Practical Applicability of the Babbush et al. Coupled-Oscillator Algorithm

Systematic evaluation of whether the exponential-speedup quantum algorithm for coupled classical oscillators can be applied to real material systems: graphene, silicon, diamond, and hexagonal BN.

Summary Metrics

Materials Evaluated

Total materials4
Best materialSilicon (0.8078)
Worst materialGraphene (0.7996)
Mean score0.8039

Algorithm Assumptions

Sparse connectivity4/4 pass
Limited init. conditions4/4 pass
Efficient oracle0/4 pass
Polylog observables4/4 pass

Observable Extraction

Total observables6
Polylog-extractable5 (83.33%)
Non-polylog1 (Phonon DOS)
Max advantage ratio0.0501

Applicability Scores

Silicon0.8078
0.8078
Diamond0.8078
0.8078
Hexagonal BN0.8006
0.8006
Graphene0.7996
0.7996

Assumption Satisfaction

Sparse Connectivity 4/4 PASS

Coupling matrix has O(1) non-zeros per row. Max degree: 17 (Si, diamond), 13 (graphene), 7 (hBN). Density: 0.0037 to 0.0866.

Limited Initial Conditions 4/4 PASS

Single-site excitation uses <1% of DOF. Gaussian wavepackets also satisfy the polylog bound for all materials.

Efficient Oracle 0/4 PASS

Degree variance (1.34-7.60) indicates boundary effects. Structured bulk lattices could enable efficient oracles with specialized construction.

Polylog Observables 4/4 PASS

83.33% of observables extractable in polylog time. Only phonon DOS requires O(N) measurements.

Scaling Comparison: Classical vs Quantum

Simulation Cost vs System Size

Quantum Advantage Ratio vs System Size

Sparsity Metrics

MaterialN atomsDensityMax DegreeMean DegreeBandwidthCondition No.
Graphene1280.08651311.07811758.3561
Silicon40960.00371715.1836574134.0418
Diamond40960.00371715.1836574134.0418
Hexagonal BN5120.011876.03126454.8958

Observable Extraction Complexity

Extraction Complexity by Material (Graphene at N=128)

Phonon Dynamics Validation

Energy Conservation Across Simulations

MaterialInit TypeEnergy DriftMax RMS Disp.Final KEFinal PE
GrapheneSingle Site0.00890.088474.565776.1015
GrapheneGaussian0.01140.2977664.9828474.2255
GrapheneEdge0.00860.3062488.2461416.0864
SiliconSingle Site0.00120.024123.811423.9692
SiliconGaussian0.00200.29773022.57002408.2690
SiliconEdge0.00200.31552242.67731728.3370

Score Breakdown by Assumption

Per-Material Assumption Scores