Efficiency of dynamical decoupling sequences in presence of pulse errors

Efficiency of dynamical decoupling sequences in presence of pulse errors

Zhihao Xiao (Ф֮ê»), Lewei He (ºÎÀÖΪ), and Wen-ge Wang (ÍõÎĸó) 111 Email address: wgwang@ustc.edu.cn Department of Modern Physics, University of Science and Technology of China, 230026 China
July 5, 2019
Abstract

For a generic dynamical decoupling sequence employing a single-axis control, we study its efficiency in the presence of small errors in the direction of the controlling pulses. In the case that the corresponding ideal dynamical-decoupling sequence produces sufficiently good results, the impact of the errors is found to scale as , with negligible first-order effect, where is the dispersion of the random errors. This analytical prediction is numerically tested in a model, in which the environment is modeled by one qubit coupled to a quantum kicked rotator in chaotic motion. In this model, with periodic pulses applied to the qubit in the environment, it is found numerically that periodic bang-bang control may outperform Uhrig dynamical decoupling.

pacs:
03.67.Pp, 03.65.Yz, 05.45.Mt

I Introduction

Dynamical decoupling (DD) has attracted lots of attention in the past years, due to its potential application in quantum information processes. The basic idea is to insert a sequence of controlling pulses within a time period of interest, such that the system of interest can be effectively decoupled from its environment. As a result, environment-induced decoherence can be effectively suppressed and initial coherence in the system can be preserved.

Several DD schemes have been proposed. For example, the so-called periodic “bang-bang ” control VL98 (); VKL99 (); Morton06 () can suppress decoherence up to the order of for a given period of coherence preservation, while the Carr-Purcell-Meiboom-Gill sequence has an efficiency Slich (); Uhrig07 (). A better result can be obtained by an approach called “concatenated dynamical decoupling” KL05 (); KL07 (), with an efficiency of the order of achieved by pulses. Recently, a remarkable progress has been made by Uhrig, showing that decoherence of a single spin, which is induced by a spin-boson bath, can be suppressed up to the order of with only pulses Uhrig07 (). Later, the Uhrig dynamical decoupling (UDD) was conjectured LWS08 () and rigorously proved to be model-independent for the pure dephasing case YL08 (). More recently, it has been found that different types of UDD sequences may be integrated to obtain better results West2010 () and, with appropriate extension, to work for a system with two spins as well MSSG10 (). Meanwhile, the efficiencies of UDD and its generalization has been beautifully demonstrated experimentally Bn09 (); Du09 (); Du11 ()

In practical application of a DD sequence, an important topic is its robustness, i.e., the influence of non-idealness of the controlling pulses in the efficiency of the DD. The non-idealness may come from finite widths of the pulses and/or small deviation of the actual directions of the pulses from their designed directions. In the case that the non-ideal parts of the pulses with finite width satisfy certain symmetry requirements, a generalized UDD can be found YL08 (). However, in a generic circumstance, accumulation effect of small imperfections in the pulses may have significant consequences when the number of pulses is not small. There have been several investigations in the impact of systematic pulse errors for some specific DD sequences, both experimentally AAPS10 (); Biercuk09 (); Wang10 (); Tyryshkin () and theoretically Wang10 (); Tyryshkin (); WD11 (); KHCM10 ().

In this work, for a generic DD employing a single-axis control, we study accumulation effects of small random errors in the direction of the controlling pulses. Analytical derivations are given in Sec.II. In particular, we show that a DD, which has a sufficiently good performance in the ideal case, is robust in the sense that the pulse errors have negligible first-order effect.

In numerical investigation, we employ a model in which there are periodic pluses in the environment, which are not directly applied to the qubit of interest. Using this model, we can test our analytical predictions, and also study another topic of interest, namely, the influence of high-frequency cutoff in the efficiency of UDD. When the spectrum of the environment has a sharp high-frequency cutoff, UDD has been found outperforming all other known DD sequences and is regarded as optimal. However, for environments with soft cutoffs in the spectra, there is no reason to expect that UDD is optimal; in fact, recently it has been shown that protocols with periodic structure, such as the Carr-Purcell-Meiboom-Gill sequence, may have a better performance for this type of environment CLND08 (); PU10 (); DWRDH1011 (); AAS11 ().

Specifically, In Sec.III, we study a model, in which the environment is simulated by a second qubit coupled to a quantum kicked rotator in chaotic motion. Previous study shows that this model, though a single-particle dynamical system, may simulate a pure-dephasing many-body bath (Caldeira-Leggett model CL83 ()), as well as some non-Markovian environments RCB06 (). Our numerical simulations show that for this type of environment the periodic bang-bang control may outperform UDD. Finally, conclusions and discussions are given in Sec.IV

Ii Analytical study of impacts of small pulse errors

In this section, we first recall essential properties of DD, then, derive expressions for the influence of small pulse errors in the efficiency of a DD that employs a single-axis control.

ii.1 Dynamical decoupling scheme

Let us first recall some essential properties of a generic DD. Consider a qubit and its environment , under a Hamiltonian

(1)

where and are the Hamiltonians of and , respectively, and is the interaction Hamiltonian. The self-Hamiltonian is assumed to be a constant, which can be taken as zero, . We further assume that the interaction Hamiltonian is commutable with , , where is the -component of the Pauli operator for the qubit .

Suppose the qubit lies initially in a state, which is not an eigenstate of the interaction Hamiltonian . Schrödinger evolution under the Hamiltonian , given by a unitary operator for a time period , may induce decoherence to the state of the qubit . The purpose of a DD is to preserve coherence in the initial state of within the time period by inserting a sequence of pulses, e.g., -pulses for the direction. Using of to indicate the instants at which pulses are applied, with totally pulses, the time-evolution operator is now written as

(2)

To measure preservation of coherence in an initial state in the -direction, one may consider measurement on the observable , which gives the signal

(3)

where indicates an eigenstate of and rotates it to an eigenstate of . After some derivation, one gets

(4)

The coherence is perfectly preserved if .

In UDD, -pulses are applied to the qubit at times , where

(5)

For UDD, . In a periodic bang-bang control of DD, pulses are applied at times with .

ii.2 expanded to the second-order term of error

As discussed in Introduction, controlling pulses in a DD may be subject to small random errors in its direction. Let us consider small random deviation in the direction for -pulses in the direction. In this case, in the ideal time-evolution operator in Eq. (2) should be replaced by

(6)

where and is a small random number with Gaussian distribution,

(7)

Here, is the dispersion of the random number, with . Then, the time evolution operator, denoted by , is written as

(8)

where we use to indicate the value of for the -th pulse. Now, the signal, denoted by , has the following expression,

(9)

To evaluate , we substitute Eq. (8) into Eq. (9) and expand the result in the power of . Up to the second-order term of , we write the signal in the following form,

(10)

Definitions of the first three terms on the right hand side of Eq. (10) will be given below, when they are treated separately.

The first term on the right hand side of Eq. (10) is obtained by considering only the contribution of in each ,

(11)

where

(12)

Here, is the time evolution operator for the case of ideal pulses in Eq. (2). Substituting Eq. (12) into Eq. (11), we find

(13)

It is seen that, for sufficiently small ,

(14)

Thus, deviation of from the ideal signal is of the order of .

Next, we calculate the second term on the right hand side of Eq. (10), which is the contribution of those multiplication terms that include only one term in each of them. Noticing that, up to the second-order contribution of , in this second term can be taken as 1, we have the following expression for it,

(15)

where

(16)

Making use of the fact that and , we find

(17)

Substituting Eq. (12) into Eq. (17), then into Eq. (15), we obtain

(18)

Typically, one has the following estimate,

(19)

hence,

(20)

Making use of Eq. (12), after simple algebra, it is found that

(21)

where

(22)

Therefore, gives a first order correction () to the ideal signal.

The first order correction also depends on the quantity . To give an estimation to , we note that , [see Eqs. (4) and (22)]. Hence,

(23)

where we have used the fact that .

Finally, we discuss the third term on the right hand side of Eq. (10), which includes all multiplication terms that have only two terms,

(24)

where

(25)

Following a procedure similar to that for , we find

(26)

Substituting the above obtained expressions of , , and into the expression of in Eq. (24), after some derivation, we obtain an expression for . Then, making use of results obtained above in Eqs.(14) and (21), finally, we find

(27)

where is of the order of 1.

Of particular interest is the case for a DD with good performance, i.e., with . In this case, the inequality (23) shows that is small. In particular, in the case that , the first order term on the right hand side of Eq. (27) can be neglected and we have

(28)

scaling as . For example, for a UDD with , is of the order of or less, hence, for a fixed , for a sufficiently large .

In concluding this section, we remark that it is straightforward to generalize the above discussions to the case of pulses with more generic random errors in their direction. In fact, in this generic case, the time-evolution operator can be obtained by replacing in Eq. (2) by

(29)

where and both and are small random numbers with Gaussian distribution. We have found results that are qualitatively the same as those discussed above for the case of in Eq. (6). In particular, we have found similar estimations as those given in the relations (27) and (28).

Iii Numerical simulations

In this section, we discuss numerical simulations we have performed for two purposes. One is to check analytical predictions given in the previous section, the other is to study the influence of kicks in the environment in the efficiency of UDD. In fact, since instant kicks in the environment may have non-negligible high-frequency components, the performance of UDD for such an environment may be not so good as that for an environment with a sharp high-frequency cutoff.

Figure 1: (Color online) Variation of the signal in Eq. (34) with time for ideal UDD and ideal periodic bang-bang control. Both UDD and periodic bang-bang control have better performance for larger values of , the number of controlling pulses. Results of UDD (circles) are better than those of the periodic bang-bang control (solid curve) for . But, for larger values of , the periodic bang-bang control outperforms UDD. Parameters are and , and the kicked rotator is in the chaotic region with . The same parameters are also used in the following figures.

iii.1 The model

We consider a model, in which there is a qubit of interest and an environment that is composed of a second qubit and a quantum kicked rotator denoted by . The qubit has interaction with only, while interacts with both and the kicked rotator . The Hamiltonian is written as

(30)

where the self-Hamiltonians are , =, and

(31)

with the period of kicking. The interaction Hamiltonians are and

(32)

Here, for clearness we write explicitly the superscript in the Pauli operator for the qubit .

This model has been studied in Ref. RCB06 (), showing that the kicked rotator, though a single-particle dynamical system, may simulate a pure-dephasing many-body bath (Caldeira-Leggett model CL83 ()), as well as some non-Markovian environments. Here, we are interested in the chaotic region of the kicked rotator, to simulate some random properties of the environment.

Figure 2: (Color online) Dependence of (solid squares) on for a UDD sequence, with and . The solid curve is a straight line, indicating a linear dependence, in agreement with the prediction of Eq. (28). For the corresponding ideal UDD, . Since , the influence of the pulse errors is profound.

The time evolution for one period is given by the unitary operator

(33)

where has been set unit. An effective Planck constant can be introduced, , where is the dimension of the Hilbert space of the kicked rotator. The classical limit is obtained by letting and while keeping fixed, and the classical counterpart is defined on a torus . The kicked rotator has a chaotic motion for larger than 6 or so.

Figure 3: Variation of with for a UDD sequence with .

iii.2 Numerical results

Let us first discuss the performance of ideal UDD and ideal periodic bang-bang control in this model. We are interested in the case that . In this case, due to the time-dependency of the Hamiltonian of the environment, Uhrig’s strategy for deriving is not applicable. It is of interest to know whether UDD is optimal or not in this case.

In numerical simulation, we calculated defined by

(34)

where is obtained by truncation of the time evolution operator in Eq. (2) at an intermediate time . Figure 1 shows that both UDD and periodic bang-bang control have better performance with increasing number of the controlling pulses. For , UDD outperforms the periodic bang-bang control in the whole time region . However, with increasing , results of the periodic bang-bang control become better than those of UDD for .

Next, we check analytical predictions given above for the impact of random errors in the direction of the controlling pulses, in particular, the behavior of in Eq. (28). We use the general form of in Eq. (29), with the same dispersion for and . An example is given in Fig.2, showing linear dependence of on . To check details of agreement with analytical predictions, we have numerically computed the corresponding ideal UDD and found that it has a good performance with . This gives [see the estimate in the inequality (23)]. Hence, for solid squares shown in the figure, as a result, Eq. (28) should hold with . Thus, results in Fig.2 indeed confirm the prediction of Eq. (28).

Figure 4: Variation of the signal with time for a UDD with non-ideal pulses, . The UDD with (right panel) gives results worse than those of the UDD with (left panel).

We have also numerically studied the case of large . In this case, higher order terms of Eq. (27) should be considered. Anyway, this expression of suggests that may be large for . Indeed, numerical results support this expectation, as shown in Fig. 3.

Figure 2 shows that for a fixed , the influence of pulse errors may become large with increasing , such that . In Fig.4, we show that for a fixed error dispersion , deviation of from 1 enlarges when is increased, where is defined by Eq. (34) with replaced by the corresponding .

Iv Conclusions and discussions

In this paper, we have analytically studied the efficiency of a generic DD with a single-axis control, when the controlling pulses are subject to small random errors in their direction. We have derived expressions for the influence of the pulse errors up to the second-order term. When the ideal DD has a sufficiently good performance, the influence has a negligible first-order effect; in this sense, good DD are relatively robust.

We have tested the above analytical predictions numerically and shown that accumulation of small pulse errors may have significant influence in the efficiency of DD. For an environment with kicks applied on some part of the environment, it has been found that the periodic bang-bang control may outperform UDD.

A natural question would concern the possibility of having negligible first-order effect of pulse errors in more generic situations, e.g., in the case of more than one layers of controlling pulses with different single-axis control in different layers. For pulse errors appearing only at certain fixed layer with single-axis control, results of this paper may be generalizable. However, the more generic situation with pulse errors in different layers, as well as the case with finite width of the pulses, are much more complex and further investigation is needed before a definite conclusion can be drawn.

ACKNOWLEDGMENTS. The authors are grateful to Jiangbin Gong for valuable discussions and suggestions. This work is partly supported by the Natural Science Foundation of China under Grant Nos. 10775123 and 10975123 and the National Fundamental Research Programme of China Grant No.2007CB925200.

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