Non-Clifford implementation of sparsification framework, sum over clifford and enhanced magic states

[riteshbhirud]

This implements sparsification framework, sum over clifford and enhanced magic states

1. Sparsification Framework (Section 5.2)

• Implements Sparsification Lemma (Lemma 6) for reducing dense stabilizer decompositions
• Achieves χ_δ(ψ) ≤ 1 + ||c||₁²/δ² bound (Theorem 1)
• Controllable approximation error via δ parameter

2. Enhanced Magic State Library (Section 5.3)

• T-gate: Optimal decomposition with ξ(T) ≈ 1.172
• CCZ gate: Optimal 8-term group decomposition with ξ(CCZ) = 16/9 ≈ 1.778
• R(θ) rotations: Parametric decomposition with ξ(R(θ)) = (cos(θ/2) + tan(π/8)sin(θ/2))²
• Implements Lifting Lemma for converting magic states to gate decompositions

3. Sum-over-Cliffords Simulation (Section 2.3.2)

• Decomposes circuits as U = Σⱼ cⱼKⱼ where Kⱼ are Clifford operations
• Multiplicative cost scaling: O(δ⁻² ∏ᵐⱼ₌₁ ξ(Vⱼ))
• Handles arbitrary mixed Clifford/non-Clifford circuits

4. Fast Metropolis Sampling (Section 4.2)

• O(kn) complexity per sample (vs naive O(k²n³))
• Auto-tuned burn-in and thinning parameters

[riteshbhirud]

opened a new PR due to merging issues with previous one. This also addresses the following comments (except for updating docs and tests based on QuantumOpticsBase) from the prev discussion: "Move everything to a module called LowRankNonClifford or a similar name
in a subfolder

Keep these public:

TGate
CCZGate
simulate_non_clifford_circuit
estimate_simulation_cost
QuantumSimulationResults
potentially will rename them

Keep consistent names and APIs with the style of use in Pauli Frame simulations, e.g. pftrajectories and getters related to extracting samples information

Rename or restructure
BitString -> Just a normal Vector{Bool} or even just a BitVector
MagicStateDecomposition -> MagicStateDecompositionCache
remove CliffordGateDecomposition if not used (and anything else not used anymore)
Use existing types and multimethods instead of building an independent type system
We can introduce a isunitary and isclifford and similar methods.

display_results should be a show method (see other existing ones)

add to this page of the docs

https://qc.quantumsavory.org/stable/allops/

have a tests that relies on purely clifford circuits and compares against mctrajectories (using Register and explicit measurements)

have a test that relies on QuantumOpticsBase (for very few qubits so it can run efficiently)"

[Krastanov]

This is awesome!!! I have a few organiziational and docstring-related comments, otherwise basically ready for merging.

Some of the docstringhelper related comments are find for skipping, just posted them for completeness.

[Krastanov]

@riteshbhirud , the test failures seem to be in the newly added tests

[riteshbhirud]

@riteshbhirud , the test failures seem to be in the newly added tests

@Krastanov Thanks for the message, it seems I forgot to reflect some changes I did in the actual code in the tests. Will do that soon.

[Krastanov]

feel free to convert to non-draft when it is ready for review

[riteshbhirud]

@Krastanov Ready for review! I addressed all points from our discussion.