QuantumComputingBench · graph-grounded exercise book
Study state and channel representations, symmetry and tensor structure through primary-source-grounded problems. Planned curriculum anchors are not local Lean closure for the complete answers.
Let V and W be finite-dimensional complex vector spaces and identify End(V tensor W) with End(V) tensor End(W). Consider the subalgebra A consisting of operators of the form a tensor 1 with a in End(V), where 1 is the id…
primary source nl only
Let a finite group act on a finite-dimensional space V with isotypical decomposition V = direct sum over alpha of (V_alpha tensor C^{n_alpha}) into pairwise inequivalent irreducible pieces V_alpha appearing with multipli…
primary source nl only
On the n-fold tensor power of V = C^d let the symmetric group S_n act by permuting tensor factors, spanning an algebra A, and let the general linear group GL(V) act diagonally by sending U to the n-fold tensor power of U…
primary source nl only
Let V be a finite-dimensional complex vector space and let the symmetric subspace of its n-fold tensor power be the set of tensors fixed by every permutation of the n factors. Prove that this symmetric subspace equals th…
primary source nl only
In the Schur-Weyl decomposition of the n-fold tensor power of C^d, only Young diagrams with at most d rows contribute. Using the realization of the unitary-group irreducible pieces by Young symmetrizers acting on tensor …
primary source nl only
Let P be the antisymmetrizing projector on the n-fold tensor power of C^d, defined as the average over S_n of the sign of a permutation times the corresponding factor permutation. Prove that if d is strictly less than n …
primary source nl only
For the fourth tensor power of C^3, determine all Young diagrams that index irreducible pieces in the Schur-Weyl decomposition, compute for each the dimension of the symmetric-group piece using hook lengths and the dimen…
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For the two-dimensional unitary group, the irreducible pieces appearing in tensor powers of C^2 are indexed by partitions lambda = (lambda_1, lambda_2) with two parts. Prove that the dimension of the piece associated wit…
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The number of standard Young tableaux of a shape lambda with n boxes equals n factorial divided by the product of all hook lengths of the diagram. Compute this number for the shape (4,2,1), displaying the hook length of …
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Two permutations in the symmetric group are conjugate if and only if they have the same cycle type. Prove the key computational identity that conjugating a single cycle by an arbitrary permutation sigma relabels its entr…
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Define the twirling map that averages an operator over all transformations of the form a unitary tensored with the same unitary, using the invariant measure on the unitary group. Prove that twirling any state produces a …
primary source nl only
A permutation-invariant operator on the n-fold tensor power of C^d, meaning one unchanged by conjugation by every factor permutation, is determined by only polynomially many parameters in n when d is held fixed. Explain …
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Let a density operator on C^d have spectrum given by a probability vector in non-increasing order, and consider the n-fold tensor power of this density operator decomposed by Schur-Weyl into blocks indexed by Young diagr…
primary source nl only
Work on C^d tensor C^d in the fixed local bases used for complex conjugation and partial transpose on the second subsystem. Write Gamma(Z)=Z^{T_B}. A state means a positive semidefinite operator of trace1. Call the invar…
primary source nl only