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 (G, .) and (H, *) be groups with identities e_G and e_H, and let phi: G -> H be a group homomorphism, meaning phi(x.y) = phi(x)*phi(y) for all x, y in G. Prove from the homomorphism property alone (without assuming p…
primary source nl only
Let (phi, V) be a finite-dimensional complex representation of a finite group G, and let <.|.> be any inner product on V. Define the averaged form <v|w>_G := (1/|G|) sum_{g in G} <phi(g) v | phi(g) w>. Prove that (a) <.|…
primary source nl only
Let (phi, V) be a representation of a finite group G over a field whose characteristic does not divide |G|, and let W be a G-invariant subspace with some (not necessarily invariant) projection P_W onto W (P_W^2 = P_W, im…
primary source nl only
The predecessor states only that a function f:V_1->V_2 satisfies f(phi_1(g)v)=phi_2(g)f(v) for every g,v, and asks to conclude f=0 or a representation isomorphism by treating its zero fiber and image as invariant VECTOR …
primary source nl only
The predecessor takes an irreducible finite-dimensional representation phi:G->GL_F(V) over an algebraically closed field F, defines a function f:V->V to be equivariant only by f(phi(g)v)=phi(g)f(v), and asks to prove f=l…
primary source nl only
Assess the unrestricted algebraic assertion that every complex irreducible representation of an abelian group is one-dimensional, without silently importing a finite-dimensional standing convention from the cited notes. …
primary source nl only
Let rho be any single-qubit density matrix (a 2x2 positive semidefinite operator with trace 1). Using the fact that the Pauli operators {1, X, Y, Z} generate an irreducible representation of the single-qubit Pauli group …
primary source nl only
Let K be an arbitrary field, V a finite-dimensional K-vector space of dimension n, G an arbitrary group, and phi:G->GL_K(V) a representation. Define the K-valued trace character chi(g)=tr_K(phi(g)). Prove conjugacy invar…
primary source nl only
Let G be a finite group and let the characters of its pairwise inequivalent complex irreducible representations W_1, ..., W_k be chi_1, ..., chi_k, orthonormal for the inner product (x, y) = (1/|G|) sum_{g} conj(x(g)) y(…
primary source nl only
Let (phi, V) be a finite-dimensional complex representation of a finite group G, and let ( phi_U, U) be a complex irreducible representation of G. Recall that Hom_G(U, V) is the space of intertwiners f: U -> V, i.e. line…
primary source nl only
Let (phi, V) be a finite-dimensional complex representation of a finite group G, and for X in L(V) define the symmetrized operator Xbar := (1/|G|) sum_{g in G} phi(g) X phi(g)^{-1}. (a) Prove that Xbar is G-invariant, i.…
primary source nl only
Let K be an arbitrary field, V a K-vector space, G an arbitrary group and phi:G->GL_K(V) a representation. No finite-dimensional or inner-product hypothesis is implicit. On the ALGEBRAIC tensor product W=V tensor_K V def…
primary source nl only
Let (phi, V) be a finite-dimensional complex representation of a finite group G. For the trivial representation, define P := (1/|G|) sum_{g in G} phi(g). Prove that (a) P is idempotent (P^2 = P), (b) im P = V^G := {|v> :…
primary source nl only
A student claims: for any representation (phi, V) of a finite group G and any X in L(V), the symmetrized operator Xbar = (1/|G|) sum_g phi(g) X phi(g)^{-1} is always block diagonal with respect to the isotypical decompos…
primary source nl only
A student asserts: the tensor product of two irreducible representations of a group G is again irreducible. Show this is false by a counterexample. Specifically, take any representation (phi, V) of G with dim V >= 2 (you…
primary source nl only
Proposition 3.22 says the decomposition of a representation into isotypical components is unique. A student over-reads this as: 'the decomposition of a representation into irreducible subrepresentations is unique.' Show …
primary source nl only