B.1 Proof details and auxiliary derivations
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B.1.1 Residue projector formula
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Let \(Q{\lvert \psi_k\rangle}=k{\lvert \psi_k\rangle}\). Then
The bracket equals one when \(k\equiv r\pmod M\) and zero otherwise. This proves Equation (3.1) and the orthogonality of the residue packets.
B.1.2 Circuit leakage telescoping
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Let \(V_j=PG_jP\). Insert and subtract one factor at a time:
Every \(G_j\) has norm one and every \(V_j\) has norm at most one. On \(P{\mathcal H}\),
Taking norms gives Proposition 3.10.
B.1.3 Hypercube quotient coupling
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Let
Every weight-\(k\) string has \(n-k\) neighbors of weight \(k+1\), while every weight-\((k+1)\) string has \(k+1\) neighbors of weight \(k\). Counting the bipartite edges yields
proving Equation (4.10).
B.1.4 Symmetric-subspace dimension
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A symmetric basis state of \(n\) qudits is indexed by occupation numbers
The stars-and-bars count is
proving Equation (4.1).
B.1.5 Bosonic and fermionic dimensions
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Bosonic occupations are weak compositions of \(N\) into \(m\) parts, giving \(\binom{N+m-1}{N}\). Fermionic occupations are \(N\)-element subsets of \(m\) modes, giving \(\binom mN\).
B.1.6 Acceptance-probability stability
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If normalized states \({\lvert \psi\rangle},{\lvert \phi\rangle}\) satisfy
and \(0\le M\le I\), then
Indeed, add and subtract \({\langle \phi\rvert} M{\lvert \psi\rangle}\) and use Cauchy–Schwarz. This converts state-vector leakage bounds into measurement-error bounds.
B.1.7 Orbit quotient invariance
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For an automorphism group \(G\), the adjacency matrix commutes with the permutation representation. The fixed subspace of the action is spanned by normalized orbit states. Therefore adjacency preserves that span. Polynomial-time simulation additionally requires direct computation of the quotient entries.