Abstract
Quantum memory is a central component for quantum information processing devices, and will be required to provide highfidelity storage of arbitrary states, long storage times and small access latencies. Despite growing interest in applying physicallayer errorsuppression strategies to boost fidelities, it has not previously been possible to meet such competing demands with a single approach. Here we use an experimentally validated theoretical framework to identify periodic repetition of a highorder dynamical decoupling sequence as a systematic strategy to meet these challenges. We provide analytic bounds—validated by numerical calculations—on the characteristics of the relevant control sequences and show that a ‘stroboscopic saturation’ of coherence, or coherence plateau, can be engineered, even in the presence of experimental imperfection. This permits highfidelity storage for times that can be exceptionally long, meaning that our deviceindependent results should prove instrumental in producing practically useful quantum technologies.
Introduction
Developing techniques for the preservation of arbitrary quantum states—that is, quantum memory—in realistic, noisy physical systems is vital if we are to bring quantumenabled applications including secure communications and quantum computation to reality. Although numerous techniques relying on both open and closedloop control have been devised to address this challenge, dynamical error suppression strategies based on dynamical decoupling (DD)^{1,2,3,4}, dynamically corrected gates (DCGs)^{5,6} and composite pulsing^{7} are emerging as a method of choice for physicallayer decoherence control in realistic settings described by nonMarkovian openquantumsystem dynamics. Theoretical and experimental studies in a variety of platforms^{8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23} have consistently pointed to dynamical error suppression as a resourceefficient approach to substantially reducing physical error rates.
Despite these impressive advances, investigations to date have largely failed to capture the typical operating conditions of any true quantum memory; namely, highfidelity storage of quantum information for arbitrarily longstorage times, with ondemand access. This would be required, for instance, in a quantum repeater, or in a quantum computer where some quantum information must be maintained with error rates deep below faulttolerant thresholds while large blocks of an algorithm are carried out on other qubits. Instead, both experiment and theory have primarily focused on two control regimes^{24}: the ‘coherencetime regime,’ where the goal is to extend the characteristic (‘1/e’ or T_{2}) decay time for coherence as long as possible, and the ‘highfidelity regime,’ where the goal is to suppress errors as low as possible for storage times short compared with T_{2} (for instance, during a single gating period). Similarly, practical constraints on control timing and access latency—of key importance to laboratory applications—have yet to be considered in a systematic way.
In this Article, we demonstrate how to realize a practically useful quantum memory via dynamical error suppression. Specifically, our studies identify the periodic repetition of a highorder DD sequence as an effective strategy for memory applications, considering realistic noise models, incorporating essential experimental limitations on available controls, and addressing the key architectural constraint of maintaining short access latencies to stored quantum information. We consider a scenario where independent qubits couple to a noisy environment, and both dephasing and depolarization errors introduced by realistic DD sequences of boundedstrength πpulses are fully accounted for. We analytically and numerically characterize the achievable longtime coherence for repeated sequences and identify conditions under which a stroboscopic ‘coherence plateau’ can be engineered, and fidelity guaranteed to a desired level at longstorage times—even in the presence of experimentally realistic constraints and imperfections. We expect that our approach will provide a practical avenue to highfidelity lowlatency quantum storage in realistic devices.
Results
Model
The salient features of our approach may be appreciated by first focusing on a singlequbit subject to dephasing. In the absence of control, we consider a model Hamiltonian of the form , where the Pauli matrix σ_{z} and _{0} define the qubit quantization axis and internal energy, respectively (we can set _{0}=0 henceforth), and B_{z}, H_{E} are operators acting on the environment Hilbert space. An exact analysis of both the free and the controlled dynamics is possible if the environment can be described in terms of either a quantum bosonic bath in thermal equilibrium (spinboson model), a weaklycoupled quantum spin bath (spinbath model), or a stationary Gaussian stochastic process (classicalnoise model)^{1,4,25,26,27,28,29,30,31}. Such dephasing models provide an accurate physical description whenever relaxation processes associated with energy exchange occur over a characteristic time scale (T_{1}) substantially longer than any typical time scale associated with the dephasing dynamics. As a result, our analysis is directly relevant to a wide range of experimentally relevant qubit systems, from trapped ions and atomic ensembles^{8,10} to spin qubits in nuclear and electron magnetic resonance and quantum dots^{12,13,14,17,31,32}.
We shall proceed by considering the effects of DD within a filterdesign framework, which generalizes the transferfunction approach widely used across the engineering community^{33} and provides a transparent and experimentally relevant picture of the controlled dynamics in the frequency domain^{8,9,24,26,34,35}. In order to more easily introduce key concepts and clearly reveal our underlying strategy, we first consider an idealized ‘bang–bang’ DD setting in which perfect instantaneous π rotations are effected by using unbounded control amplitudes. As we move forward, we will relax these unphysical constraints, and demonstrate how similar results may be obtained with experimentally realistic controls.
In such an idealized control scenario, a DD sequence may be specified in terms of the pulsetiming pattern , where we also define t_{0}≡0, t_{n+1}≡T_{p} as the sequence duration, and we take all the interpulse intervals (t_{j+1}−t_{j}) to be lowerbounded by a minimum interval τ (ref. 28). The control propagator reads , with y_{p}(t) being a piecewiseconstant function that switches between ±1 whenever a pulse is applied. The effect of DD on qubit dephasing may be evaluated exactly in terms of a spectral overlap of the control modulation and the noise power spectral density, S(ω) (refs 26, 34), which is determined by the Fourier transform of the twotime noise correlation function^{30}. Typically, S(ω) has a powerlaw behaviour at low frequencies, and decays to zero beyond an upper cutoff ω_{c}, that is, S(ω)∝ω^{s}f(ω, ω_{c}), and the ‘rolloff function’ f specifies the highfrequency behaviour, f=Θ(ω−ω_{c}) corresponding to a ‘hard’ cutoff. Let denote the Fourier transform of y_{p}(t), which is given by (refs 4, 26). The filter function (FF) of the sequence p is given by , and the bang–bangcontrolled qubit coherence decays as ${e}^{{\chi}_{p}}$, where the decoupling error at time t=T_{p}, and the case n=0 recovers free evolution over [0, T_{p}].
In this framework, the applied DD sequence behaves like a ‘highpass’ filter, suppressing errors arising from slowly fluctuating (lowfrequency) noise. Appropriate construction of the sequence then permits the bulk of the noise power spectrum to be efficiently suppressed, and coherence preserved. For a given sequence p, this effect is captured quantitatively through the order of error suppression α_{p}, determined by the scaling of the FF near ω=0, that is, , for a sequencedependent prefactor A_{bb}. A high multiplicity of the zero at ω=0 leads to a perturbatively small value of χ_{p} as long as ω_{c}. In principle, one may thus achieve lowerror probabilities over a desired storage time T_{s} simply by using a highorder DD sequence, such as concatenated DD (CDD; ref. 3), or Uhrig DD (UDD; ref. 4), with the desired storage time T_{s}≡T_{p}.
Quantum memory requirements
Once we attempt to move beyond this idealized scenario in order to meet the needs of a practically useful, longtime quantum memory, several linked issues arise. First, perturbative DD sequences are not generally viable for highfidelity longtime storage as they require arbitrarily fast control (τ→0). Real systems face systematic constraints mandating τ>0, and as a result, increasing α_{p} necessitates extension of T_{p}, placing an upper bound on highfidelity storage times^{27,28,36}. (For instance, a UDD_{n} sequence achieves α_{p}=n with n pulses, applied at .) For fixed T_{p}, increasing α_{p} implies increasing n, at the expense of shrinking τ as τ≡t_{1}=O(T_{p}/n^{2}). If τ>0 is fixed, and α_{p} is increased by lengthening T_{p}, eventually the perturbative corrections catch up, preventing further error reduction. Second, potentially useful numerical DD approaches, such as randomized DD^{37,38} or optimized ‘bandwidthadapted’ DD^{28}, become impractical as the configuration space of all possible DD sequences over which to search grows exponentially with T_{s}. Third, DD exploits interference pathways between controlmodulated trajectories, meaning that midsequence interruption (t<T_{p}) typically result in significantly suboptimal performance (Fig. 1). However, a stored quantum state in a practical quantum memory must be accessible not just at a designated final retrieval time but at intermediate times also, at which it may serve as an input to a quantum protocol.
Addressing all such issues requires a systematic approach to DD sequence construction. Here, we identify a ‘modular’ approach to generate lowerror, lowlatency DD sequences for longtime storage out of shorter blocks: periodic repetition of a base, highorder DD cycle.
Quantum memory via periodic repetition
The effect of repetition for an arbitrary sequence is revealed by considering the transformation properties of the FF under sequence combination. Consider two sequences, p_{1} and p_{2}, joined to form a longer one, denoted p_{1}+p_{2}, with propagator ${y}_{{p}_{1}+{p}_{2}}$(t). In the Fourier space we have Let now [p]^{m} denote the sequence resulting from repeating p, of duration T_{p}, m times, with T_{s}=mT_{p}. Computing by iteration, the following exact expression is found:
Equation (1) describes dephasing dynamics under arbitrary multipulse control, generalizing special cases in which this strategy is implicitly used for simple base sequences (periodic DD, p={τ, τ} (ref. 27) and Carr–Purcell, p={τ, 2τ, τ}), and showing similarities with the intensity pattern due to an mline diffraction grating^{31}. The singlecycle FF, F_{p}(ω), is multiplied by a factor that is rapidly oscillating for large m and develops peaks scaling with at multiples of the ‘resonance frequency,’ ω_{res}=2π/T_{p}, introduced by the periodic modulation (see Fig. 2 for an illustration).
After many repeats, the DD error is determined by the interplay between the order of error suppression of the base sequence, the noise power behaviour at low frequencies and the size of noise contributions at the resonance frequencies. The case of a hard upper frequency cutoff at ω_{c} is the simplest to analyse. Applying the Riemann–Lebesgue lemma removes the oscillating factor, resulting in the following asymptotic expression:
provided that ${\text{\chi}}_{{\text{[}p\text{]}}^{\infty}}$ is finite. The meaning of this exact result is remarkable: for small m, the DD error initially increases as (m^{2}χ_{p}), until coherence stroboscopically saturates to a nonzero residual plateau value (), and no further decoherence occurs. Mathematically, the emergence of this coherence plateau requires that simple conditions be obeyed by the chosen base sequence relative to the characteristics of the noise:
which correspond to removing the singularity of the integrand in equation (2) at 0 and ω_{res}, respectively. Thus, judicious selection of a base sequence, fixing α_{p} and T_{p}, can guarantee indefinite saturation of coherence in principle. Moreover, as for all m, the emergence of coherence saturation in the infinitetime limit stroboscopically guarantees highfidelity throughout longstorage times. By construction, this approach also guarantees that access latency is capped at the duration of the base sequence, with ; sequence interrupts at intermediate times that are multiples of T_{p} are thus permitted in the plateau regime without degradation of error suppression.
Additional insight into the above phenomenon may be gained by recalling that for free dephasing dynamics (α_{p}=0), the possibility of nonzero asymptotic coherence is known to occur for supraOhmic (s>1) bosonic environments^{25,27}, consistent with equation (3). The onset of a plateau regime in the controlled dynamics may then be given an intuitive interpretation by generalizing the analysis carried out in Hodgson et al.^{27} for periodic DD: if the conditions in equation (3) are obeyed, the lowfrequency (longtime) behaviour becomes effectively supraohmic by action of the applied DD sequence and, after a shorttime transient, the dephasing dynamics ‘oscillate in phase’ with the periodically repeated blocks. For sufficiently small T_{p}, the ‘differential’ DD error accumulated over each cycle in this steady state is very small, leading to the stroboscopic plateau. Interestingly, that phase noise of a local oscillator can saturate at long times under suitable spectral conditions has also long been appreciated in the precision oscillator community^{33}.
In light of the above considerations, the occurrence of a coherence plateau may be observed even for subOhmic noise spectra (s<1), as typically encountered, for instance, in both spin qubits (s=−2, as in Fig. 1) and trapped ions (s=−1, see ref. 39). Numerical calculations of the DD error using such realistic noise spectra demonstrate both the plateau phenomenon and the natural emergence of periodically repeated sequences as an efficient solution for longtime storage, also confirming the intuitive picture given above. In these calculations, we employ a direct bandwidthadapted DD search up to time T_{s}, by enforcing additional sequencing constraints. Specifically, we turn to Walsh DD, wherein pulse patterns are given by the Walsh functions, to provide solutions that are efficient in the complexity of sequencing^{29}. Walsh DD comprises familiar DD protocols, such as spin echo, Carr–Purcell and CDD, along with more general protocols, including repetitions of shorter sequences.
Starting with a free evolution of duration τ, all possible Walsh DD sequences can be recursively built out of simpler ones within Walsh DD, doubling in length with each step. Further, as all interpulse intervals in Walsh DD protocols are constrained to be integer multiples of τ, there are Walsh DD sequences that stop at time T_{s}, a very small subset of all possible digital sequences, enabling an otherwise intractable bandwidthadapted DD numerical minimization of the spectral overlap integral .
Representative results are shown in Fig. 3, where for each T_{s} all Walsh DD sequences with given τ are evaluated and those with the lowest error are selected. The choice of τ sets the minimum achievable error and also determines whether a plateau is achievable, as, for a given T_{s}, it influences the available values of T_{p} and α_{p}. As T_{s} grows, the best performing sequences (shown) are found to consist of a few concatenation steps (increasing α_{p} of the base sequence to obey equation (3)), followed by successive repetitions of that fixed cycle. Once the plateau is reached, increasing the number of repetitions does not affect the calculated error, indicating that stroboscopic sequence interrupts would be permitted without performance degradation. Beside providing a direct means of finding highfidelity longtime DD schemes, these numerical results support our key analytic insights as to use of periodic sequence design.
Realistic effects
For clarity, we have thus far relied on a variety of simplifications, including an assumption of pure phase decoherence and perfect π rotations. However, as we next show, our results hold in much less idealized scenarios of interest to experimentalists. We begin by considering realistic control limitations. Of greatest importance is the inclusion of errors due to finite pulse duration, as they will grow with T_{s} if not appropriately compensated. Even starting from the dephasingdominated scenario we consider, applying real DD pulses with duration τ_{π}>0 introduces both dephasing and depolarization errors, the latter along, say, the y axis if control along x is used for pulsing. As a result, the conditions given in equation (3) can no longer guarantee a coherence plateau in general: simply incorporating ‘primitive’ uncorrected πpulses into a highorder DD sequence may contribute a net depolarizing error substantial enough to make a plateau regime inaccessible. This intuition may be formalized, and new conditions for the emergence of a coherence plateau determined, by exploiting a generalized multiaxis FF formalism^{35,40}, in which both environmental and finitewidth errors may be accounted for, to the leading order, by adding in quadrature the z and y components of the ‘control vector’ that are generated in the nonideal setting (see Methods).
The end result of this procedure may be summarized in a transparent way: to the leading order, the total FF can be written as , where F_{p}(ω) is the FF for the bang–bang DD sequence previously defined and F_{pul}(ω) depends on the details of the pulse implementation. Corrections in the prefactors A_{bb}, A_{pul} arise from higherorder contributions. The parameter α_{pul} captures the error suppression properties of the pulses themselves, similar to the sequence order of error suppression α_{p}. A primitive pulse results in α_{pul}=1 due to the dominant uncorrected ydepolarization. An expression for the asymptotic DD error may then be obtained starting from equation (1) and separating . An additional constraint thus arises by requiring that both the original contribution of equation (2) and be finite. Thus, in order to maintain a coherence plateau in the longtime limit we now require
We demonstrate the effects of pulsewidth errors in Fig. 4c. When using primitive π_{x}pulses (α_{pul}=1), the depolarizing contribution due to F_{pul}(ω) dominates the total value of . For the dephasing spectrum we consider, s=−2, the condition for maintenance of a plateau using primitive pulses is not met, and the total error grows unboundedly with m after a maximum plateau duration T_{max}≡m_{max}T_{p} (m_{max} may be estimated by requiring that , along lines similar to those discussed in the Methods section). The unwanted depolarizing contribution can, however, by suppressed by appropriate choice of a higherorder ‘corrected’ pulse, such as a DCG^{5,6}, already shown to provide efficient error suppression in the presence of nonMarkovian timedependent noise^{35}. For a firstorder DCG, the dominant error contribution is cancelled, resulting in α_{pul}=2, as illustrated in Fig. 4a; incorporating DCGs into the base DD sequence thus allows the coherence plateau to be restored. For small values of τ_{π}, the error contribution remains small and the plateau error is very close to that obtained in the bang–bang limit. Increasing τ_{π} leads this error contribution to grow, and the plateau saturates at a new higher value.
‘Hardwareadapted’ DCGs additionally provide a means to ensure robustness against control imperfections (including rotationangle and/or offresonance errors) while incorporating realistic control constraints. For instance, sequences developed for singlettriplet spin qubits^{41} can simultaneously achieve insensitivity against nuclearspin decoherence and charge noise in the exchange control fields, with inclusion of finite timing resolution and pulse rise times. A quantitative performance analysis may be carried out in principle through appropriate generalization of the FF formalism introduced above. Thus, the replacement of loworder primitive pulses with higherorder corrected pulses provides a straightforward path toward meeting the conditions for a coherence plateau with realistic DD sequences. These insights are also supported by recent DD nuclear magnetic resonance experiments^{31,32}, that have demonstrated the ability to largely eliminate the effects of pulse imperfections in long pulse trains.
Another experimentally realistic and important control imperfection is limited timing precision. The result of this form of error is either premature or delayed memory access at time T′_{s}=mT_{p}±δt, offset relative to the intended one. Qualitatively, the performance degradation resulting from such accesstiming errors may be expected to be similar to the one suffered by a highorder DD sequence under pulsetiming errors, analysed previously^{24}. A rough sensitivity estimate may be obtained by adding an uncompensated ‘freeevolution’ period of duration δt following the mth repeat of the sequence, with the resulting FF being determined accordingly. In this case, the effective order of suppression transitions α_{p}→0, appropriate for free evolution, at a crossover frequency determined by the magnitude of the timing jitter. In order to guarantee the desired (plateau) fidelity level, it is necessary that the total FF—including timing errors—still meets the requirements set in equation (4). In general, this is achievable for supraOhmic spectra with s>1. When these conditions are not met, the resulting error can be much larger than the plateau value if the jitter is appreciable. Therefore, access timing places a constraint on a system designer to ensure that quantum memories are clocked with lowjitter, highresolution systems. Considering the situation analysed in Fig. 3 with τ=1 μs and ~1.3 × 10^{−9}, we estimate that access jitter of order 1.5 ps may be tolerated before the total measured error exceeds the bound of 2. As current digital delay generators allow for subps timing resolution and ps jitter, the requisite timing accuracy is nevertheless within reach with existing technologies.
We next address different aspects of the assumed noise model. Consider first the assumption of a hard spectral cutoff in bounding the longstorage time error. If such an assumption is not obeyed (hence residual noise persists beyond ω_{c}), it is impossible to fully avoid the singular behaviour introduced by the periodic modulation as m→∞. Contributions from the resonating region ω≈ω_{res} are amplified with m, and, similar to pulseerrors, cause to increase unboundedly with time and coherence to ultimately decay to zero. Nonetheless, a very large number of repetitions, m_{max}, may still be applied before such contributions become important (note that this is the case in the previous figures, where we assume a soft Gaussian cutoff). We lowerbound m_{max} by considering a scenario in which a plateau is preserved with a hard cutoff and estimating when contributions to error for frequencies ω>ω_{c} become comparable to the plateau error. For simplicity, we assume that noise for ω>ω_{c} falls in the passband of the FF and that at ω=ω_{c}, the noise powerlaw changes from ω^{s}→ω^{−r}, with r>0. Treating such a case with s=−2 and using again repeated CDD_{4} with τ=1 μs as in Fig. 3, we find that as long as r is sufficiently large, the plateau error ~10^{−9} can persist for m_{max}10^{4}–10^{6} repetitions (that is, up to a storage time of over 10 s), before the accumulated error due to highfrequency contributions exceeds the plateau coherence (see Methods). This makes it possible to engineer a coherence plateau over an intermediate range of T_{s}, which can still be exceptionally long from a practical standpoint, depending on the specific rolloff behaviour of S(ω) at frequencies beyond ω_{c}.
Lastly, we turn to consideration of more general opensystem models. For instance, consider a system–bath interaction, which includes both a dominant dephasing component and an ‘offaxis’ perturbation, resulting in energy relaxation with a characteristic time scale T_{1}. Then the initial dephasing dynamics, including the onset of a coherence plateau, will not be appreciably modified so long as these two noise sources are uncorrelated and there is a sufficient separation of time scales. If , and the maximum error per cycle is kept sufficiently small, the plateau will persist until uncorrected T_{1} errors dominate . We reiterate that in many experimentally relevant settings—notably, both trappedion and spin qubits—T_{1} effects may indeed be neglected up to very longstorage times. Ultimately, stochastic error sources due, for instance, to spontaneous emission processes and/or Markovian noise (including white control noise) may form a limiting mechanism. In such circumstances, the unfavourable exponential scaling of Markovian errors with storage time poses a problem for highfidelity storage through DD alone. Given a simple exponential decay with timeconstant T_{M} and assuming that equation (4) is met, we may estimate a maximum allowed plateau duration as . Thus, even with T_{M}=100 s, a plateau at =10^{−5} would terminate after T_{max}=1 ms. Our results thus confirm that guaranteeing highfidelity quantum memory through DD alone requires Markovian noise sources to be minimized, or else motivates the combination of our approach with quantum error correction protocols.
Discussion
The potential performance provided by our approach is quite remarkable. Besides the illustrative error calculations we have already presented, we find that many other interesting scenarios arise where extremely lowerror rates can be achieved in realistic noise environments for leading quantum technologies. For instance, ytterbium ion qubits, of direct relevance to applications in quantum repeaters, could allow longtime, lowerror coherence plateaus at the time scale of hours, based on bare freeinductiondecay (1/e) times of order seconds^{39}. Calculations using a common 1/ω noise power spectrum with CDD_{2}, a Gaussian highfrequency cutoff near 100 Hz, τ=1 ms and DCG operations with τ_{π}=10 μs, give an estimate of the plateau error rate of 2.5 × 10^{−9}. This kind of error rate—and the corresponding access latency of just 4 ms—has the potential to truly enable viable quantum memories for repeater applications. Similarly, the calculations shown throughout the manuscript rely on the wellcharacterized noise power spectrum associated with nuclearspin fluctuations in spin qubits. Appropriate sequence construction and timing selection^{41} permits the analytical criteria set out in equation (3) to be met, and similar error rates to be achieved, subject to the limits of Markovian noise processes as described above.
In summary, we have addressed a fundamental and timely problem in quantum information processing—determining a means to effectively produce a practically useful highfidelity quantum memory, by using dynamical error suppression techniques. We have identified the key requirements towards this end, and developed a strategy for sequence construction based on repetition of highorder DD base sequences. Our results allow analytical bounding of the longtime error rates and identify conditions in which a maximum error rate can be stroboscopically guaranteed for long times with small access latencies, even in the presence of limited control. We have validated these insights and analytic calculations using an efficient search over Walsh DD sequences assuming realistic noise spectra. The results of our numerical search bear similarity to an analytically defined strategy established in Hodgson et al.^{27} for optimizing longtime storage in a supraOhmic excitonic qubit.
From a practical perspective, our analyses help set technological targets on parameters such as errorperpulse, timing resolution and Markovian noise strengths required to achieve the full benefits of our approach to quantum memory. This work also clearly shows how a system designer may calculate the impact of such imperfections for a specific platform, bound performance and examine technological tradeoffs in attempting to reach a target memory fidelity and storage time. As the role of optimization in any particular setting is limited to finding a lowerror sequence of duration T_{p} to be repeated up to T_{s}, our framework dramatically reduces the complexity of finding highperformance DD protocols.
Future work will characterize the extent to which similar strategies may be employed to tackle more generic quantum memory scenarios. For instance, recent theoretical methods permit consideration of noise correlations across different spatial directions^{40} in general nonMarkovian singlequbit environments for which T_{2} and T_{1} may be comparable. In such cases, multiaxis DD sequences such as XY4 (ref. 2) may be considered from the outset in order to suppress phase and energy relaxation, as experimentally demonstrated recently^{42}. Likewise, we remark that our approach naturally applies to multiple qubits subject to dephasing from independent environments. As expressions similar to the spectral overlap integral still determine the decay rates of different coherence elements^{43}, exact DD can be achieved by simply replacing individual with collective π pulses, and conditions similar to equation (2) may then be separately envisioned to ensure that each coherence element saturates, again resulting in a guaranteed highstorage fidelity. Addressing the role of correlated dephasing noise and/or other realistic effects in multiqubit longtime storage represents another important extension of this work.
Methods
Inclusion of pulse errors
Consider a base sequence p of total duration T_{p}, including both freeevolution periods and control pulses with nonzero duration τ_{π}, where the center of the jth pulse occurs at time t_{j}≡δ_{j}T_{p}, with δ_{j}[0, 1]. FFs that incorporate, to leading order in T_{p}, errors due to both dephasing dynamics and nonideal pulses are derived following^{40}. The total FF, F(ω)=F_{p}(ω)+F_{pul}(ω), may be expressed as
where r_{z(y)} are, respectively, the total z(y) components of the control vector for pure dephasing in the relevant quadrature, determined by the togglingframe Hamiltonian associated with the control sequence. In the ideal bang–bang limit, , where for example, α_{p}=4, for CDD_{4}. In general, the total contributions to the FF are
where and we incorporate pulse contributions through .
For primitive pulses with a rectangular profile, and Ω≡π/τ_{π}, direct calculation yields^{35}:
For the threesegment firstorder DCG we employ, one finds instead^{35,40}:
where . Starting from these expressions and suitably Taylorexpanding around ω=0, one may then show that the dominant pulse contributions arise from r_{y}(ω) in the uncorrected case, with α_{pul}=1 and A_{pul}=−T_{p}τ_{π}/π, whereas they arise from r_{z}(ω) in the DCG case, with α_{pul}=2 and
Assuming a noise power spectrum with a hard cutoff, S(ω)=g(ω/ω_{c})^{s} × Θ(ω−ω_{c}), the following expression for the (leading order) total asymptotic DD error, , is obtained:
leading to the plateau conditions quoted in equation (4).
Effect of a soft spectral cutoff
Consider, again, a highorder DD sequence which is implemented with realistic pulses and is repeated m times. Then the leading contribution to the DD is given by
where the FF F(ω) is computed as described above and S(ω)=g(ω/ω_{c})^{s} f(ω, ω_{c}). While this integral converges nicely if we assume a sharp highfrequency cutoff, this is rarely encountered in reality. For a soft spectral cutoff, we can break the error integral up into two (low frequency versus high frequency) contributions, say, . We wish to estimate how many repeats of the base sequence are permitted under conditions otherwise leading to a plateau, before corrections due to the highfrequency tail dominate the error behaviour and destroy the plateau. Assume that the conditions given in equation (4) are obeyed, and let the maximum number of allowed repetitions be denoted by m_{max}. Then m_{max} may be determined by requiring that .
As, for every m, we have , a lower bound for m_{max} may be obtained by estimating m* such that . We may therefore simply identify with the hardcutoff asymptotic value given in equation (9). In order to obtain an explicit expression for the highfrequency contribution, we assume that the noise power above ω_{c} also takes a powerlaw form, S(ω)=g(ω/ω_{c})^{r}, formally corresponding to a rolloff f=(ω/ω_{c})^{r−s}, with power r>0. (Note that other possible choices of f, such as exponential or Gaussian rolloffs, may be treated along similar lines, at the expense of more complicated integrals). Thus, we may write
where we have set the FF to the maximum value of the peaks in the passband. This value increases with pulse number and sequence order and must be calculated explicitly for a particular base sequence. For sufficiently large m, the oscillatory factor in the integrand may be approximated in terms of a Dirac comb,
This allows us to write
where we have exploited the fact that 0<ω_{c}<2π/T_{p} and ζ(s) denotes the Riemann zeta function.
The error due to the soft rolloff at high frequencies thus increases linearly with m (hence T_{s}=mT_{p}), as intuition suggests. As the zeta function is decreasing with r and attains its maximum value at r=0, corresponding to an infinite white noise floor, we obtain the following upper bound (recall that ζ(2)=π^{2}/6):
By equating and using Equations 9, 10, 11, 12, 13, 14, we finally arrive at the desired lowerbound:
The above estimate can be applied, in particular, to the specific situation analysed in the main text: base sequence CDD_{4} with τ=1 μs, DCG implementations with τ_{π}≤10 ns, and s=−2. In this case T_{p}≈16 μs, α_{p}=4, , , and one can effectively neglect the contribution to m_{max} due to pulse errors to within the accuracy of this lower bound. Let x≡T_{p}ω_{c}/2π which, by the assumed plateau condition, ranges within [0,1]. Then we may rewrite
implying that, for instance, at least 10^{5} repetitions are allowed at x=0.001 if r≥6, and at least 10^{4} at x=0.01 if r≥8. At the value x=0.16, corresponding to ω_{c}/2π as used in the main text, r18 ensures m_{max}10^{4} hence a storage time of about T_{s}≈0.1 s with error as low as 10^{−9}. As demonstrated by the data in Fig. 4, T_{s} is in fact in excess of 1 s under the assumed Gaussian cutoff, which is realistic for this system. In general, we have verified by direct numerical evaluation of the error integral in Equation (10) that, although qualitatively correct, the lower bound in Equation (16) can significantly underestimate the achievable plateau length (for example, at x=0.16, a storage time T_{s}≈0.1 s is reached already at r15). Altogether, this analysis thus indicates that highfrequency tails do not pose a practically significant limitation provided that the noise falls off sufficiently fast, as anticipated.
Additional information
How to cite this article: Khodjasteh, K. et al. Designing a practical highfidelity longtime quantum memory. Nat. Commun. 4:2045 doi: 10.1038/ncomms3045 (2013).
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Acknowledgements
Work supported by the US ARO under contract No. W911NF1110068, the US NSF under Grant No. PHY0903727 (to L.V.), the IARPA QCS program under contract No. RC051S4 (to L.V.), the IARPA MQCO program (to M.J.B.), and the ARC Centre for Engineered Quantum Systems, CE110001013. We thank SeungWoo Lee for a critical reading of the manuscript.
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L.V. formulated the problem. K.K. and L.V. established analytical error bounds and plateau conditions in the ideal case. M.J.B. led the analysis of realistic effects and systematic impacts, with FF calculations being carried out by J.S., D.H. and T.J.G. All authors jointly validated the results and prepared the manuscript.
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Khodjasteh, K., Sastrawan, J., Hayes, D. et al. Designing a practical highfidelity longtime quantum memory. Nat Commun 4, 2045 (2013). https://doi.org/10.1038/ncomms3045
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