Phase Estimation Algorithm Research Articles

Ghost Beam Suppression in Deep Frequency Modulation Interferometry for Compact On-Axis Optical Heads.

We present a compact optical head design for wide-range and low noise displacement sensing using deep frequency modulation interferometry (DFMI). The on-axis beam topology is realised in a quasi-monolithic component and relies on cube beamsplitters and beam transmission through perpendicular surfaces to keep angular alignment constant when operating in air or in a vacuum, which leads to the generation of ghost beams that can limit the phase readout linearity. We investigated the coupling of these beams into the non-linear phase readout scheme of DFMI and implemented adjustments of the phase estimation algorithm to reduce this effect. This was done through a combination of balanced detection and the inherent orthogonality of beat signals with different relative time-delays in deep frequency modulation interferometry, which is a unique feature not available for heterodyne, quadrature or homodyne interferometry.

Bridge Dynamic Cable-Tension Estimation with Interferometric Radar and APES-Based Time-Frequency Analysis

Dynamic cable-tension is an important bridge-health indicator. However, it is difficult to be measured precisely and efficiently. A remote bridge dynamic cable-tension measurement method is proposed. It uses an interferometric radar sensor, a time-frequency analysis technique, and a tension estimation approach based on a string-vibration-equation. One radar can measure the displacements of multiple cables aligned on one side of a bridge, at the same time. By solving the string vibration equation, each cable-tension is calculated from its fundamental frequency, which is obtained by time-frequency analyzing a short section of the cable’s whole displacement vector in an overlapped-piecewise manner. An adaptive amplitude and phase estimation (APES) algorithm is used to solve the frequency resolution deterioration problem due to the short duration. Simulations and field experiments with a K band interferometric radar validate that the proposed method is superior to traditional cable-tension measurements in terms of precision, robustness, and efficiency. The proposed method is of great application value in measuring and monitoring large cable-stayed bridges and cable-suspended bridges.

Quantum Solver of Contracted Eigenvalue Equations for Scalable Molecular Simulations on Quantum Computing Devices.

The accurate computation of ground and excited states of many-fermion quantum systems is one of the most consequential, contemporary challenges in the physical and computational sciences whose solution stands to benefit significantly from the advent of quantum computing devices. Existing methodologies using phase estimation or variational algorithms have potential drawbacks such as deep circuits requiring substantial error correction or nontrivial high-dimensional classical optimization. Here, we introduce a quantum solver of contracted eigenvalue equations, the quantum analog of classical methods for the energies and reduced density matrices of ground and excited states. The solver does not require deep circuits or difficult classical optimization and achieves an exponential speed-up over its classical counterpart. We demonstrate the algorithm though computations on both a quantum simulator and two IBM quantum processing units.

Analysis of phase estimation for star QAM coherent optical packet switched system

The development of data centers, 5th generation wireless network remote network education and cloud computing is increasingly demanding optical packet switched (OPS) optical interconnect networks. The blind phase estimation algorithm is one of the important phase estimation algorithms in the star QAM OPS network. However, the blind phase estimation algorithm can cause system performance degradation rapidly in the case of low signal to noise ratio (SNR). To our best knowledges, the principle and processing mechanism of the problem have not been analyzed and studied in detail. In this paper, the blind phase estimation is analyzed for the first time based on SNR and phase noise probability model at low optical signal to noise ratio. Based on the analysis, the common single differential and doubly differential coding schemes are explained for blind phase estimation. Then, the performance of the blind phase estimation + single differential and blind phase estimation + doubly differential algorithms for star QAM OPS system be compared in detail, including time-varying phase processing capability, transmission performance and tolerance to laser linewidth. The experiments be performed to verify the performance of the blind phase estimation + single differential and blind phase estimation + doubly differential algorithms.

A Phase Estimation Algorithm for Quantum Speed-Up Multi-Party Computing

Security and privacy issues have attracted the attention of researchers in the field of IoT as the information processing scale grows in sensor networks. Quantum computing, theoretically known as an absolutely secure way to store and transmit information as well as a speed-up way to accelerate local or distributed classical algorithms that are hard to solve with polynomial complexity in computation or communication. In this paper, we focus on the phase estimation method that is crucial to the realization of a general multi-party computing model, which is able to be accelerated by quantum algorithms. A novel multi-party phase estimation algorithm and the related quantum circuit are proposed by using a distributed Oracle operator with iterations. The proved theoretical communication complexity of this algorithm shows it can give the phase estimation before applying multi-party computing efficiently without increasing any additional complexity. Moreover, a practical problem of multi-party dating investigated shows it can make a successful estimation of the number of solution in advance with zero communication complexity by utilizing its special statistic feature. Sufficient simulations present the correctness, validity and efficiency of the proposed estimation method.

Accurate determination of random pattern phase in structured illumination microscopy with low modulation depth

The initial phase of structured illumination is an important parameter in structured illumination microscopy (SIM). Its estimation accuracy directly affects the reconstruction quality of SIM super-resolution images. However, when the modulation of the system is less than 0.02, the current phase estimation algorithm will cause an estimation error above 0.2 rad. An algorithm based on multi-image correlation processing in frequency domain (MCF) is proposed to solve this problem. Simulation and experimental results show that the MCF algorithm greatly improves both the initial phase estimation accuracy and the reconstruction quality of super-resolution images for low modulation SIM systems with a random phase shift. This means that the MCF algorithm can extend the scope of the SIM technology, especially for low modulation systems or systems lacking precise phase-shifting components.

Toward Quantum Computing for High-Energy Excited States in Molecular Systems: Quantum Phase Estimations of Core-Level States.

This paper explores the utility of the quantum phase estimation (QPE) algorithm in calculating high-energy excited states characterized by the promotion of electrons occupying core-level shells. These states have been intensively studied over the last few decades, especially in supporting the experimental effort at light sources. Results obtained with QPE are compared with various high-accuracy many-body techniques developed to describe core-level states. The feasibility of the quantum phase estimator in identifying classes of challenging shake-up states characterized by the presence of higher-order excitation effects is discussed. We also demonstrate the utility of the QPE algorithm in targeting excitations from specific centers in a molecule. Lastly, we discuss how the lowest-order Trotter formula can be applied to reducing the complexity of the ansatz without affecting the error.

Qudits and High-Dimensional Quantum Computing

Qudit is a multi-level computational unit alternative to the conventional 2-level qubit. Compared to qubit, qudit provides a larger state space to store and process information, and thus can provide reduction of the circuit complexity, simplification of the experimental setup and enhancement of the algorithm efficiency. This review provides an overview of qudit-based quantum computing covering a variety of topics ranging from circuit building, algorithm design, to experimental methods. We first discuss the qudit gate universality and a variety of qudit gates including the pi/8 gate, the SWAP gate, and the multi-level controlled-gate. We then present the qudit version of several representative quantum algorithms including the Deutsch-Jozsa algorithm, the quantum Fourier transform, and the phase estimation algorithm. Finally we discuss various physical realizations for qudit computation such as the photonic platform, iron trap, and nuclear magnetic resonance.

Quantum algorithm for the nonlinear dimensionality reduction with arbitrary kernel

Dimensionality reduction (DR) techniques play an extremely critical role in the data mining and pattern recognition field. However, most DR approaches involve large-scale matrix computations, which cause too high running complexity to implement in the big data scenario efficiently. The recent developments in quantum information processing provide a novel path to alleviate this problem, where a potential quantum acceleration can be obtained comparing with the classical counterpart. Nevertheless, existing proposals for quantum DR methods faced the common dilemma of the nonlinear generalization owing to the intrinsic linear limitation of quantum computation. In this paper, an architecture to simulate the arbitrary nonlinear kernels on a universal quantum computer is illustrated and further propose the quantum kernel principal component analysis (QKPCA) algorithm. The key idea is employing the truncated Taylor expansion to approximate the arbitrary nonlinear kernel within the fixed error and then constructing the corresponding Hamiltonian simulation for the quantum phase estimation algorithm. It is demonstrated theoretically that the QKPCA is qualified for the nonlinear DR task while the exponential speedup is also maintained. In addition, this research has the potential ability to develop other quantum DR approaches and existing linear quantum machine learning models.

Optimizing quantum phase estimation for the simulation of Hamiltonian eigenstates

We revisit quantum phase estimation algorithms for the purpose of obtaining the energy levels of many-body Hamiltonians and pay particular attention to the statistical analysis of their outputs. We introduce the mean phase direction of the parent distribution associated with eigenstate inputs as a new post-processing tool. By connecting it with the unknown phase, we find that if used as its direct estimator, it exceeds the accuracy of the standard majority rule using one less bit of resolution, making evident that it can also be inverted to provide unbiased estimation. Moreover, we show how to directly use this quantity to accurately find the energy levels when the initialized state is an eigenstate of the simulated propagator during the whole time evolution, which allows for shallower algorithms. We then use IBM Q hardware to carry out the digital quantum simulation of three toy models: a two-level system, a two-spin Ising model and a two-site Hubbard model at half-filling. Methodologies are provided to implement Trotterization and reduce the variability of results in noisy intermediate scale quantum computers.

The quantum version of the shifted power method and its application in quadratic binary optimization

In this paper, we present a direct quantum adaptation of the classical shifted power method. The method is very similar to the iterative phase estimation algorithm; however, it does not require any initial estimate of an eigenvector, and as in the classical case its convergence and the required number of iterations are directly related to the eigengap. If the amount of the gap is in the order of $1/poly(n)$, then the algorithm can converge to the dominant eigenvalue in $O(poly(n))$ time. The method can be potentially used for solving any eigenvalue related problem and finding minimum/maximum of a data set in lieu of Grover's search algorithm. In addition, if the solution space of an optimization problem with $n$ parameters is encoded as the eigenspace of an $2^n$ dimensional unitary operator in $O(poly(n))$ time and the eigengap is not too small, then the solution for such a problem can be found in $O(poly(n))$. As an example, using the quantum gates, we show how to generate the solution space of the quadratic unconstrained binary optimization as the eigenvectors of a diagonal unitary matrix and find the solution for the problem.

Robust and Low-Complexity Principal Component-Based Phase Estimation Algorithm for Probabilistically Shaped Square-QAM Systems

An optimized principal component-based phase estimation (OPCPE) algorithm is proposed for probabilistically shaped (PS) square-quadrature amplitude modulation (QAM) systems. The optimization method is based on mirroring and partially scaling the received constellation. Numerical simulation validation shows that the proposed phase recovery algorithm outperforms the well-known blind phase search (BPS) algorithm at low signal-to-noise ratio (SNR) and solves the incompatibility between principal component-based phase estimation (PCPE) and PS at high SNR. It's worth mentioning that the proposed algorithm retains the advantages of low cycle slip rate (CSR) and low computational complexity of PCPE. Furthermore, the performance of a hybrid implementation combining the proposed algorithm and BPS is also investigated. When the shaping parameter for a PS-64QAM system is 0.03, by using the proposed simple and hybrid methods, the SNR required to reach the 0.8 normalized generalized mutual information (NGMI) threshold can be reduced by about 0.74 dB and 0.98 dB, respectively, benchmarking the widely used two-stage BPS (2S-BPS) algorithm. To the best of our knowledge, this is the first time to robustly apply PCPE algorithm to PS systems, and the proposed design has stable GMI performance, strong cycle slip tolerance and low computational complexity, which makes it a suitable candidate for practical probabilistically-shaped transmissions.

Entanglement in Phase Estimation Algorithm and Quantum Counting Algorithm

For some certain problems, quantum algorithms are theoretically able to solve them quickly than classical algorithms. But the role of entanglement in achieving the quantum computational speedup is not fully understood. By theoretical analysis and numerical calculation of four practical use cases, we investigate the entanglement features of the quantum states employed in quantum phase estimation algorithm and quantum counting algorithm. The results show that whether these two algorithms generate entanglement depend on whether the input quantum state of the second register is a superposition state of the eigenstates.

A Multireference Quantum Krylov Algorithm for Strongly Correlated Electrons.

We introduce a multireference selected quantum Krylov (MRSQK) algorithm suitable for quantum simulation of many-body problems. MRSQK is a low-cost alternative to the quantum phase estimation algorithm that generates a target state as a linear combination of non-orthogonal Krylov basis states. This basis is constructed from a set of reference states via real-time evolution; thus, avoiding the numerical optimization of parameters. An efficient algorithm for the evaluation of the off-diagonal matrix elements of the overlap and Hamiltonian matrices is discussed and a selection procedure is introduced to identify a basis of orthogonal references that ameliorates the linear dependency problem. Preliminary benchmarks on linear H6, H8, and BeH2 indicate that MRSQK can predict the energy of these systems accurately using very compact Krylov bases.

Amplitude estimation without phase estimation

This paper focuses on the quantum amplitude estimation algorithm, which is a core subroutine in quantum computation for various applications. The conventional approach for amplitude estimation is to use the phase estimation algorithm, which consists of many controlled amplification operations followed by a quantum Fourier transform. However, the whole procedure is hard to implement with current and near-term quantum computers. In this paper, we propose a quantum amplitude estimation algorithm without the use of expensive controlled operations; the key idea is to utilize the maximum likelihood estimation based on the combined measurement data produced from quantum circuits with different numbers of amplitude amplification operations. Numerical simulations we conducted demonstrate that our algorithm asymptotically achieves nearly the optimal quantum speedup with a reasonable circuit length.

Controlled extreme wave generation using an improved focusing method

Dispersive focusing technology is one of the most widely used methods to simulate extreme waves in the laboratory. However, such methods that are based on the linear dispersion relation are not robust. The focusing waves are always generated at unexpected positions owing to nonlinear effects. Hence, in this paper, an improved focusing method named the All Phase Correction Method (APCM) is developed to produce accurate focusing waves at a predefined position. An improved phase estimation algorithm and second-order wavemaker theory is employed to iteratively correct the phases and amplitudes of the generated waves. Different types of experiments were conducted in a wave flume using this method. The results indicate that the present method works well for accurately generating extreme events. Both the component phases and the focal positions match well with the target values. The influences of wave steepness, spectral bandwidth and spectral type on the correction procedure were investigated extensively. The analysis revealed that with the increase in wave steepness and narrowness of spectral width, the phase deviation increased. Thus, more correction iterations are needed to achieve convergence of iteration. In addition, the spectral type also influences the wave correction procedure. The focal position deviations for the constant-steepness spectrum and the constant-amplitude spectrum are significantly smaller than that of the JONSWAP spectrum, and the correlation coefficient converges faster.

Analysis and Application of Taylor-Kalman Filters Under a Distorted Grid Condition

The Taylor-Kalman filter (TKF) is a linear filter that can be modeled using either an accurately modeled TKF (known as the modified TKF) or an approximately modeled TKF. The higher-order TKF can not only adapt to the small frequency deviation condition but also prevents certain harmonics leakage. It has been increasingly widely applied in power grid synchronization and other power system fields. However, to the best of authors’ knowledge, systematical analyses of the impact of different models on the estimation performance and the methods for extraction of the shifted grid frequency have not been reported to date. The aim of this paper is to examine the TKF-based phase estimation algorithms (PEA) with regard to aspects such as the steady state, dynamic state and computational cost. Comparisons were carried out to evaluate the effects of the model and the order of TKF on the dynamic response and steady-state error. A strong tracking algorithm was also introduced to enhance the dynamic response. Several approaches for reducing the computational burden are given. Finally, combined with the moving average filter (MAF), which is a typical low-pass filter, an application example of TKF-based PEA was developed, and its performance was verified by experiments.

A probabilistic spin annihilation method for quantum chemical calculations on quantum computers.

A probabilistic spin annihilation method based on the quantum phase estimation algorithm is presented for quantum chemical calculations on quantum computers. This approach can eliminate more than one spin component from the spin contaminated wave functions by single operation. Comparison with the spin annihilation operation on classical computers is given.

Quantum simulations of excited states with active-space downfolded Hamiltonians.

Many-body techniques based on the double unitary coupled cluster (DUCC) ansatz can be used to downfold electronic Hamiltonians into low-dimensional active spaces. It can be shown that the resulting dimensionality reduced Hamiltonians are amenable for quantum computing. Recent studies performed for several benchmark systems using phase estimation (PE) algorithms for quantum computers demonstrated that these formulations can recover a significant portion of ground-state dynamical correlation effects that stem from the electron excitations outside of the active space. These results have also been confirmed in studies of ground-state potential energy surfaces using quantum simulators. In this letter, we study the effectiveness of the DUCC formalism in describing excited states. We also emphasize the role of the PE formalism and its stochastic nature in discovering/identifying excited states or excited-state processes in situations when the knowledge about the true configurational structure of a sought after excited state is limited or postulated (due to the specific physics driving excited-state processes of interest). In this context, we can view PE algorithms as an engine for verifying various hypotheses for excited-state processes and providing statistically meaningful results that correspond to the electronic state(s) with the largest overlap with a postulated configurational structure. We illustrate these ideas on examples of strongly correlated molecular systems, characterized by small energy gaps and high density of quasidegenerate states around the Fermi level.

An Absolute Phase Estimation Method for Interferometry Imagery

Accurate interpretation of interferometric image requires an extremely challenging task based on actual phase reconstruction for incomplete noise observation. In spite of the establishment of comprehensive solutions, until now, a guaranteed means of solution method is yet to exist. The initially observed interferometric image is formed by 2π-periodic phase image that wrapped within (-π, π]. Such inverse problem is further corrupted by noise distortion and leads to the degradation of interferometric image. In order to overcome this, an effective algorithm that enables noise suppression and absolute phase reconstruction of interferometric phase image is proposed. The proposed method incorporates an improved order statistical filter that is able to adjust or vary on its filtering rate by adapting to phase noise level of relevant interferometric image. Performance of proposed method is evaluated and compared with other existing phase estimation algorithms. The comparison is based on a series of computer simulated and real interferometric data images. The experiment results illustrate the effectiveness and competency of the proposed method.