Abstract
Unwanted interaction between a quantum system and its fluctuating environment leads to decoherence and is the primary obstacle to establishing a scalable quantum information processing architecture. Strategies such as environmental and materials engineering, quantum error correction and dynamical decoupling can mitigate decoherence, but generally increase experimental complexity. Here we improve coherence in a qubit using realtime Hamiltonian parameter estimation. Using a rapidly converging Bayesian approach, we precisely measure the splitting in a singlettriplet spin qubit faster than the surrounding nuclear bath fluctuates. We continuously adjust qubit control parameters based on this information, thereby improving the inhomogenously broadened coherence time from tens of nanoseconds to >2 μs. Because the technique demonstrated here is compatible with arbitrary qubit operations, it is a natural complement to quantum error correction and can be used to improve the performance of a wide variety of qubits in both meteorological and quantum information processing applications.
Introduction
Hamiltonian parameter estimation is a rich field of active experimental and theoretical research that enables precise characterization and control of quantum systems^{1}. For example, magnetometry schemes employing Hamiltonian learning have demonstrated dynamic range and sensitivities exceeding those of standard methods^{2,3}. Such applications focused on estimating parameters that are quasistatic on experimental timescales. However, the effectiveness of Hamiltonian learning also offers exciting prospects for estimating fluctuating parameters responsible for decoherence in quantum systems.
The quantum system that we study is a singlettriplet (S−T_{0}) qubit^{4,5} which is formed by two gatedefined lateral quantum dots (QDs) in a GaAs/AlGaAs heterostructure (Fig. 1a, Supplementary Fig. 1), similar to that of refs 6, 7. The qubit can be rapidly initialized in the singlet state S› in ≈20 ns and read out with 98% fidelity in ≈1 μs (refs 8, 9; Supplementary Fig. 2). Universal quantum control is provided by two distinct drives^{10}: the exchange splitting, J, between S› and T_{0}›, and the magnetic field gradient, ΔB_{z}, due to the hyperfine interaction with host Ga and As nuclei. The Bloch sphere representation for this qubit can be seen in Fig. 1b. In this work, we focus on qubit evolution around ΔB_{z} (Fig. 2a). Due to statistical fluctuations of the nuclei, ΔB_{z} varies randomly in time, and consequently oscillations around this field gradient decay in a time (ref. 4). A nuclear feedback scheme relying on dynamic nuclear polarization^{11} can be employed to set the mean gradient, (g*μ_{B}ΔB_{z}/h≈60 MHz in this work) as well as reduce the variance of the fluctuations. Here, g*≈−0.44 is the effective gyromagnetic ratio in GaAs, μ_{B} is the Bohr magneton and h is Planck’s constant. In what follows, we adopt units where g*μ_{B}/h=1. The nuclear feedback relies on the avoided crossing between the S› and T_{+}› states. When the electrons are brought adiabatically through this crossing, their total spin changes by Δm_{s}=±1, which is accompanied by a nuclear spin flip to conserve angular momentum. With the use of this feedback, the coherence time improves to (ref. 11; Fig. 2b), limited by the low nuclear pumping efficiency^{10}. Crucially, the residual fluctuations are considerably slower than the timescale of qubit operations^{12}.
In this work we employ techniques from Hamiltonian estimation to prolong the coherence of a qubit by more than a factor of 30. Importantly, our estimation protocol, which is based on recent theoretical work^{13}, requires relatively few measurements (≈100) which we perform rapidly enough (total time ≈100 μs) to resolve the qubit splitting faster than its characteristic fluctuation time. We adopt a paradigm in which we separate experiments into ‘estimation’ and ‘operation’ segments, and we use information from the former to optimize control parameters for the latter in realtime. Our method dramatically prolongs coherence without using complex pulse sequences such as those required for nonidentity dynamically decoupled operations^{14}.
Results
Rotating frame S−T_{0} qubit
To take advantage of the slow nuclear dynamics, we introduce a method that measures the fluctuations and manipulates the qubit on the basis of precise knowledge but not precise control of the environment. We operate the qubit in the rotating frame of ΔB_{z}, where qubit rotations are driven by modulating J at the frequency (refs 15, 16). This is in contrast to traditional modes of operation of the S−T_{0} qubit, which rely on DC voltage pulses. To measure Rabi oscillations, the qubit is adiabatically prepared in the ground state of ΔB_{z} (ψ›=↑↓›), and an oscillating J is switched on (Fig. 2e), causing the qubit to precess around J in the rotating frame. In addition, we perform a Ramsey experiment (Fig. 2c) to determine , and as expected, we observe the same decay (Fig. 2d) as Fig. 2b. More precisely, the data in Fig. 2d represent the average of 1,024 experimental repetitions of the same qubit operation sequence immediately following nuclear feedback. The feedback cycle resets ΔB_{z} to its mean value (60 MHz) with residual fluctuations of between experimental repetitions. However, within a given experimental repetition, ΔB_{z} is approximately constant. Therefore we present an adaptive control scheme where, following nuclear feedback, we quickly estimate ΔB_{z} and tune to prolong qubit coherence (Fig. 3a).
Bayesian estimation
To estimate ΔB_{z} , we repeatedly perform a series of singleshot measurements after allowing the qubit to evolve around ΔB_{z} (using DC pulses) for some amount of time (Fig. 2a). Rather than fixing this evolution time to be constant for all trials, we make use of recent theoretical results in Hamiltonian parameter estimation^{13,16,17} and choose linearly increasing evolution times, t_{k}=kt_{samp}, where k=1,2,…N. We choose the sampling time t_{samp} such that the estimation bandwidth is several times larger than the magnitude of the residual fluctuations in ΔB_{z}, roughly 10 MHz. With a Bayesian approach to estimate ΔB_{z} in realtime, the longer evolution times (large k) leverage the increased precision obtained from earlier measurements to provide improved sensitivity, allowing the estimate to outperform the standard limit associated with repeating measurements at a single evolution time. Denoting the outcome of the kth measurement as m_{k} (either S› or T_{0}›), we define P(m_{k}ΔB_{z}) as the conditional probability for m_{k} given a value ΔB_{z}. We write
where r_{k}=1 (−1) for m_{k}=S›(T_{0}›), and α=0.25 and β=0.67 are parameters determined by the measurement error and axis of rotation on the Bloch sphere (see Methods). Since we assume that earlier measurement outcomes do not affect later ones (that is, that there is no measurement backaction), we write the conditional probability for ΔB_{z} given the results of N measurements as:
Using Bayes’ rule, that is, P(ΔB_{z}m_{k})=P(m_{k}ΔB_{z})P(ΔB_{z})/P(m_{k}), and equation (1), we can rewrite equation (3) as:
where N is a normalization constant and P_{0}(ΔB_{z}) is a prior distribution to which the algorithm is empirically insensitive, and which we take to be a constant over the estimation bandwidth. After the last measurement, we find the value of ΔB_{z} that maximizes the posterior distribution P(ΔB_{z}m_{N},m_{N−1},...m_{1}).
Adaptive control
We implement this algorithm in realtime on a fieldprogrammable gate array (FPGA), computing P(ΔB_{z}) for 256 values of ΔB_{z} between 50 and 70 MHz. With each measurement m_{k}, the readout signal is digitized and passed to the FPGA, which computes P(ΔB_{z}) and updates an analogue voltage that tunes the frequency of a voltage controlled oscillator (Fig. 1a; Supplementary Note 1, Supplementary Fig. 3). Following the Nth sample, nearly matches ΔB_{z}, and since the nuclear dynamics are slow, the qubit can be operated with long coherence without any additional complexity. To quantify how well the FPGA estimate matches ΔB_{z}, we perform a Ramsey experiment (deliberately detuned to observe oscillations) with this realtime tracking of ΔB_{z} and find optimal performance for N≈120, with a maximum experimental repetition rate, limited by the FPGA, of 250 kHz and a sampling time t_{samp}=12 ns. Under these conditions, and making a new estimate after every 42 Ramsey experiments, we observe , a 30fold increase in coherence (Fig. 3b). We note that these data are taken with the same pulse sequence as those in Fig. 2d. To further compare qubit operations with and without this technique, we measure Ramsey fringes for ≈250 s (Fig. 3d), and histogram the observed Ramsey detunings. With adaptive control we observe a stark narrowing of the observed frequency distribution, consistent with this improved coherence (Fig. 3c).
Discussion
Although the estimation scheme employed here is theoretically predicted to improve monotonically with N (ref. 13), we find that there is an optimum (N≈120), after which slowly decreases with increasing N (Fig. 4a). A possible explanation for this trend is fluctuation of the nuclear gradient during the estimation period. To investigate this, we obtain time records of ΔB_{z} using the Bayesian estimate and find that its variance increases linearly in time at the rate of (6.7±0.7 kHz)^{2} μs^{−1} (Fig. 4c). The observed linear behaviour suggests a model where the nuclear gradient diffuses, which can arise, for example, from dipolar coupling between adjacent nuclei. Using the measured diffusion of ΔB_{z}, we simulate the performance of the Bayesian estimate as a function of N (see Methods). Given that the simulation has no free parameters, we find good agreement with the observed , indicating that indeed, diffusion limits the accuracy with which we can measure ΔB_{z} (Fig. 4a).
This model suggests that increasing the rate of measurements during estimation will improve the accuracy of the Bayesian estimate. Because our FPGA limits the repetition rate of qubit operations to 250 kHz, we demonstrate the effect of faster measurements through software postprocessing with the same Bayesian estimate. To do so, we first use the same estimation sequence, but for the operation segment, we measure the outcome after evolving around ΔB_{z} for a single evolution time, t_{evo}, rather than performing a rotating frame Ramsey experiment, and we repeat this experiment a total of N_{tot} times. In processing, we perform the Bayesian estimate of each ΔB_{z,i}, sort the data by adjusted time (for i=1,2,…N_{tot}), and average together points of similar τ to observe oscillations (see Methods). We fit the decay of these oscillations to extract and the precision of the Bayesian estimate, . For the same operation and estimation parameters, we find that extracted from software postprocessing agrees with that extracted from adaptive control Supplementary Fig. 4, Supplementary Note 2. Using a repetition rate as high as 667 kHz, we show coherence times above 2,800 ns, corresponding to an error of σ_{ΔBz}=80 kHz (Fig. 4d), indicating that improvements are easily attainable by using faster (commercially available) FPGAs.
In addition, we use this postprocessing to examine the effect of this technique on the duty cycle of experiments as well as the stability of the ΔB_{z} estimate. To do so we introduce a delay T_{delay} between the estimation of ΔB_{z} and the single evolution measurement performed in place of the operation. We find , where c=0.99 (Fig. 4c), consistent with diffusion of ΔB_{z}. Indeed, this dependence underscores the potential of adaptive control, since it demonstrates that after a single estimation sequence, the qubit can be operated for >1 ms with . Thus, adaptive control need not significantly reduce the experimental duty cycle.
In this work, we have used realtime adaptive control on the basis of Hamiltonian parameter estimation of a S−T_{0} spin qubit to prolong from 70 ns to >2 μs. Dephasing due to nuclear spins has long been considered a significant obstacle to quantum information processing using semiconductor spin qubits^{18}, and elimination of nuclear spins is an active and fruitful area of research^{19,20,21}. However, here we have shown that with a combination of nuclear feedback, rotating frame S−T_{0} spin resonance, and realtime Hamiltonian estimation, we are able to achieve ratios of coherence times to operation times in excess of 200 without recourse to dynamical decoupling^{12,22,23}. If the same adaptive control techniques were applied to gradients as high as 1 GHz (ref. 10), ratios exceeding 4,000 would be possible, and longer coherence times may be attainable with more sophisticated techniques^{13}. Though the observed coherence times are still smaller than the Hahn echo time, (ref. 12), the method we have presented is straightforward to implement, compatible with arbitrary qubit operations, and general to all qubits that suffer from nonMarkovian noise. Looking ahead, it is likely, therefore, to have a key role in realistic quantum error correction efforts^{24,25,26,27}, where even modest improvements in baseline error rate greatly diminish experimental complexity and enhance prospects for a scalable quantum information processing architecture.
Methods
Bayesian estimate
We wish to calculate the probability that the nuclear magnetic field gradient has a certain value, ΔB_{z}, given a particular measurement record comprising N measurements. We follow the technique in Sergeevich et al.^{13} with slight modifications. Writing the outcome of the kth measurement as m_{k}, we write this probability distribution as
To arrive at an expression for this distribution, we will write down a model for the dynamics of the system, that is, P(m_{N},m_{N−1},...m_{1}ΔB_{z}). Using Bayes’ rule we can relate the two equations as
First, we seek a model that can quantify P(m_{N},m_{N−1},...m_{1}ΔB_{z}) that accounts for realistic errors in the system, namely measurement error, imperfect state preparation and error in the axis of rotation around the Bloch sphere. For simplicity, we begin with a model that accounts only for measurement error. Denoting the error associated with measuring a S› (T_{0}›) as η_{S} (η_{T}), we write
We combine these two equations and write
where r_{k}=1 (−1) for m_{k}=S›(T_{0}›) and α and β are given by
Next, we generalize the model to include the effects of imperfect state preparation, and the presence of nonzero J during evolution, which renders the initial state nonorthogonal to the axis of rotation around the Bloch sphere (see above). We assume that the angle of rotation around the Bloch sphere lies somewhere in the x–z plane and makes an angle θ with the z axis. We define δ=cos^{2} (θ). Next, we include imperfect state preparation by writing the density matrix ρ_{init}=(1−ε)S› ‹S)+εT_{0}› ‹T_{0}. With this in hand, we can write down the model
Using the same notation for r_{k}=1 (−1) for m_{k}=S›(T_{0}›), we rewrite this in one equation as
where we now have
We find the best performance for α=0.25 and β=0.67, which is consistent with known values for qubit errors.
We next turn our attention to implementing Bayes’ rule to turn this model into a probability distribution for ΔB_{z}. First, we assume that all measurements are statistically independent, allowing us to write
We next use Bayes rule (6) and rewrite this equation as
Using our model (13) we can rewrite this as
where N is a normalization constant, and P_{0}(ΔB_{z}) is a prior distribution for ΔB_{z} which we take to be a constant over the estimation bandwidth, and to which the estimator is empirically insensitive. With this formula, it is simple to see that the posterior distribution for ΔB_{z} can be updated in real time with each successive measurement. After the Nth measurement, we choose the value for ΔB_{z}, which maximizes the posterior distribution (18).
Simulation with diffusion
We simulate the performance of our software scaling and hardware (FPGA) estimates of ΔB_{z} using the measured value of the diffusion rate. We assume that ΔB_{z} obeys a random walk, but assume that during a single evolution time t_{k}, ΔB_{z} is static. This assumption is valid when , where is the diffusion rate of ΔB_{z}. For an estimation of ΔB_{z} with N different measurements, we generate a random walk of N different values for ΔB_{z} (using the measured diffusion), simulate the outcome of each measurement, and compute the Bayesian estimate of ΔB_{z} using the simulated outcomes. By repeating this procedure 4,096 times, and using the mean squared error, MSE=‹(ΔB_{z}−ΔB_{z}^{estimated})^{2}› as a metric for performance, we can find the optimal number of measurements to perform. To include the entire error budget of the FPGA apparatus, we add to this MSE the error from the phase noise of the VCO, the measured voltage noise on the analogue output controlling the VCO, and the diffusion of ΔB_{z} during the ‘operation’ period of the experiment.
Software postprocessing
The estimate of ΔB_{z} can be independently verified using software analysis. In this experiment, we use the same method to estimate ΔB_{z} as in the adaptive control experiment, but in the operation segment perform oscillations around ΔB_{z} for verification. We choose m different evolution times and measure each n times for a total of N_{tot}=m × n measurements of ΔB_{z}. In the ith experiment (i=1,2,…N_{tot}), we evolve for a time t_{evo,i}, accumulating phase φ_{i}=ΔB_{z,i}t_{evo,i}. Because we make a precise measurement of ΔB_{z} at the start of each experiment, we can employ it to rescale the time, t_{evo,i}, so that the phase accumulated for a given time is constant using the equation,
This sets φ_{i}(τ_{i})=‹ΔB_{z}›τ_{i}, with residual error arising from inaccuracy in the estimate of ΔB_{z,i}. The data are then sorted by τ, and points of similar τ are averaged using a Gaussian window with σ_{τ}=0.5 ns≪T≈16 ns, where T is the period of the oscillations.
Additional information
How to cite this article: Shulman, M. D. et al. Suppressing qubit dephasing using realtime Hamiltonian estimation. Nat. Commun. 5:5156 doi: 10.1038/ncomms6156 (2014).
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Acknowledgements
We acknowledge James MacArthur for building the correlated double sampler. This research was funded by the United States Department of Defense, the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), and the Army Research Office grant (W911NF1110068 and W911NF1110068). All statements of fact, opinion or conclusions contained herein are those of the authors and should not be construed as representing the official views or policies either expressly or implied of of IARPA, the ODNI, or the U.S. Government. S.P.H was supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. A.C.D. acknowledges discussions with Matthew Wadrop regarding extracting diffusion constants. A.C.D. and S.D.B. acknowledge support from the ARC via the Centre of Excellence in Engineering Quantum Systems (EQuS) project number CE110001013. This work was performed in part at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Infrastructure Network (NNIN), which is supported by the National Science Foundation under NSF award no. ECS0335765. CNS is a part of Harvard University.
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V.U. prepared the crystal M.D.S. fabricated the sample, J.M.N. programmed the FPGA, M.D.S, S.P.H., J.M.N., S.D.B, A.C.D, and A.Y. carried out the experiment, analyzed the data, and wrote the paper.
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Supplementary Figures 14 and Supplementary Notes 12 (PDF 868 kb)
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Shulman, M., Harvey, S., Nichol, J. et al. Suppressing qubit dephasing using realtime Hamiltonian estimation. Nat Commun 5, 5156 (2014). https://doi.org/10.1038/ncomms6156
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DOI: https://doi.org/10.1038/ncomms6156
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