eq. (3) with n↑↓ = 8，n↑↑= 5 (dashed)， n↑1 = 13， n↑ =0 (dot-dashed). The transition at九 isshown only in七heformer c部 e.τ 10 τ c ? ? ?? ??0.1 We model the dynamical noise by supposing that both ゐ(t)and Jij(t) ra吋omlychange after each time interval T and are constant otherwise. For J = d = 0 the spectrum of the Hamiltonian is composed of n + 1 degenerate levels， with interlevel spacing 2ム0，corresponding to the energy required to flip a single qubit. We study the case 0く d，J<<ム0，in which the degeneracies are resolved and the spectrum is composed by η+ 1 bands. In this limit the coupling between different bands is very weak and each state is effectively coupled to O(n) other states inside the band. Weωsume free boundary conditions and express al energies in uni tsムo(五 1). We study the behavior of the error (that is the logarithm of the五delity[4]) (2) Et(r) = -lnFt(r) = -ln I (w[T e叶イι叫where T denotes time ordering. The behavior of E will be studied at fixed t as a function of T， the inverse仕equencyof the noise characterizing the fiuctuations of d and J. The initial state I宙)is taken to be an eigenstate of oY) (j = 1，・・・川)and we concentrate on the central band of zero total magnetization， characterized by the highest density of states， and for which one expects the effect of noise to be most pronounced. An exact calculation of the error to order J2 can be carried out explicitly [5]. We ??? ???「?「量子系およびマクロ系におけるカオスと非線形動力学」only give here the approximate expression at a fixed time t 三7:? ?， ， ， ， ， ， ， ????? ?←?←????????? ? ?、? ? ?、，??っ ?????????? ， ， 、 、? ??7 く 7p7pくアく 7cア >7c， J ~ 8 7 > 7c， Jく 8/η(al regimes) (al regimes) (ergodic regime) (FGR regime) (3) where nc = n↑↓ +η↑ is the total number of linksヲn↑i(n↑↓) being the number of nearest-neighbor parallel (antiparallel) pairs in the initial state. Notice that， unlike niL and n↑， nc does not depend on the initial state I宙)but only on the geometry of the spin lattice. In Fig. 1 we show the behavior of Et(7) for different values of 8. In the stαtic situation (large 7， so that 8's and J's can be considered constant) system (1) is characterized by two distinct dynamical regimes: the Fermi Golden Rule (FGR) (Jく Jc)and the ergodic regime (J > Jc)， where Jc rv 8/η[2， 3].The FGR holds below threshold (weak coupling J ~ 8) and is characterized by a Lorentzian local density of states. The ergodic regime takes place in the strong-coupling regime 8 ~ J， when al the levels inside the band participate to the dynamics [6] and the local density of states coincides with the (Gaussian) density of states. The fidelity 九(ァ)[仕omwhich the error (2) is computed] is always the Fourier transform of the local density of states [6]. When 7 becomes smaller， the imperfections become dynαm'tcαl and different regimes emerge as a function of the仕equency7-1. The tr.ansition at 7 = 7c occurs when the error starts deviating from the linear behavior given by Eq. (3). As 7く九the two distinct (ergodic and FGR) behaviors characterizi時 thestatic case (c∞om pared i泊nFig. 1 only for the sets with n↑↓= 8) cannot be resolved anymore. The additional kink at 7 竺 7p=π/4ムosets in when the single spin dynamics starts to play a role. As a global feature， the error tends to vanish linearly with 7. In concl usionヲbelowa given (frequency) threshold， the errors can be considered static， and thus can be corrected by using any of the known methods. One observes in this case two different dynamical regimes. Above this threshold these regimes become unresolved. The difference between these regimes， found for static imper-fections， holds also in the quωi-static case. On the other hand， unitary dynamical errors average to zero in the high仕equencycase. Our results can be relevant in the context of the strategies that have been proposed during the last few years in order to suppress decoherence [7]. These results are independent of the form and the size of the quantum computer. They remain valid under quite general conditions on the system Hamiltonian [5]， allowing a more general application of these findings. Our results show that it is crucial to optimize the computing timescale， by ??ょっ????研究会報告This work was partially supported by the European Community under contracts IST-SQUBIT， IST-SQUBIT2 and RTN-Nanoscale Dynamics， and by the bilateral ltalian-Japanese project 15C1 on “Quantum Information and Computation" of the ltalian Ministry for Foreign Affairs. [1] M.A. Nielsen and I.1. Chuang， Qωntum computαtionαnd quαntum informαtion， (Cam-bridge Univ. Press， 2000). [2] B. Georgeot and D.L. Shepel)叩 sky，Phys. Rev. E 62， 3504 (2000); 62， 6366 (2000). [3] G. Bene凶， G.Cωati， S.Monta時ero，D.L. Shepelyansky， Phys. Rev. Lett. 87， 227901 (2001); S. Mo∞n陥碍e町ro，G. Bene凶， and R. Fazio， Phys. Rev. Lett. 91， 187901 (2003). [4] A. Peres， Phys. Rev. A 30 1610 (1984). [5] P. Facchi， S.Montangero， R.Fazio and S. Pascazio，句ynamicalImperfections in quan-tum computers，" quant-phj0407098. [6] V.V. Flambau瓜 Aust.J. Phys. 53， N4， (2000). [7] A. Ekert， C.Macchiavello， Acta Phys. Polo孔 A93，63 (1998); quant-phj9904070. ? ? ? 】っ ????

Dynamical versus static imperfections in quantum computers(6) Approaches from mathematical science and quantum information, Chaos and Nonlinear Dynamics in Quantum-Mechanical and Macroscopic Systems)

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Dynamical versus static imperfections in quantum computers(6) Approaches from mathematical science and quantum information, Chaos and Nonlinear Dynamics in Quantum-Mechanical and Macroscopic Systems)

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