QuantumComputingBench · graph-grounded exercise book
Reason from a target amplitude family to normalization, structural representations and a preparation construction. Inspect sparse and zero boundaries before transferring an argument to a circuit interface.
Fix a real point a and a natural number k, and prescribe target values v(0), v(1), ..., v(k). Define a degree-at-most-k real polynomial P by taking the local Taylor polynomial centred at a whose coefficient of (x-a)^j is…
local source supported
Let p and q be real polynomials and a a real point. Suppose (x-a)^{k+1} divides the difference p-q. Prove that for every order j with 0 <= j <= k, the j-th derivatives of p and q agree at a: (d^j p/dx^j)(a) = (d^j q/dx^j…
local source supported
Consider the generating power series obtained as the product of the exponential series exp(X) and the series 1/(1-X)^{k+1}. Let its degree-r coefficient be a_{k,r}. In the coordinate t, define A_k(t) as the truncation of…
local source supported
Two real polynomials p and q both have degree at most 2k+1, and both satisfy the same two-point jet data: all derivatives through order k agree at the node -1, and all derivatives through order k agree at the node 0. Pro…
local source supported
A colleague claims that the two-point jet data 'all derivatives through order k agree at -1 and at 0' pins down a real polynomial uniquely, with no restriction on its degree. Show this is false by exhibiting an explicit …
nl bridge from inspected source
Let the generating series be C(X) = exp(X) * (1 - X)^{-(k+1)} as formal power series over the reals. Prove that its coefficient of X^r is exactly sum_{m=0}^{r} C(k + r - m, k) / m!, where C(.,.) is the binomial coefficie…
local source supported
For natural k define the canonical real source coefficient polynomial by a_{k,r}=sum_{m=0}^r C(k+r-m,k)/m! for0<=r<=k and A_k(t)=sum_{r=0}^k a_{k,r}t^r. Equivalently it is the degree-k truncation of the formal series exp…
local source supported
For natural numbers k and i with 0 <= i <= k, work over the real polynomial ring and define B_{d,r}(t)=C(d,r)*t^r*(1-t)^(d-r) for 0 <= r <= d. The predecessor described the endpoint-factor term t^i*(1-t)^(k+1) as having …
local source supported
A degree-d polynomial is held as a coefficient row c(0),...,c(d) in the Bernstein basis, so that its value at t is sum_{r=0}^{d} c(r) B_{d,r}(t). One de Casteljau step with parameter s replaces the row c by the row with …
local source supported
Let a degree-d polynomial be given by Bernstein coefficients c(0),...,c(d), all lying in a common interval [lo, hi]. Suppose we restrict the polynomial to a subinterval [u, v] with 0 <= u < v <= 1 by two successive de Ca…
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Let q be any real polynomial, start and step any real numbers, and count a natural number. Set d=natDegree(q): this is the usual degree when q is nonzero, and d=0 for the zero polynomial. This natural-degree convention i…
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On the exponential tail of the target, the sampled amplitude at x_j = start + step*j is exp(x_j), and the discrete squared mass over count samples is E = sum_{j=0}^{count-1} exp(x_j)^2. Give the exact closed form for E, …
local source supported
Let k,n be natural numbers and L>0. The actual real source target is f_k(p)=HermitePolynomial.smoothInitial(k,p): on the splice interval[-1,0] it equals the source Hermite bridge S_k, whose order-zero right endpoint iden…
local source supported
Consider a bivariate real array F(x,y) = q(u(x) + v(y)), where q is a real polynomial of degree at most d and u, v are arbitrary real-valued functions of the row index x and column index y respectively. Prove that F fact…
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Work with real arrays on a finite row set X and column set Y, allowing empty sets. An array factors through width w, a natural number, if F=AB with A of shape X×w and B of shape w×Y. Given factor witnesses, construct fac…
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The full target initial datum is a piecewise function: an exponential left tail for p < -1, a degree-<=2k+1 polynomial bridge for -1 <= p < 0, and an exponential right tail for p >= 0. Sample it on an integer-indexed gri…
nl bridge from inspected source
Fix natural k,N and real L>0. The actual Hermite sample is a_j=HermiteStatePreparation.sampledAmplitude k N L j=HermitePolynomial.smoothInitial k(-pi L+j*(2pi L/2^N)), for0<=j<2^N. Let Z=sqrt(sum_j a_j^2)>0 and |psi>=sum…
nl bridge from inspected source
A gluing principle states: if two real functions f and g are each C^k and all their derivatives through order k agree at a cut point c, then the spliced function (equal to f for x < c and g for x >= c) is C^k. A student …
nl bridge from inspected source
For each natural smoothing order k, let S_k(p) be the actual polynomial (HermitePolynomial.sourceInterpolant k).eval p. To fix this family unambiguously, put t=p+1, a_{k,r}=sum_{m=0}^r binom(k+r-m,k)/m!, A_k(t)=sum_{r=0}…
local source supported
To locate where the left exponential tail meets the polynomial bridge on a grid of 2^n cells, one performs an exact-comparison binary search carrying real interval endpoints: starting from the real interval [lower, lower…
local source supported