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arXiv:1011.3646 · Physical Review A

Quantum abacus for counting and factorizing numbers

M. V. Suslov, Gordey B. Lesovik, G. Blatter

Published May 19, 2011• 17 citations
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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.

DOI
10.1103/physreva.83.052317

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Implementations of the quantum phase estimation and quantum counting algorithms with qudits are proposed. The construction of the basic building blocks of the algorithms is described in detail and the fidelity of the quantum counting algorithm with qubits and qudits is simulated numerically. Along with the exponential increase of the size of the target register, the simulations with qudits demonstrate significantly higher probability of finding the number of solutions than qubits and more consistent performance for a different number of solutions.
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We propose a quantum circuit that creates a pure state corresponding to the quantum superposition of all prime numbers less than 2n2^n2n, where nnn is the number of qubits of the register. This Prime state can be built using Grover's algorithm, whose oracle is a quantum implementation of the classical Miller-Rabin primality test. The Prime state is highly entangled, and its entanglement measures encode number theoretical functions such as the distribution of twin primes or the Chebyshev bias. This algorithm can be further combined with the quantum Fourier transform to yield an estimate of the prime counting function, more efficiently than any classical algorithm and with an error below the bound that allows for the verification of the Riemann hypothesis. Arithmetic properties of prime numbers are then, in principle, amenable to experimental verifications on quantum systems.
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We propose a quantum circuit that creates a pure state corresponding to the quantum superposition of all prime numbers less than 2^n, where n is the number of qubits of the register. This Prime state can be built using Grover's algorithm, whose oracle is a quantum implementation of the classical Miller-Rabin primality test. The Prime state is highly entangled, and its entanglement measures encode number theoretical functions such as the distribution of twin primes or the Chebyshev bias. This algorithm can be further combined with the quantum Fourier transform to yield an estimate of the prime counting function, more efficiently than any classical algorithm and with an error below the bound that allows for the verification of the Riemann hypothesis. We also propose a Twin Prime state to measure the number of twin primes and another state to test the Goldbach conjecture. Arithmetic properties of prime numbers are then, in principle, amenable to experimental verifications on quantum systems.