Approximate equivalence checking

[TeWas]

Description

Introducing approximate equivalence checking by computing the Frobenius inner product between 𝑈 and 𝑉 and comparing the result to a predefined threshold. This enhancement allows for a more flexible comparison that accounts for near-equivalent circuits. The DDAlternatingChecker and DDEquivalenceChecker have been adapted to incorporate this approach.

Additionally, the HybridSchrodingerFeynmanChecker has been added. The checker divides a circuit horizontally into two halves: a lower part and an upper part. This is achieved by cutting controlled gates that cross the middle line according to the Schmidt decomposition.

By leveraging key trace equalities - specifically,

• $tr[L \otimes U] = tr[L] \cdot tr[U]$
• $tr[L + U] = tr[L] + tr[U]$

we can treat the lower and upper circuit parts, as well as the summands from the Schmidt decomposition, independently. This enables parallel trace computation, allowing to check the equivalence of larger, yet shallow circuits.

Fixes #442

Checklist:

• The pull request only contains commits that are related to it.
• I have added appropriate tests and documentation.
• I have made sure that all CI jobs on GitHub pass.
• The pull request introduces no new warnings and follows the project's style guidelines.

[codecov]

Codecov Report

Attention: Patch coverage is 94.77912% with 13 lines in your changes missing coverage. Please review.

Project coverage is 95.4%. Comparing base (472e756) to head (b92d53e).
Report is 20 commits behind head on main.

Files with missing lines	Patch %	Lines
...c/checker/dd/DDHybridSchrodingerFeynmanChecker.cpp	93.5%	9 Missing ⚠️
...e/checker/dd/DDHybridSchrodingerFeynmanChecker.hpp	91.6%	2 Missing ⚠️
src/checker/dd/DDEquivalenceChecker.cpp	87.5%	2 Missing ⚠️

Additional details and impacted files

@@           Coverage Diff           @@
##            main    #443     +/-   ##
=======================================
+ Coverage   95.3%   95.4%   +0.1%     
=======================================
  Files         40      42      +2     
  Lines       1835    2057    +222     
  Branches     224     261     +37     
=======================================
+ Hits        1749    1964    +215     
- Misses        86      93      +7     

Flag	Coverage Δ
cpp	95.9% <94.7%> (+0.1%)	⬆️
python	93.8% <100.0%> (+<0.1%)	⬆️

Files with missing lines	Coverage Δ
include/Configuration.hpp	0.0% <ø> (ø)
include/EquivalenceCheckingManager.hpp	100.0% <100.0%> (ø)
include/checker/EquivalenceChecker.hpp	100.0% <100.0%> (ø)
src/Configuration.cpp	100.0% <100.0%> (ø)
src/EquivalenceCheckingManager.cpp	93.5% <100.0%> (+1.0%)	⬆️
src/checker/dd/DDAlternatingChecker.cpp	98.7% <100.0%> (+<0.1%)	⬆️
src/mqt/qcec/configuration.py	100.0% <100.0%> (ø)
...e/checker/dd/DDHybridSchrodingerFeynmanChecker.hpp	91.6% <91.6%> (ø)
src/checker/dd/DDEquivalenceChecker.cpp	94.8% <87.5%> (-0.7%)	⬇️
...c/checker/dd/DDHybridSchrodingerFeynmanChecker.cpp	93.5% <93.5%> (ø)

... and 2 files with indirect coverage changes