Quantum abacus for counting and factorizing numbers

Journal Article · Sun May 15 00:00:00 EDT 2011 · Physical Review. A

DOI:https://doi.org/10.1103/PHYSREVA.83.052317· OSTI ID:21546734

Suslov, M V ^[1]; Lesovik, G B ^[2]; Blatter, G ^[3]

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1. Moscow Institute of Physics and Technology, Institutskii per. 9, 141700 Dolgoprudny, Moscow District (Russian Federation)
2. L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, 117940 Moscow (Russian Federation)
3. Theoretische Physik, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zuerich (Switzerland)

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{sup 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.

• OSTI ID: 21546734
• Journal Information: Physical Review. A, Vol. 83, Issue 5; Other Information: DOI: 10.1103/PhysRevA.83.052317; (c) 2011 American Institute of Physics; ISSN 1050-2947
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
97 MATHEMATICAL METHODS AND COMPUTING
ALGORITHMS
ANALOG-TO-DIGITAL CONVERTERS
ELECTRIC POTENTIAL
FOURIER TRANSFORMATION
HILBERT SPACE
INTERACTIONS
PARTICLES
QUANTUM COMPUTERS
QUBITS
SEMICLASSICAL APPROXIMATION
SPIN
ANGULAR MOMENTUM
APPROXIMATIONS
BANACH SPACE
CALCULATION METHODS
COMPUTERS
ELECTRONIC EQUIPMENT
EQUIPMENT
INFORMATION
INTEGRAL TRANSFORMATIONS
MATHEMATICAL LOGIC
MATHEMATICAL SPACE
PARTICLE PROPERTIES
QUANTUM INFORMATION
SPACE
TRANSFORMATIONS