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# Hilbert Space Truncation

**Hilbert space truncation** is the finite approximation to an infinite Hilbert space. Each qubit is embedded in a harmonic oscillator Hilbert space (charge or phase basis) with infinite dimension. Numerically, we truncate to a finite cutoff: $n_{\text{cut}}$ for transmons means we keep basis states $|0\rangle, |1\rangle, \ldots, |n_{\text{cut}}\rangle$.

Truncation error: if the true spectrum extends beyond $n_{\text{cut}}$, we lose information about higher levels and matrix elements.

```python
from scqubits import Transmon

# Truncation at different cutoffs
transmon_small = Transmon(EJ=15.0, EC=0.3, ncut=10)
transmon_large = Transmon(EJ=15.0, EC=0.3, ncut=50)

# Frequencies should converge
print(transmon_small.f_01())   # Slightly off
print(transmon_large.f_01())   # More accurate

# Check convergence: increase ncut until results stabilize
for ncut in [10, 20, 30, 40, 50]:
    transmon = Transmon(EJ=15.0, EC=0.3, ncut=ncut)
    print(f"ncut={ncut}: f_01={transmon.f_01():.6f}")
```

## Choosing Cutoff

**Too small**: inaccurate eigenfrequencies, missing matrix elements

**Too large**: computationally expensive (diagonalization scales as $n^3$)

**Practical rule**: increase $n_{\text{cut}}$ until observable quantities converge. For transmons, $n_{\text{cut}} \sim 30–50$ usually suffices. For high-frequency qubits or with strong drives, use larger cutoffs.

scqubits handles this automatically but warns if truncation is suspect.