We develop several algorithms for performing quantum phase estimation based on basic measurements and classical post-processing. We present a pedagogical review of quantum phase estimation and simulate the algorithm to numerically determine its scaling in circuit depth and width. We show that the use of purely random measurements requires a number of measurements that is optimal up to constant factors, albeit at the cost of exponential classical post-processing; the method can also be used to improve classical signal processing. We then develop a quantum algorithm for phase estimation that yields an asymptotic improvement in runtime, coming within a factor of log* of the minimum number of measurements required while still requiring only minimal classical post-processing. The corresponding quantum circuit requires asymptotically lower depth and width (number of qubits) than quantum phase estimation.
We develop several algorithms for performing quantum phase estimation based on basic measurements and classical post-processing. We present a pedagogical review of quantum phase estimation and simulate the algorithm to numerically determine its scaling in circuit depth and width. We show that the use of purely random measurements requires a number of measurements that is optimal up to constant factors, albeit at the cost of exponential classical post-processing; the method can also be used to improve classical signal processing. We then develop a quantum algorithm for phase estimation that yields an asymptotic improvement in runtime, coming within a factor of log* of the minimum number of measurements required while still requiring only minimal classical post-processing. The corresponding quantum circuit requires asymptotically lower depth and width (number of qubits) than quantum phase estimation.
We develop several algorithms for performing quantum phase estimation based on basic measurements and classical post-processing. We present a pedagogical review of quantum phase estimation and simulate the algorithm to numerically determine its scaling in circuit depth and width. We show that the use of purely random measurements requires a number of measurements that is optimal up to constant factors, albeit at the cost of exponential classical post-processing; the method can also be used to improve classical signal processing. We then develop a quantum algorithm for phase estimation that yields an asymptotic improvement in runtime, coming within a factor of log∗ of the minimum number of measurements required while still requiring only minimal classical post-processing. The corresponding quantum circuit requires asymptotically lower depth and width (number of qubits) than quantum 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 Quantum effects like entanglement and coherent amplification can be used to drastically enhance the accuracy of quantum parameter estimation beyond classical limits. However, challenges such as decoherence and time-dependent errors hinder Heisenberg-limited amplification. We introduce Quantum Signal-Processing Phase Estimation algorithms that are robust against these challenges and achieve optimal performance as dictated by the Cramér-Rao bound. These algorithms use quantum signal transformation to decouple interdependent phase parameters into largely orthogonal ones, ensuring that time-dependent errors in one do not compromise the accuracy of learning the other. Combining provably optimal classical estimation with near-optimal quantum circuit design, our approach achieves a standard deviation accuracy of 10−4 radians for estimating unwanted swap angles in superconducting two-qubit experiments, using low-depth ( < 10) circuits. This represents up to two orders of magnitude improvement over existing methods. Theoretically and numerically, we demonstrate the optimality of our algorithm against time-dependent phase errors, observing that the variance of the time-sensitive parameter φ scales faster than the asymptotic Heisenberg scaling in the small-depth regime. Our results are rigorously validated against the quantum Fisher information, confirming our protocol’s ability to achieve unmatched precision for two-qubit gate learning.
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.
Quantum phase estimation (QPE) is fundamental to quantum algorithms for eigenvalue problems, chemistry simulation, and number-theoretic applications. However, canonical QPE implementations employ controlled unitary operations and quantum Fourier transform, incurring substantial circuit depth and control overhead that severely limit applicability on nearterm quantum hardware. This work develops a novel class of control-free phase estimation algorithms that eliminate both controlled unitaries and QFT by leveraging signal processing principles. Our approach applies uncontrolled powers of the unitary U = e −iHτ to an eigenstate or eigenstate-rich state, measures in multiple bases to capture interference patterns, and reconstructs eigenphases via classical post-processing techniques including Prony's method, compressed sensing, and harmonic inversion. We establish that for eigenstates or states with sparse eigendecomposition, our algorithms achieve phase estimation precision ϵ using Õ(1/ϵ) applications of U , matching standard QPE query complexity up to logarithmic factors while dramatically reducing circuit depth and eliminating control overhead. We provide detailed error analysis characterizing robustness to measurement noise, finite sampling, and state preparation errors. Applications to electronic structure estimation in small molecules and variational quantum eigensolvers demonstrate the practical viability of control-free QPE on both NISQ and early fault-tolerant quantum computers. Our results suggest that signal-processing-based phase estimation may offer more efficient implementations of quantum algorithms on realistic hardware.
Quantum phase estimation (QPE) is a key ingredient to some of the most important results of quantum computing. In this paper we examine the QPE algorithm through the looking glass of signal processing. This perspective helps to reveal a relation between the distribution of quantum phase estimates and the shape of the discrete Fourier transform (DFT) of a pure complex sinusoid. Guided by spectral shaping theory, we design a class of quantum circuits that are the quantum equivalents to classical windowing operators, for a family of window functions that have compact support in the DFT. These circuits have complexity linear in the number of target qubits, strictly lower than QPE itself. Using these circuits we are able to reshape QPE's performance characteristics. In particular, the asymptotic marginal cost for 2x reduction in fail rate (2x improvement in reliability) is reduced from one qubit to 1/3 qubit or lower, which can help reduce the qubit budget for a preset accuracy-reliability target, especially in high-reliability scenarios. This makes a demonstration of the value of signal processing principles in designing quantum algorithms.