Scaling and efficient classical simulation of the quantum Fourier transform

2017 ◽  
Vol 17 (1&2) ◽  
pp. 1-14
Author(s):  
Kieran J. Woolfe ◽  
Charles D. Hill ◽  
Lloyd C. L. Hollenberg
Keyword(s):  
Fourier Transform ◽  
Matrix Product ◽  
Product State ◽  
Quantum Fourier Transform ◽  
Numerical Evidence ◽  
Matrix Product State ◽  
Product Operator ◽  
Classical Simulation ◽  
Schmidt Rank

We provide numerical evidence that the quantum Fourier transform can be efficiently represented in a matrix product operator with a size growing relatively slowly with the number of qubits. Additionally, we numerically show that the tensors in the operator converge to a common tensor as the number of qubits in the transform increases. Together these results imply that the application of the quantum Fourier transform to a matrix product state with n qubits of maximum Schmidt rank χ can be simulated in O(n (log(n))2 χ 2 ) time. We perform such simulations and quantify the error involved in representing the transform as a matrix product operator and simulating the quantum Fourier transform of periodic states.

scholarly journals Tangent-space methods for truncating uniform MPS

2021 ◽  
Vol 4 (1) ◽  
Author(s):  
Bram Vanhecke ◽  
Maarten Van Damme ◽  
Jutho Haegeman ◽  
Laurens Vanderstraeten ◽  
Frank Verstraete
Keyword(s):  
Tangent Space ◽  
Computational Cost ◽  
Infinite Matrix ◽  
Matrix Product ◽  
Product State ◽  
Matrix Product State ◽  
Product Operator ◽  
Tensor Network ◽  
Network Simulations ◽  
Central Bottleneck

An essential primitive in quantum tensor network simulations is the problem of approximating a matrix product state with one of a smaller bond dimension. This problem forms the central bottleneck in algorithms for time evolution and for contracting projected entangled pair states. We formulate a tangent-space based variational algorithm to achieve this goal for uniform (infinite) matrix product states. The algorithm exhibits a favourable scaling of the computational cost, and we demonstrate its usefulness by several examples involving the multiplication of a matrix product state with a matrix product operator.

scholarly journals Geometric entanglement from matrix product state representations

2011 ◽  
Vol 13 (9) ◽  
pp. 093041 ◽  
Author(s):  
Bing-Quan Hu ◽  
Xi-Jing Liu ◽  
Jin-Hua Liu ◽  
Huan-Qiang Zhou
Keyword(s):  
Matrix Product ◽  
Product State ◽  
Matrix Product State

scholarly journals Optimising Matrix Product State Simulations of Shor's Algorithm

Quantum ◽  
2019 ◽  
Vol 3 ◽  
pp. 116 ◽  
Author(s):  
Aidan Dang ◽  
Charles D. Hill ◽  
Lloyd C. L. Hollenberg
Keyword(s):  
Dimensional Structure ◽  
Matrix Product ◽  
Product State ◽  
One Dimensional ◽  
Matrix Product State ◽  
Shor's Algorithm ◽  
The One ◽  
High Level ◽  
Space Requirements ◽  
Factoring Algorithm

We detail techniques to optimise high-level classical simulations of Shor's quantum factoring algorithm. Chief among these is to examine the entangling properties of the circuit and to effectively map it across the one-dimensional structure of a matrix product state. Compared to previous approaches whose space requirements depend on r, the solution to the underlying order-finding problem of Shor's algorithm, our approach depends on its factors. We performed a matrix product state simulation of a 60-qubit instance of Shor's algorithm that would otherwise be infeasible to complete without an optimised entanglement mapping.

scholarly journals Matrix Product State–Based Quantum Classifier

2019 ◽  
Vol 31 (7) ◽  
pp. 1499-1517 ◽  
Author(s):  
Amandeep Singh Bhatia ◽  
Mandeep Kaur Saggi ◽  
Ajay Kumar ◽  
Sushma Jain
Keyword(s):  
Performance Metrics ◽  
Quantum Computer ◽  
Binary Classification ◽  
Learning Ability ◽  
Matrix Product ◽  
Product State ◽  
Data Set ◽  
Matrix Product State ◽  
Tensor Network ◽  
Network States

Interest in quantum computing has increased significantly. Tensor network theory has become increasingly popular and widely used to simulate strongly entangled correlated systems. Matrix product state (MPS) is a well-designed class of tensor network states that plays an important role in processing quantum information. In this letter, we show that MPS, as a one-dimensional array of tensors, can be used to classify classical and quantum data. We have performed binary classification of the classical machine learning data set Iris encoded in a quantum state. We have also investigated its performance by considering different parameters on the ibmqx4 quantum computer and proved that MPS circuits can be used to attain better accuracy. Furthermore the learning ability of an MPS quantum classifier is tested to classify evapotranspiration (ET[Formula: see text]) for the Patiala meteorological station located in northern Punjab (India), using three years of a historical data set (Agri). We have used different performance metrics of classification to measure its capability. Finally, the results are plotted and the degree of correspondence among values of each sample is shown.

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