Multiqubit Clifford groups are unitary 3-designs
By Huangjun Zhu. 2015
212 citations recorded by the source corpus.
Unitary $t$-designs are a ubiquitous tool in many research areas, including randomized benchmarking, quantum process tomography, and scrambling. Despite the intensive efforts of many researchers, little is known about unitary $t$-designs with $t\geq3$ in the literature. We show that the multiqubit Clifford group in any even prime-power dimension is not only a unitary 2-design, but also a 3-design. Moreover, it is a minimal 3-design except for dimension~4. As an immediate consequence, any orbit of pure states of the multiqubit Clifford group forms a complex projective 3-design; in particular, the set of stabilizer states forms a 3-design. In addition, our study is helpful to studying higher moments of the Clifford group, which are useful in many research areas ranging from quantum information science to signal processing. Furthermore, we reveal a surprising connection between unitary 3-designs and the physics of discrete phase spaces and thereby offer a simple explanation of why no discrete Wigner function is covariant with respect to the multiqubit Clifford group, which is of intrinsic interest to studying quantum computation.
Subfields: Artificial Intelligence, Materials Chemistry, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/14fb69b12c1c19c3ff1953e6a95c910c2f59611e
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In Artificial Intelligence
- A. Nahum
- Hsin-Yuan Huang
- V. Khemani
- Matthew Fisher
- Richard Kueng
- John Preskill
- Nicholas Hunter-Jones
- S. Vijay
In Materials Chemistry
In Numerical Analysis
Citations in the corpus
- Random Quantum Circuits (2022, background): “This follows from the fact that the Clifford group forms a unitary 3-design (234, 235), and hence exactly reproduces averages involving the first three moments of the unitary group.”
- Linear growth of quantum circuit complexity (2021, background): “[16, 59, 60] and [43]), which we choose to be t.”
- Unitary designs from statistical mechanics in random quantum circuits. (2019, background): “But for general k , very few constructions of exact k -designs are known, with the exception of k = 2 and k = 3 [17–21].”
- Predicting many properties of a quantum system from very few measurements (2020, background): “The n -qubit Clifford group Cl(2 n ) also comprises a unitary 3-design [45, 67, 69].”