2021
He, Ran; Ai, Ming-Zhong; Cui, Jin-Ming; Huang, Yun-Feng; Han, Yong-Jian; Li, Chuan-Feng; Guo, Guang-Can; Sierra, G.; Creffield, C. E.
Riemann zeros from Floquet engineering a trapped-ion qubit Journal Article
In: npj Quantum Information, vol. 7, no. 1, 2021.
@article{He_2021,
title = {Riemann zeros from Floquet engineering a trapped-ion qubit},
author = {Ran He and Ming-Zhong Ai and Jin-Ming Cui and Yun-Feng Huang and Yong-Jian Han and Chuan-Feng Li and Guang-Can Guo and G. Sierra and C. E. Creffield},
url = {https://arxiv.org/abs/2102.06936},
doi = {10.1038/s41534-021-00446-7},
year = {2021},
date = {2021-07-01},
urldate = {2021-07-01},
journal = {npj Quantum Information},
volume = {7},
number = {1},
publisher = {Springer Science and Business Media LLC},
abstract = {The non-trivial zeros of the Riemann zeta function are central objects in number theory. In particular, they enable one to reproduce the prime numbers. They have also attracted the attention of physicists working in Random Matrix Theory and Quantum Chaos for decades. Here we present an experimental observation of the lowest non-trivial Riemann zeros by using a trapped ion qubit in a Paul trap, periodically driven with microwave fields. The waveform of the driving is engineered such that the dynamics of the ion is frozen when the driving parameters coincide with a zero of the real component of the zeta function. Scanning over the driving amplitude thus enables the locations of the Riemann zeros to be measured experimentally to a high degree of accuracy, providing a physical embodiment of these fascinating mathematical objects in the quantum realm.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The non-trivial zeros of the Riemann zeta function are central objects in number theory. In particular, they enable one to reproduce the prime numbers. They have also attracted the attention of physicists working in Random Matrix Theory and Quantum Chaos for decades. Here we present an experimental observation of the lowest non-trivial Riemann zeros by using a trapped ion qubit in a Paul trap, periodically driven with microwave fields. The waveform of the driving is engineered such that the dynamics of the ion is frozen when the driving parameters coincide with a zero of the real component of the zeta function. Scanning over the driving amplitude thus enables the locations of the Riemann zeros to be measured experimentally to a high degree of accuracy, providing a physical embodiment of these fascinating mathematical objects in the quantum realm.
2020
García-Martín, D.; Ribas, E.; Carrazza, S.; Latorre, J. I.; Sierra, G.
The Prime state and its quantum relatives Journal Article
In: Quantum, vol. 4, pp. 371, 2020.
@article{Garc_a_Mart_n_2020,
title = {The Prime state and its quantum relatives},
author = {D. García-Martín and E. Ribas and S. Carrazza and J. I. Latorre and G. Sierra},
url = {https://arxiv.org/abs/2005.02422},
doi = {10.22331/q-2020-12-11-371},
year = {2020},
date = {2020-12-01},
urldate = {2020-12-01},
journal = {Quantum},
volume = {4},
pages = {371},
publisher = {Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften},
abstract = {The Prime state of n qubits, |P_n⟩, is defined as the uniform superposition of all the computational-basis states corresponding to prime numbers smaller than 2n. This state encodes, quantum mechanically, arithmetic properties of the primes. We first show that the Quantum Fourier Transform of the Prime state provides a direct access to Chebyshev-like biases in the distribution of prime numbers. We next study the entanglement entropy of |Pn⟩ up to n=30 qubits, and find a relation between its scaling and the Shannon entropy of the density of square-free integers. This relation also holds when the Prime state is constructed using a qudit basis, showing that this property is intrinsic to the distribution of primes. The same feature is found when considering states built from the superposition of primes in arithmetic progressions. Finally, we explore the properties of other number-theoretical quantum states, such as those defined from odd composite numbers, square-free integers and starry primes. For this study, we have developed an open-source library that diagonalizes matrices using floats of arbitrary precision.},
keywords = {},
pubstate = {published},
tppubtype = {article}
}
The Prime state of n qubits, |P_n⟩, is defined as the uniform superposition of all the computational-basis states corresponding to prime numbers smaller than 2n. This state encodes, quantum mechanically, arithmetic properties of the primes. We first show that the Quantum Fourier Transform of the Prime state provides a direct access to Chebyshev-like biases in the distribution of prime numbers. We next study the entanglement entropy of |Pn⟩ up to n=30 qubits, and find a relation between its scaling and the Shannon entropy of the density of square-free integers. This relation also holds when the Prime state is constructed using a qudit basis, showing that this property is intrinsic to the distribution of primes. The same feature is found when considering states built from the superposition of primes in arithmetic progressions. Finally, we explore the properties of other number-theoretical quantum states, such as those defined from odd composite numbers, square-free integers and starry primes. For this study, we have developed an open-source library that diagonalizes matrices using floats of arbitrary precision.