Quantum abacus for counting and factorizing numbers

Abstract

We generalize the binary quantum counting algorithm of Lesovik, Suslov, and Blatter [Phys. Rev. A 82, 012316 (2010)] to higher counting bases. The algorithm makes use of qubits, qutrits, and qudits to count numbers in a base-2, base-3, or base-d representation. In operating the algorithm, the number n < N = d K is read into a K-qudit register through its interaction with a stream of n particles passing in a nearby wire; this step corresponds to a quantum Fourier transformation from the Hilbert space of particles to the Hilbert space of qudit states. An inverse quantum Fourier transformation provides the number n in the base-d representation; the inverse transformation is fully quantum at the level of individual qudits, while a simpler semiclassical version can be used on the level of qudit registers. Combining registers of qubits, qutrits, and qudits, where d is a prime number, with a simpler single-shot measurement allows us to find the powers of 2, 3, and other primes d in the number n. We show that the counting task naturally leads to the shift operation and an algorithm based on the quantum Fourier transformation. We discuss possible implementations of the algorithm using quantum spin-d systems, d-well systems, and their emulation with spin-1/2 or double-well systems. We establish the analogy between our counting algorithm and the phase estimation algorithm and make use of the latter's performance analysis in stabilizing our scheme. Applications embrace a quantum metrological scheme to measure voltage (an analog to digital converter) and a simple procedure to entangle multiparticle states.

Quantum-enhanced magnetometry by phase estimation algorithms with a single artificial atom

“…in the so-called quantum abacus. 24 The Kitaev algorithm starts from a minimal delay τ = τ 0 and determines the most significant bit b K−1 in its first step, further proceeding with the less significant bits b K−2 , . .…”

Optimal noninvasive measurement of full counting statistics by a single qubit

“…Another remark concerns the relation of measuring the FCS to the problem of quantum counting 31,32 , where a qubit register of K qubits with coupling strengths λ j = π 2 j−1 , j = 1, . .…”

“…Specific results in the form of tables are given for the measurement of the third-order cumulant ⟨Q 3 ⟩. In section IV we present a summary, emphasize our main results, and add some concluding remarks on the use of different types of qubits and the relation to quantum counting 31,32 .…”

Quantum metrology with a transmon qutrit

“…Repeating this Ramsey cycle several times, one can accumulate enough statistics and extract the value of the field H, see Refs. [15,17].…”

“…To start with, we consider a situation where the magnetic field assumes only one of three values H ∈ {0, h, 2h}, h > 0; the task then is to unambiguously distinguish between these three alternatives via a singleshot measurement of the state (2); such a one-shot discrimination is indeed possible as was shown in Ref. [17] within the context of the quantum counting problem. We expose the initial balanced state |ψ 0 during a specific time interval τ 0 to the field such that the phase φ = µhτ 0 / assumes the value φ = 2π/3.…”