Entanglement in Shor's Factoring Algorithm

AbstractWe report a proof-of-concept demonstration of a quantum order-finding algorithm for factoring the integer 21. Our demonstration involves the use of a compiled version of the quantum phase estimation routine, and builds upon a previous demonstration. We go beyond this work by using a configuration of approximate Toffoli gates with residual phase shifts, which preserves the functional correctness and allows us to achieve a complete factoring of $$N=21$$ N = 21 . We implemented the algorithm on IBM quantum processors using only five qubits and successfully verified the presence of entanglement between the control and work register qubits, which is a necessary condition for the algorithm’s speedup in general. The techniques we employ may be useful in carrying out Shor’s algorithm for larger integers, or other algorithms in systems with a limited number of noisy qubits.

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Optimising Matrix Product State Simulations of Shor's Algorithm

Quantum ◽

10.22331/q-2019-01-25-116 ◽

2019 ◽

Vol 3 ◽

pp. 116 ◽

Cited By ~ 2

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.

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Fast equivalence -- checking for quantum circuits

Quantum Information and Computation ◽

10.26421/qic10.9-10-1 ◽

2010 ◽

Vol 10 (9&10) ◽

pp. 721-734

Author(s):

Shigeru Yamashita ◽

Igor L. Markov

Keyword(s):

Formal Verification ◽

Quantum Algorithms ◽

Quantum Circuits ◽

Equivalence Checking ◽

Sat Solvers ◽

Reversible Circuits ◽

Verification Tools ◽

Verification Methodology ◽

Verification Techniques ◽

Factoring Algorithm

We perform formal verification of quantum circuits by integrating several techniques specialized to particular classes of circuits. Our verification methodology is based on the new notion of a reversible miter that allows one to leverage existing techniques for simplification of quantum circuits. For reversible circuits which arise as runtime bottlenecks of key quantum algorithms, we develop several verification techniques and empirically compare them. We also combine existing quantum verification tools with the use of SAT-solvers. Experiments with circuits for Shor's number-factoring algorithm, containing thousands of gates, show improvements in efficiency by four orders of magnitude.

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A quantum circuit for Shor's factoring algorithm using 2n+2 qubits

Quantum Information and Computation ◽

10.26421/qic6.2-4 ◽

2006 ◽

Vol 6 (2) ◽

pp. 184-192

Author(s):

Y. Takahashi ◽

N. Kunihiro

Keyword(s):

Quantum Circuit ◽

Factoring Algorithm

We construct a quantum circuit for Shor's factoring algorithm that uses 2n+2 qubits, where n is the length of the number to be factored. The depth and size of the circuit are O(n^3) and O(n^3\log n), respectively. The number of qubits used in the circuit is less than that in any other quantum circuit ever constructed for Shor's factoring algorithm. Moreover, the size of the circuit is about half the size of Beauregard's quantum circuit for Shor's factoring algorithm, which uses 2n+3 qubits.

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