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Method to Factorize Fifteen with Josephson Charge Qubits

Juha J. Vartiainen, A. O. Niskanen, Mikio Nakahara, Martti M. Salomaa

Published Aug 29, 2003• 0 citations
OpenAlex

Abstract

We investigate both the physical and algorithmic requirements for an implementation of Shor's factorization algorithm on a hypothetical Josephson charge qubit register. We represent an implementation which is optimal for small number of available qubits, but still provides a general method to factorize integer without a prior knowledge of the number to be factorized. Furthermore, our strategy is easily generalized to other realizations and parameter values. In order to meet the stringent requirements set by short decoherence time, we accelerate the algorithm by decomposing it into tailored two- and three-qubit gates and find their physical realizations through the numerical optimization.

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We investigate both the physical and algorithmic requirements for an implementation of Shor's factorization algorithm on a hypothetical Josephson charge qubit register. We represent an implementation which is optimal for small number of available qubits, but still provides a general method to factorize integer without a prior knowledge of the number to be factorized. Furthermore, our strategy is easily generalized to other realizations and parameter values. In order to meet the stringent requirements set by short decoherence time, we accelerate the algorithm by decomposing it into tailored two- and three-qubit gates and find their physical realizations through the numerical optimization.
We investigate the physical implementation of Shor's factorization algorithm on a Josephson charge qubit register. While we pursue a universal method to factor a composite integer of any size, the scheme is demonstrated for the number 21. We consider both the physical and algorithmic requirements for an optimal implementation when only a small number of qubits are available. These aspects of quantum computation are usually the topics of separate research communities; we present a unifying discussion of both of these fundamental features bridging Shor's algorithm to its physical realization using Josephson junction qubits. In order to meet the stringent requirements set by a short decoherence time, we accelerate the algorithm by decomposing the quantum circuit into tailored two- and three-qubit gates and we find their physical realizations through numerical optimization.
We 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 [Formula: see text]. 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.
We try to minimize the number of qubits needed to factor an integer of n bits using Shor's algorithm on a quantum computer. We introduce a circuit which uses 2n+3 qubits and O(n^3 lg(n)) elementary quantum gates in a depth of O(n^3) to implement the factorization algorithm. The circuit is computable in polynomial time on a classical computer and is completely general as it does not rely on any property of the number to be factored. Keywords: Factorization, quantum circuits, modular arithmetics
Quantum-circuit optimization is essential for any practical realization of quantum computation, in order to beat decoherence. We present a scheme for implementing the final stage in the compilation of quantum circuits, i.e. for finding the actual physical realizations of the individual modules in the quantum-gate library. We find that numerical optimization can be efficiently utilized in order to generate the appropriate control-parameter sequences which produce the desired three-qubit modules within the Josephson charge-qubit model. Our work suggests ways in which one can in fact considerably reduce the number of gates required to implement a given quantum circuit, hence diminishing idle time and significantly accelerating the execution of quantum algorithms.