Quantum phase estimation is a core task in quantum technologies ranging from metrology to quantum computing, where it appears as a key subroutine in various algorithms. Here, we quantitatively connect the performance of phase estimation protocols with quantum coherence. To achieve this, we construct and characterize resource theories of quantum networks that cannot generate coherence. Given multiple copies of a unitary encoding an unknown phase and access to a fixed coherent state, we estimate the phase using such networks. For a unified and general approach, we assess the quality of the estimate using a generic cost function that penalizes deviations from the true value. We determine the minimal average cost that can be achieved in this manner and explicitly derive optimal protocols. From this, we construct a family of coherence measures that directly connect a state's coherence with its value for phase estimation, demonstrating that every bit of coherence helps. This establishes coherence as a resource that quantifies the performance of phase estimation, and, thus, of any quantum technology relying on it as a subroutine.
Quantum phase estimation is a core task in quantum technologies ranging from metrology to quantum computing, where it appears as a key subroutine in various algorithms. Here, we quantitatively connect the performance of phase estimation protocols with quantum coherence. To achieve this, we construct and characterize resource theories of quantum networks that cannot generate coherence. Given multiple copies of a unitary encoding an unknown phase and access to a fixed coherent state, we estimate the phase using such networks. For a unified and general approach, we assess the quality of the estimate using a generic cost function that penalizes deviations from the true value. We determine the minimal average cost that can be achieved in this manner and explicitly derive optimal protocols. From this, we construct a family of coherence measures that directly connect a state's coherence with its value for phase estimation, demonstrating that every bit of coherence helps. This establishes coherence as a resource that quantifies the performance of phase estimation, and, thus, of any quantum technology relying on it as a subroutine.
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.
The detection and characterization of quantum coherence is of fundamental importance both in the foundations of quantum theory as well as for the rapidly developing field of quantum technologies, where coherence has been linked to quantum advantage. Typical approaches for detecting coherence employ {\it coherence witnesses} -- observable quantities whose expectation value can be used to certify the presence of coherence. By design, coherence witnesses are only able to detect coherence for some, but not all, possible states of a quantum system. In this work we construct protocols capable of detecting the presence of coherence in an {\it unknown} pure quantum state ∣ψ⟩. Having access to m copies of an unknown pure state ∣ψ⟩ we show that the sample complexity of any experimental procedure for detecting coherence with constant probability of success ≥2/3 is Θ(c(∣ψ⟩)−1), where c(∣ψ⟩) is the geometric measure of coherence of ∣ψ⟩. However, assuming access to the unitary Uψ which prepares the unknown state ∣ψ⟩, and its inverse Uψ†, we devise a coherence detecting protocol that employs amplitude-amplification {\it a la} Grover, and uses a quadratically smaller number O(c(∣ψ⟩)−1/2) of samples. Furthermore, by augmenting amplitude amplification with phase estimation we obtain an experimental estimation of upper bounds on the geometric measure of coherence within additive error ε with a sample complexity that scales as O(1/ε) as compared to the O(1/ε2) sample complexity of Monte Carlo estimation methods. The average number of samples needed in our amplitude estimation protocol provides a new operational interpretation for the geometric measure of coherence. Finally, we also derive bounds on the amount of noise our protocols are able to tolerate.
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].
We consider performing phase estimation under the following conditions: we are given only one copy of the input state, the input state does not have to be an eigenstate of the unitary, and the state must not be measured. Most quantum estimation algorithms make assumptions that make them unsuitable for this 'coherent' setting, leaving only the textbook approach. We present novel algorithms for phase, energy, and amplitude estimation that are both conceptually and computationally simpler than the textbook method, featuring both a smaller query complexity and ancilla footprint. They do not require a quantum Fourier transform, and they do not require a quantum sorting network to compute the median of several estimates. Instead, they use block-encoding techniques to compute the estimate one bit at a time, performing all amplification via singular value transformation. These improved subroutines accelerate the performance of quantum Metropolis sampling and quantum Bayesian inference.
Quantum phase estimation is one of the fundamental primitives that underpins many quantum algorithms, including Shor’s algorithm for efficiently factoring large numbers. Due to its significance as a subroutine, in this work, we consider the coherent version of the phase estimation problem, where given an arbitrary input state and black-box access to unitaries U and controlled- U , the goal is to estimate the phases of U in superposition. Most existing phase estimation algorithms involve intermediary measurements that disrupt coherence. Only a couple of algorithms, including the standard quantum phase estimation algorithm, consider this coherent setting. However, the standard algorithm only succeeds with a constant probability. To boost this success probability, one can employ the coherent median technique, resulting in an algorithm with asymptotically optimal query complexity (the total number of calls to U and controlled- U ). However, this coherent median technique requires a large number of ancilla qubits and a computationally expensive quantum sorting network. To address this, in this work, we propose an improved version of the standard algorithm called the tapered quantum phase estimation algorithm, which leverages tapering (or window) functions commonly used in classical signal processing. Our algorithm achieves the asymptotically optimal query complexity without requiring the expensive coherent median technique to boost success probability. Moreover, we find the absolutely optimal taper—not only in the asymptotic scaling but in terms of exact performance. We provide an efficiently preparable ancilla state based on an approximation of the optimal taper, which incurs at most a factor-of-two increase in the probability of error, thereby maintaining near-optimal performance in practice. In the Appendixes, we give an explicit construction of the taper state preparation circuit. Finally, we derive an error bound for coherent quantum phase estimation when the phase estimate is used as a control and subsequently uncomputed.