Tutorial: Gate-based superconducting quantum computing
Sangil Kwon, Akiyoshi Tomonaga, Gopika Lakshmi Bhai, Simon J. Devitt, Jaw-Shen Tsai
介绍理解和构建基于门的超导量子计算系统的基本概念元素。
在本教程中，我们介绍基本概念元素以理解和构建基于门的超导量子计算系统。
In this tutorial, we introduce basic conceptual elements to understand and build a gate-based superconducting quantum computing system.
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In this tutorial we review the basic building blocks of Quantum Information Processing with cold trapped atomic-ions. We mainly focus on methods to implement single-qubit rotations and two-qubit entangling gates, which form a universal set of quantum gates. Different ion qubit choices and their respective gate implementations are described.
Tutorial on Superconducting Quantum Circuits: From Basics to Applications
本教程为本科生提供超导量子电路的基础知识，涵盖超导性、约瑟夫森效应、transmon量子比特及其哈密顿量推导，通过数值模拟展示强耦合 regime 的相干能量交换。
With the increasing rise of publicly available high level quantum computing languages, the field of Quantum Computing has reached an important milestone of separation of software from hardware. Consequently, the study of Quantum Algorithms is beginning to emerge as university courses and disciplines around the world, spanning physics, math, and computer science departments alike. As a continuation to its predecessor: "Introduction to Coding Quantum Algorithms: A Tutorial Series Using Qiskit", this tutorial series aims to help understand several of the most promising quantum algorithms to date,…
Quantum algorithms often benefit from the ability to execute multi-qubit (>2) gates. To date such multi-qubit gates are typically decomposed into single- and two-qubit gates, particularly in superconducting qubit architectures. The ability to perform multi-qubit operations in a single step could vastly improve the fidelity and execution time of many algorithms. Here, we propose a single shot method for executing an i-Toffoli gate, a three-qubit gate gate with two control and one target qubit, using currently existing superconducting hardware. We show numerical evidence for a process fidelity o…
We propose a way for realizing a two-qubit controlled phase gate with superconducting quantum interference devices (SQUIDs) coupled to a superconducting resonator. In this proposal, the two lowest levels of each SQUID serve as the logical states and two intermediate levels of each SQUID are used for the gate realization. We show that neither adjustment of SQUID level spacings during the gate operation nor uniformity in SQUID parameters is required by this proposal. In addition, this proposal does not require the adiabatic passage or a second-order detuning and thus the gate is much faster.
Nonadiabatic holonomic quantum computation has received increasing attention due to its robustness against control errors as well as high-speed realization. The original protocol of nonadiabatic holonomic one-qubit gates has been experimentally demonstrated with superconducting transmon qutrit. However, the original protocol requires two noncommuting gates to realize an arbitrary one-qubit gate, which doubles the exposure time of gates to error sources and therefore makes the gates vulnerable to environment-induced decoherence. Single-shot protocol was subsequently proposed to realize an arbit…
We present "Diagrams of States", a way to graphically represent and analyze how quantum information is elaborated during the execution of quantum circuits. This introductory tutorial illustrates the basics, providing useful examples of quantum computations: elementary operations in single-qubit, two-qubit and three-qubit systems, immersions of gates on higher dimensional spaces, generation of single and multi-qubit states, procedures to synthesize unitary, controlled and diagonal matrices. To perform the analysis of quantum processes, we directly derive diagrams of states from physical impleme…
We propose how to realize a three-step controlled-phase gate of one superconducting qubit simultaneously controlling n qubits selected from N qubits in a cavity (1<n<N). The operation time of this gate is independent of the number n of qubits involved in the gate operation. This phase gate controlling at once n qubits is insensitive to the initial state of the cavity mode and can be used to produce an analogous CNOT gate simultaneously acting on n qubits.
A scheme for coupling superconducting charge qubits via a one-dimensional superconducting transmission line resonator is proposed. The qubits are working at their optimal points, where they are immune to the charge noise and possess long decoherence time. Analysis on the dynamical time evolution of the interaction is presented, which is shown to be insensitive to the initial state of the resonator field. This scheme enables fast gate operation and is readily scalable to multiqubit scenario.
Fast robust two-qubit gate operation with low susceptibility to crosstalk are the key to scalable quantum information processing. Parametrically driven gate is inherently insensitive to crosstalk while superadiabatic control can speed up the gate without losing accuracy. We propose and experimentally implement superadiabatic two-qubit gates using parametric modulation on superconducting quantum circuits. Our results demonstrate the preservation of adiabaticity at a gate speed close to the quantum limit, in addition to robustness against control instability. We demonstrate a CZ gate with error …
High-quality-factor 3D cavities in superconducting circuits are ideal candidates for bosonic logical qubits as their fidelity is limited only by the low photon loss rate. However, the transmon qubits that are used to manipulate bosonic qubits result in the emergence of additional relaxation and dephasing channels. In this work, a numerical study is performed to elucidate the effect of the various loss channels on the performance of logical gates on a bosonic qubit. A gate error model is developed that encapsulates the loss mechanisms for arbitrary gate operations and predicts experimentally ac…
Quantum gates based on geometric phases possess intrinsic noise-resilience features and therefore attract much attention. However, the implementations of previous geometric quantum computation typically require a long pulse time of gates. As a result, their experimental control inevitably suffers from the cumulative disturbances of systematic errors due to excessive time consumption. Here, we experimentally implement a set of noncyclic and nonadiabatic geometric quantum gates in a superconducting circuit, which greatly shortens the gate time. And also, we experimentally verify that our univers…
Superconducting qubits are a promising candidate for building a quantum computer. A continued challenge for fast yet accurate gates to minimize the effects of decoherence. Here we apply numerical methods to design fast entangling gates, specifically the controlled Z, in an architecture where two qubits are coupled via a resonator. We find that the gates can be sped up by a factor of two and reach any target fidelity. We also discuss how systematic errors arising from experimental conditions affect the pulses and how to remedy them, providing a strategy for the experimental implementation of ou…
The superconducting circuit companion -- an introduction with worked examples
本教程面向物理背景入门者，系统讲解超导量子比特电路分析方法，从电路图到量子化哈密顿量，附单/多量子比特实例，还补充高阶电路设计技术。
Pulse Design of Baseband Flux Control for Adiabatic Controlled-Phase Gates in Superconducting Circuits
本文针对超导量子计算机中易出错的二比特门瓶颈，研究了绝热受控相位门的实现，将其控制轨迹设计表述为脉冲设计问题，并在模拟中证明基于 Chebyshev 的轨迹较 Slepian 轨迹能将门非保真度平均降低 23.3%。
Based on a quantum analysis of two capacitively coupled current-biased Josephson junctions, we propose two fundamental two-qubit quantum logic gates. Each of these gates, when supplemented by single-qubit operations, is sufficient for universal quantum computation. Numerical solutions of the time-dependent Schroedinger equation demonstrate that these operations can be performed with good fidelity.
High-quality two-qubit gate operations are crucial for scalable quantum information processing. Often, the gate fidelity is compromised when the system becomes more integrated. Therefore, a low-error-rate, easy-to-scale two-qubit gate scheme is highly desirable. Here, we experimentally demonstrate a new two-qubit gate scheme that exploits fixed-frequency qubits and a tunable coupler in a superconducting quantum circuit. The scheme requires less control lines, reduces crosstalk effect, simplifies calibration procedures, yet produces a controlled-Z gate in 30ns with a high fidelity of 99.5%, der…
We present a scheme to achieve maximally entangled states, controlled phase-shift gate, and SWAP gate for two superconducting-quantum-interference-device (SQUID) qubits, by placing SQUIDs in a microwave cavity. We also show how to transfer quantum information from one SQUID qubit to another. In this scheme, no transfer of quantum information between the SQUIDs and the cavity is required, the cavity field is only virtually excited and thus the requirement on the quality factor of the cavity is greatly relaxed.
High-fidelity three-qubit iToffoli gate for fixed-frequency superconducting qubits
研究固定频率超导量子比特中高保真三比特 iToffoli 门的实现，基于交叉共振效应并同步施加微波脉冲，达到 98.26% 的过程保真度，为多比特门开发开辟了新路径。
Robust gates with spin-locked superconducting qubits
本文提出将自旋锁定（连续动力学解耦）与基于耦合器的transmon CZ门结合的理论方案，通过解析和数值结果证明其能有效降低门误差，助力未来容错量子计算。
Superconducting circuits with coupler architecture receive considerable attention due to their advantages in tunability and scalability. Although single-qubit gates with low error have been achieved, high-fidelity two-qubit gates in coupler architecture are still challenging. This paper pays special attention to examining the gate error sources and primarily concentrates on the related physical mechanism of ZZ parasitic couplings using a systematic effective Hamiltonian approach. Benefiting from the effective Hamiltonian, we provide simple and straightforward insight into the ZZ parasitic coup…
High fidelity two-qubit gates are fundamental for scaling up the superconducting number. We use two qubits coupled via a frequency-tunable coupler which can adjust the coupling strength, and demonstrate the CZ gate using two different schemes, adiabatic and diabatic methods. The Clifford based Randomized Benchmarking (RB) method is used to assess and optimize the CZ gate fidelity. The fidelity of adiabatic and diabatic CZ gates are 99.53(8)% and 98.72(2)%, respectively. We also analyze the errors induced by the decoherence. Comparing to 30 ns duration time of adiabatic CZ gate, the duration ti…
Based on a `shortcut-to-adiabaticity' (STA) scheme, we theoretically design and experimentally realize a set of high-fidelity single-qubit quantum gates in a superconducting Xmon qubit system. Through a precise microwave control, the qubit is driven to follow a fast `adiabatic' trajectory with the assistance of a counter-diabatic field and the correction of derivative removal by adiabatic gates. The experimental measurements of quantum process tomography and interleaved randomized benchmarking show that the process fidelities of our STA quantum gates are higher than 94.9% and the gate fideliti…
Present-day quantum computers suffer from various noises or errors such as gate error, relaxation, dephasing, readout error, and crosstalk. Besides, they offer a limited number of qubits with restrictive connectivity. Therefore, quantum programs running these computers face resilience issues and low output fidelities. The noise in the cloud-based access of quantum computers also introduces new modes of security and privacy issues. Furthermore, quantum computers face several threat models from insider and outsider adversaries including input tampering, program misallocation, fault injection, Re…
Crosstalk-robust superconducting two-qubit geometric gates using tunable couplers
本文提出一种基于耦合器的超导双量子位几何门方案，通过引入参数自由度，有效抑制串扰并实现快速门操作，展示了在超导量子计算中实现鲁棒高效双量子位门的可行性。
We present a Superconducting Planar ARchitecture for Quantum Simulations (SPARQS) intended to implement a scalable qubit layout for quantum simulators. To this end, we describe the iFREDKIN gate as a controlled entangler for the simulation of Fermionic systems that is advantageous if it can be directly implemented. Using optimal control, we show that and how this gate can be efficiently implemented in the SPARQS circuit, making it a promising platform and control scheme for quantum simulations. Such a quantum simulator can be built with current quantum technologies to advance the design of mol…
A Hardware-Efficient Mølmer-Sørensen Gate for Superconducting Quantum Computers
本文在IBM超导量子处理器上实现了硬件高效的Mølmer-Sørensen门，通过量子过程层析成像表征其性能，达到92.47%的过程保真度，与原生CX门（93.02%）相当，并验证了贝尔态制备的成功率，表明非原生纠缠门可优化至与硬件原生操作相当的性能。
In this paper, we propose a scheme to implement two-qubit logic gates with a controllable and selective interaction in a scalable superconducting circuit of charge qubits. A nanomechanical resonator is used as a data bus to connect qubits. It is indicated that a controllable interaction between qubits can be obtained by making use of the data bus. It is shown that a selective interaction between qubits can be realized when many qubits are involved in the system under our consideration.
We have implemented a Walsh-Hadamard gate, which performs a quantum Fourier transform, in a superconducting qutrit. The qutrit is encoded in the lowest three energy levels of a capacitively shunted flux device, operated at the optimal flux-symmetry point. We use an efficient decomposition of the Walsh-Hadamard gate into two unitaries, generated by off-diagonal and diagonal Hamiltonians respectively. The gate implementation utilizes simultaneous driving of all three transitions between the three pairs of energy levels of the qutrit, one of which is implemented with a two-photon process. The gat…
We propose an approach to realize an $n$-qubit controlled-$U$ gate with superconducting quantum interference devices (SQUIDs) in cavity QED. In this approach, the two lowest levels of a SQUID represent the two logical states of a qubit while a higher-energy intermediate level serves the gate manipulation. Our method operates essentially by creating a single photon through one of the control SQUIDs and then performing an arbitrary unitrary transformation on the target SQUID with the assistance of the cavity photon. In addition, we show that the method can be applied to implement an $n $% -qubit…
Demonstrating a universal logical gate set in error-detecting surface codes on a superconducting quantum processor
本研究针对超导系统中表面码逻辑量子比特两量子比特门缺失的问题，用处理器\textit{Wukong}在距离-2表面码上实现通用逻辑门集合，含横向CNOT与任意单量子比特旋转门，还容错制备逻辑贝尔态，证实纠缠，推动超导容错量子计算发展。
SU(4) gate design via unitary process tomography: its application to cross-resonance based superconducting quantum devices
本文提出利用CR哈密顿量传播子与Weyl腔几何结构，在超导量子器件上实现无回波的连续R_{ZZ}(θ)门，并验证了其在SU(4)门设计中的保真度与时间优势。
Using geometric phases to realize noise-resilient quantum computing is an important method to enhance the control fidelity. In this work, we experimentally realize a universal nonadiabatic geometric quantum gate set in a superconducting qubit chain. We characterize the realized single- and two-qubit geometric gates with both quantum process tomography and randomized benchmarking methods. The measured average fidelities for the single-qubit rotation gates and two-qubit controlled-Z gate are 0.9977(1) and 0.977(9), respectively. Besides, we also experimentally demonstrate the noise-resilient fea…
Holonomies, arising from non-Abelian geometric transformations of quantum states in Hilbert space, offer a promising way for quantum computation. These holonomies are not commutable and thus can be used for the realization of a universal set of quantum logic gates, where the global geometric feature may result in some noise-resilient advantages. Here we report the first on-chip realization of a non-Abelian geometric controlled-Not gate in a superconducting circuit, which is a building block for constructing a holonomic quantum computer. The conditional dynamics is achieved in an all-to-all con…
Recent experiments with silicon qubits demonstrated strong coupling of a microwave resonator to the spin of a single electron in a double quantum dot, opening up the possibility of long-range spin-spin interactions. We present our theoretical calculation of effective interactions between distant quantum dot spins coupled by a resonator, and propose a protocol for fast, high-fidelity two-qubit gates consistent with experimentally demonstrated capabilities. Our simulations show that, in the presence of noise, spin-spin entangling gates significantly outperform cavity-mediated gates on charge qub…
We present a way to realize a $n$-qubit controlled phase gate with superconducting quantum interference devices (SQUIDs) by coupling them to a superconducting resonator. In this proposal, the two logical states of a qubit are represented by the two lowest levels of a SQUID. An intermediate level of each SQUID is utilized to facilitate coherent control and manipulation of quantum states of the qubits. It is interesting to note that a $n$-qubit controlled phase gate can be achieved with $n$ SQUIDs by successively applying a $π/2$ Jaynes-Cummings pulse to each of the $n-1$ control SQUIDs before a…
Efficient two-qutrit gates in superconducting circuits using parametric coupling
针对耦合 qutrit 门因耗时长而保真度低的问题，提出基于两次部分态交换、局部操作和参数耦合的两 qutrit CZ 门方案。
The possibility to utilize different types of two-qubit gates on a single quantum computing platform adds flexibility in the decomposition of quantum algorithms. A larger hardware-native gate set may decrease the number of required gates, provided that all gates are realized with high fidelity. Here, we benchmark both controlled-Z (CZ) and exchange-type (iSWAP) gates using a parametrically driven tunable coupler that mediates the interaction between two superconducting qubits. Using randomized benchmarking protocols we estimate an error per gate of $0.9\pm0.03\%$ and $1.3\pm0.4\%$ fidelity for…
Robust, fast and high-fidelity composite single-qubit gates for superconducting transmon qubits
针对超导 transmon 量子比特的相干误差问题，提出基于复合脉冲的控制方法，实现鲁棒快速的高保真单量子比特门，并经仿真和四种独立技术验证。
We present a way for implementing an n-qubit controlled-rotation gate with three-level superconducting qubit systems in cavity QED. The two logical states of a qubit are represented by the two lowest levels of each system while a higher-energy level is used for the gate implementation. The method operates essentially by preparing a $W$ state conditioned on the states of the control qubits, creating a single photon in the cavity mode, and then performing an arbitrary rotation on the states of the target qubit with assistance of the cavity photon. It is interesting to note that the basic operati…
A Tutorial for Characterizing Transmon Qubits
本文提供可调谐Transmon量子比特表征与优化的完整工作流程教程，涵盖低温设置、参数放大器操作、磁通甜点识别、脉冲校准和读出优化等步骤。
Any quantum computational network can be constructed with a sequence of the two-qubit diagonal quantum gates and one-qubit gates in two-state quantum systems. The universal construction of these quantum gates in the quantum systems and of the quantum computational networks with these gates may be achieved with the help of the operator algebra structure of Hamiltonians of the systems and the properties of the multiple-quantum operator algebra subspaces of the Liouville operator space and the specific properties of the quantum algorithm corresponding to the quantum network. As an example, the tw…
We demonstrate fast two-qubit gates using a parity-violated superconducting qubit consisting of a capacitively-shunted asymmetric Josephson-junction loop under a finite magnetic flux bias. The second-order nonlinearity manifesting in the qubit enables the interaction with a neighboring single-junction transmon qubit via first-order inter-qubit sideband transitions with Rabi frequencies up to 30~MHz. Simultaneously, the unwanted static longitudinal~(ZZ) interaction is eliminated with ac Stark shifts induced by a continuous microwave drive near-resonant to the sideband transitions. The average f…
A scheme for implementing 2-qubit quantum controlled phase gate (QCPG) is proposed with two superconducting quantum interference devices (SQUIDs) in a cavity. The gate operations are realized within the two lower flux states of the SQUIDs by using a quantized cavity field and classical microwave pulses. Our scheme is achieved without any type of measurement, does not use the cavity mode as the data bus and only requires a very short resonant interaction of the SQUID-cavity system. As an application of the QCPG operation, we also propose a scheme for generating the cluster states of many SQUIDs…
Minimal-Energy Optimal Control of Tunable Two-Qubit Gates in Superconducting Platforms Using Continuous Dynamical Decoupling
本研究通过连续动态去耦与变分最小能量最优控制相结合，实现超导平台中高保真度的可调双量子比特门设计。
High-fidelity and robust quantum manipulation is the key for scalable quantum computation. Therefore, due to the intrinsic operational robustness, quantum manipulation induced by geometric phases is one of the promising candidates. However, the longer gate time for geometric operations and more physical-implementation difficulties hinder its practical and wide applications. Here, we propose a simplified implementation of universal holonomic quantum gates on superconducting circuits with experimentally demonstrated techniques, which can remove the two main challenges by introducing the time-opt…
Stark shift on a superconducting qubit in circuit quantum electrodynamics (QED) has been used to construct universal quantum entangling gates on superconducting resonators in previous works. It is a second-order coupling effect between the resonator and the qubit in the dispersive regime, which leads to a slow state-selective rotation on the qubit. Here, we present two proposals to construct the fast universal quantum gates on superconducting resonators in a microwave-photon quantum processor composed of multiple superconducting resonators coupled to a superconducting transmon qutrit, that is,…
Cross-resonance (CR) gate has emerged as a promising scheme for fault-tolerant quantum computation with fixed-frequency qubits. We experimentally implement entangling CR gate by using a microwave-only control in a tunable coupling superconducting circuit, where the tunable coupler provides extra degrees of freedom to verify optimal condition for constructing CR gate. By developing three-qubit CR Hamiltonian tomography protocol, we systematically investigate the dependency of gate fidelities on spurious qubit interactions and present the first experimental approach to the evaluation of the pert…
One-step implementation of nonadiabatic geometric fSim gate in superconducting circuits
本文提出一种在超导电路中一步实现非绝热几何fSim门的新方案，通过结合非绝热全息CP门与非循环几何iSWAP门，耗时减半且对参数波动和环境干扰具有鲁棒性，易于实验实现。
Demonstration of deterministic SWAP gate between superconducting and frequency-encoded microwave-photon qubits
该研究演示了超导原子量子比特与微波光子量子比特之间的确定性SWAP门，利用单光子拉曼相互作用实现高保真度双向量子态传输，为分布式量子计算提供关键元件。
Quantum SWAP gate realized with CZ and iSWAP gates in a superconducting architecture
本研究在超导架构中利用CZ和iSWAP门实现了高效的量子SWAP门，提升了量子电路编译效率。
Geometric phases are robust against certain types of local noises, and thus provide a promising way towards high-fidelity quantum gates. However, comparing with the dynamical ones, previous implementations of nonadiabatic geometric quantum gates usually require longer evolution time, due to the needed longer evolution path. Here, we propose a scheme to realize nonadiabatic geometric quantum gates with short paths based on simple pulse control techniques, instead of deliberated pulse control in previous investigations, which can thus further suppress the influence from the environment induced n…
Quantum protocols based on adiabatic evolution are remarkably robust against imperfections of control pulses and system uncertainties. While adiabatic protocols have been successfully implemented for quantum operations such as quantum state transfer and single-qubit gates, their use for geometric two-qubit gates remains a challenge. In this paper, we propose a general scheme to realize robust geometric two-qubit gates in multi-level qubit systems where the interaction between the qubits is mediated by an auxiliary system (such as a bus or coupler). While our scheme utilizes Stimulated Raman Ad…
Quantum information processing with superconducting circuits: realizing and characterizing quantum gates and algorithms in open quantum systems
本论文研究超导电路中开放量子系统的一步式量子门实现方案，提出对谐振器衰减不敏感的高保真X旋转门和纠缠门，并用于实现Grover算法及表征纠缠门性能。
Fast CZ Gate via Energy-Level Engineering in Superconducting Qubits with a Tunable Coupler
本文提出通过可调耦合器的能级工程实现快速高保真CZ门，有效抑制非谐振偏移影响，拓展电路执行深度。
Building large-scale superconducting quantum computers requires two complimentary elements: scalable wiring techniques and multiplex architectures. In our previous work [Béjanin et al., Phys. Rev. Applied 6, 044010 (2016)], we have introduced and characterized a truly vertical interconnect named the quantum socket. In this paper, we exercise the quantum socket using high-coherence flux-tunable Xmon transmon qubits. In particular, we test potential qubit heating and one-qubit gate performance. We observe no heating effects and time-stable gate fidelities in excess of 99.9%. We then propose and …
Hardware-Efficient Quantum Random Access Memory Design with a Native Gate Set on Superconducting Platforms
针对超导平台上的bucket-brigade QRAM，本文提出原生门集{iSCZ, C-iSCZ}以替代{SWAP, CSWAP}，从而消除分解开销并降低电路深度与门数。
ZZ-Interaction-Free Single-Qubit-Gate Optimization in Superconducting Qubits
提出并实验验证了一种基于微扰分析半解析优化脉冲的方法，在超导transmon量子比特上实现了无ZZ交互的单量子比特门操作，有效克服了量子比特频率波动问题且无需额外硬件开销。
Conditional quantum oscillations are investigated for quantum gate operations in superconducting flux qubits. We present an effective Hamiltonian which describes a conditional quantum oscillation in two-qubit systems. Rabi-type quantum oscillations are discussed in implementing conditional quantum oscillations to quantum gate operations. Two conditional quantum oscillations depending on the states of control qubit can be synchronized to perform controlled-gate operations by varying system parameters. It is shown that the conditional quantum oscillations with their frequency synchronization mak…
The experimental optimization of a two-qubit controlled-Z (CZ) gate is realized following two different data-driven gradient ascent pulse engineering (GRAPE) protocols in the aim of optimizing the gate operator and the output quantum state, respectively. For both GRAPE protocols, the key computation of gradients utilizes mixed information of the input Z-control pulse and the experimental measurement. With an imperfect initial pulse in a flattop waveform, our experimental implementation shows that the CZ gate is quickly improved and the gate fidelities subject to the two optimized pulses are ar…
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