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.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.

arXiv:1712.07311QuantumJan 25, 201939 citations

Simulations with matrix product states

Ulrich Schollwöck

AIP conference proceedingsJan 1, 20120 citations

Peter Shor on the genesis of Shor’s algorithm

David Zierler, Ryan Dahn

Physics TodayMar 10, 20254 citations

Shor's Algorithm: Factoring and Cryptanalysis

Bhupendra Singh, Mohankumar Mylsamy, Thamaraimanalan Thangarajan

“Shor's Algorithm: Factoring and Cryptanalysis” focuses on Shor's Algorithm, particularly its application to factoring and cryptanalysis. It begins with an overview of Shor's algorithm, detailing the classical extraction of periods from measured values and the quantum gates required for the Quantum Fourier Transform (QFT). Further, this chapter explores the applications and limitations of Shor's algorithm, including its impact on RSA encryption. A literature review follows, discussing Shor's algorithm in the context of quantum computation, its effectiveness on RSA, and the effects of imperfections on its factorization process. The section also includes an experimental study of Shor's algorithm using IBM Q, followed by a practical implementation of factoring numbers N=21 and N=35. Further, it covers resource estimation, quantum simulations, and different types of simulators in IBM Qiskit, including state vector, QASM, Matrix Product State (MPS), and stabilizer simulators, providing insights into their use cases and the limits of classical simulation.“Shor's Algorithm: Factoring and Cryptanalysis” focuses on Shor's Algorithm, particularly its application to factoring and cryptanalysis. It begins with an overview of Shor's algorithm, detailing the classical extraction of periods from measured values and the quantum gates required for the Quantum Fourier Transform (QFT). Further, this chapter explores the applications and limitations of Shor's algorithm, including its impact on RSA encryption. A literature review follows, discussing Shor's algorithm in the context of quantum computation, its effectiveness on RSA, and the effects of imperfections on its factorization process. The section also includes an experimental study of Shor's algorithm using IBM Q, followed by a practical implementation of factoring numbers N=21 and N=35. Further, it covers resource estimation, quantum simulations, and different types of simulators in IBM Qiskit, including state vector, QASM, Matrix Product State (MPS), and stabilizer simulators, providing insights into their use cases and the limits of classical simulation.

CRC Press eBooksNov 14, 20250 citations

Simulations of Shor's Algorithm using Matrix Product States

Wang, David S., Charles D. Hill, Lloyd C. L. Hollenberg

We show that under the matrix product state formalism the states produced in Shor's algorithm can be represented using O(max( $4lr^2$ , $2^{2l}$ )) space, where l is the number of bits in the number to factorise, and r is the order and the solution to the related order-finding problem. The reduction in space compared to an amplitude formalism approach is significant, allowing simulations as large as 42 qubits to be run on a single processor with 32GB RAM. This approach is readily adapted to a distributed memory environment, and we have simulated a 45 qubit case using 8 cores with 16GB RAM in approximately one hour.We show that under the matrix product state formalism the states produced in Shor's algorithm can be represented using O(max($4lr^2$, $2^{2l}$)) space, where l is the number of bits in the number to factorise, and r is the order and the solution to the related order-finding problem. The reduction in space compared to an amplitude formalism approach is significant, allowing simulations as large as 42 qubits to be run on a single processor with 32GB RAM. This approach is readily adapted to a distributed memory environment, and we have simulated a 45 qubit case using 8 cores with 16GB RAM in approximately one hour.

arXiv:1501.07644Jan 30, 20152 citations

An Optimization of Shor's Algorithm Based on Quantum Parallelism

Shuaishuai Zhu, Ming Yuan

Lecture notes in electrical engineeringJan 1, 20260 citations

We show that under the matrix product state formalism the states produced in Shor's algorithm can be represented using O(max(

4lr^2

2^{2l}

)) space, where l is the number of bits in the number to factorise, and r is the order and the solution to the related order-finding problem. The reduction in space compared to an amplitude formalism approach is significant, allowing simulations as large as 42 qubits to be run on a single processor with 32GB RAM. This approach is readily adapted to a distributed memory environment, and we have simulated a 45 qubit case using 8 cores with 16GB RAM in approximately one hour.