Compressed Gate Characterization for Quantum Devices with Time-Correlated Noise

<sec>Spin qubits in silicon-based semiconductor quantum dots have become one of the prominent candidates for realizing fault-tolerant quantum computing due to their long coherence time, good controllability, and compatibility with modern advanced integrated circuit manufacturing processes. In recent years, due to the remarkable advances in silicon-based materials, the structure of quantum dot and its fabrication process, and qubit manipulation technology, the great progress of high-fidelity state preparation and readout, single- and two-qubit gates of spin qubits for silicon based semiconductor quantum computation has been achieved. The control fidelities for single- and two-qubit gates all exceed 99%—fault tolerance threshold required by the surface code known for its exceptionally high tolerance to errors. In this paper, we briefly introduce the basic concepts of silicon-based semiconductor quantum dots, discuss the state-of-art technologies used to improve the fidelities of single- and two-qubit gates, and finally highlight the research directions that should be focused on.</sec><sec>This paper is organized as follows. Firstly, we introduce three major types of quantum dot (QD) devices fabricated on different silicon-based substrates, including Si/SiGe heterojunction and Si/SiO<sub>2</sub>. The spin degree of electron or nucleus hosted in QD can be encoded into spin qubits. Electron spin qubits can be thermally initialized to ground state by using an electron reservoirs, and can be read out by spin-charge conversion mechanism: energy-selective readout (Elzerman readout) with reservoirs or Pauli spin blockade (PSB) without reservoirs. Additionally, high fidelity single-shot readout has been demonstrated by using radio-frequency gate reflectometry combined with the PSB, which has unique advantages in large-scale qubit array. To coherently control the spin qubits, electron dipole resonance (ESR) or electron dipole spin resonance (EDSR) for electron and nuclear magnetic resonance (NMR) for nucleus are introduced. With the help of isotope purification greatly improving the dephasing time of qubit and fast single-qubit manipulation based on EDSR, fidelity above 99.9% can be reached. For the two-qubit gates based on exchange interaction between electron spins, the strength of interaction <inline-formula><tex-math id="M1">\begin{document}$ J $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20221900_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20221900_M1.png"/></alternatives></inline-formula> combined with Zeeman energy difference <inline-formula><tex-math id="M2">\begin{document}$ \Delta {E}_{z} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20221900_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20221900_M2.png"/></alternatives></inline-formula> determines the energy levels of system, which lead to the different two-qubit gates, such as controlled-Z (CZ), controlled-Rotation (CROT), and the square root of the SWAP gate (<inline-formula><tex-math id="M3">\begin{document}$ \sqrt{\rm{S}\rm{W}\rm{A}\rm{P}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20221900_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20221900_M3.png"/></alternatives></inline-formula>) gates. In order to improve the fidelity of two-qubit gates, a series of key technologies is used experimentally, they being isotope purification, symmetry operation, careful Hamiltonian engineering, and gate set tomography. Fidelity of two-qubit gates exceeding 99% has been demonstrated for electron spin qubits in Si/SiGe quantum dots and nuclear spin qubits in donors. These advances have pushed the silicon-based spin qubit platform to become a major stepping stone towards fault-tolerant quantum computation. Finally, we discuss the future study of spin qubits, that is, how to effectively expand the number of qubits, and many other problems to be explored and solved.</sec>

Quantum computation requires high-fidelity single-qubit and two-qubit gates on a scalable platform. Silicon spin qubits are a promising platform toward realization of this goal. In this paper we show how to perform single-qubit and controlled-Z (cz) gates in a linear chain of three spin qubits with always-on exchange coupling, which is relevant for certain dot- and donor-based silicon devices. We also show how to make the cz gate robust against both charge noise and pulse length error using a two-tone pulse shaping method. The robust pulse maintains a fidelity of 99.99% at 3.5% fluctuations in exchange or pulse amplitude, which is an improvement over the uncorrected pulses where this fidelity can only be maintained for fluctuations in exchange up to 2% or up to 0.2% in amplitude.

Shuttling spins with high fidelity is a key requirement to scale up semiconducting quantum computers, enabling qubit entanglement over large distances and favoring the integration of control electronics on-chip. To decouple the spin from the unavoidable charge noise, state-of-the-art spin shuttlers try to minimize the inhomogeneity of the Zeeman field. However, this decoupling is challenging in otherwise promising quantum computing platforms such as hole spin qubits in silicon and germanium, characterized by a large spin-orbit interaction and an electrically tunable qubit frequency. In this work, we show that, surprisingly, the large inhomogeneity of the Zeeman field stabilizes the coherence of a moving spin state, thus also enabling high-fidelity shuttling in these systems. We relate this enhancement in fidelity to the deterministic dynamics of the spin that filters out the dominant low-frequency contributions of the charge noise. By simulating several different scenarios and noise sources, we show that this is a robust phenomenon generally occurring at large field inhomogeneity. By appropriately adjusting the motion of the quantum dot, we also design realistic protocols enabling faster and more coherent spin shuttling. Our findings are generally applicable to a wide range of setups and could pave the way toward large-scale quantum processors. Published by the American Physical Society 2024

A crucial task to obtain optimal and reliable quantum devices is to quantify their overall performance. The average fidelity of quantum gates is a particular figure of merit that can be estimated efficiently by randomized benchmarking (RB). However, the concept of gate-fidelity itself relies on the crucial assumption that noise behaves in a predictable, time-local, or so-called Markovian manner, whose breakdown can naturally become the leading source of errors as quantum devices scale in size and depth. We analytically show that error suppression techniques such as dynamical decoupling (DD) and Pauli-twirling can operationally Markovianize RB: (i) fast DD reduces non-Markovian RB to an exponential decay plus longer-time corrections, while on the other hand, (ii) Pauli-twirling generally does not affect the average, but (iii) it always suppresses the variance of such RB outputs. We demonstrate these effects numerically with a qubit noise model. Our results show that simple and efficient error suppression methods can simultaneously tame non-Markovian noise and allow for standard and reliable gate quality estimation, a fundamentally important task in the path toward fully functional quantum devices.

High-fidelity single- and multiqubit operations compose the backbone of quantum information processing. This fidelity is based on the ability to couple single- or two-qubit levels in an extremely coherent and precise manner. A necessary condition for coherent quantum evolution is a highly stable local oscillator driving these transitions. Here we study the effect of fast noise, that is, noise at frequencies much higher than the local oscillator linewidth, on the fidelity of one- and two-qubit gates in a trapped-ion system. We analyze and measure the effect of fast noise on single-qubit operations, including resonant $\ensuremath{\pi}$ rotations and off-resonant sideband transitions. We further numerically analyze the effect of fast phase noise on the M\o{}lmer-S\o{}rensen two-qubit gate. We find a unified and simple way to estimate the performance of all of these operations through a single parameter given by the noise power spectral density at the qubit response frequency. While our analysis focuses on phase noise and on trapped-ion systems, it is relevant for other sources of fast noise as well as for other qubit systems in which spinlike qubits are coupled by a common bosonic field. Our analysis can help in guiding the design of quantum hardware platforms and gates, improving their fidelity towards fault-tolerant quantum computing.

Fault-tolerant quantum computers that can solve hard problems rely on quantum error correction1. One of the most promising error correction codes is the surface code2, which requires universal gate fidelities exceeding an error correction threshold of 99 per cent3. Among the many qubit platforms, only superconducting circuits4, trapped ions5 and nitrogen-vacancy centres in diamond6 have delivered this requirement. Electron spin qubits in silicon7-15 are particularly promising for a large-scale quantum computer owing to their nanofabrication capability, but the two-qubit gate fidelity has been limited to 98 per cent owing to the slow operation16. Here we demonstrate a two-qubit gate fidelity of 99.5 per cent, along with single-qubit gate fidelities of 99.8 per cent, in silicon spin qubits by fast electrical control using a micromagnet-induced gradient field and a tunable two-qubit coupling. We identify the qubit rotation speed and coupling strength where we robustly achieve high-fidelity gates. We realize Deutsch-Jozsa and Grover search algorithms with high success rates using our universal gate set. Our results demonstrate universal gate fidelity beyond the fault-tolerance threshold and may enable scalable silicon quantum computers.

The 1-D diffusion coefficient associated with a charged atom fluctuating in an ion-channel binding pocket is statistically analyzed. More specifically, unconstrained and constrained molecular dynamics simulations of potassium in gramicidin A are studied. Time domain transition density based inference methods are used to fit simple stochastic differential equations and also to carry out frequentist goodness of fit tests. Particular attention is paid to varying the time between adjacent time series observations due to the well-known "non-Markovian noise" that can appear in this system due to inertia and other unresolved coordinates influencing the dynamics. Different types of non-Markovian noise are shown by the goodness of fit tests to be statistically significant on vastly different time scales. On intermediate scales, a Markovian model is not rejected by the tests; models calibrated at these intermediate scales demonstrate a predictive capability for some physical quantities. However, in this intermediate regime, ergodic sampling does not occur over the length of a time series, but a local diffusion coefficient is deemed statistically acceptable for the observed raw data. It is demonstrated that a linear mixed effects model can be used to summarize the variation induced by slow unresolved degrees of freedom acting as a non-Markovian noise source. The utility of quantitative criteria for assessing low-dimensional stochastic models calibrated from time series generated by high-dimensional biomolecular systems is briefly discussed. Less coarse-grained data summaries of this type show promise for better understanding the kinetic signature of unresolved degrees of freedom in time series coming from simulations and single-molecule experiments.

Rapid progress in semiconductor spin qubits has enabled experimental demonstrations of a two-qubit logic gate. Understanding spin decoherence in a two-qubit logic gate is necessary for optimal qubit operation. We study spin decoherence due to 1/f charge noise for two electrons in a double quantum dot used for a two-qubit controlled-phase gate. In contrast to the usual belief, spin decoherence can be dominated by the tunneling noise from 1/f charge noise instead of the detuning noise. Tunneling noise can dominate because the effect of tunneling noise on the spin qubit is first order in the charge admixture; while the effect of the detuning noise is only second order. The different orders of contributions result in different detuning dependence of the decoherence, which provides a way to identify the noise source. We find that decoherence in a recent two-qubit experiment was dominated by the tunneling noise from 1/f charge noise. The results illustrate the importance of considering tunneling noise to design optimal operation of spin qubits.

Recent progress in atomistic modelling and simulations of donor spin qubits in silicon

In recent advancements of quantum computing utilizing spin qubits, it has been demonstrated that this platform possesses the potential for implementing two-qubit gates with fidelities exceeding 99.5%. However, as with other qubit platforms, it is not feasible to completely turn qubit couplings off. This study aims to investigate the impact of coherent error matrices in gate set tomography by employing a double quantum dot. We evaluate the infidelity caused by residual exchange between spins and compare various mitigation approaches, including the use of adjusted timing through simple drives, considering different parameter settings in the presence of charge noise. Furthermore, we extend our analysis to larger arrays of exchange-coupled spin qubits to provide an estimation of the expected fidelity. In particular, we demonstrate the influence of residual exchange on a single-qubit Y gate and the native two-qubit SWAP gate in a linear chain. Our findings emphasize the significance of accounting for residual exchange when scaling up spin-qubit devices and highlight the tradeoff between the effects of charge noise and residual exchange in mitigation techniques. Published by the American Physical Society 2024

The current advancement in quantum computers has been focusing on increasing the number of qubits and enhancing their fidelity. However, the available quantum devices, known as Intermediate Scale Quantum (NISQ) computers, still suffer from different sources of noise that impact their reliability. Thus, practical noise modeling is of great importance in the development of quantum error mitigation approaches. In this paper, we propose a Machine Learning (ML)-based scheme to predict the output fidelity of the quantum circuit executed on NISQ devices. We show the benefit of using Graph Neural Network (GNN)-based models compared to traditional ML-based models in capturing the quantum circuit structure in addition to its gates’ features, which enable characterizing unpredicted quantum circuit errors. We use different metrics to measure the fidelity of the quantum circuit output. Our experimental results using different quantum algorithms executed on IBM Q Guadalupe quantum computer show the high prediction accuracy of our ML reliability models. Our results also show that our models can guide the single-qubit gate rescheduling to improve the output fidelity of the quantum circuit without the need for prior execution of dedicated calibration circuits.

Spin qubits in silicon are strong contenders for the realization of a practical quantum computer, having demonstrated single- and two-qubit gates with fidelities above the fault-tolerant threshold, and entanglement of three qubits. However, maintaining high-fidelity operations while increasing the qubit count remains challenging and therefore only two-qubit algorithms have been executed. Here we utilize a four-qubit silicon processor with all control fidelities above the fault-tolerant threshold and demonstrate a three-qubit Grover’s search algorithm with a ~95% probability of finding the marked state. Our processor is made of three phosphorus atoms precision-patterned into isotopically pure silicon, which localise one electron. The long coherence times of the qubits enable single-qubit fidelities above 99.9% for all qubits. Moreover, the efficient single-pulse multi-qubit operations enabled by the electron–nuclear hyperfine interaction facilitate controlled-Z gates between all pairs of nuclear spins with fidelities above 99% when using the electron as an ancilla. These control fidelities, combined with high-fidelity non-demolition readout of all nuclear spins, allow the creation of a three-qubit Greenberger–Horne–Zeilinger state with 96.2% fidelity. Looking ahead, coupling neighbouring nuclear spin registers, as the one shown here, via electron–electron exchange may enable larger, fault-tolerant quantum processors.

Quantum system characterization techniques represent the front line in the identification and mitigation of noise in quantum computing, but can be expensive in terms of quantum resources and time to repeatedly employ. Another challenging aspect is that parameters governing the performance of various operations tend to drift over time, and monitoring these is hence a difficult task. One of the most promising characterization techniques, gate set tomography (GST), provides a self-consistent estimate of the completely positive, trace-preserving (CPTP) maps for a complete set of gates, as well as preparation and measurement operators. We develop a method for performance optimization seeded by tomography (POST), which couples the power of GST with a classical optimization routine to achieve a consistent gate improvement in just a short number of steps within a given calibration cycle. By construction, the POST procedure finds the best available gate operation given the hardware, and is therefore robust to the effects of drift. Further, in comparison to other quantum error mitigation techniques, it builds upon a one-time application of GST. To demonstrate the performance of this method on a real quantum computer, we map out the operations of six qubit pairs on the superconducting \emph{ibmq\_poughkeepsie} quantum device. Under the restriction of logical-only control, we monitor the performance of the POST approach on a chosen CNOT gate over a period of six weeks. In this time, we achieve a consistent improvement in gate fidelity, averaging a fidelity increase of 21.1\% as measured by randomized benchmarking. The POST approach should find wide applicability as it is hardware agnostic, and can be applied at the upper logical level or at a deeper pulse control level.

We demonstrate that exciting possible realizations of quantum Floquet matter are within reach for modern silicon spin qubits based in quantum dots, most notably the discrete time crystal (DTC). This is significant given that spin qubits have fallen behind other qubit architectures in terms of size and control. However, silicon spin qubits are especially well suited to this task as the charge noise that usually foils gate operations can now be leveraged as an asset in this time-crystal realization. We illustrate differences between prethermal phenomena and true time-crystalline spatiotemporal order. We demonstrate that even for a spin chain of four qubits, rich regime structures can be established by observing signatures of the discrete time crystal and the Floquet symmetry-protected topological regime both distinct from the thermal regime. We also analyze the persistence of these signatures at longer chain lengths, showing that the DTC lifetime grows exponentially with the system length and that these signatures may even be detectable for chains as small as three qubits. We also discuss the effects of longer pulse durations and the effectiveness of pulse sequences for converting the exchange interaction to an Ising model. Our theoretical predictions are well suited for immediate experimental implementations using currently existing quantum dot spin qubit systems.

Scalable quantum processors require high-fidelity universal quantum logic operations in a manufacturable physical platform. Donors in silicon provide atomic size, excellent quantum coherence and compatibility with standard semiconductor processing, but no entanglement between donor-bound electron spins has been demonstrated to date. Here we present the experimental demonstration and tomography of universal one- and two-qubit gates in a system of two weakly exchange-coupled electrons, bound to single phosphorus donors introduced in silicon by ion implantation. We observe that the exchange interaction has no effect on the qubit coherence. We quantify the fidelity of the quantum operations using gate set tomography (GST), and we use the universal gate set to create entangled Bell states of the electrons spins, with fidelity 91.3 ± 3.0%, and concurrence 0.87 ± 0.05. These results form the necessary basis for scaling up donor-based quantum computers.