QuantumComputingBench · graph-grounded exercise book
Follow the roles of input states, operator access, polynomial transformations and output interpretation. Source-only algorithm questions keep their assumptions and query/gate/readout boundaries explicit.
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A POVM P = {P_i : i in [2^m]} is implemented through block encodings as follows. For each outcome i we are given a unitary U_i that is a (beta, a)-block-encoding of sqrt(P_i), all sharing the SAME ancilla size a and the …
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Continue the POVM-by-block-encoding construction of Lemma 9.15: after applying W = sum_i |i><i| tensor U_i to |u> |0^a> |psi> and successfully postselecting the block-encoding ancilla on 0^a, the (renormalized) joint sta…
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Two normalized but possibly non-orthogonal pure states |psi> and |phi> are given with overlap c := <psi|phi>, |c| < 1, and we are promised the unknown state is one of them. Consider the three-element POVM with P_psi = (1…
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For the unambiguous-discrimination POVM with P_psi = (1/(1+|c|)) |phi_perp><phi_perp|, P_phi = (1/(1+|c|)) |psi_perp><psi_perp|, P_? = I - P_psi - P_phi, where c = <psi|phi>, |c| < 1, and |psi_perp>, |phi_perp> are defin…
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Keep the original POVM setting of Lemma9.15: L=2^m outcomes, P_i>=0 with sum_i P_i=I, K_i=sqrt(P_i), and controlled selector W=sum_i|i><i| tensor U_i. Preserve the rule that success is the designated block-ancilla zero e…
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Keep the original real-matrix, reversible arithmetic and controlled-rotation assumptions, and the declared convention A~_ij=cos(pi theta~_ij). Let t=(i,j) denote the combined retained index, q_t=theta~_ij be the ACTUALLY…
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Two input models for the entries of a matrix A appear in the sparse block-encoding discussion: the BIT oracle O~_A |0^{d'}>|i>|j> = |A~_ij>|i>|j>, which writes a classical fixed-point string into a register, and the AMPL…
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Let A be a diagonal N x N matrix (a 1-sparse matrix whose only nonzero entries are A_ii), with ||A||_max <= 1. Using the simplified amplitude oracle O_A |0>|i> = (A_ii |0> + sqrt(1 - |A_ii|^2) |1>) |i>, set U_A := O_A. P…
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A Hermitian matrix A does NOT automatically have a Hermitian block encoding: even for a Hermitian diagonal A and even when the approximation error is exactly 0, a generic (alpha, m, 0)-block-encoding U_A need not satisfy…
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The block-encoding notion generalizes beyond computational-basis subspaces. Fix rank-N and rank-N' orthogonal projectors Pi, Pi' on an ambient space C^{MN} (M = 2^m), with orthonormal bases {|phi_j>}_{j in [N]} for range…
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Let U_A in HBE_{1,m}(A) be a Hermitian block encoding of a Hermitian A with ||A|| <= 1, spectral decomposition A = sum_i lambda_i |v_i><v_i|, so that U_A |0^m>|v_i> = lambda_i |0^m>|v_i> + sqrt(1-lambda_i^2)|perp_i> with…
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