Fast quantum modular exponentiation

Abstract

We present a detailed analysis of the impact on quantum modular exponentiation of architectural features and possible concurrent gate execution. Various arithmetic algorithms are evaluated for execution time, potential concurrency, and space tradeoffs. We find that to exponentiate an n-bit number, for storage space 100n (twenty times the minimum 5n), we can execute modular exponentiation two hundred to seven hundred times faster than optimized versions of the basic algorithms, depending on architecture, for n = 128. Addition on a neighboronly architecture is limited to O(n) time while non-neighbor architectures can reach O(log n), demonstrating that physical characteristics of a computing device have an important impact on both real-world running time and asymptotic behavior. Our results will help guide experimental implementations of quantum algorithms and devices.

Tensor-product versus geometric-product coding

“…A situation where geometric projections play a simplifying role in GA analogues of quantum algorithms is the problem of deleting intermediate 'carry bits' in quantum adder networks [12,13,14,15]. A glimpse at the network proposed in [12] shows that the number of gates could be reduced by almost a half if one did not insist on performing this task in a reversible way.…”

Quantum arithmetics via computation with minimized external control: The half-adder

“…Likewise to a classical computer, a quantum adder can be constructed from quantum halfadders to provide a way to add two unknown quantum states together. These operations are the building blocks of quantum arithmetics [17][18][19], which is a key component (together with the quantum Fourier transform) for implementing Shor's algorithm, capable of factoring numbers in polynomial timemuch faster than is possible on a classical computer [20].…”

“…In this paper we focus on developing a logic unit that implements the quantum half-adder gate, a recurring transformation for quantum arithmetics [17][18][19], and, as such, a key component in the Shor algorithm [20]. Because of its importance, different schemes have been proposed to implement half-adders in different experimental systems, such as linear optics [21], nanographene molecules [22], 1D cellular automaton [23], and superconducting [24] or atomic [25] systems.…”

Checking equivalence of quantum circuits and states

“…The relative-phase differences can be canceled out if every pair of these gates in the circuit is strategically placed [20]. Since circuit minimization is being pursued for a number of key quantum arithmetic circuits with many Toffoli gates, such as modular exponentiation [22,9,18,17], this optimization could reduce the number of gates even further. The inner product and matrix product may be used to determine such equivalences, but in this work, we present new decision-diagram (DD) algorithms to accomplish the task more efficiently.…”