Z_Stream is our in-house algorithm for computing the exact partition function of a 1-D periodic Ising chain. It,
- halves the configuration space via global spin-flip symmetry
- streams the remaining states across all CPU-cores
- bins identical energy levels on the fly
- reinserts a symmetry factor to rebuild the full spectrum
Why we care
Accurate values underpin free-energy, entropy and phase-diagram studies. Classical exact summation caps out at ~26 spins; Monte-Carlo loses the absolute normalisation. Z_Stream gives ground-truth results, yet keeps RAM growth polynomial through real-time binning.
Current snapshot (8 cores, Python prototype)
| N | Wall time | Peak RAM |
|---|---|---|
| 10 | 0.07s | 0.07 MB |
| 20 | 0.93s | 0.08 MB |
| 26 | 74.40s | 0.09 MB |
| 30 | 1525.00s | 0.09 MB |
| 32 | 6228.00s | 0.09 MB |
Time still scales ≈ c 2^N ; memory only aN^b.
Where we’re heading – classical × quantum hybrid
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Massively-parallel classical front-end – SIMD / GPU kernels (CUDA & ROCm) for energy evaluation. – Dynamic load balancing over heterogeneous clusters.
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Quantum co-processor back-end – Map energy-bin counting to a quantum amplitude-estimation task, replacing classical sampling with Grover-style scaling. – Hybrid workflow: CPU/GPU generates candidate bit-strings; a near-term quantum processor verifies and aggregates amplitudes.
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Variational rewrite – Treat as a generative overlap between a parameterised quantum state and an energy-diagonal operator; optimise the state on quantum hardware while classically accumulating the bin counts.
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Error-mitigated extrapolation – Use classical Z_Stream as a noise-free baseline to calibrate small-N quantum runs, then extrapolate to larger N where classical enumeration fails.
Scientific & industrial implications
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Statistical-physics benchmarking – Hybrid Z_Stream lets us push exact into regimes previously reserved for Monte-Carlo.
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Quantum algorithm test-bed – Provides a deterministic ground truth for validating NISQ hardware.
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Materials discovery & optimisation – Accurate free energies accelerate search in spintronic and magnetocaloric compounds.
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Education & outreach – A tangible example of how classical/quantum co-design extends computational frontiers.
Z_Stream is under active development. Expect faster kernels, a quantum plug-in, and occasional dragons. Follow the journey or join the effort on GitHub – contributions, benchmarks and bold ideas are always welcome!
Sources
> Classical exhaustive enumeration scales as .> Even a 1-D Ising chain hits the wall around :> configurations. > Storing each spin as one byte already needs > GB, > and the Boltzmann-weight table doubles that. > See e.g. A. D. Sokal, Monte Carlo Methods in Statistical Mechanics (Les Houches, 1998), > Sec. II.3, or the scaling discussion in > T. Neuhaus & K. Heinisch, “Exact enumeration of Ising configurations”, > Comput. Phys. Commun. 184 (2013) 1719–1727.
W. Janke, “Monte Carlo Methods in Statistical Physics”, Les Houches Lecture Notes (1998), Sec. 3.2, esp. p. 97–100. D. P. Landau & K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, 5 th ed. (Cambridge Univ. Press, 2021), Chap. 9 “Free-Energy Methods”.