Davide Castaldo, Soran Jahangiri, Agostino Migliore et al. · Quantum Science and Technology · 2024
Abstract The simulation of electronic properties is a pivotal issue in modern electronic structure theory, driving significant efforts over the past decades to develop protocols for computing energy derivatives. In this work, we address this problem by developing a strategy to integrate the quantum phase estimation algorithm within a fully differentiable framework. This is accomplished by devising a smooth estimator able to tackle arbitrary initial states. We provide analytical expressions to characterize the statistics and algorithmic cost of this estimator. Furthermore, we provide numerical evidence that the estimation accuracy is retained when an arbitrary state is considered and that it exceeds the one of standard majority rule. We explicitly use this procedure to estimate chemically relevant quantities, demonstrating our approach through ground-state and triplet excited state geometry optimization with simulations involving up to 19 qubits. This work paves the way for new quantum algorithms that combine interference methods and quantum differentiable programming.
Cited by 2Open access
Davide Castaldo, Soran Jahangiri, Agostino Migliore et al. · arXiv (Cornell University) · 2024
The simulation of electronic properties is a pivotal issue in modern electronic structure theory, driving significant efforts over the past decades to develop protocols for computing energy derivatives. In this work, we address this problem by developing a strategy to integrate the quantum phase estimation algorithm within a fully differentiable framework. This is accomplished by devising a smooth estimator able to tackle arbitrary initial states. We provide analytical expressions to characterize the statistics and algorithmic cost of this estimator. Furthermore, we provide numerical evidence that the estimation accuracy is retained when an arbitrary state is considered and that it exceeds the one of standard majority rule. We explicitly use this procedure to estimate chemically relevant quantities, demonstrating our approach through ground-state and triplet excited state geometry optimization with simulations involving up to 19 qubits. This work paves the way for new quantum algorithms that combine interference methods and quantum differentiable programming.
Cited by 0Open access
Scott Shepard, John Abraham · Quantum 2.0 Conference and Exhibition · 2022
Means for extracting higher-order phase information and sub-Heisenberg accuracies within an SU(2) interferometer utilizing coherent-state light are discussed.
Cited by 0
Kianna Wan, Mario Berta, Earl T. Campbell · Physical Review Letters · 2022
Phase estimation is a quantum algorithm for measuring the eigenvalues of a Hamiltonian. We propose and rigorously analyze a randomized phase estimation algorithm with two distinctive features. First, our algorithm has complexity independent of the number of terms L in the Hamiltonian. Second, unlike previous L-independent approaches, such as those based on qDRIFT, all algorithmic errors in our method can be suppressed by collecting more data samples, without increasing the circuit depth.
Cited by 81Open access
Joseph G. Smith, C. H. W. Barnes, David R. M. Arvidsson-Shukur · Physical review. A/Physical review, A · 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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>N</a:mi><a:mi>tot</a:mi></a:msub></a:math> times, the variance of the estimates can scale as <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>O</b:mi><b:mo>(</b:mo><b:mn>1</b:mn><b:mo>/</b:mo><b:msubsup><b:mi>N</b:mi><b:mi>tot</b:mi><b:mn>2</b:mn></b:msubsup><b:mo>)</b:mo></b:mrow></b:math>, compared to the best “classical” strategy with <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>O</c:mi><c:mo>(</c:mo><c:mn>1</c:mn><c:mo>/</c:mo><c:msub><c:mi>N</c:mi><c:mi>tot</c:mi></c:msub><c:mo>)</c:mo></c:mrow></c:math>. 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 <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>O</d:mi><d:mo>(</d:mo><d:mn>1</d:mn><d:mo>/</d:mo><d:msubsup><d:mi>N</d:mi><d:mi>tot</d:mi><d:mn>2</d:mn></d:msubsup><d:mo>)</d:mo></d:mrow></d:math>. 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
Cited by 15Open access
A. N. Prokopenya · Lecture notes in computer science · 2015
Cited by 5
Priyanka Sharma, Rukhsar Zafar, Amit Kumar Sharma · Lecture notes in electrical engineering · 2025
Cited by 0
Theerapat Tansuwannont, Surachate Limkumnerd, Sujin Suwanna et al. · Open Physics · 2019
Abstract Quantum algorithm is an algorithm for solving mathematical problems using quantum systems encoded as information, which is found to outperform classical algorithms in some specific cases. The objective of this study is to develop a quantum algorithm for finding the roots of n th degree polynomials where n is any positive integer. In classical algorithm, the resources required for solving this problem increase drastically when n increases and it would be impossible to practically solve the problem when n is large. It was found that any polynomial can be rearranged into a corresponding companion matrix, whose eigenvalues are roots of the polynomial. This leads to a possibility to perform a quantum algorithm where the number of computational resources increase as a polynomial of n. In this study, we construct a quantum circuit representing the companion matrix and use eigenvalue estimation technique to find roots of polynomial.
Cited by 3Open access
Luca Pezzè, Augusto Smerzi · INO Open Portal · 2021
Quantum phase estimation (QPE) is one of the most important subroutines in quantum computing. In general applications, current QPE algorithms either suffer an exponential time overload or require a set of--notoriously quite fragile--GHZ states. These limitations have so far prevented the demonstration of QPE beyond proof of principles. Here we propose a new QPE algorithm based on a cascade of Gaussian spin states (GSSs) and a suitable adaptive measurement protocol. GSSs are renownedly resilient and have been created experimentally in a variety of platforms, from hundreds of ions up to millions of cold and ultracold neutral atoms. We show that our protocol achieves a QPE sensitivity overcoming previous schemes, including that obtained with GHZ states, scales linearly with time, and is robust against certain sources of noise and decoherence, such as detection noise, in particular. Our work paves the way toward an efficient implementation of the QPE, as well as applications of atomic squeezed states for quantum computation.
Cited by 29Open access
Wan, Kianna, Berta, Mario, Campbell, Earl T. · arXiv (Cornell University) · 2021
Phase estimation is a quantum algorithm for measuring the eigenvalues of a Hamiltonian. We propose and rigorously analyse a randomized phase estimation algorithm with two distinctive features. First, our algorithm has complexity independent of the number of terms L in the Hamiltonian. Second, unlike previous L-independent approaches, such as those based on qDRIFT, all sources of error in our algorithm can be suppressed by collecting more data samples, without increasing the circuit depth.
Cited by 1Open access
Tansuwannont, Theerapat, Limkumnerd, Surachate, Suwanna, Sujin et al. · arXiv (Cornell University) · 2015
Quantum algorithm is an algorithm for solving mathematical problems using quantum systems encoded as information, which is found to outperform classical algorithms in some specific cases. The objective of this study is to develop a quantum algorithm for finding the roots of nth degree polynomials where n is any positive integer. In classical algorithm, the resources required for solving this problem increase drastically when n increases and it would be impossible to practically solve the problem when n is large. It was found that any polynomial can be rearranged into a corresponding companion matrix, whose eigenvalues are roots of the polynomial. This leads to a possibility to perform a quantum algorithm where the number of computational resources increase as a polynomial of n. In this study, we construct a quantum circuit representing the companion matrix and use eigenvalue estimation technique to find roots of polynomial.
Cited by 0Open access
S. G. Tan, Son-Hsien Chen, Yaocheng Yang et al. · ArXiv.org · 2025
We propose an approach to measure the quantum phase of an electron in a non-Abelian system using the algorithm of Quantum Phase Estimation (QPE). The discrete-path systems were previously studied in the context of square or rectangular rings. Present focus is on measuring the quantum phases. The merit of the algorithm approach is two-fold. First off, it eliminates the need for an interferometric set up. Quantum phase is measured by reading off of measurable qubit states of the QPE modules. Secondly, the QPE works by subjecting the quantum state to a sequence of quantum computing operations that eventually map the phase information into measurable qubit states. All the operations are realizable by standard quantum computer gates and algorithms, placing the new effort within the reach of standard quantum computational framework.
Cited by 0Open access
Xue Wen · 2025
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.
Cited by 0
Joseph G. Smith, C. H. W. Barnes, David R. M. Arvidsson-Shukur · Physical review. A/Physical review, A · 2025
The quantum-phase-estimation algorithm (QPEA) is widely used to find estimates of unknown phases. The original algorithm relied on an input state in a uniform superposition of all possible bit strings. However, it is known that other input states can reduce certain Bayesian risks of the final estimate. Here, we derive a method to find the risk-minimizing input state for any risk. These states are represented by an eigenvector of a Toeplitz matrix with elements given by the Fourier coefficients of the loss function of interest. We show that, while the true optimal state does not have a closed form for a general loss function, it is well approximated by a state with a cosine form. When the cosine frequency is chosen appropriately, these states outperform the original QPEA and achieve the optimal theoretical quantum-advantage scaling for three common risks. Furthermore, we prove that the uniform input state is suboptimal for any reasonable loss function. Finally, we design methods to mitigate the impact of depolarizing noise on the performance of QPEA.
Cited by 0Open access
Garcia-Mata, Ignacio, Shepelyansky, Dima L. · arXiv (Cornell University) · 2007
We study numerically the effects of static imperfections and residual couplings between qubits for the quantum phase estimation algorithm with two qubits. We show that the success probability of the algorithm is affected significantly more by static imperfections than by random noise errors in quantum gates. An improvement of the algorithm accuracy can be reached by application of the Pauli-random-error-correction method (PAREC).
Cited by 0Open access
Joseph G. Smith, C. H. W. Barnes, David R. M. Arvidsson-Shukur · ArXiv.org · 2025
The quantum-phase-estimation algorithm (QPEA) is widely used to find estimates of unknown phases. The original algorithm relied on an input state in a uniform superposition of all possible bit strings. However, it is known that other input states can reduce certain Bayesian risks of the final estimate. Here, we derive a method to find the risk-minimizing input state for any risk. These states are represented by an eigenvector of a Toeplitz matrix with elements given by the Fourier coefficients of the loss function of interest. We show that, while the true optimal state does not have a closed form for a general loss function, it is well approximated by a state with a cosine form. When the cosine frequency is chosen appropriately, these states outperform the original QPEA and achieve the optimal theoretical quantum-advantage scaling for three common risks. Furthermore, we prove that the uniform input state is suboptimal for any reasonable loss function. Finally, we design methods to mitigate the impact of depolarizing noise on the performance of QPEA.
Cited by 0Open access
Ammar Daskin · arXiv (Cornell University) · 2017
In this brief paper, we go through the theoretical steps of the spectral clustering on quantum computers by employing the phase estimation and the amplitude amplification algorithms. We discuss circuit designs for each step and show how to obtain the clustering solution from the output state. In addition, we introduce a biased version of the phase estimation algorithm which significantly speeds up the amplitude amplification process. The complexity of the whole process is analyzed: it is shown that when the circuit representation of a data matrix of order $N$ is produced through an ancilla based circuit in which the matrix is written as a sum of $L$ number of Householder matrices; the computational complexity is bounded by $O(2^mLN)$ number of quantum gates. Here, $m$ represents the number of qubits (e.g., 6) involved in the phase register of the phase estimation algorithm.
Cited by 4Open access
Charles Woodrum, David E. Weeks · 2023
Quantum phase estimation is one of the most fundamental and useful quantum computing algorithms available to researchers, but traditional methods for estimating the phase of a unitary operator acting on its eigenvalue leave information about the phase contained in all of the output from a quantum computer unused. Mean squared error (MSE) for such a scheme is 0.03 for simulated data with and without noise. Machine learning with classical methods, like linear regression (MSE=0.011 for noisy data and 0.0043 for noise-free data) and random forests (MSE=0.01 for noisy data and 0.006 for noise-free data) increase the accuracy of the estimation drastically without requiring additional qubits like the traditional approach. However, neural networks reduce this number further (MSE=0.0077) in the case of noisy data and with noise-free data (MSE=0.003). Examining the performance of neural networks with different modifications also yields insight into their performance on quantum computing output that could be generalized to output from other algorithms and on quantum computers with different numbers of qubits.
Cited by 1
D. V. Babukhin, A. A. Zhukov, W. V. Pogosov · AIP conference proceedings · 2020
Methods of processing quantum data become more important as quantum computing devices improve their quality towards fault tolerant universal quantum computers. These methods include discrimination and filtering of quantum states given as an input to the device that may find numerous applications in quantum information technologies. In the present paper, we address a scheme of a classification of input states, which is nondestructive and deterministic for certain inputs, while probabilistic, in general case. This can be achieved by incorporating phase estimation algorithm into the hybrid quantum-classical computation scheme, where quantum block is trained classically. We perform proof-of-principle implementation of this idea using superconducting quantum processor of IBM Quantum Experience. Another aspect we are interested in is a mitigation of errors occurring due to the quantum device imperfections. We apply a series of heuristic tricks at the stage of classical postprocessing in order to improve raw experimental data and to recognize patterns in them. These ideas may find applications in other realization of hybrid quantum-classical computations with noisy quantum machines
Cited by 0
Wenbin Yu, Hao Feng, Yinsong Xu et al. · Computers, materials & continua/Computers, materials & continua (Print) · 2021
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.
Cited by 3Open access