Quantum computing is the field that studies computation using quantum mechanical systems, exploiting properties such as superposition or entanglement. The focus of this project will be on quantum phase estimation algorithms. The formalism commonly used to represent quantum algorithms as quantum circuits composed by qubits and quantum gates will be presented. Then, we will be able to explain the quantum Fourier transform (that is exponentially faster than its classical analogue) and three quantum phase estimation algorithms: Kitaev’s phase estimation (KPEA), the phase estimation algorithm based on QFT (PEA) and the iterative quantum phase estimation algorithm based on a semiclassical Fourier transform (IPEA). We will use the Qiskit framework in order to run simulations of such algorithms, and even run them in real IBM Q Experience quantum computers (specifically, we will use the 5 qubit computer ibmq_essex). Statistical post-processing techniques will be essential in order to extract a phase estimation from the results obtained. We will use two post-processing techniques: the majority and the average rule. The performance of these methods applied to the results of several simulations and executions in ibmq_essex will be studied. At last, we will talk about the possible applications of phase estimation algorithms. Phase estimation is used as a subroutine in the field of quantum simulation, allowing us to obtain several magnitudes of a quantum system such as the energy levels. We will present an efficient quantum algorithm for factoring called Shor’s algorithm. The cost of factorization with a classical computer increases exponentially with the size of the input, so this is a example of a task that cannot be solved efficiently on a classical computer, but can be solved efficiently by a quantum computer. Phase estimation is also used in the field of quantum metrology. Specifically, an N-V center can be used to measure magnetic fields with high sensitivity and spatial resolution operating at room temperature thanks to a measuring protocol based on phase estimation.
Quantum computing is the field that studies computation using quantum mechanical systems, exploiting properties such as superposition or entanglement. The focus of this project will be on quantum phase estimation algorithms. The formalism commonly used to represent quantum algorithms as quantum circuits composed by qubits and quantum gates will be presented. Then, we will be able to explain the quantum Fourier transform (that is exponentially faster than its classical analogue) and three quantum phase estimation algorithms: Kitaev’s phase estimation (KPEA), the phase estimation algorithm based on QFT (PEA) and the iterative quantum phase estimation algorithm based on a semiclassical Fourier transform (IPEA). We will use the Qiskit framework in order to run simulations of such algorithms, and even run them in real IBM Q Experience quantum computers (specifically, we will use the 5 qubit computer ibmq_essex). Statistical post-processing techniques will be essential in order to extract a phase estimation from the results obtained. We will use two post-processing techniques: the majority and the average rule. The performance of these methods applied to the results of several simulations and executions in ibmq_essex will be studied. At last, we will talk about the possible applications of phase estimation algorithms. Phase estimation is used as a subroutine in the field of quantum simulation, allowing us to obtain several magnitudes of a quantum system such as the energy levels. We will present an efficient quantum algorithm for factoring called Shor’s algorithm. The cost of factorization with a classical computer increases exponentially with the size of the input, so this is a example of a task that cannot be solved efficiently on a classical computer, but can be solved efficiently by a quantum computer. Phase estimation is also used in the field of quantum metrology. Specifically, an N-V center can be used to measure magnetic fields with high sensitivity and spatial resolution operating at room temperature thanks to a measuring protocol based on phase estimation.
Abstract This guide is based on the IBM Quantum course Phase Estimation and Factoring and focuses on one of the most important breakthroughs in quantum computing: the quantum phase estimation (QPE) algorithm and its application to Shor’s factoring algorithm. The lesson begins with the phase estimation problem, Solving this problem efficiently is a cornerstone of many quantum algorithms. Spectral Theorem and Mathematical Foundations The discussion starts with the spectral theorem from linear algebra. In the context of this lesson, it is applied to unitary matrices, which can be diagonalized in an orthonormal eigenbasis. This property makes it possible to extract phase information using interference effects and controlled unitary operations. Later, the same mathematical framework extends to Hermitian operators, which are central to quantum mechanics and Hamiltonian simulation. Quantum Phase Estimation (QPE) The Quantum Phase Estimation algorithm combines: Controlled unitary operations Superposition and interference The Quantum Fourier Transform (QFT) The QFT plays a crucial role by transforming phase information encoded in amplitudes into measurable computational basis states. Importantly, the QFT can be implemented efficiently with a quantum circuit whose complexity scales polynomially with the number of qubits. From Phase Estimation to Shor’s Algorithm By applying QPE to a specific modular exponentiation operator, one obtains Shor’s algorithm, an efficient quantum algorithm for integer factorization. Shor’s algorithm demonstrates an exponential speedup over the best-known classical factoring algorithms and has profound implications for cryptography, particularly for RSA-based systems. Significance This lesson illustrates a major milestone in quantum computing: It connects abstract linear algebra (spectral theorem) to concrete algorithmic advantage. It shows how quantum interference enables extraction of hidden periodicity. It demonstrates a real-world cryptographic impact through efficient factorization. Through hands-on experimentation on IBM Q, theoretical concepts are translated into working quantum circuits, reinforcing the deep relationship between mathematics, algorithm design, and quantum computational power.
The Quantum Fourier Transform (QFT)andQuantumPhaseEstimation (QPE)are twofundamental algorithmic frameworks in quantum computation, and they provide the mathematical basis of many exponential-speedup quan tum algorithms, such as Shor’s factoring algorithm and quantum simulation techniques. In this work we present a detailed theoretical and computational study of the Quantum Fourier Transform and its integration in the Quan tum Phase Estimation. The discussion begins with the linear algebraic structure of the Hilbert spaces, the unitary transformations, and the complex vector representations that enable quantum interference and superposition. The QFT is studied as the unitary transform of computational basis states in the amplitudes in the frequency domain by phase rotations and tensor-product decompositions. The efficient circuit implementation in terms of Hadamard and controlled phase gates of this transform is discussed. The advantage of the transform in terms of polynomial time complexity over the classical discrete Fourier transform in some quantum scenarios is discussed.The QFT is implemented and the Quantum Phase Estimation algorithm is proposed to obtain the eigenphase information of the unitary operators. In the report the mathematical representation of QPE is derived by using the controlled unitary evolution, the inverse Fourier transforms and the measurement theory in finite dimensional Hilbert spaces. We discuss the probabilistic interpretation of phase extraction in detail and the convergence properties of the bi nary phase approximation. We also discuss the relation of QPE to eigenvalue problems in quantum mechanics and its applications to Hamiltonian simulation, quantum chemistry, cryptography, and hidden subgroup prob lems.Implementation aspects of these algorithms are studied using quantum circuit representations and oracle based computation models. We also discuss error sensitivity, qubit precision requirements, decoherence limita tions and practical constraints of Noisy Intermediate-Scale Quantum (NISQ) hardware. The work synthesizes the mathematical foundations, computational complexity analysis and physical realizations of the QFT and the QPE, demonstrating how the Fourier-based quantum transformations provide the essential mechanism underlying modern quantum algorithmic speedups and scalable quantum information processing.
It is suggested that computers exploiting quantum phenomena can solve tasks with an efficiency surpassing that achievable by classical computers. One example of this quantum advantage arises in the task of phase estimation. Here, an unknown eigenvalue of a unitary evolution is estimated, a critical subroutine for many quantum algorithms and quantum metrology protocols. For a fixed amount of resources, the utilisation of quantum coherence and entanglement can lead to a quadratic increase in precision of the final estimate compared to classical statistical limits. In systems affected by depolarising noise, including current quantum hardware, this improvement becomes a constant factor. While numerous phase estimation algorithms have been proposed claiming to achieve these theoretical limits, the lack of well-defined criteria for both precision and resource utilisation casts doubt on these results. Further issues arise due to a lack of point identification (the existence of multiple equally likely estimates) that some algorithms have. I aim to address these issues in this thesis. I employ statistical inference, the theoretical foundation for interpreting any quantum measurement, in order to find rigorous definitions for precision and resource use. By combining these definitions with previous work in the field of quantum metrology, I am able to derive theoretical precision limits that one can hope to achieve. I then outline several phase estimation algorithms that can achieve these limits in different contexts. These algorithms that I describe are split into three major groups. Algorithms in the first group focus on the process of point identification; one set of measurements provides a precise estimate of the unknown phase before an additional set of measurements removes any ambiguities present. This second set of measurements can be either decided before any measurement is taken or during measurements in an iterative fashion. I show that the balancing of resources used in each set of measurements can lead to optimally scaling precision both with and without depolarising noise. These algorithms are based upon two of my peer-reviewed articles, [Phys. Rev. A 106, 062615 (2022)] and [Phys. Rev. A 109, 042412 (2024)]. The second group of algorithms that I introduce use the principles of Bayesian experimental design. Here, previous measurement outcomes are used to select future measurements recursively. These algorithms require large classical computing times to select each measurement but boast high precision in certain metrics. The final group of algorithms exploit the oscillating nature of phases by using a discrete Fourier transform (DFT). The DFT can be implemented using either classical or quantum hardware, the latter of which leads to the famous quantum phase estimation algorithm (QPEA). In this thesis, I show that the QPEA performs worse than some classical precision limits. However, I also suggest some modifications to the QPEA circuit that can lead to optimal precision scaling. These results and modifications are described in one of my articles awaiting publication [arXiv:2505.00764].
Quantum phase estimation (QPE) is one of the core algorithms for quantum computing. It has been extensively studied and applied in a variety of quantum applications such as the Shor's factoring algorithm, quantum sampling algorithms and the calculation of the eigenvalues of unitary matrices. The QPE algorithm has been combined with Kitaev's algorithm and the inverse quantum Fourier transform (IQFT) which are utilized as a fundamental component of such quantum algorithms. In this paper, we explore the computational challenges of implementing QPE algorithms on noisy intermediate-scale quantum (NISQ) machines using the IBM Q Experience (e.g., the IBMQX4, 5-qubit quantum computing hardware platform). Our experimental results indicate that the accuracy of finding the phase using these QPE algorithms is severely constrained by the NISQ computer's physical characteristics such as coherence time and error rates. To mitigate these physical limitations, we propose implementing a modified solution by reducing the number of controlled rotation gates and phase shift operations, thereby increasing the accuracy of the finding phase in near-term quantum computers.
Quantum computing can provide speedups in solving many problems as the evolution of a quantum system is described by a unitary operator in an exponentially large Hilbert space. Such unitary operators change the phase of their eigenstates and make quantum algorithms fundamentally different from their classical counterparts. Based on this unique principle of quantum computing, we develop an algorithmic toolbox, quantum phase processing, that can directly apply arbitrary trigonometric transformations to eigenphases of a unitary operator. The quantum phase processing circuit is constructed simply, consisting of single-qubit rotations and controlled unitaries, typically using only one ancilla qubit. Besides the capability of phase transformation, quantum phase processing in particular can extract the eigeninformation of quantum systems by simply measuring the ancilla qubit, making it naturally compatible with indirect measurement. Quantum phase processing complements another powerful framework known as quantum singular value transformation and leads to more intuitive and efficient quantum algorithms for solving problems that are particularly phase related. As a notable application, we propose a quantum phase estimation algorithm without quantum Fourier transform, which requires the fewest ancilla qubits and matches the best performance so far. We further exploit the power of our method by investigating a plethora of applications in Hamiltonian simulation, entanglement spectroscopy, and quantum entropy estimation, demonstrating improvements or optimality for almost all cases.
Quantum computing over the past few decades has experienced a huge boost, due in part to the discovery of algorithms such as the Shor's algorithm which promises a significant reduction in complexity compared to its analogues of classical computing. The main objective of this thesis is to introduce improvements in some algorithms based on the quantum Fourier transform, more specifically in the phase estimation algorithm and the order-finding algorithm. In addition to this, the problem of efficient generation of quantum states is also studied, where as a result a method has been developed for the efficient generation of quantum states.