Abstract
The errorrobust and short composite operations named ConCatenated Composite Pulses (CCCPs), developed as highprecision unitary operations in quantum information processing (QIP), are derived from composite pulses widely employed in nuclear magnetic resonance (NMR). CCCPs simultaneously compensate for two types of systematic errors, which was not possible with the known composite pulses in NMR. Our experiments demonstrate that CCCPs are powerful and versatile tools not only in QIP but also in NMR.
Introduction
Nuclear magnetic resonance (NMR) is widely used for chemical analysis of various molecules by pharmaceutical companies^{1} owing to highly developed NMR techniques^{2}. Some of these advanced techniques in NMR have been transferred to quantum information processing (QIP)^{3} because NMR manipulations are regarded as controlling and measuring quantum objects, called spins. We have been working on transferring one of the existing NMR techniques, a composite pulse^{4,5,6,7} that realises a reliable single spin rotation with erroneous pulses, to QIP. There are two types of composite pulses in NMR: One compensates for pulselength errors (PLEs), whereas the other compensates for offresonance errors (OREs). PLEs correspond to rotation angle errors in the dynamics of the qubit on a Bloch sphere, and OREs to rotation axis errors. We have successfully developed an errorrobust and shortpulselength composite operation (pulses), named ConCatenated Composite Pulses (CCCPs), by combining the abovementioned two types of composite pulses in an effort to develop highprecision unitary operations^{8,9} for QIP. This third type of composite pulses simultaneously compensates for the two types of errors (PLEs and OREs in NMR) at the cost of operation time, which was not possible with the known composite pulses in NMR.
The purpose of this paper is to feedback our achievement for QIP to NMR. CCCPs are able to lead significant signal strength improvement without any changes in the hardware settings.
Let us briefly review the principle of composite pulses compensating PLEs or OREs in NMR^{6}. Throughout this paper, the system is a nucleus with spin 1/2 (in short, a spin) in a static magnetic field along the zaxis. An ideal rotation operation of the spin without errors is given as
where θ is the rotation angle, \({\boldsymbol{n}}(\phi )=(\cos \phi ,\sin \phi ,0)\) is the rotation axis in the xyplane, and σ = (σ_{x}, σ_{y}, σ_{z}) is the Pauli matrices. This rotation may be realized by a radiofrequency pulse in NMR, the frequency of which is the same as the Larmor frequency of the spin.
We consider a realistic pulse in which a PLE and/or an ORE are present. The firstorder terms of the errors are discussed since we are interested in the cases where the errors are small. The rotation operator \({R^{\prime} }_{\varepsilon }(\theta ,\phi )\) associated with a pulse under a PLE is given as
where ε is the strength of the PLE, which is unknown — but constant and small. Higherorder terms beyond the first order in ε are suppressed in the second equality. This type of error cannot be avoided because of inhomogeneity in the B_{1} field^{10}. By contrast, the rotation operator \({R}_{f}^{^{\prime} }(\theta ,\phi )\) associated with a pulse under an ORE is given as
where f is the strength of the ORE. OREs are caused whenever the Larmor frequency of the spin is not the same as the transmitter frequency. Therefore, OREs cannot be avoided in NMR measurements because of the chemical shifts of spins. As with a PLE, f is unknown — but constant and small. Therefore, when both a PLE and an ORE are present, the rotation associated with a pulse is given as
The second line is an approximation when both ε and f are small.
The NMR community has developed a technique to overcome PLEs or OREs by combining several pulses^{3,6,7}. Given a target rotation R(θ, ϕ), we can find an equivalent rotation sequence that is equal to the target R(θ, ϕ) in a case without errors, as follows:
Here, R(θ_{i}, ϕ_{i}) is the ith rotation associated with the ith pulse, and N denotes the number of pulses. The point of the decomposition (5) is
if a PLE and/or an ORE exist. This nonequality is caused by the noncommutativity among R(θ_{i}, ϕ_{i}). Therefore, by appropriately tuning the parameters \({\{{\theta }_{i},{\phi }_{i}\}}_{i=1}^{N}\) in Eq. (5), we may be able to obtain a sequence that (i) virtually works as the target R(θ, ϕ) when there are no errors, and (ii) is less sensitive to the systematic errors. Indeed, various pulse sequences have been designed^{4,5,6,11,12,13,14} in such a way that Eq. (5) has no firstorder terms of errors if only one of ε and f exists^{6}. We state that such a pulse sequence without the firstorder term of ε (f) is robust against PLEs (OREs).
We now present two typical composite pulses that are robust against either PLEs or OREs: Broad Band 1 (BB1)^{11}, and Compensation for OffResonance with a Pulse SEquence (CORPSE)^{12}. See more details in Methods. BB1 is designed in order to compensate for a PLE and behaves as
under both a PLE and an ORE. BB1 filters out the PLE but leaves the ORE unchanged, which we call the residual error preserving property (REPP) with respect to ORE. In contrast to BB1, CORPSE is a composite pulse robust against OREs and behaves as
Thus, CORPSE possesses REPP with respect to PLE. Not all composite pulses have REPP, which was not known before ref. ^{15}.
We show how to design a CCCP that compensates for both a PLE and an ORE simultaneously by taking advantage of REPP, with BB1 and CORPSE as an example^{15,16}. BB1 is robust against PLEs, and CORPSE is robust against OREs and has the REPP with respect to the PLE. Therefore, we replace all pulses in BB1 with CORPSE. This CCCP is called CORPSEinBB1, or CinBB in short. The number of pulses in CinBB is 4 × 3 = 12. The number of pulses in CinBB can be further reduced to N = 6, and the resulting CCCP is called the reduced CinBB (RCinBB). See Methods and ref. ^{15} for further details. Another interesting approach to tackle both PLEs and OREs was discussed by Jones^{17}, in which composite pulses were designed to compensate for higherorder error terms of both PLEs and OREs simultaneously. The rotation angle θ is, however, fixed to π in these composite pulses. See the review by Merrill and Brown^{18} on composite pulses including CCCPs.
The signal after a single square π/2pulse is shown as the dashed lines in Fig. 1. This single square pulse has a constant B_{1} during the period of τ_{p} and B_{1}τ_{p} is π/2. Its rotation axis in the Bloch sphere is, here, the yaxis, and thus the magnetization after the π/2pulse is in parallel to the xaxis if there are no errors. Figure 1(a) shows the normalized signal as a function of ε (the dotted curve); ε as small as ε = 0.1 leads to a significant signal reduction. Figure 1(b) shows that the magnetization after the single square π/2pulse deviates from the xaxis and its deviation appears to be proportional to f. Then, let us consider the signal after the RCinBB π/2pulse which consists of six square pulses (see Methods for details). The solid line in Fig. 1(a) shows that one obtains a larger signal for a wide range of ε with the RCinBB π/2pulse than with the single square π/2pulse. On the other hand, the solid line in Fig. 1(b) shows that the magnetization after the RCinBB π/2pulse is close to the xaxis for a wider range of f than after the single square π/2pulse.
Results
Simulations of NMR experiments
Let us take into account a nonunitary time development caused by a spin–spin relaxation with a characteristic time T_{2} in simulating NMR experiments. We introduce this effect as a phase flip channel^{19}. In the case of singlespin experiments,
where \({p}_{ss}(\Delta )=(1+\exp (\Delta /{T}_{2}))/2\approx 1\Delta /2{T}_{2}\) and Ad(ξ, ρ) = ξ^{†}ρξ with an arbitrary unitary operator ξ. Δ is a small time interval. The subscript ss denotes "spinspin”.
The time evolution during a pulse is simulated as follows:
where U_{pulse} is a unitary operation generated by the pulse. Note that τ_{p} is the total pulse duration and is assumed to be small. Therefore, we employ the SuzukiTrotter formula, which ensures the decomposition of the dynamical evolution into the form of pure relaxation process followed by the application of the composite pulse^{20}.
We examine Hahn echo experiments^{2} with two pulses which are affected by fluctuating PLEs and OREs. Their means are \(\bar{\varepsilon }=\bar{f}=0.1\) and their standard deviations are both 0.08. Although these values may be unreasonably large for modern NMR spectrometers, simulated results show that the echo signals with RCinBB pulses do not fluctuate, as shown in Fig. 2. Simulations of the Hahn echo experiments as a function of the error strengths are summarized in Fig. 3. The Hahn echo experiments with two single square pulses (Fig. 3a) are strongly affected by PLEs, whereas those with RCinBB pulses are robust against these errors (Fig. 3b). It turns out that a composite pulse robust against PLEs is sufficient for obtaining a correct T_{2} even when both PLEs and OREs are present.
Let us examine twodimensional (2D) shiftCOrrelation SpectroscopY (COSY) experiments, one of the most important NMR measurement methods^{1}, with two interacting spins. The interaction is a scalar coupling in a weak coupling limit^{2}. The simulations during the evolution and detection periods^{1} are done as follows:
where T_{2,i} is the spinspin relaxation time of the ith spin. Equation (11) is again, similarly to Eq. (10), based on the SuzukiTrotter formula^{20}: The first equation in Eq. (11) describes the spinspin relaxation channel and the second one is the time development generated by the spinspin interaction.
ρ(t + nδ) can be obtained by iterating the above operations n times. Note that p_{ss}(δ) ≈ 1 − δ/(2T_{2,i}) for the ith spin because δ is sufficiently small compared to T_{2,i}. During a pulse, the time development is simulated similarly to the case of singlespin experiments. Simulations are summarized in Fig. 4 in the case that the chemical shifts of these spins are 1 and 4 ppm and J = 0.5 ppm. In COSY experiments, spurious peaks called axial peaks sometimes appear owing to the inaccuracy of the first pulse^{1}. We are able to reproduce these axial peaks in the simulation of the COSY experiments with two single square pulses (ε = f = 0.1), as shown in Fig. 4a. By contrast, no axial peaks appear in the simulation with RCinBB pulses in Fig. 4b.
Experimental demonstration of NMR measurements with CCCPs
The advantages of CCCPs in NMR are demonstrated in the following experiments. The singlepulse experiments were carried out using 300 mM ^{13}Clabelled chloroform in acetoned_{6} at 25 °C. We examined the performance of a composite π/2pulse applied to ^{13}C and compared the result with that of a single square pulse^{21}. It is clear that the RCinBB pulses are more advantageous than single square pulses in terms of PLE, as shown in Fig. 5. This is also demonstrated in the corresponding numerical calculations, shown in Fig. 1(a). We also examined the RCinBB pulse in terms of the ORE, as shown in Fig. 6 (see also Fig. 1(b)). The RCinBB composite pulse is clearly more advantageous than the square pulse when −0.3 < f < 0.8. Although the spectra with the RCinBB composite pulses corresponding to f < −0.5 and 1.0 < f are more distorted than those of the single square pulses, such large f ’s are not relevant in usual experiments. See ref. ^{15} for details.
The advantage of the RCinBB π/2pulse in NMR is also evaluated, as shown in Fig. 7. We applied two successive (RCinBB, CORPSE, BB1, or square) π/2pulses to the thermal equilibrium state. A pair of successive π/2pulses is equivalent to a single πpulse without errors and should lead to no signal. Therefore, the observed residual signals are measures of errors in these pulses. The advantage of the RCinBB π/2pulse is clear from the fact that the two successive RCinBB π/2pulses lead to small signals in wider ranges of both PLEs (ε) and OREs (f ).
The advantage of the RCinBB pulse in NMR is also evaluated in the case of πpulses, as shown in Fig. 8. We applied a (RCinBB, CORPSE, BB1, or square) πpulse to the thermal equilibrium state. An ideal π pulse should lead to no signal. The advantage of the π RCinBB pulse is clear from the fact that the RCinBB πpulses lead to small signals in wider ranges of both PLEs (ε) and OREs (f). These experiments were carried out as in the case of Fig. 7. It is interesting to note that the behaviours as a function of ε of CORPSE and square pulses are identical, which indicates that CORPSE has REPP with respect to PLE in the whole range of ε in Figs. 7 and 8. On the other hand, the REPP of BB1 with respect to ORE is only valid for small \(\leftf\right\).
Next, we performed COSY experiments of 300 mM 3chloro2,4,5,6tetrafluorobenzotrifluoride in benzened_{6}. We utilized ^{19}F at 2, 4, 5, and 6 as the target nuclear spin. T_{1}’s are between 0.6 and 1.0 s, whereas T_{2}’s are ~0.3 s. We chose this molecule for the following reasons. First, ^{19}F signals of the molecule are widely spread, as shown in Fig. 9. Second, the spectrum pattern is complex enough to examine the performance of the pulses, despite of its simple molecular structure.
Here, the pulse duration of a single square π/2pulse is 12.4 μs, which corresponds to a B_{1} strength of 20 kHz in frequency. The total duration of the RCinBB pulse is 16.1 × 12.4 μs = 2.00 × 10^{2} μs, which is almost instantaneous compared with the inverse of the interaction strength in frequency. Therefore, the replacement of a square pulse with the RCinBB pulse should not cause problems for most applications of liquid state NMR measurements.
Since the B_{1} strength is 20 kHz, it is comparable to the frequency difference between the highest (−160 ppm) and the lowest (−118 ppm) peaks at 11.7 T (488 MHz for ^{19}F and 500 MHz for ^{1}H); see Fig. 9. In the case of square pulses, the correlation peak between −118.0 ppm (f_{1}) and −126.5 ppm (f_{2}), and that between −126.5 ppm (f_{1}) and −118.0 ppm (f_{2}), are hardly visible. As shown in Fig. 10, however, these have much higher intensities in the case of the RCinBB pulses. In addition, the advantage of the RCinBB pulses is much more clearly demonstrated in the onedimensional (1D) spectra in Fig. 11. The phases of peaks obtained with square pulses are highly distorted. This may be one of the biggest reasons why the above correlation peaks are almost invisible.
Discussion
Composite pulses have been developed in the NMR community and are widely employed. Our proposed composite operations, CCCPs, directly descend from these and have been developed as robust unitary operations for QIP. We feedback our achievements to NMR: We applied CCCPs to liquidstate NMR spectroscopy and demonstrated improved NMR sensitivity compared to standard 1D and 2D NMR measurements with square pulses. The proposed CCCPs are robust against two systematic errors, the PLE and ORE in NMR, at the cost of execution time.
We demonstrated the advantage of the RCinBB pulses over the BB1, CORPSE, and square pulses in 1D and 2D (COSY) experiments. In terms of the compensation of PLEs and OREs, the replacement of single square pulses with CCCPs, such as the RCinBB pulses, should be widely utilized in other experiments in liquidstate NMR. On the other hand, the application to solidstate NMR (SSNMR) may be limited, because shorter pulses are favourable in SSNMR in general and much longer CCCPs might be unacceptable in most cases. In the case of SSNMR, COMI, II, and III pulses^{22} are often employed as wideband (robust against f) pulses. These pulses employ only 0° and 180° phase pulses and thus the requirement of the electronics is less demanding compared with our proposed CCCPs. We believe, however, that advances in electronics can now allow use of CCCPs even in SSNMR experiments.
CCCPs consist of simple spin rotation pulses and thus they are technically easy to implement although they are not optimal in terms of quantum control theory^{23,24,25,26}. As mentioned before, composite pulses are widely used in NMR experiments, and we hope that CCCPs will be employed instead of these known composite or single square pulses because of their advantage. We believe that CCCPs should be useful for magnetic resonance imaging, too. Furthermore, CCCPs might be applied to positron g/2 measurements through the use of a ^{3}HeNMR probe^{27} in which the inhomogeneity of an excitation field may be large. Also, because the pulse sequence in nonlinear optical spectroscopy has been inspired by the NMR pulse techniques^{28}, CCCPs may be applicable in such optical systems in order to enhance the accuracy of optical spectroscopy.
Material and Methods
BB1
BB1^{11} is an N = 4 composite pulse robust against PLEs. The parameters are as follows:
BB1 under both a PLE and an ORE results in
CORPSE
CORPSE^{12} is an N = 3 composite pulse robust against OREs. Its parameters are
where n_{1}, n_{2}, and n_{3} are nonnegative integers. In particular, when we take n_{1} = n_{3} = 0 and n_{2} = 1, the execution time is minimized. In this case, CORPSE is referred to as short CORPSE. Another notable case takes place when n_{1} − n_{2} + n_{3} = 0. In this case, with both a PLE and an ORE, CORPSE results in
Reduced CORPSE in BB1
RCinBB^{15} is given as follows:
Table 1 shows parameters of π/2 and πpulses of the above three composite pulses.
300 mM ^{13}Clabelled chloroform in acetoned _{6}
^{13}Clabelled chloroform was purchased from Cambridge Isotopes. To the 300 mM ^{13}Clabelled chloroform acetoned_{6} solution, 4 mM of iron(III) acetylacetonate was added. Resulting T_{1} (^{13}C) and T_{2} (^{13}C) were ~ 6 s and 200 ms, respectively, while T_{1} (^{1}H) and T_{2} (^{1}H) were both ~200 ms.
2% HDO in D_{2}O
To the solventmixture composed of 594 μL of D_{2}O and 6 μL of H_{2}O, 2 mg of CuCl_{2} was added, resulting in T_{1} (^{1}H) and T_{2} (^{1}H) of ~50 ms at 9.7 T. Note that the solvent mixing causes 2 % HDO solution, due to the HD chemical exchange.
300 mM 3chloro2,4,5,6tetrafluorobenzotrifluoride in benzened _{6}
3chloro2,4,5,6tetrafluorobenzotrifluoride was diluted with benzened_{6} to 300 mM solution.
NMR measurements
All the NMR experiments described in this article were measured on a JNMECA500 spectrometer (working at 11.7 T) or JNMECZ400S spectrometers (working at 9.4 T) (JEOL RESONANCE Inc.). The 2% HDO sample was measured at 9.4 T (400 MHz for ^{1}H), and the other samples at 11.7 T (500 MHz for ^{1}H and 488 MHz for ^{19}F). The measurements were carried out at 25 °C. A 5 mm ({^{1}H, ^{19}F}X) broadband (BB) probe was used (11.7 T), and 5 mm ROYAL probes were used (9.4 T). We took 1/4 of the square 2πpulse duration as the pulse duration of a square π/2pulse (11.7 T). Instead, the nonlinear least square curve fitting method^{29} was used (9.4 T). During the ^{13}C observing experiments, ^{1}H are decoupled by WALTZ16 decoupling trains. The acquired 2D timedomain data were processed as follows. For both t_{1} and t_{2} periods, the shifted sinebell window function was multiplied. For t_{1}, zerofilling was done once. These data were then Fouriertransformed.
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Acknowledgements
The authors acknowledge valuable discussions with Masahiro Kitagawa and Makoto Negoro. M.B. thanks the Institute for Molecular Science and the National Institutes of Natural Sciences for their hospitality. This work is partially supported by JSPS KAKENHI, Grant Numbers 24320008, 25400422, 25800181, 26400422, 16K05492, 17K05082, and 19K14636, the DAIKO Foundation, the Collaborative Research Project of the Laboratory for Materials and Structures, the Institute of Innovative Research, Tokyo Institute of Technology, the Joint Studies Program of the Institute for Molecular Science, JST PRESTO (feasibility study of specific research proposal) Grant Number JPMJPR19MB, and JST CREST, Grant Number JPMJCR1774.
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M.B. conducted numerical simulations with theoretical support from T.I., Y.K., M.N., and Y.S. M.B., Y.K. and N.N. conducted experiments. All authors contributed to writing the manuscript.
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Bando, M., Ichikawa, T., Kondo, Y. et al. Concatenated Composite Pulses Applied to LiquidState Nuclear Magnetic Resonance Spectroscopy. Sci Rep 10, 2126 (2020). https://doi.org/10.1038/s41598020588239
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