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.

Quantum CFD Simulations for Heat Transfer Applications

2020 ◽  
Author(s):  
Chao Lu ◽  
Zhao Hu ◽  
Bei Xie ◽  
Ning Zhang
Keyword(s):  
Heat Transfer ◽  
Quantum Computing ◽  
Linear System ◽  
Linear Equation ◽  
Linear Systems ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Quantum Phase ◽  
Cfd Simulations ◽  
Analytical Result

Abstract In this paper, computational heat transfer (CHT) equations were solved using the state-of-art quantum computing (QC) technology. The CHT equations can be discretized into a linear equation set, which can be possibly solved by a QC system. The linear system can be characterized by Ax = b. The A matrix in this linear system is a Hermitian matrix. The linear system is then solved by using the HHL algorithm, which is a quantum algorithm to solve a linear system. The quantum circuit requires an Ancilla qubit, clock qubits, qubits for b and a classical bit to record the result. The process of the HHL algorithm can be described as follows. Firstly, the qubit for b is initialized into the phase as desire. Secondly, the quantum phase estimation (QPE) is used to determine the eigenvalues of A and the eigenvalues are stored in clock qubits. Thirdly, a Rotation gate is used to rotate the inversion of eigenvalues and information is passed to the Ancilla bit to do Pauli Y-rotation operation. Fourthly, revert the whole processes to untangle qubits and measure all of the qubits to output the final results for x. From the existing literature, a few 2 × 2 matrices were successfully solved with QC technology, proving the possibility of QC on linear systems [1]. In this paper, a quantum circuit is designed to solve a CHT problem. A simple 2 by 2 linear equation is modeled for the CHT problem and is solved by using the quantum computing. The result is compared with the analytical result. This result could initiate future studies on determining the quantum phase parameters for more complicated QC linear systems for CHT applications.

scholarly journals Quantum Circuit Design of Toom 3-Way Multiplication

2021 ◽  
Vol 11 (9) ◽  
pp. 3752
Author(s):  
Harashta Tatimma Larasati ◽  
Asep Muhamad Awaludin ◽  
Janghyun Ji ◽  
Howon Kim
Keyword(s):  
Circuit Design ◽  
Quantum Circuit ◽  
Lower Number ◽  
Unique Property ◽  
Large Numbers ◽  
Toffoli Gates ◽  
Asymptotic Complexity ◽  
Lower Depth ◽  
The Cost ◽  
Division Operation

In classical computation, Toom–Cook is one of the multiplication methods for large numbers which offers faster execution time compared to other algorithms such as schoolbook and Karatsuba multiplication. For the use in quantum computation, prior work considered the Toom-2.5 variant rather than the classically faster and more prominent Toom-3, primarily to avoid the nontrivial division operations inherent in the latter circuit. In this paper, we investigate the quantum circuit for Toom-3 multiplication, which is expected to give an asymptotically lower depth than the Toom-2.5 circuit. In particular, we designed the corresponding quantum circuit and adopted the sequence proposed by Bodrato to yield a lower number of operations, especially in terms of nontrivial division, which is reduced to only one exact division by 3 circuit per iteration. Moreover, to further minimize the cost of the remaining division, we utilize the unique property of the particular division circuit, replacing it with a constant multiplication by reciprocal circuit and the corresponding swap operations. Our numerical analysis shows that the resulting circuit indeed gives a lower asymptotic complexity in terms of Toffoli depth and qubit count compared to Toom-2.5 but with a large number of Toffoli gates that mainly come from realizing the division operation.

scholarly journals Quantum Phase Estimation Algorithm for Finding Polynomial Roots

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 839-849
Author(s):  
Theerapat Tansuwannont ◽  
Surachate Limkumnerd ◽  
Sujin Suwanna ◽  
Pruet Kalasuwan
Keyword(s):  
Quantum Algorithm ◽  
Estimation Algorithm ◽  
Quantum Circuit ◽  
Quantum Systems ◽  
Phase Estimation ◽  
Companion Matrix ◽  
Polynomial Roots ◽  
Mathematical Problems ◽  
Phase Estimation Algorithm ◽  
Computational Resources

AbstractQuantum 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.

scholarly journals A 2D nearest-neighbor quantum architecture for factoring in polylogarithmic depth

2013 ◽  
Vol 13 (11&12) ◽  
pp. 937-962
Author(s):  
Paul Pham ◽  
Krysta M. Svore
Keyword(s):  
Nearest Neighbor ◽  
Upper Bounds ◽  
Phase Estimation ◽  
Constant Depth ◽  
Shor's Algorithm ◽  
Circuit Depth ◽  
Quantum Architecture ◽  
The Cost ◽  
2D Model

We present a 2D nearest-neighbor quantum architecture for Shor's algorithm to factor an $n$-bit number in $O(\log^3n)$ depth. Our implementation uses parallel phase estimation, constant-depth fanout and teleportation, and constant-depth carry-save modular addition. We derive upper bounds on the circuit resources of our architecture under a new 2D model which allows a classical controller and parallel, communicating modules. We provide a comparison to all previous nearest-neighbor factoring implementations. Our circuit results in an exponential improvement in nearest-neighbor circuit depth at the cost of a polynomial increase in circuit size and width.

scholarly journals Factoring using 2n+2 qubits with Toffoli based modular multiplication

2017 ◽  
Vol 17 (7&8) ◽  
pp. 673-684
Author(s):  
Thomas Haner ◽  
Martin Roetteler ◽  
Krysta M. Svore
Keyword(s):  
Hardware Implementation ◽  
Quantum Algorithm ◽  
Modular Multiplication ◽  
Time Costs ◽  
Space And Time ◽  
Gate Count ◽  
Circuit Depth ◽  
The Cost ◽  
Shor’S Quantum Algorithm

We describe an implementation of Shor’s quantum algorithm to factor n-bit integers using only 2n+2 qubits. In contrast to previous space-optimized implementations, ours features a purely Toffoli based modular multiplication circuit. The circuit depth and the overall gate count are in O(n 3 ) and O(n 3 log n), respectively. We thus achieve the same space and time costs as Takahashi et al. [1], while using a purely classical modular multiplication circuit. As a consequence, our approach evades most of the cost overheads originating from rotation synthesis and enables testing and localization of some faults in both, the logical level circuit and an actual quantum hardware implementation. Our new (in-place) constant-adder, which is used to construct the modular multiplication circuit, uses only dirty ancilla qubits and features a circuit size and depth in O(n log n) and O(n), respectively.

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.

Determination of the minimum number of technological bases of a part

2020 ◽  
pp. 73-75
Author(s):  
B.M. Bazrov ◽  
T.M. Gaynutdinov
Keyword(s):  
Cost Price ◽  
Labor Input ◽  
Part Surface ◽  
Technological Base ◽  
Size Accuracy ◽  
Minimum Number ◽  
The Cost ◽  
Input Cost ◽  
Selection Of

The selection of technological bases is considered before the choice of the type of billet and the development of the route of the technological process. A technique is proposed for selecting the minimum number of sets of technological bases according to the criterion of equality in the cost price of manufacturing the part according to the principle of unity and combination of bases at this stage. Keywords: part, surface, coordinating size, accuracy, design and technological base, labor input, cost price. [email protected]

A Harmonic Source Location Method Based on PMU Measurements in Distribution Network

2015 ◽  
Vol 775 ◽  
pp. 409-414
Author(s):  
Bing Jun Li ◽  
Su Quan Zhou ◽  
Xiao Xiang Lun
Keyword(s):  
Power System ◽  
State Estimation ◽  
Distribution Network ◽  
Measured Data ◽  
Optimal Measurement ◽  
Location Method ◽  
Measurement Placement ◽  
Minimum Number ◽  
Harmonic Source ◽  
The Cost

It is of great importance to identify the location of the harmonic sources for the harmonic governance in the power system. Applied with optimal measurement placement (OMP) and harmonic state estimation (HSE), this paper presents a novel process based on PMU measurements to locate the harmonic sources in the distribution network. Considering the cost and the observability, the OMP can provide a scheme of the measurement placement with the minimum number of PMU measurements. In order to simplify the HSE equation, the measured data are converted to the form of voltage by the method proposed in this paper.By solving the HSE equation, the location and magnitude of the harmonic source are evaluated. The methodology is applied to the IEEE 33-bus system, and the obtained results are properly analyzed.

scholarly journals Connectivity matrix model of quantum circuits and its application to distributed quantum circuit optimization

2021 ◽  
Vol 20 (7) ◽  
Author(s):  
Ismail Ghodsollahee ◽  
Zohreh Davarzani ◽  
Mariam Zomorodi ◽  
Paweł Pławiak ◽  
Monireh Houshmand ◽  
...  
Keyword(s):  
Quantum Computation ◽  
Quantum Computer ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Connectivity Matrix ◽  
Circuit Optimization ◽  
Novel Approach ◽  
The Matrix ◽  
Minimum Number ◽  
Two Phases

AbstractAs quantum computation grows, the number of qubits involved in a given quantum computer increases. But due to the physical limitations in the number of qubits of a single quantum device, the computation should be performed in a distributed system. In this paper, a new model of quantum computation based on the matrix representation of quantum circuits is proposed. Then, using this model, we propose a novel approach for reducing the number of teleportations in a distributed quantum circuit. The proposed method consists of two phases: the pre-processing phase and the optimization phase. In the pre-processing phase, it considers the bi-partitioning of quantum circuits by Non-Dominated Sorting Genetic Algorithm (NSGA-III) to minimize the number of global gates and to distribute the quantum circuit into two balanced parts with equal number of qubits and minimum number of global gates. In the optimization phase, two heuristics named Heuristic I and Heuristic II are proposed to optimize the number of teleportations according to the partitioning obtained from the pre-processing phase. Finally, the proposed approach is evaluated on many benchmark quantum circuits. The results of these evaluations show an average of 22.16% improvement in the teleportation cost of the proposed approach compared to the existing works in the literature.

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.

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