Accelerate Jordan-Wigner transform

[vincentmr]

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---

Context:
Jordan-Wigner transforms can take a while to compute. Most of the time is spent in __init__ of the various Pauli gates.

Description of the Change:
Cache Pauli gate objects.

Benefits:
Faster transforms (e.g. 160->38 sec. for generating the molecular Hamiltonian of an H2 molecule in the "cc-pvdz" basis).

Possible Drawbacks:
Deepcopy necessary?

Related GitHub Issues:

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[codecov]

Codecov Report

Merging #4046 (608cd8a) into master (1b6706b) will decrease coverage by 0.01%.
The diff coverage is 100.00%.

@@            Coverage Diff             @@
##           master    #4046      +/-   ##
==========================================
- Coverage   99.73%   99.73%   -0.01%     
==========================================
  Files         345      345              
  Lines       30787    30784       -3     
==========================================
- Hits        30704    30701       -3     
  Misses         83       83              

Impacted Files	Coverage Δ
pennylane/pauli/utils.py	100.00% <100.00%> (ø)
pennylane/qchem/observable_hf.py	98.83% <100.00%> (-0.07%)	⬇️

[soranjh]

Thanks @vincentmr for the improvement! Left a suggestion and also a timing comparison for computing molecular Hamiltonians (the molecules are H2, H3+, LiH, H2O).

[vincentmr]

Thanks @vincentmr for the improvement! Left a suggestion and also a timing comparison for computing molecular Hamiltonians (the molecules are H2, H3+, LiH, H2O).

This is nice, thanks for the timing data.

[vincentmr]

For the record, after the current changes, the biggest bottlenecks are _pauli_mult called in jordan_wigner and simplify called in qubit_observable.

[soranjh]

For the record, after the current changes, the biggest bottlenecks are _pauli_mult called in jordan_wigner and simplify called in qubit_observable.

Thanks @vincentmr. The qchem module will be upgraded to use more efficient Pauli arithmetic.

[obliviateandsurrender]

Thanks @vincentmr, overall looks good! Does caching have any ill-effect on memory profiles for building larger observables? It might be nice to have some memory-benchmarks too.

[vincentmr]

Thanks @vincentmr, overall looks good! Does caching have any ill-effect on memory profiles for building larger observables? It might be nice to have some memory-benchmarks too.

Calling _get_pauli_map for 4 up to 32 wires, the time for the first call grows linearly, but then it is pretty much constant.

mprof: Sampling memory every 0.1s
running new process
time(4)=0.0002491474151611328
time(4)=4.76837158203125e-07
time(8)=0.00027823448181152344
time(8)=7.152557373046875e-07
time(12)=0.0004432201385498047
time(12)=4.76837158203125e-07
time(16)=0.0005674362182617188
time(16)=4.76837158203125e-07
time(20)=0.0006365776062011719
time(20)=4.76837158203125e-07
time(24)=0.0007550716400146484
time(24)=4.76837158203125e-07
time(28)=0.0008456707000732422
time(28)=4.76837158203125e-07
time(32)=0.0014255046844482422
time(32)=9.5367431640625e-07

There is no noticeable differences between the memory profiles for any wire count. They all look like (on my laptop)

[obliviateandsurrender]