Quantum errors in noisy environments remain a major obstacle to advancing quantum information technology. In this work, we expand a recently developed geometric framework, originally utilized for analyzing noise accumulation and creating dynamical error-correcting gates at the control pulse level, to now study noise dynamics at the quantum circuit level. Through a geometric perspective, we demonstrate how circuit noise robustness can be enhanced using twirling techniques. Additionally, we show that circuits modified by random twirling correspond to random walk trajectories in this geometric framework, and provide a fresh perspective on randomized compiling by analytically deriving the perturbative expression for the resultant Pauli noise channel. We also illustrate that combining robustness optimization strategies at both the control pulse and circuit levels can significantly boost overall circuit fidelity even further through numerical examples. This research illuminates pathways to achieving noise-resistant quantum control beyond mere optimization of control pulses.
Quantum errors in noisy environments remain a major obstacle to advancing quantum information technology. In this work, we expand a recently developed geometric framework, originally utilized for analyzing noise accumulation and creating dynamical error-correcting gates at the control pulse level, to now study noise dynamics at the quantum circuit level. Through a geometric perspective, we demonstrate how circuit noise robustness can be enhanced using twirling techniques. Additionally, we show that circuits modified by random twirling correspond to random walk trajectories in this geometric framework, and provide a fresh perspective on randomized compiling by analytically deriving the perturbative expression for the resultant Pauli noise channel. We also illustrate that combining robustness optimization strategies at both the control pulse and circuit levels can significantly boost overall circuit fidelity even further through numerical examples. This research illuminates pathways to achieving noise-resistant quantum control beyond mere optimization of control pulses.
Abstract Quantum errors in noisy environments remain a major obstacle to advancing quantum information technology. Standard quantum error correction requires massive ancillary qubit overhead, motivating the need for hardware-efficient mitigation strategies. In this work, we propose a framework for dynamical quantum error correction at the circuit level, requiring no logical encoding or ancillary qubits. By extending a geometric framework-originally developed for creating dynamical error-correcting gates at the control pulse level—to the discrete dynamics of digital circuits, we map the accumulation of coherent errors to trajectories in a high-dimensional error space. We demonstrate that inserting deterministically optimized twirling gate sequences actively shapes these trajectories, utilizing destructive interference to keep the accumulated error bounded with error scaling O ( 1 ) . This deterministic path-shaping suppresses circuit errors fundamentally differently than the stochastic random-walk behavior of standard randomized compiling with error scaling O ( N ) . Furthermore, we show that this circuit-level dynamical correction synergizes with pulse-level robust control, providing an analytical bridge between continuous noise dynamics and discrete quantum compilation. This research illuminates pathways to achieving highly noise-resistant quantum circuits prior to the era of fault tolerance.
Quantum information processing faces a significant hurdle: noise. Different noise sources induce varying errors in quantum operations depending on the underlying dynamics. To gain a deeper understanding of these error mechanisms, we introduce the concept of quantum error evolution diagrams (QEEDs). QEEDs establish a dual correspondence between driven noisy quantum dynamics and geometric space curves, providing quantitative geometric metrics to assess the severity of the noises. This framework enables the design of universal robust quantum gates that correct errors induced by generic noise. Additionally, we present a protocol for constructing a universal set of single- and two-qubit robust quantum gates with simple and smooth control pulses of arbitrary length and fidelities exceeding 99.99% across a wide range of noise strengths. Our work provides new insights into the geometric nature of noisy quantum dynamics and paves the way for developing strategies to dynamically correct quantum errors.
Group twirling is crucial in quantum information processing, particularly in randomized benchmarking and randomized compiling. While protocols based on Pauli twirling have been effectively crafted to transform arbitrary noise channels into Pauli channels for Clifford gates—thereby facilitating efficient benchmarking and mitigating worst-case errors—the lack of practical twirling groups for multiqubit non-Clifford gates remains a challenge. To address this gap, we study the issue of finding twirling groups for generic quantum gates, focusing on a widely used circuit structure in randomized benchmarking or randomized compiling. Specifically, for multiqubit controlled phase gates, which are essential in quantum algorithms and directly implementable in practice, we determine optimal twirling groups within the realm of classically replaceable unitary operations. Contrasting with the local Pauli twirling group for Clifford gates, the optimal groups for such gates contain nonlocal operations, highlighting the overhead of tailoring noise in global non-Clifford contexts. We design new benchmarking procedures for multiqubit controlled phase gates based on the optimal twirling groups. Our simulation results show that our scheme can improve the precision and accuracy of benchmarking in small-scale systems. Published by the American Physical Society 2024
Implementing high-fidelity quantum control and suppressing the unwanted environmental noise has been one of the essential challenges in developing quantum information technologies. In the past, driving pulse sequences based on Dirac delta functions or square wave functions, such as Hahn spin echo or CPMG, have been developed to dynamically correcting the noise effects. However, implementing these ideal pulses with high fidelity is a challenging task in real experiments. In this thesis, we provide a new and simple method to explore the entire solution space of driving pulse shapes that suppress environmental noise in the evolution of the system. In this method, any single-qubit phase gate that is first-order robust against quasi-static transversal noise corresponds to a closed curve on a two-dimensional plane, and more general first-order robust single-qubit gates correspond to closed three-dimensional space curves. Second-order robust gates correspond to closed curves having the property that their projection onto any two-dimensional planes shall enclose a zero net area. The driving pulse shapes that implement the gates can be determined by the curvature, torsion, and the length of the curve. By utilizing the framework it is possible to obtain globally optimal solutions in pulse shaping in respect of experimental constraints by mapping them into geometrical optimization problems. One such problem we solved is to prove that the fastest possible single-qubit phase gates that are second-order noise-resistant shall be implemented using sign-flipping square functions. Since square waves are not experimentally feasible, we provide a method to smooth these pulses with minimal loss in gate speed while maintaining the robustness, based on the geometrical framework. This framework can also be useful in diagnosing the noise-cancellation properties of pulse shapes generated from numerical methods such as GRAPE. We show that this method for pulse shaping can significantly improve the fidelity of single-qubit gates through numerical simulation.
Quantum computers are poised to radically outperform their classical counterparts by manipulating coherent quantum systems. A realistic quantum computer will experience errors due to the environment and imperfect control. When these errors are even partially coherent, they present a major obstacle to performing robust computations. Here, we propose a method for introducing independent random single-qubit gates into the logical circuit in such a way that the effective logical circuit remains unchanged. We prove that this randomization tailors the noise into stochastic Pauli errors, which can dramatically reduce error rates while introducing little or no experimental overhead. Moreover, we prove that our technique is robust to the inevitable variation in errors over the randomizing gates and numerically illustrate the dramatic reductions in worst-case error that are achievable. Given such tailored noise, gates with significantly lower fidelity---comparable to fidelities realized in current experiments---are sufficient to achieve fault-tolerant quantum computation. Furthermore, the worst-case error rate of the tailored noise can be directly and efficiently measured through randomized benchmarking protocols, enabling a rigorous certification of the performance of a quantum computer.
The primary aim of the thesis is to introduce and investigate an original protocol for the design and the optimization of parameterized noisy quantum channels, with the goal of mitigating hardware-induced noise. Significant attention is also devoted to the design of noise models, which are essential for emulation purposes. The protocol begins with the derivation of a noise model for a hardware gate, represented as a quantum channel, that can be simulated classically. This task is achieved through process tomography, a procedure that enables the characterization of the channel. It should be noted that such simulation is feasible on classical computers, since the optimization task concerns low-dimensional gates typically acting on 2–3 qubits. The noisy channel is then engineered by embedding it into a probabilistic ensemble of circuits, in which additional single-qubit parameterized gates — assumed to be noiseless — are introduced. Through this procedure, the noisy channel becomes parameterized and can be optimized according to an appropriate metric. Following a worst-case, agnostic perspective, we choose the diamond distance between the noisy channel and the target unitary channel as the cost function. This metric is generally considered the most suitable for distinguishing quantum channels. The resulting optimal ensemble of circuits can be implemented on real hardware under fixed calibration conditions, yielding a channel that approximates the target more closely than the original noisy channel. This approach can be interpreted as a noise mitigation strategy based on pre-processing design and optimization: rather than correcting errors after the computation, less noisy channels are employed during the computation itself. In this context, the protocol — within the noise-aware variational quantum compiling framework — shares with the randomized compiling the principle of circuit dressing to tailor noise during execution. The methodology is validated on two toy models of the single-qubit identity with amplitude damping noise, combining both analytical and numerical approaches. The analysis highlights the role of convexity in the space of quantum channels, with explicit demonstrations of optimal convex combinations and connections to encoding-decoding and dynamical decoupling strategies. The analysis of these simple models is intended as a proof-of-principle of the protocol in a controlled and well-characterized environment. In addition, this thesis introduces the design of random noise models that are constrained to maintain a fixed diamond distance from a target quantum channel. The controlled addition of random noise to a unitary channel represents a significant and — to our knowledge — previously unexplored approach within noise modeling. Indeed, in principle, the currently available built-in noise models may limit the expressivity of the emulation. The task was achieved by enriching the noise spectrum through the addition of random Kraus operators and small systematic rotations. To assess the effects of hardware-induced noise, we proposed an experimental design aimed at simulating the time evolution of a triangular Ising model in a transverse field. Our approach employs Trotterization, adapted to the topology and connectivity of a certain quantum processing unit (QPU) profile. Building on the noisy native gates of the QPU profile, we propose the emulation, engineering and optimization of a noisy CZ gate, subsequently employed within sequences of non-native gates. Using the designed noise models, we perform several emulations aimed at evaluating the effectiveness of the protocol in the presence of progressively dominant systematic rotations, and at comparing it with Pauli twirling, a technique of randomized compiling.