Sign-operator representation of a Boolean function. Let f:{0,1}n{0,1} be a Boolean function that admits representation by a Boolean circuit C with n inputs and m gates. (a) What does the two- qubit circuit X[20]H[20](CX)H[20]X[20] do to the basis vectors 00 and 10? (b) Present a matrix VfC 2n+w2n+w that (i) satisfies Vfx0w=(1)f(x)x0w for all x{0,1}n, and (ii) factors into a product of O(m) extensions of at-most-threequbit quantum gates with w=O(m). Hints: Recall the one-qubit Hadamard and Pauli-X gates with H=21[1111] and X=[0110]. For part (a), direct calculation suffices. For part (b), transform C one gate at time into a reversible circuit. Use ideas in previous week's problem set to implement a transformation x0wxf(x)0w1, then apply part (a) to produce the sign (1)f(x)xf(x)0w1, and finally uncompute to obtain (1)f(x)x0w..

1. Sign-operator representation of a Boolean function. Let f:{0,1}n{0,1} be a Boolean function that
admits representation by a Boolean circuit C with n inputs and m gates. (a) What does the two-
qubit circuit X[20]H[20](CX)H[20]X[20] do to the basis vectors 00 and 10? (b) Present a matrix VfC
2n+w2n+w that (i) satisfies Vfx0w=(1)f(x)x0w for all x{0,1}n, and (ii) factors into a product of O(m)
extensions of at-most-threequbit quantum gates with w=O(m). Hints: Recall the one-qubit
Hadamard and Pauli-X gates with H=21[1111] and X=[0110]. For part (a), direct calculation
suffices. For part (b), transform C one gate at time into a reversible circuit. Use ideas in previous
week's problem set to implement a transformation x0wxf(x)0w1, then apply part (a) to produce the
sign (1)f(x)xf(x)0w1, and finally uncompute to obtain (1)f(x)x0w.