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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Let A be a Hermitian n-qubit operator and let U be a unitary on one flag ancilla and the system whose exact clean block is A: J†UJ=A, where J maps v to |0>v. Do not assume U is Hermitian. Set P=JJ† and R=2P-I. For a unit…
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On the invariant plane attached to a value t in the closed interval from minus one to one, the qubitization iterate acts as the plane rotation M = [[t, -sqrt(1-t^2)],[sqrt(1-t^2), t]]. Prove that the k-th power of M has …
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Let A be a general (not necessarily normal) n-qubit operator with singular value decomposition sending each right singular vector v_i to sigma_i times the left singular vector w_i. A block-encoding unitary U realizes A i…
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Consider the two-by-two operator A that sends the first basis vector to zero and the second basis vector to the first (the strictly upper-triangular nilpotent). A tempting inference says: since the only eigenvalue of A i…
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A unitary U is written in two-by-two block form with top-left block B occupying the first p coordinates (p at most q, total p+q). Prove that the singular values of the top-left block B all lie in the closed interval from…
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In the cosine-sine view of qubitization the flag reflection is built from the single-qubit operator Z acting on the ancilla flag, and the large outer unitaries that diagonalize the construction are block-diagonal with re…
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To build a block encoding of a linear combination of even Chebyshev transforms, sum over k from 0 to K-1 of coefficients times the degree-2k Chebyshev transform, one needs a select operation that, conditioned on an index…
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The pure qubitization-plus-linear-combination construction directly produces block encodings of polynomials of DEFINITE parity (all-even or all-odd Chebyshev content). Suppose instead you are given a general real polynom…
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A single-ancilla block encoding alone does not specify its factor or error: with J|psi>=|0>_s|psi>, an exact (kappa,1)-block encoding satisfies J†U J=A/kappa for known kappa>0, while an approximate one only satisfies ||A…
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Suppose an operator A maps an N-dimensional input space into an N'-dimensional output space and is accessed through a single unitary U on a larger space together with two projectors: an input projector of rank N and an o…
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Take the smallest nontrivial subspace-encoding example: a four-dimensional space, the identity as the accessing unitary, an input projector that keeps the first qubit in its zero state, and an output projector obtained b…
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Keep the original promise of exactly one marked basis state among N states, with no exclusion of the positive endpoint N=1. Put g=|marked>, psi=N^(-1/2) sum_x |x>, R_g=I-2|g><g| and R_psi=2|psi><psi|-I, with Grover itera…
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For the single-marked search with N basis states, after choosing the number of iterations to be the nearest integer at or below a quarter of pi over the half-angle minus one half, prove that the failure probability (the …
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Keep the original assumptions: M>=1 marked states among N basis states, M known in advance, so the allowed domain includes M=N. Let Π_mark project onto the marked basis states, psi be uniform, and G=(2|psi><psi|-I)(I-2Π_…
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View Grover's single-marked search through qubitization beyond the computational basis. Let the uniform state and the marked state define rank-one projectors, and let the reflection about the uniform state play the role …
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Let k be a natural number, m=2k+1, and T_m the first-kind Chebyshev polynomial, with T_j(cos beta)=cos(j beta). Grover's normalized-overlap calculation uses T_(2k+1)(a)=(-1)^k sin((2k+1)arcsin(a)) for a in[0,1], where ar…
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Keep the original setting N=2^n, a unique marked input and classical worst-case query count, including the unbiased-algorithm assumption. Assess/refute the predecessor's claim that success greater than1/2 on every input …
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The marking information for search is first given as a bit oracle that adds the function value into an answer qubit modulo two. Prove that running this bit oracle with the answer qubit prepared in the minus state convert…
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Keep the original assumption of a small positive initial success probability p for a state psi=U|0> and an orthogonal good projector Π, and keep the original query-level cost convention. Assess/refute the predecessor's a…
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