scholarly journals Quantum algorithm for matrix functions by Cauchy's integral formula

2020 ◽  
Vol 20 (1&2) ◽  
pp. 14-36
Author(s):  
Souichi Takahira ◽  
Asuka Ohashi ◽  
Tomohiro Sogabe ◽  
Tsuyoshi S. Usuda
Keyword(s):  
Integral Formula ◽  
Quantum Algorithm ◽  
Trapezoidal Rule ◽  
Phase Estimation ◽  
Matrix Functions ◽  
Output State ◽  
Cauchy’S Integral Formula ◽  
Hamiltonian Simulation ◽  
Eigenvalue Estimation ◽  
And Function

For matrix A, vector b and function f, the computation of vector f(A)b arises in many scientific computing applications. We consider the problem of obtaining quantum state |f> corresponding to vector f(A)b. There is a quantum algorithm to compute state |f> using eigenvalue estimation that uses phase estimation and Hamiltonian simulation e^{\im A t}. However, the algorithm based on eigenvalue estimation needs \poly(1/\epsilon) runtime, where \epsilon is the desired accuracy of the output state. Moreover, if matrix A is not Hermitian, \e^{\im A t} is not unitary and we cannot run eigenvalue estimation. In this paper, we propose a quantum algorithm that uses Cauchy's integral formula and the trapezoidal rule as an approach that avoids eigenvalue estimation. We show that the runtime of the algorithm is \poly(\log(1/\epsilon)) and the algorithm outputs state |f> even if A is not Hermitian.

scholarly journals Quantum algorithms for Gibbs sampling and hitting-time estimation

2017 ◽  
Vol 17 (1&2) ◽  
pp. 41-64
Author(s):  
Anirban Narayan Chowdhury and ◽  
Rolando D. Somma
Keyword(s):  
Space Dimension ◽  
Gibbs State ◽  
Quantum Algorithms ◽  
Stochastic Matrix ◽  
Quantum Algorithm ◽  
Time Estimation ◽  
Hitting Time ◽  
Output State ◽  
Hamiltonian Simulation ◽  
Linear Systems Of Equations

We present quantum algorithms for solving two problems regarding stochastic processes. The first algorithm prepares the thermal Gibbs state of a quantum system and runs in time almost linear in p Nβ/Z and polynomial in log(1/epsilon), where N is the Hilbert space dimension, β is the inverse temperature, Z is the partition function, and epsilon is the desired precision of the output state. Our quantum algorithm exponentially improves the complexity dependence on 1/epsilon and polynomially improves the dependence on β of known quantum algorithms for this problem. The second algorithm estimates the hitting time of a Markov chain. For a sparse stochastic matrix P, it runs in time almost linear in 1/(epsilon ∆3/2 ), where epsilon is the absolute precision in the estimation and ∆ is a parameter determined by P, and whose inverse is an upper bound of the hitting time. Our quantum algorithm quadratically improves the complexity dependence on 1/epsilon and 1/∆ of the analog classical algorithm for hitting-time estimation. Both algorithms use tools recently developed in the context of Hamiltonian simulation, spectral gap amplification, and solving linear systems of equations.

scholarly journals Meromorphic or Holomorphic Completion of a Reinhardt Domain

1970 ◽  
Vol 38 ◽  
pp. 1-12 ◽  
Author(s):  
Eiichi Sakai
Keyword(s):  
Meromorphic Function ◽  
Power Series ◽  
Meromorphic Functions ◽  
Integral Formula ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Reinhardt Domain ◽  
Continuity Theorem ◽  
Cauchy’S Integral Formula ◽  
Long Time

In the theory of functions of several complex variables, the problem about the continuation of meromorphic functions has not been much investigated for a long time in spite of its importance except the deeper result of the continuity theorem due to E. E. Levi [4] and H. Kneser [3], The difficulty of its investigation is based on the following reasons: we can not use the tools of not only Cauchy’s integral formula but also the power series and there are indetermination points for the meromorphic function of many variables different from one variable. Therefore we shall also follow the Levi and Kneser’s method and seek for the aspect of meromorphic completion of a Reinhardt domain in Cn.

Quantum algorithm for measuring the energy of n qubits with unknown pair-interactions

2002 ◽  
Vol 2 (3) ◽  
pp. 198-207
Author(s):  
D. Janzing
Keyword(s):  
Quantum State ◽  
Quantum Computer ◽  
Quantum Algorithm ◽  
Pair Interaction ◽  
Phase Estimation ◽  
Solid States ◽  
Energy Gaps ◽  
Large N ◽  
Conditional Transformation ◽  
Essential Assumption

The well-known algorithm for quantum phase estimation requires that the considered unitary is available as a conditional transformation depending on the quantum state of an ancilla register. We present an algorithm converting an unknown n-qubit pair-interaction Hamiltonian into a conditional one such that standard phase estimation can be applied to measure the energy. Our essential assumption is that the considered system can be brought into interaction with a quantum computer. For large n the algorithm could still be applicable for estimating the density of energy states and might therefore be useful for finding energy gaps in solid states.

scholarly journals On the Surface Fields of a Small Circular Loop Antenna Placed on Plane Stratified Earth

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
Mauro Parise
Keyword(s):  
Analytical Method ◽  
Integral Formula ◽  
Integral Representations ◽  
Loop Antenna ◽  
Circular Loop ◽  
Time Harmonic ◽  
Hankel Functions ◽  
Cauchy’S Integral Formula ◽  
Em Field ◽  
Surface Fields

An analytical method is presented which makes it possible to derive exact explicit expressions for the time-harmonic surface fields excited by a small circular loop antenna placed on the top surface of plane layered earth. The developed procedure leads to casting the complete integral representations for the EM field components into forms suitable for application of Cauchy’s integral formula. As a result, the surface fields are expressed as sums of Hankel functions. Numerical simulations are performed to show the validity and accuracy of the proposed solution.

Cauchy’s Integral Formula

2011 ◽  
pp. 111-115
Author(s):  
Ravi P. Agarwal ◽  
Kanishka Perera ◽  
Sandra Pinelas
Keyword(s):  
Integral Formula ◽  
Cauchy’S Integral Formula

scholarly journals Enhancing the Quantum Linear Systems Algorithm Using Richardson Extrapolation

2022 ◽  
Vol 3 (1) ◽  
pp. 1-37
Author(s):  
Almudena Carrera Vazquez ◽  
Ralf Hiptmair ◽  
Stefan Woerner
Keyword(s):  
Necessary Conditions ◽  
Linear Equations ◽  
Circuit Complexity ◽  
Quantum Algorithm ◽  
Richardson Extrapolation ◽  
Classical Simulation ◽  
Hamiltonian Simulation ◽  
Systems Of Linear Equations ◽  
Amplitude Amplification ◽  
Theoretical Results

We present a quantum algorithm to solve systems of linear equations of the form Ax = b , where A is a tridiagonal Toeplitz matrix and b results from discretizing an analytic function, with a circuit complexity of O (1/√ε, poly (log κ, log N )), where N denotes the number of equations, ε is the accuracy, and κ the condition number. The repeat-until-success algorithm has to be run O (κ/(1-ε)) times to succeed, leveraging amplitude amplification, and needs to be sampled O (1/ε 2 ) times. Thus, the algorithm achieves an exponential improvement with respect to N over classical methods. In particular, we present efficient oracles for state preparation, Hamiltonian simulation, and a set of observables together with the corresponding error and complexity analyses. As the main result of this work, we show how to use Richardson extrapolation to enhance Hamiltonian simulation, resulting in an implementation of Quantum Phase Estimation (QPE) within the algorithm with 1/√ε circuits that can be run in parallel each with circuit complexity 1/√ ε instead of 1/ε. Furthermore, we analyze necessary conditions for the overall algorithm to achieve an exponential speedup compared to classical methods. Our approach is not limited to the considered setting and can be applied to more general problems where Hamiltonian simulation is approximated via product formulae, although our theoretical results would need to be extended accordingly. All the procedures presented are implemented with Qiskit and tested for small systems using classical simulation as well as using real quantum devices available through the IBM Quantum Experience.

scholarly journals Complex Clifford analysis and domains of holomorphy

1990 ◽  
Vol 48 (3) ◽  
pp. 413-433 ◽  
Author(s):  
John Ryan
Keyword(s):  
Clifford Analysis ◽  
Integral Formula ◽  
Complex Variables ◽  
Analytic Surface ◽  
Higher Dimension ◽  
Cauchy’S Integral Formula ◽  
Domains Of Holomorphy ◽  
Huygens Principle ◽  
Real Analytic ◽  
Real Analytic Surface

AbstractIntegrals related to Cauchy's integral formula and Huygens' principle are used to establish a link between domains of holomorphy in n complex variables and cells of harmonicity in one higher dimension. These integrals enable us to determine domains to which analytic functions on real analytic surface in Rn+1 may be extended to solutions to a Dirac equation.

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