ABSTRACT Quantum computing utilizes the underlying principles of quantum mechanics to perform computations with unmatched performance capabilities. Rather than using classical bits, it operates on qubits, which can exist in superposition and entangled states. This enables the solution of problems that are considered intractable for classical computers. However, since qubits are realized by physical systems such as the spins of electrons, they are highly sensitive to environmental disturbances and hardware imperfections. To achieve reliable scaling and practical application in the future, addressing these errors is of utmost importance. Different classes of errors exist, such as coherent and incoherent errors, caused by imperfections in quantum operations or the decoherence of quantum states. They are inherently different, as they arise from either a lack of precision or intrinsic randomness. Current literature struggles to provide a unified framework that models both types of errors simultaneously. In this paper, an approach based on possibility theory—a theory of imprecise probabilities—is presented to model quantum uncertainty. Possibility theory is particularly useful for systems affected by both epistemic and aleatoric uncertainty, that is, uncertainty due to limited knowledge and uncertainty due to inherent randomness, respectively. By exploring noisy quantum algorithms within a possibilistic framework, different statements about robustness can be derived without requiring prior assumptions about the underlying noise model. Moreover, a possibilistic model enables the derivation of sampling criteria for guaranteed statistical performance and provides insight into the number of measurements required—an important consideration, given that such resources are costly in practice.
ABSTRACT Quantum computing utilizes the underlying principles of quantum mechanics to perform computations with unmatched performance capabilities. Rather than using classical bits, it operates on qubits, which can exist in superposition and entangled states. This enables the solution of problems that are considered intractable for classical computers. However, since qubits are realized by physical systems such as the spins of electrons, they are highly sensitive to environmental disturbances and hardware imperfections. To achieve reliable scaling and practical application in the future, addressing these errors is of utmost importance. Different classes of errors exist, such as coherent and incoherent errors, caused by imperfections in quantum operations or the decoherence of quantum states. They are inherently different, as they arise from either a lack of precision or intrinsic randomness. Current literature struggles to provide a unified framework that models both types of errors simultaneously. In this paper, an approach based on possibility theory—a theory of imprecise probabilities—is presented to model quantum uncertainty. Possibility theory is particularly useful for systems affected by both epistemic and aleatoric uncertainty, that is, uncertainty due to limited knowledge and uncertainty due to inherent randomness, respectively. By exploring noisy quantum algorithms within a possibilistic framework, different statements about robustness can be derived without requiring prior assumptions about the underlying noise model. Moreover, a possibilistic model enables the derivation of sampling criteria for guaranteed statistical performance and provides insight into the number of measurements required—an important consideration, given that such resources are costly in practice.
Quantum decay, also known as decoherence, represents a significant challenge in the realization of practical quantum computers. The loss of quantum information due to interactions with the environment introduces errors and ultimately limits the computational power. This paper presents a probabilistic assessment model for evaluating the decay probability within quantum computing processes. The model leverages quantum mechanics and statistical analysis to quantify error probabilities, considering factors such as environmental noise and qubit interactions. We develop a framework for estimating the impact of decay on quantum computations, providing insights for designing more robust quantum algorithms and hardware. The core contribution lies in offering a quantifiable approach to assess and mitigate the risks associated with quantum decay, ultimately contributing to the advancement of quantum computing technology. The model incorporates key parameters including qubit fidelity, environmental noise levels, and the duration of quantum operations, allowing for a dynamic evaluation of the overall decay probability. The results generated by this model can be directly applied to optimize quantum algorithms and hardware designs, leading to improved performance and reliability. ---
A mathematical model of computation is an idealized abstraction. We design algorithms and perform analysis on the assumption that the mathematical operations we specify will be carried out exactly, and without error. Physical devices that implement an abstract model of computation are imperfect and of limited precision. For example, when a digital circuit is implemented on a physical circuit board, unwanted electrical noise in the environment may cause components to behave differently than expected, and may cause voltage levels (bit-values) to change. These sources of error must be controlled or compensated for, or else the resulting loss of efficiency may reduce the power of the information-processing device. If individual steps in a computation succeed with probability p, then a computation involving t sequential steps will have a success probability that decreases exponentially as pt. Although it may be impossible to eliminate the sources of errors, we can devise schemes to allow us to recover from errors using a reasonable amount of additional resources. Many classical digital computing devices use error-correcting codes to perform detection of and recovery from errors. The theory of error-correcting codes is itself a mathematical abstraction, but it is one that explicitly accounts for errors introduced by the imperfection and imprecision of realistic devices. This theory has proven extremely effective in allowing engineers to build computing devices that are resilient against errors. Quantum computers are more susceptible to errors than classical digital computers, because quantum mechanical systems are more delicate and more difficult to control. If large-scale quantum computers are to be possible, a theory of quantum error correction is needed. The discovery of quantum error correction has given researchers confidence that realistic large-scale quantum computing devices can be built despite the presence of errors. We begin by considering fundamental concepts for error correction in a classical setting. We will focus on three of these concepts: (a) the characterization of the error model, (b) the introduction of redundancy through encoding, and (c) an error recovery procedure. We will later see that these concepts generalize quite naturally for quantum error correction.
Errors occurring on noisy hardware pose a key challenge to reliable quantum computing. Existing techniques such as error correction, mitigation, or suppression typically separate the error handling from the algorithm analysis and design. In this paper, we develop an alternative, algorithm-centered framework for understanding and improving the robustness against errors. For a given quantum algorithm and error model, we derive worst-case fidelity bounds which can be efficiently computed to certify the robustness. We consider general error models including coherent and (Markovian) incoherent errors and allowing for set-based error descriptions to address uncertainty or time-dependence in the errors. Our results give rise to guidelines for robust algorithm design and compilation by optimizing our theoretical robustness measure. We demonstrate the practicality of the framework with numerical results on algorithm analysis and robust optimization, including the robustness analysis of a 50-qubit modular adder circuit.
Modern society increasingly depends on ever more capable computing devices. Yet some important problems remain beyond the reach of classical computers due to their inherent computational complexity. At the same time, the energy required to power this growing computational demand is an important concern. Over the past decade, quantum computers have received increased attention, not as a replacement for classical systems, but as a complementary technology with the potential to tackle complex problems more efficiently and significantly boost computational performance. This is made possible by leveraging the principles of quantum mechanics. To achieve this, quantum computers rely on a fundamental basic block called a qubit. There are several different technologies for realizing qubits, and no single approach has yet established itself as the definitive choice. In this work, we focus on silicon spin qubits in quantum dots, which use the intrinsic spin states of an electron as the computational basis states. This platform offers notable advantages including long coherence times, compatibility with the existing semiconductor industry and scalability. However, this technology is still very immature and substantial progress is still required. Just like for the design of classical electronics, simulations can play a crucial role by reducing development cycles, lowering costs, and improving overall efficiency. Throughout this thesis, we will propose a comprehensive simulation framework that allows to study and design spin qubit systems focusing on key aspects of the system architecture. First, a spin qubit compact model is presented that enables the co-simulation of spin qubits together with the classical CMOS control and readout circuitry that will significantly impact the performance of the qubits. We show that the model is capable of including all elements to build an elementary quantum gate set and illustrate its use by co-simulating a spin qubit coupled to a readout resonator with a complete cryo-CMOS readout chip design in a commercial electronic design automation tool. Next, we discuss a hybrid-Hamiltonian simulation approach that is aimed at simulating spin qubit microarchitectures that are suited for running quantum error correction protocols, which protect the encoded information against the errors that unavoidably happen due to the inherently noisy physical qubit devices. This simulation approach incorporates several novel strategies to tackle the challenge of the required computing power that grows exponentially with the size of the simulated system. Furthermore, additional functionality is incorporated to evaluate and track the performance of the implemented error correction protocols. Finally, we propose a characterization protocol for constructing a correlated noise model of the charge noise that is an important factor affecting spin qubit performance. The sources of charge noise are modelled as two-level fluctuators and the entire qubit array is characterized by dividing it into patches and running batches of benchmarking quantum gate sequences. The resulting noise model enables to scale the simulations to larger system sizes and to study the system performance and hardware demands of error correction protocols. Together, these models form a multi-layer simulation framework that enables detailed analysis and optimization of future spin qubit microarchitectures, with particular emphasis on error correction protocols. As such, the framework contributes to the realization of practical large-scale quantum computing.
This paper presents a novel approach to simulating quantum computing algorithms using Monte Carlo methods, specifically designed to account for the inherent probabilistic nature of quantum systems. Traditional simulation methods often rely on deterministic models, which fail to capture the fundamental uncertainty associated with quantum mechanics. Our method employs a stochastic simulation engine that accurately models the probabilistic behavior of quantum gates and measurements. The core of the approach involves generating a large number of samples, each representing a possible outcome of a quantum computation. By analyzing the statistical distribution of these samples, we can approximate the expected results of the algorithm. This technique offers a practical way to evaluate the performance of quantum algorithms, especially those with high levels of uncertainty, and provides a valuable tool for algorithm design and optimization. The method avoids the limitations of purely deterministic simulations and offers a more realistic representation of quantum computation. Key aspects include the development of a robust sampling strategy and the efficient calculation of probabilities. The approach is demonstrated through a detailed explanation and illustrative examples, focusing on the conceptual framework and the underlying mathematical principles.
This review is designed to introduce mathematicians and computational scientists to quantum computing (QC) through the lens of uncertainty quantification (UQ) by presenting a mathematically rigorous and accessible narrative for understanding how noise and intrinsic randomness shape quantum computational outcomes in the language of mathematics. By grounding quantum computation in statistical inference, we highlight how mathematical tools such as probabilistic modeling, stochastic analysis, Bayesian inference, and sensitivity analysis, can directly address error propagation and reliability challenges in today's quantum devices. We also connect these methods to key scientific priorities in the field, including scalable uncertainty-aware algorithms and characterization of correlated errors. The purpose is to narrow the conceptual divide between applied mathematics, scientific computing and quantum information sciences, demonstrating how mathematically rooted UQ methodologies can guide validation, error mitigation, and principled algorithm design for emerging quantum technologies, in order to address challenges and opportunities present in modern-day quantum high performance and fault-tolerant quantum computing paradigms.