BUG: PeriodIndex does not survive JSON serialization roundtrip

[jmg-duarte]

• I have checked that this issue has not already been reported.

• I have confirmed this bug exists on the latest version of pandas.

• I have confirmed this bug exists on the master branch of pandas.

Reproducible Example

import pandas as pd
df = pd.DataFrame(
  {"column": range(10, 20, 2)},
  index=pd.PeriodIndex(
    year=[2020, 2020, 2020, 2020, 2021], quarter=[1, 2, 3, 4, 1]
  ),
)

df_from_json = pd.read_json(df.to_json(orient='table'), orient='table')
assert df.equals(df_from_json)

Issue Description

The index should make the roundtrip.

Possible related issue: #32665

Expected Behavior

From my point of view, this issue has two parts, the title and example report on the overall issue which is the failure to roundtrip the index when using JSON to serialize the DataFrame.

The second part of the issue is that when reading from JSON, the index gets serialized to a DatetimeIndex with freq=None.
In the JSON however, the frequency is there, see the output of df.to_json(orient='table') for the previous example:

{'data': [{'column': 10, 'index': '2020-01-01T00:00:00.000Z'},
          {'column': 12, 'index': '2020-04-01T00:00:00.000Z'},
          {'column': 14, 'index': '2020-07-01T00:00:00.000Z'},
          {'column': 16, 'index': '2020-10-01T00:00:00.000Z'},
          {'column': 18, 'index': '2021-01-01T00:00:00.000Z'}],
 'schema': {'fields': [{'freq': 'Q-DEC', 'name': 'index', 'type': 'datetime'},
                       {'name': 'column', 'type': 'integer'}],
            'pandas_version': '0.20.0',
            'primaryKey': ['index']}}

Notice how the index field has the freq correctly set.

Installed Versions

INSTALLED VERSIONS
------------------
commit           : 945c9ed766a61c7d2c0a7cbb251b6edebf9cb7d5
python           : 3.8.5.final.0
python-bits      : 64
OS               : Darwin
OS-release       : 20.6.0
Version          : Darwin Kernel Version 20.6.0: Wed Jun 23 00:26:31 PDT 2021; root:xnu-7195.141.2~5/RELEASE_X86_64
machine          : x86_64
processor        : i386
byteorder        : little
LC_ALL           : None
LANG             : en_GB.UTF-8
LOCALE           : en_GB.UTF-8
pandas           : 1.3.4

numpy            : 1.21.3

pytz             : 2021.3

dateutil         : 2.8.2

pip              : 21.3.1

setuptools       : 58.3.0

Cython           : None

pytest           : 6.2.5

hypothesis       : None

sphinx           : None

blosc            : None

feather          : None

xlsxwriter       : None

lxml.etree       : None

html5lib         : 1.1

pymysql          : None

psycopg2         : None

jinja2           : None

IPython          : 7.30.0

pandas_datareader: None

bs4              : None

bottleneck       : None

fsspec           : None

fastparquet      : None

gcsfs            : None

matplotlib       : None

numexpr          : None

odfpy            : None

openpyxl         : None

pandas_gbq       : None

pyarrow          : 5.0.0

pyxlsb           : None

s3fs             : None

scipy            : None

sqlalchemy       : 1.4.26

tables           : None

tabulate         : None

xarray           : None

xlrd             : None

xlwt             : None

numba            : None

[jmg-duarte]

I have a fix for this. I'm just trying to setup pandas in development mode on my computer and failing (see #47737).

I've also tried regular pip but pandas requires pg_config for some reason?

Is there a way I can avoid setting up the more complex parts of the development environment so I can just test the JSON part?