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].
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 algorithms are critical subroutines in many applications for quantum computers and in quantum-metrology protocols. These algorithms estimate the unknown strength of a unitary evolution. By using coherence or entanglement to sample the unitary N tot times, the variance of the estimates can scale as O ( 1 / N tot 2 ) , compared to the best “classical” strategy with O ( 1 / N tot ) . The original algorithm for quantum phase estimation cannot be implemented on near-term hardware as it requires large-scale entangled probes and fault-tolerant quantum computing. Therefore, alternative algorithms have been introduced that rely on coherence and statistical inference. These algorithms produce quantum-boosted phase estimates without interprobe entanglement. This family of phase-estimation algorithms have, until now, never exhibited the possibility of achieving optimal scaling O ( 1 / N tot 2 ) . Moreover, previous works have not considered the effect of noise on these algorithms. Here, we present a coherence-based phase-estimation algorithm which can achieve the optimal quadratic scaling in the mean absolute error and the mean squared error. In the presence of noise, our algorithm produces errors that approach the theoretical lower bound. The optimality of our algorithm stems from its adaptive nature: Each step is determined, iteratively, using a Bayesian protocol that analizes the results of previous steps. Published by the American Physical Society 2024
Quantum-phase-estimation algorithms are critical subroutines in many applications for quantum computers and in quantum-metrology protocols. These algorithms estimate the unknown strength of a unitary evolution. By using coherence or entanglement to sample the unitary Ntot times, the variance of the estimates can scale as O(1/Ntot2), compared to the best ``classical'' strategy with O(1/Ntot). The original algorithm for quantum phase estimation cannot be implemented on near-term hardware as it requires large-scale entangled probes and fault-tolerant quantum computing. Therefore, alternative algorithms have been introduced that rely on coherence and statistical inference. These algorithms produce quantum-boosted phase estimates without inter-probe entanglement. This family of phase-estimation algorithms have, until now, never exhibited the possibility of achieving optimal scaling O(1/Ntot2). Moreover, previous works have not considered the effect of noise on these algorithms. Here, we present a coherence-based phase-estimation algorithm which can achieve the optimal quadratic scaling in the mean absolute error and the mean squared error. In the presence of noise, our algorithm produces errors that approach the theoretical lower bound. The optimality of our algorithm stems from its adaptive nature: Each step is determined, iteratively, using a Bayesian protocol that analyses the results of previous steps.
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.
In this thesis I study optimal strategies to estimate or characterize quantum states and processes (quantum channels). I consider the problem of estimating a phase encoded on a gaussian state. Phase estimation is at the heart of many quantum metrology applications, such as gravitational wave detectors or clock synchronization. I address the problem to the phase estimation fidelity for pure and mixed gaussian states and study how temperature affects the precision of estimation. Although the case of pure states has been studied before, very little is known about estimation in mixed states. The interest in mixed gaussian states relies on the fact that these are the states one has in real experiments with light due to the unavoidable presence of noise. Displaced and squeezed thermal states are studied. For the case of displaced states, I find that fidelity estimation gets worse when the temperature increases. On the contrary, for squeezed thermal states I find the surprising result that a larger temperature yields better estimation. This apparently paradoxical feature, also appears when I derive the estimation accuracy that can be obtained when an asymptotically large number of copies of the state is available. I also study quantum benchmarks that can be applied to a wide variety of protocols, such us quantum storage or quantum teleportation. A very important question in QIT is to know whether a particular quantum protocol can be realized by classical means with the same efficiency. After all, QIT lives from the promise of this quantum advantage. This question becomes specially relevant in experimental (real) implementations of quantum schemes. In real experiments there are always losses and the ideal quantum protocols become imperfect. It is then necessary to assess whether the same experimental result could be obtained by classical (less costly) means. So, for example, in a particular teleportation experiment -which involves generation of entanglement, complicated Bell measurements, fighting decoherence, etc.- one might ask whether one could achieve the same (i.e. mapping the state of a system to a second system located in a different place or time) by measuring the system, transmitting the collected information to the other location, and finally preparing the system accordingly. It is hence of paramount importance to give bounds on the efficiency of these measure and prepare (or entanglement breaking) channels. For this purpose we propose a threshold based on a phase-covariant family of states. Besides of being a family with a straightforward group-theoretical structure, which gives obvious computational advantages, this family is interesting because it can be tested experimentally also in a straightforward way. A crucial feature for the practical use of our benchmark is that it does not require pure (ideal) input states. I finally consider the problem of estimating the parameter of the amplification channel. The amplification channel is interesting by itself but also by the fact that the Hawking-Unruh effect can be formally modeled by it. The Unruh-Hawking effect is a relativistic effect and as such it is hard to measure and detect in current experiments. I investigate whether quantum strategies can provide an improvement over classical protocols in the estimation of the amplification parameter. If the field is inertially prepared in the vacuum one could measure its temperature as detected by a noninertial observer and extract information about the parameter. But I would like to know if the vacuum state is the optimal probe state in an inertial frame for such a task or a better sensitivity could be achieved if the field were in a state of nonzero energy (mean photon number) from an inertial perspective. The aim is to determine the optimal input states, so that, by optimally measuring the output, one can construct an estimator for amplification channel parameter which is unbiased and has the minimum possible variance.
In an idealistic setting, quantum metrology protocols allow to sense physical parameters with mean squared error that scales as 1/N2 with the number of particles involved---substantially surpassing the 1/N-scaling characteristic to classical statistics. A natural question arises, whether such an impressive enhancement persists when one takes into account the decoherence effects that are unavoidable in any real-life implementation. In this thesis, we resolve a major part of this issue by describing general techniques that allow to quantify the attainable precision in metrological schemes in the presence of uncorrelated noise. We show that the abstract geometrical structure of a quantum channel describing the noisy evolution of a single particle dictates then critical bounds on the ultimate quantum enhancement. Our results prove that an infinitesimal amount of noise is enough to restrict the precision to scale classically in the asymptotic N limit, and thus constrain the maximal improvement to a constant factor. Although for low numbers of particles the decoherence may be ignored, for large N the presence of noise heavily alters the form of both optimal states and measurements attaining the ultimate resolution. However, the established bounds are then typically achievable with use of techniques natural to current experiments. In this work, we thoroughly introduce the necessary concepts and mathematical tools lying behind metrological tasks, including both frequentist and Bayesian estimation theory frameworks. We provide examples of applications of the methods presented to typical qubit noise models, yet we also discuss in detail the phase estimation tasks in Mach-Zehnder interferometry both in the classical and quantum setting---with particular emphasis given to photonic losses while analysing the impact of decoherence.
Quantum computers have great potential to uncover new and unexpected physical phenomena.They can simulate classically intractable many-particle quantum systems and probe them in a programmable, computational manner.They can also process experimental quantum data directly, enabling us to probe unseen aspects of physical systems that cannot be reached by any classical means.In both roles, refining quantum algorithms toward their fundamental limits expands and sharpens the reach of quantum computers.Among quantum algorithms, quantum estimation is of central importance because it elicits the physical properties of interest from quantum systems.Even in an ideal, noiseless setting, however, obtaining precise estimates of such properties is significantly resource-intensive, unless one leverages coherent quantum operations.While many quantum-enhanced (i.e., higher performance than any classical means enabled by quantum effects) estimation algorithms have been proposed under reasonable access assumptions, it is still an ongoing quest to design quantum algorithms achieving ultimate estimation efficiency allowed by quantum mechanics.In addition, rapid experimental progress makes us envision an early fault-tolerant regime with large-scale yet not fully error-corrected quantum computers.Toward this regime, it is also crucial to develop quantum estimation algorithms that elicit as much information of interest as possible from a quantum system subject to noise.In these contexts, this thesis advances quantum-enhanced algorithms for estimating physical properties in two respects: Heisenberg-limited multi-observable estimation and single-observable estimation for the early faulttolerant quantum computing (FTQC) regime.First, we provide an adaptive quantum algorithm for estimating expectation values of multiple observables simultaneously in an ideal (noiseless) setting.Given common access to block-encoded observables, a state-preparation unitary, and its inverse, this algorithm achieves a double quantum enhancement in query complexity to the state preparation and its inverse: the mean-squared-error (MSE) scaling at the Heisenberg limit and sublinear scaling in the number of observables.It also essentially matches the query lower bound in the small-additive-error regime (not in MSE), previously derived in quantum complexity theory.In addition, the proposed algorithm significantly reduces the number of ancilla qubits and the classical computation for circuit synthesis, compared to prior art for this task.The double enhancement and the circuit simplification come from a novel algorithm design that integrates key ideas from adaptive phase estimation, quantum amplification, and quantum gradient estimation.Next, we focus on expectation value estimation for a single observable in the early FTQC regime.In this regime, available coherent resources are mostly consumed for a state preparation unitary, and the dominant noise stems from that costly unitary and its inverse.Under a specific model reflecting these aspects, we derive the quantum-enhanced precision limit, that is, the minimal quantum Cramér-Rao bound over all possible estimation strategies in the model.Then, by designing an adaptive optimization based on Fisher information, we propose a quantum-enhanced estimation algorithm, with rigorous estimation guarantees, that asymptotically achieves the quantum Cramér-Rao bound, when the target system size is sufficiently large.Finally, we numerically demonstrate that the asymptotic behavior holds with only a few hundred measurements per step and a few tens of qubits.