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arXiv:1607.08093 · Physical Review A

Benchmarking of dynamically corrected gates for the exchange-only spin qubit in a 1/f noise environment

Chengxian Zhang, Xu-Chen Yang, Xin Wang

Published Oct 17, 2016• 8 citations
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Abstract

We study theoretically the responses of the dynamically corrected gates to time-dependent noises in the exchange-only spin-qubit system. We consider $1/f$ noises having spectra proportional to $1/{\ensuremath{\omega}}^{\ensuremath{\alpha}}$, where the exponent $\ensuremath{\alpha}$ indicates the strength of correlation within the noise. The quantum gate errors due to noises are extracted from a numerical simulation of randomized benchmarking and are compared between the application of uncorrected operations and that of dynamically corrected gates robust against the hyperfine noise. We have found that for $\ensuremath{\alpha}\ensuremath{\gtrsim}1.5$, the dynamically corrected gates offer a considerable reduction in the gate error and such a reduction is approximately two orders of magnitude for the experimentally relevant noise exponent. On the other hand, no improvement of the gate fidelity is provided for $\ensuremath{\alpha}\ensuremath{\lesssim}1.5$. This critical value of ${\ensuremath{\alpha}}_{c}\ensuremath{\approx}1.5$ is larger than that for the cases for the singlet-triplet qubits. The filter transfer functions corresponding to the dynamically corrected gates are also computed and compared to those derived from uncorrected pulses. Our results suggest that the dynamically corrected gates are useful measures to suppress the hyperfine noise when operating the exchange-only qubits.

DOI
10.1103/physreva.94.042323

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•Jan 1, 2019•0 citations
We study theoretically the responses of the dynamically corrected gates to time-dependent noises in the exchange-only spin-qubit system. We consider 1/f1/f1/f noises having spectra proportional to 1/\ensuremathω\ensuremathα1/{\ensuremath{\omega}}^{\ensuremath{\alpha}}1/\ensuremathω\ensuremathα, where the exponent \ensuremathα\ensuremath{\alpha}\ensuremathα indicates the strength of correlation within the noise. The quantum gate errors due to noises are extracted from a numerical simulation of randomized benchmarking and are compared between the application of uncorrected operations and that of dynamically corrected gates robust against the hyperfine noise. We have found that for \ensuremathα\ensuremath≳1.5\ensuremath{\alpha}\ensuremath{\gtrsim}1.5\ensuremathα\ensuremath≳1.5, the dynamically corrected gates offer a considerable reduction in the gate error and such a reduction is approximately two orders of magnitude for the experimentally relevant noise exponent. On the other hand, no improvement of the gate fidelity is provided for \ensuremathα\ensuremath≲1.5\ensuremath{\alpha}\ensuremath{\lesssim}1.5\ensuremathα\ensuremath≲1.5. This critical value of \ensuremathαc\ensuremath≈1.5{\ensuremath{\alpha}}_{c}\ensuremath{\approx}1.5\ensuremathαc​\ensuremath≈1.5 is larger than that for the cases for the singlet-triplet qubits. The filter transfer functions corresponding to the dynamically corrected gates are also computed and compared to those derived from uncorrected pulses. Our results suggest that the dynamically corrected gates are useful measures to suppress the hyperfine noise when operating the exchange-only qubits.
Semiconductor quantum dot spin qubits are promising candidates for quantum computing. In these systems, the dynamically corrected gates offer considerable reduction of gate errors and are therefore of great interest both theoretically and experimentally. They are, however, designed under the static-noise model and may be considered as low-frequency filters. In this work, we perform a comprehensive theoretical study of the response of a type of dynamically corrected gates, namely the supcode for singlet-triplet qubits, to realistic 1/f noises with frequency spectra 1/ω(α). Through randomized benchmarking, we have found that supcode offers improvement of the gate fidelity for α 1 and the improvement becomes exponentially more pronounced with the increase of the noise exponent in the range 1 α ≤ 3 studied. On the other hand, for small α, supcode will not offer any improvement. The δJ-supcode, specifically designed for systems where the nuclear noise is absent, is found to offer additional error reduction than the full supcode for charge noises. The computed filter transfer functions of the supcode gates are also presented.
We present composite pulse sequences that perform fault-tolerant two-qubit gate operations on exchange-only quantum-dot spin qubits in various experimentally relevant geometries. We show how to perform dynamically corrected two-qubit gates in exchange-only systems with the leading hyperfine error term canceled. These pulse sequences are constructed to conform to the realistic experimental constraint of strictly non-negative couplings. We establish that our proposed pulse sequences lead to several orders of magnitude improvement in the gate fidelity compared with their uncorrected counterparts. Together with single-qubit dynamically corrected gates, our results enable noise-resistant universal quantum operations with exchange-only qubits.
We present a comprehensive theoretical treatment of supcode, a method for generating dynamically corrected quantum gate operations, which are immune to random noise in the environment, by using carefully designed sequences of soft pulses. supcode enables dynamical error suppression even when the control field is constrained to be positive and uniaxial, making it particularly suited to counteracting the effects of noise in systems subject to these constraints such as singlet-triplet qubits. We describe and explain in detail how to generate supcode pulse sequences for arbitrary single-qubit gates and provide several explicit examples of sequences that implement commonly used gates, including the single-qubit Clifford gates. We develop sequences for noise-resistant two-qubit gates for two exchange-coupled singlet-triplet qubits by cascading robust single-qubit gates, leading to a 35% reduction in gate time compared to previous works. This cascade approach can be scaled up to produce gates for an arbitrary-length spin qubit array and is thus relevant to scalable quantum computing architectures. To more accurately describe real spin qubit experiments, we show how to design sequences that incorporate additional features and practical constraints such as sample-specific charge noise models and finite pulse rise times. We provide a detailed analysis based on randomized benchmarking to show how supcode gates perform under realistic $1/{f}^{\ensuremath{\alpha}}$ noise and find a strong dependence of gate fidelity on the exponent $\ensuremath{\alpha}$, with best performance for $\ensuremath{\alpha}>1$. Our supcode sequences can therefore be used to implement robust universal quantum computation while accommodating the fundamental constraints and experimental realities of singlet-triplet qubits.
We present a set of experimentally feasible pulse sequences that implement any single-qubit gate on a singlet-triplet spin qubit and demonstrate that these new sequences are up to three times faster than existing sequences in the literature. We show that these sequences can be extended to incorporate built-in dynamical error correction, yielding gates that are robust to both charge and magnetic field noise and up to twice as fast as previous dynamically corrected gate schemes. We present a thorough comparison of the performance of our new sequences with that of several existing ones using randomized benchmarking, considering both quasistatic and $1/{f}^{\ensuremath{\alpha}}$ noise models. We provide our results both as a function of evolution time and as a function of the number of gates, which respectively yield both an effective coherence time and an estimate of the number of gates that can be performed within this coherence time. We determine which set of pulse sequences gives the best results for a wide range of noise strengths and power spectra. Overall, we find that the traditional, slower sequences perform best when there is no field noise or when the noise contains significant high-frequency components; otherwise, our new, fast sequences exhibit the best performance.
In addition to magnetic field and electric charge noise adversely affecting spin-qubit operations, performing single-qubit gates on one of multiple coupled singlet-triplet qubits presents a new challenge: crosstalk, which is inevitable (and must be minimized) in any multiqubit quantum computing architecture. We develop a set of dynamically corrected pulse sequences that are designed to cancel the effects of both types of noise (i.e., field and charge) as well as crosstalk to leading order, and provide parameters for these corrected sequences for all 24 of the single-qubit Clifford gates. We then provide an estimate of the error as a function of the noise and capacitive coupling to compare the fidelity of our corrected gates to their uncorrected versions. Dynamical error correction protocols presented in this work are important for the next generation of singlet-triplet qubit devices where coupling among many qubits will become relevant.
A key ingredient for fault-tolerant quantum computers are sufficiently accurate logic gates on single and multiple qubits in the presence of decohering noise. In this thesis, we theoretically develop and experimentally demonstrate such high-fidelity quantum gates for semiconductor-based quantum computing. Specifically, we consider a qubit encoded in the singlet and one triplet state of two electron spins in GaAs. While its potential for optical coupling makes GaAs an attractive material, highly auto-correlated magnetic field noise from fluctuating nuclear spins leads to considerably lower coherence times than for Si-based devices, where nuclear spins can be removed by isotopic purification. In addition to noise, the control methods used in earlier experiments are based on unrealistic approximations. For these reasons, fidelities well above 99 %, as required by current quantum error correction (QEC) schemes, have not been obtained before this thesis. To tackle these issues, we extend the filter function formalism to describe quantum gates and processes in the presence of experimentally relevant non-Markovian noise. Since this formalism can consider all relevant noise sources in a computationally efficient manner, it can be used to find optimal control pulses by numerical optimization, leading to predicted single-qubit gate fidelities of 99.9 %. Furthermore, we deal with the considerable control challenges associated with this qubit type by experimental calibration of the optimized control pulses. To this end, we extend our previous experimental gate set calibration (GSC) routine to remove systematic gate errors on an arbitrary number of qubits. We apply the numerically optimized single-qubit control pulses to our GaAs sample and experimentally calibrate them with GSC. This procedure yields an average gate fidelity of 99.50 ± 0.04 % and a low leakage rate of 0.13 ± 0.03 % out of the computational subspace, characterized by randomized benchmarking. We also optimize realistic two-qubit control pulses, considering current control hardware as well as interqubit Coulomb and exchange coupling that cannot be fully turned off. Using measured noise spectra, we show that two-qubit gate fidelities of 99.90 % can be reached in GaAs, while 99.99 % can be achieved in Si. These results demonstrate that high-fidelity gates can be realized even in the presence of nuclear spins as in GaAs.