scholarly journals Quantum phase estimation with arbitrary constant-precision phase shift operators

2012 ◽  
Vol 12 (9&10) ◽  
pp. 864-875
Author(s):  
Hamed Ahmadi ◽  
Chen-Fu Chiang
Keyword(s):  
Phase Shift ◽  
Arbitrary Constant ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Phase ◽  
Phase Estimation ◽  
Shift Operators ◽  
Physical Implementation ◽  
Alternative Approach ◽  
Original Approach

While Quantum phase estimation (QPE) is at the core of many quantum algorithms known to date, its physical implementation (algorithms based on quantum Fourier transform (QFT) ) is highly constrained by the requirement of high-precision controlled phase shift operators, which remain difficult to realize. In this paper, we introduce an alternative approach to approximately implement QPE with arbitrary constant-precision controlled phase shift operators. The new quantum algorithm bridges the gap between QPE algorithms based on QFT and Kitaev's original approach. For approximating the eigenphase precise to the nth bit, Kitaev's original approach does not require any controlled phase shift operator. In contrast, QPE algorithms based on QFT or approximate QFT require controlled phase shift operators with precision of at least Pi/2n. The new approach fills the gap and requires only arbitrary constant-precision controlled phase shift operators. From a physical implementation viewpoint, the new algorithm outperforms Kitaev's approach.

scholarly journals Photonic scheme of quantum phase estimation for quantum algorithms via cross-Kerr nonlinearities under decoherence effect

2019 ◽  
Vol 27 (21) ◽  
pp. 31023 ◽  
Author(s):  
Changho Hong ◽  
Jino Heo ◽  
Min-Sung Kang ◽  
Jingak Jang ◽  
Hyun-Jin Yang ◽  
...  
Keyword(s):  
Quantum Algorithms ◽  
Quantum Phase ◽  
Phase Estimation ◽  
Decoherence Effect

Faster phase estimation

2014 ◽  
Vol 14 (3&4) ◽  
pp. 306-328
Author(s):  
Krysta M. Svore ◽  
Matthew B. Hastings ◽  
Michael Freedman
Keyword(s):  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Quantum Phase ◽  
Phase Estimation ◽  
Post Processing ◽  
Minimum Number ◽  
Circuit Depth ◽  
Lower Depth ◽  
The Cost ◽  
Random Measurements

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.

Algorithms With Superpolynomial Speed-Up

2006 ◽  
Author(s):  
Phillip Kaye ◽  
Raymond Laflamme ◽  
Michele Mosca
Keyword(s):  
Quantum Algorithms ◽  
Discrete Logarithm ◽  
Quantum Algorithm ◽  
Estimation Algorithm ◽  
Discrete Logarithm Problem ◽  
Phase Estimation ◽  
Post Process ◽  
Classical Algorithm ◽  
Speed Up ◽  
Hadamard Gate

In this chapter we examine one of two main classes of algorithms: quantum algorithms that solve problems with a complexity that is superpolynomially less than the complexity of the best-known classical algorithm for the same problem. That is, the complexity of the best-known classical algorithm cannot be bounded above by any polynomial in the complexity of the quantum algorithm. The algorithms we will detail all make use of the quantum Fourier transform (QFT). We start off the chapter by studying the problem of quantum phase estimation, which leads us naturally to the QFT. Section 7.1 also looks at using the QFT to find the period of periodic states, and introduces some elementary number theory that is needed in order to post-process the quantum algorithm. In Section 7.2, we apply phase estimation in order to estimate eigenvalues of unitary operators. Then in Section 7.3, we apply the eigenvalue estimation algorithm in order to derive the quantum factoring algorithm, and in Section 7.4 to solve the discrete logarithm problem. In Section 7.5, we introduce the hidden subgroup problem which encompasses both the order finding and discrete logarithm problem as well as many others. This chapter by no means exhaustively covers the quantum algorithms that are superpolynomially faster than any known classical algorithm, but it does cover the most well-known such algorithms. In Section 7.6, we briefly discuss other quantum algorithms that appear to provide a superpolynomial advantage. To introduce the idea of phase estimation, we begin by noting that the final Hadamard gate in the Deutsch algorithm, and the Deutsch–Jozsa algorithm, was used to get at information encoded in the relative phases of a state. The Hadamard gate is self-inverse and thus does the opposite as well, namely it can be used to encode information into the phases. To make this concrete, first consider H acting on the basis state |x⟩ (where x ∊ {0, 1}). It is easy to see that You can think about the Hadamard gate as having encoded information about the value of x into the relative phases between the basis states |0⟩ and |1⟩.

scholarly journals Frequentist and Bayesian Quantum Phase Estimation

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 628 ◽  
Author(s):  
Yan Li ◽  
Luca Pezzè ◽  
Manuel Gessner ◽  
Zhihong Ren ◽  
Weidong Li ◽  
...  
Keyword(s):  
Phase Shift ◽  
Unknown Parameter ◽  
Quantum Noise ◽  
The State ◽  
Quantum Phase ◽  
Phase Estimation ◽  
True Value ◽  
Interferometric Phase ◽  
Fixed Parameter ◽  
Cramer Rao Bound

Frequentist and Bayesian phase estimation strategies lead to conceptually different results on the state of knowledge about the true value of an unknown parameter. We compare the two frameworks and their sensitivity bounds to the estimation of an interferometric phase shift limited by quantum noise, considering both the cases of a fixed and a fluctuating parameter. We point out that frequentist precision bounds, such as the Cramér–Rao bound, for instance, do not apply to Bayesian strategies and vice versa. In particular, we show that the Bayesian variance can overcome the frequentist Cramér–Rao bound, which appears to be a paradoxical result if the conceptual difference between the two approaches are overlooked. Similarly, bounds for fluctuating parameters make no statement about the estimation of a fixed parameter.

scholarly journals Uniform approximation by (quantum) polynomials

2011 ◽  
Vol 11 (3&4) ◽  
pp. 215-225
Author(s):  
Andrew Drucker ◽  
Ronald de Wolf
Keyword(s):  
Approximation Theory ◽  
Uniform Approximation ◽  
Quantum Algorithms ◽  
Continuous Functions ◽  
Quantum Phase ◽  
Phase Estimation ◽  
Classical Theorem ◽  
Quantitative Version ◽  
Quantum Counting ◽  
Approximation Of Continuous Functions

We show that quantum algorithms can be used to re-prove a classical theorem in approximation theory, Jackson's Theorem, which gives a nearly-optimal quantitative version of Weierstrass's Theorem on uniform approximation of continuous functions by polynomials. We provide two proofs, based respectively on quantum counting and on quantum phase estimation.

scholarly journals Practical Quantum Computing

2021 ◽  
Vol 2 (1) ◽  
pp. 1-35
Author(s):  
Adrien Suau ◽  
Gabriel Staffelbach ◽  
Henri Calandra
Keyword(s):  
Wave Equation ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Equation Solving ◽  
Small Quantum ◽  
Quantum Wave ◽  
Variational Algorithms ◽  
The One

In the last few years, several quantum algorithms that try to address the problem of partial differential equation solving have been devised: on the one hand, “direct” quantum algorithms that aim at encoding the solution of the PDE by executing one large quantum circuit; on the other hand, variational algorithms that approximate the solution of the PDE by executing several small quantum circuits and making profit of classical optimisers. In this work, we propose an experimental study of the costs (in terms of gate number and execution time on a idealised hardware created from realistic gate data) associated with one of the “direct” quantum algorithm: the wave equation solver devised in [32]. We show that our implementation of the quantum wave equation solver agrees with the theoretical big-O complexity of the algorithm. We also explain in great detail the implementation steps and discuss some possibilities of improvements. Finally, our implementation proves experimentally that some PDE can be solved on a quantum computer, even if the direct quantum algorithm chosen will require error-corrected quantum chips, which are not believed to be available in the short-term.

scholarly journals Deep neural networks for quantum circuit mapping

2021 ◽  
Author(s):  
Giovanni Acampora ◽  
Roberto Schiattarella
Keyword(s):  
Neural Networks ◽  
Deep Neural Networks ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Machine Learning Techniques ◽  
Quantum Computers ◽  
Physical Constraints ◽  
Proper Mapping ◽  
Correct Execution

AbstractQuantum computers have become reality thanks to the effort of some majors in developing innovative technologies that enable the usage of quantum effects in computation, so as to pave the way towards the design of efficient quantum algorithms to use in different applications domains, from finance and chemistry to artificial and computational intelligence. However, there are still some technological limitations that do not allow a correct design of quantum algorithms, compromising the achievement of the so-called quantum advantage. Specifically, a major limitation in the design of a quantum algorithm is related to its proper mapping to a specific quantum processor so that the underlying physical constraints are satisfied. This hard problem, known as circuit mapping, is a critical task to face in quantum world, and it needs to be efficiently addressed to allow quantum computers to work correctly and productively. In order to bridge above gap, this paper introduces a very first circuit mapping approach based on deep neural networks, which opens a completely new scenario in which the correct execution of quantum algorithms is supported by classical machine learning techniques. As shown in experimental section, the proposed approach speeds up current state-of-the-art mapping algorithms when used on 5-qubits IBM Q processors, maintaining suitable mapping accuracy.

scholarly journals Quantum phase estimation using a multi-headed cat state

2015 ◽  
Vol 32 (6) ◽  
pp. 1186 ◽  
Author(s):  
Su-Yong Lee ◽  
Chang-Woo Lee ◽  
Hyunchul Nha ◽  
Dagomir Kaszlikowski
Keyword(s):  
Quantum Phase ◽  
Phase Estimation
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