Checkpoint: Estimate Simulability

This checkpoint pulls the module together: given the entanglement structure of a state, estimate the bond dimension an exact matrix product state would need — the single number that decides whether a classical tensor-network simulation is feasible.

The rule you will apply

The exact MPS bond dimension across a cut equals the Schmidt rank at that cut: the number of non-zero Schmidt coefficients $\lambda_\alpha$ when you write

|\psi\rangle = \sum_{\alpha} \lambda_\alpha\, |u_\alpha\rangle_L \otimes |v_\alpha\rangle_R .

A coefficient that is (numerically) zero contributes nothing and does not count. The overall bond dimension a faithful MPS needs is the largest Schmidt rank over all cuts of the chain. If that number stays small as the system grows, the state is efficiently simulable; if it grows exponentially, it is not.

The state in this problem

Consider a 4-qubit state cut down the middle, into the left pair $\{0,1\}$ and the right pair $\{2,3\}$ . Suppose its Schmidt decomposition across that cut has the squared Schmidt coefficients (the eigenvalues of the reduced density matrix)

\{\lambda_\alpha^2\} = \{\,0.5,\; 0.5,\; 0,\; 0\,\} .

Two of the four possible Schmidt terms carry weight; the other two vanish. So the Schmidt rank — and hence the exact bond dimension across this cut — is $2$ . The entanglement entropy is $S = -\sum_\alpha \lambda_\alpha^2 \log_2 \lambda_\alpha^2 = -(0.5\log_2 0.5 + 0.5\log_2 0.5) = 1$ bit, consistent with $S = \log_2 \chi = \log_2 2 = 1$ .

Your task

The reduced-density-matrix eigenvalues across the middle cut are given to you as an array. Count how many are non-negligible (above a tolerance such as $10^{-9}$ ) and return that count — the exact bond dimension $\chi$ for this cut. For the spectrum above the answer is $2$ .

Try it

Compute and return the bond dimension. The grader checks the returned number.

Run your code to see the quantum state.

After running, you should return the integer $2$ : a state with two non-zero Schmidt coefficients across a cut needs an MPS bond dimension of exactly $2$ there.