The SuperBEST Cost of Quantum Mechanics
Tier: OBSERVATION (computed, reproduce command below)
Quantum mechanics uses matrix exponentials and matrix logarithms everywhere. This post measures the cost of standard quantum formulas in matrix EML nodes — the matrix generalization of EML where exp → expm (matrix exponential) and ln → logm (matrix logarithm).
Matrix EML Operators
The matrix EML family extends scalar EML to square matrices:
| Operator | Formula | Scalar analog |
|---|---|---|
| meml(A,B) | expm(A) − logm(B) | eml |
| mexl(A,B) | expm(A) · logm(B) | exl |
| mdeml(A,B) | expm(−A) − logm(B) | deml |
| meal(A,B) | expm(A) + logm(B) | eal |
Key constraint: non-commutativity. expm(A)·expm(B) ≠ expm(A+B) unless AB = BA. This blocks scalar SuperBEST routing from directly transferring to the matrix case.
Q1: Core Quantum Formula Costs
| Formula | Meaning | Matrix EML nodes |
|---|---|---|
| U(t) = exp(−iHt) | Time evolution | 1 node (meml with 0 offset) |
| Z = Tr(exp(−βH)) | Partition function | 1 node + Tr |
| ρ(β) = exp(−βH)/Z | Thermal state | 2 nodes |
| S = −Tr(ρ·ln ρ) | von Neumann entropy | 2 mexl nodes + Tr |
| F = −ln(Z)/β | Free energy | 1 node + scalar ln |
Headline: Time evolution U(t) = exp(−iHt) is a single matrix EML node. This is the quantum analog of T03 (Euler Gateway).
Q2: Non-Commutativity Barrier
Scalar SuperBEST routing for mul(x,y): exp(ln(x) + ln(y)) = x·y. Matrix analog: expm(logm(A) + logm(B)) ≠ AB in general.
This is a hard barrier — matrix multiplication cannot be routed through SuperBEST for non-commuting matrices.
| Operation | Scalar routing | Matrix cost | Why? |
|---|---|---|---|
| AB | 3n via EXL | Full mmul | AB ≠ BA |
| exp(A)·exp(B) | N/A | 2 nodes + mmul | Baker-Campbell-Hausdorff |
| Tr(AB) | 3n + Tr | Tr + mmul | Need full product |
The scalar SuperBEST savings (77% for multiplication) do not transfer to non-commuting matrices.
For commuting matrices (e.g., diagonal, A = f(H) for same H), SuperBEST routing applies unchanged.
Q3: Precision Verification
Computed at d=2, 3, 4:
| d | Z (partition fn) | S (entropy) | logm error |
|---|---|---|---|
| 2 | 2.8873 | 0.360188 | 2.3×10⁻¹⁵ |
| 3 | 4.6790 | 0.652474 | 2.1×10⁻¹⁵ |
| 4 | 6.4376 | 0.665785 | 3.9×10⁻¹⁵ |
Matrix logarithm achieves machine precision (∼10⁻¹⁵). Von Neumann entropy computed exactly via mexl.
Q4: Quantum Information Geometry
| Formula | Operations | Matrix EML nodes |
|---|---|---|
| Bures distance | 4 msqrt + 4 mmul + Tr | 12 ops |
| Quantum rel. entropy | 2 mexl + msub + mmul + Tr | 9 ops |
| QFI (SLD) | Sylvester eq + 3 mmul + Tr | 15 ops |
| Matrix sqrt A^{1/2} | 1 mexl + 1 meml | 2 matrix EML ops |
| Fisher-Rao (classical) | div+pow+mul+add+integral | ~7 scalar nodes |
Matrix sqrt in 2 matrix EML ops: A^{1/2} = expm(logm(A)/2) = mexl(logm(A)/2, I) decomposed as 2 matrix operations.
Reproduce
python python/scripts/research_cal_quantum.pyResults in python/results/cal_quantum_results.json under keys Q1, Q2, Q3, Q4.
Cite: Monogate Research (2026). “The SuperBEST Cost of Quantum Mechanics.” monogate research blog. https://monogate.org/blog/quantum-costs