How smooth is quantum complexity?
The “quantum complexity” of a unitary operator measures the difficulty of its construction from a set of elementary quantum gates. While the notion of quantum complexity was first introduced as a quantum generalization of the classical computational complexity, it has since been argued to hold a fun...
| Main Authors: | , |
|---|---|
| Format: | Journal article |
| Language: | English |
| Published: |
Springer Nature
2021
|
| _version_ | 1826310070373187584 |
|---|---|
| author | Bulchandani, VB Sondhi, S |
| author_facet | Bulchandani, VB Sondhi, S |
| author_sort | Bulchandani, VB |
| collection | OXFORD |
| description | The “quantum complexity” of a unitary operator measures the difficulty of its construction from a set of elementary quantum gates. While the notion of quantum complexity was first introduced as a quantum generalization of the classical computational complexity, it has since been argued to hold a fundamental significance in its own right, as a physical quantity analogous to the thermodynamic entropy. In this paper, we present a unified perspective on various notions of quantum complexity, viewed as functions on the space of unitary operators. One striking feature of these functions is that they can exhibit non-smooth and even fractal behaviour. We use ideas from Diophantine approximation theory and sub-Riemannian geometry to rigorously quantify this lack of smoothness. Implications for the physical meaning of quantum complexity are discussed. |
| first_indexed | 2024-03-07T07:45:12Z |
| format | Journal article |
| id | oxford-uuid:4318e3f3-f164-483d-beda-bff9c968c60a |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:45:12Z |
| publishDate | 2021 |
| publisher | Springer Nature |
| record_format | dspace |
| spelling | oxford-uuid:4318e3f3-f164-483d-beda-bff9c968c60a2023-06-05T12:38:43ZHow smooth is quantum complexity?Journal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:4318e3f3-f164-483d-beda-bff9c968c60aEnglishSymplectic ElementsSpringer Nature2021Bulchandani, VBSondhi, SThe “quantum complexity” of a unitary operator measures the difficulty of its construction from a set of elementary quantum gates. While the notion of quantum complexity was first introduced as a quantum generalization of the classical computational complexity, it has since been argued to hold a fundamental significance in its own right, as a physical quantity analogous to the thermodynamic entropy. In this paper, we present a unified perspective on various notions of quantum complexity, viewed as functions on the space of unitary operators. One striking feature of these functions is that they can exhibit non-smooth and even fractal behaviour. We use ideas from Diophantine approximation theory and sub-Riemannian geometry to rigorously quantify this lack of smoothness. Implications for the physical meaning of quantum complexity are discussed. |
| spellingShingle | Bulchandani, VB Sondhi, S How smooth is quantum complexity? |
| title | How smooth is quantum complexity? |
| title_full | How smooth is quantum complexity? |
| title_fullStr | How smooth is quantum complexity? |
| title_full_unstemmed | How smooth is quantum complexity? |
| title_short | How smooth is quantum complexity? |
| title_sort | how smooth is quantum complexity |
| work_keys_str_mv | AT bulchandanivb howsmoothisquantumcomplexity AT sondhis howsmoothisquantumcomplexity |