Shaped Pulses Boost Quantum Computing’s Accuracy and Reliability

A thorough overview of pulse-shaping techniques applicable to transmon qubits is presented by Animesh Patra and Ankur Raina from the Department of EECS. The techniques address multiple sources of error that hinder high-fidelity quantum computation. The work uniquely combines physical insight, analytical modelling using the Magnus expansion, and practical guidance on hardware implementation, including arbitrary waveform generators and local oscillators, to offer a unified framework for both newcomers and experienced researchers in superconducting quantum computing. By detailing techniques such as derivative removal by adiabatic gate (DRAG), the research clarifies how to suppress unwanted off-resonant excitations and enhance the accuracy of single and two-qubit gate operations.

Finite bandwidth limitations induce qubit excitation leakage

Derivative removal by adiabatic gate, or DRAG, is a pulse-shaping technique central to minimising errors in transmon qubit control. It addresses the problem of unwanted off-resonant excitations, where energy can leak into higher, unintended energy levels, reducing fidelity when microwave pulses drive a qubit. Transmon qubits, unlike ideal two-level systems, possess a finite anharmonicity, the spacing between energy levels decreases as one moves up the ladder. This anharmonicity, while crucial for addressing individual qubits, also means that microwave pulses designed for the 0-to-1 transition can inadvertently excite the qubit to higher levels, such as the 1-to-2 transition. DRAG tackles this by adding a carefully calculated ‘quadrature’ component to the pulse, a signal offset in phase proportional to the rate of change of the main pulse. This quadrature component isn’t merely an arbitrary addition; it’s derived from a detailed analysis of the qubit’s Hamiltonian and the desired control operations.

A subtle adjustment cancels out terms in the Magnus expansion, a mathematical technique approximating complex quantum processes, that cause unwanted transitions. The Magnus expansion provides a systematic way to decompose a complex time-dependent Hamiltonian into a series of simpler, time-independent terms, allowing for a more tractable analysis of the qubit’s evolution under the influence of the applied pulse. By carefully shaping the pulse and incorporating the DRAG component, researchers can effectively suppress the terms in the Magnus expansion that contribute to off-resonant excitations. This effectively smooths the pulse and confines the excitation to the desired qubit states. Finite bandwidth in microwave pulses causes energy leakage into unintended qubit levels, thereby reducing control fidelity. Real-world microwave sources aren’t perfect; they have a limited bandwidth, meaning they can’t perfectly reproduce arbitrarily fast changes in the pulse shape. This limitation introduces spectral leakage, spreading the pulse’s energy across a wider range of frequencies and increasing the probability of driving off-resonant transitions. The approach accounts for anharmonicity and deviations from ideal behaviour, offering improvements over simpler pulse shapes like square pulses due to their broader frequency spectra.

Optimised pulse shaping achieves fivefold reduction in transmon qubit error rates

Transmon qubit error rates, originating from off-resonant excitations, have fallen by a factor of five through optimised pulse shaping. This improvement surpasses the previously achievable threshold for reliable quantum computation, where error rates needed to fall below 1% for scalable algorithms. Achieving fault-tolerant quantum computation requires extremely low error rates, typically on the order of 10^-3 or lower. Previously, consistently achieving such low error rates proved elusive due to the complexities of hardware limitations and pulse design. The team’s framework integrates analytical modelling using the Magnus expansion with practical considerations for arbitrary waveform generators and IQ mixing, offering a unified approach to qubit control. The Magnus expansion isn’t just a theoretical tool; it informs the design of the DRAG pulse, allowing researchers to predict and mitigate the effects of various error sources.

Digital-to-analogue conversion (DUC) sharply improves pulse fidelity for transmon qubits by resampling and scaling in-phase and quadrature signals using a numerically controlled oscillator, minimising phase and frequency drift between the channels. The process of converting a digital pulse representation into an analogue microwave signal is prone to imperfections. DUC techniques address these imperfections by carefully resampling and scaling the in-phase (I) and quadrature (Q) components of the pulse, ensuring that they are accurately aligned in both time and amplitude. Minimising phase and frequency drift between the I and Q channels is crucial for maintaining the desired pulse shape and preventing unwanted distortions. Accurate matching of I and Q signals is ensured through this digitisation process, preventing quadrature imbalance and maintaining orthogonality, both important for precise qubit control. Quadrature imbalance occurs when the and Q signals are not perfectly matched in amplitude or phase, leading to errors in the qubit’s state manipulation. Maintaining orthogonality between the I and Q signals ensures that they represent independent degrees of freedom, allowing for precise control over the qubit’s state. Furthermore, the team demonstrated the decomposition of a complex CNOT gate, essential for entanglement, into a cross-resonance (CR) gate and single-qubit gates, showing a practical approach to building quantum algorithms. The CNOT gate, a fundamental building block of quantum circuits, requires precise control over two qubits. Decomposing it into simpler gates, such as the CR gate and single-qubit rotations, can simplify the control process and reduce the overall error rate.

Extending pulse-shaping techniques to multi-qubit superconducting architectures

The analysis concentrates on the DRAG technique and the cross-resonance gate, demonstrably reducing error through optimised pulse shapes. Although other pulse-shaping methods exist, each with its own strengths and weaknesses, such as optimal control theory and GRAPE (Gradient Ascent Pulse Engineering), a thorough comparison remains outstanding. Each technique offers a different approach to pulse design, with varying levels of complexity and computational cost. Optimal control theory aims to find the pulse shape that maximises the fidelity of a given gate operation, while GRAPE uses a gradient-based optimisation algorithm to iteratively refine the pulse shape. It is important to acknowledge that these techniques were demonstrated on simplified systems, and extending these findings to larger, more complex quantum processors presents a significant challenge. As the number of qubits increases, the complexity of the control system grows exponentially, making it more difficult to maintain high fidelity and coherence.

A clear understanding of single and two-qubit gate control, alongside practical considerations of hardware limitations like arbitrary waveform generators, provides a key foundation for scaling up superconducting quantum computers. This delivers a valuable resource for newcomers and experienced researchers alike, accelerating progress in the field. The analysis highlights how limited bandwidth introduces unwanted energy leakage by examining pulse envelopes and their spectral properties, motivating the use of techniques employing a quadrature component to refine pulse shapes. The spectral properties of a pulse determine its susceptibility to noise and its ability to selectively address the desired qubit transitions. Analytical tools, such as the Magnus expansion, clarify how errors accumulate during gate operations, offering a pathway to optimise control fidelity. Understanding the error accumulation process is crucial for developing strategies to mitigate errors and improve the overall performance of the quantum computer.

A comprehensive analysis of pulse-shaping techniques for transmon qubits provides a unified framework for optimising control and minimising errors. This work demonstrates how carefully designed microwave pulses, accounting for factors like bandwidth limitations and hardware imperfections in devices such as arbitrary waveform generators, are essential for high-fidelity qubit control. By employing methods like the derivative removal by adiabatic gate technique and utilising analytical tools such as the Magnus expansion, researchers can better understand and suppress unwanted excitations. The authors suggest further comparison of pulse-shaping methods, including optimal control theory and GRAPE, remains an area for future investigation.