Details
| Originalsprache | Englisch |
|---|---|
| Aufsatznummer | 052203 |
| Fachzeitschrift | Journal of mathematical physics |
| Jahrgang | 58 |
| Ausgabenummer | 5 |
| Publikationsstatus | Veröffentlicht - 1 Mai 2017 |
Abstract
Unitary 2-designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries, often referred to as Haar random unitaries. They are used in a wide variety of theoretical and practical quantum information protocols and also have been used to model the dynamics in complex quantum many-body systems. Here, we show that unitary 2-designs can be approximately implemented by alternately repeating random unitaries diagonal in the Pauli-Z basis and Pauli-X basis. We also provide a converse about the number of repetitions needed to achieve unitary 2-designs. These results imply that the process after l repetitions achieves a Θ(d-l)-approximate unitary 2-design. Based on the construction, we further provide quantum circuits that efficiently implement approximate unitary 2-designs. Although a more efficient implementation of unitary 2-designs is known, our quantum circuit has its own merit that it is divided into a constant number of commuting parts, which enables us to apply all commuting gates simultaneously and leads to a possible reduction of an actual execution time. We finally interpret the result in terms of the dynamics generated by time-dependent Hamiltonians and provide for the first time a random disordered time-dependent Hamiltonian that generates a unitary 2-design after switching interactions only a few times.
ASJC Scopus Sachgebiete
- Physik und Astronomie (insg.)
- Statistische und nichtlineare Physik
- Mathematik (insg.)
- Mathematische Physik
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in: Journal of mathematical physics, Jahrgang 58, Nr. 5, 052203, 01.05.2017.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Unitary 2-designs from random X- and Z-diagonal unitaries
AU - Nakata, Yoshifumi
AU - Hirche, Christoph
AU - Morgan, Ciara
AU - Winter, Andreas
PY - 2017/5/1
Y1 - 2017/5/1
N2 - Unitary 2-designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries, often referred to as Haar random unitaries. They are used in a wide variety of theoretical and practical quantum information protocols and also have been used to model the dynamics in complex quantum many-body systems. Here, we show that unitary 2-designs can be approximately implemented by alternately repeating random unitaries diagonal in the Pauli-Z basis and Pauli-X basis. We also provide a converse about the number of repetitions needed to achieve unitary 2-designs. These results imply that the process after l repetitions achieves a Θ(d-l)-approximate unitary 2-design. Based on the construction, we further provide quantum circuits that efficiently implement approximate unitary 2-designs. Although a more efficient implementation of unitary 2-designs is known, our quantum circuit has its own merit that it is divided into a constant number of commuting parts, which enables us to apply all commuting gates simultaneously and leads to a possible reduction of an actual execution time. We finally interpret the result in terms of the dynamics generated by time-dependent Hamiltonians and provide for the first time a random disordered time-dependent Hamiltonian that generates a unitary 2-design after switching interactions only a few times.
AB - Unitary 2-designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries, often referred to as Haar random unitaries. They are used in a wide variety of theoretical and practical quantum information protocols and also have been used to model the dynamics in complex quantum many-body systems. Here, we show that unitary 2-designs can be approximately implemented by alternately repeating random unitaries diagonal in the Pauli-Z basis and Pauli-X basis. We also provide a converse about the number of repetitions needed to achieve unitary 2-designs. These results imply that the process after l repetitions achieves a Θ(d-l)-approximate unitary 2-design. Based on the construction, we further provide quantum circuits that efficiently implement approximate unitary 2-designs. Although a more efficient implementation of unitary 2-designs is known, our quantum circuit has its own merit that it is divided into a constant number of commuting parts, which enables us to apply all commuting gates simultaneously and leads to a possible reduction of an actual execution time. We finally interpret the result in terms of the dynamics generated by time-dependent Hamiltonians and provide for the first time a random disordered time-dependent Hamiltonian that generates a unitary 2-design after switching interactions only a few times.
UR - http://www.scopus.com/inward/record.url?scp=85030154583&partnerID=8YFLogxK
U2 - 10.1063/1.4983266
DO - 10.1063/1.4983266
M3 - Article
AN - SCOPUS:85030154583
VL - 58
JO - Journal of mathematical physics
JF - Journal of mathematical physics
SN - 0022-2488
IS - 5
M1 - 052203
ER -