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Functions will be denoted by capitol letters (such as F ), and sets by capitol script letters
(such as X ). We denote vectors with bold lower-case letters (such as v), and the components of a
vector v ∈ A n by v i , i ∈ [n]. We denote matrices with bold capital letters (such as M), and the
components of a matrix M ∈ A m×n by M i,j , i ∈ [m], j ∈ [n]. Given a function F : X → Y and a
vector v ∈ X n , let F (v) denote the vector (F (v 1 ), F (v 2 ), ..., F (v k )). Let F ([n]) denote the vector
(F (1), F (2), ..., F (n)).
Given a vector space V, let dim V be the dimension of V, or the number of vectors in any basis for
V. Given a set of vectors {v 1 , ..., v k }, let span{v 1 , ..., v k } denote the space of all linear combinations
of vectors in {v 1 , ..., v k }. Given a subspace S of an inner-product space V , and a vector v ∈ V ,
define proj S v as the orthogonal projection of v onto S, that is, the vector w ∈ S such that |v − w|
is minimized.
Given a matrix M, we define the rank, denoted rank(M), to be the size of the largest subset of
rows (equivalently, columns) of M that are linearly independent.
Given a function F : X → Y and a subset S ⊆ X , the restriction of F to S is the function
F S : S → Y where F S (x) = F (x) for all x ∈ S. A distribution D on the set of functions F from
X to Y induces a distribution D S on the set of functions from S to Y, where we sample from D S
by first sampling a function F from D, and outputting F S . We say that D is k-wise independent
if, for each set S of size at most k, each of the distributions D S are truly random distributions on
functions from S to Y. A set F of functions from X to Y is k-wise independent if the uniform
distribution on F is k-wise independent.
2.1 Quantum Computation
The quantum system A is a complex Hilbert space H with inner product 〈·|·〉. The state of a
quantum system is given by a vector |ψ〉 of unit norm (〈ψ|ψ〉 = 1). Given quantum systems H 1
and H 2 , the joint quantum system is given by the tensor product H 1 ⊗ H 2 . Given |ψ 1 〉 ∈ H 1 and
|ψ 2 〉 ∈ H 2 , the product state is given by |ψ 1 〉|ψ 2 〉 ∈ H 1 ⊗ H 2 . Given a quantum state |ψ〉 and an
orthonormal basis B = {|b 0 〉, ..., |b d−1 〉} for H, a measurement of |ψ〉 in the basis B results in a
value b i with probability |〈b i |ψ〉| 2 , and the state |ψ〉 is collapsed to the state |b i 〉. We let b i ← |ψ〉
denote the distribution on b i obtained by sampling |ψ〉.
A unitary transformation over a d-dimensional Hilbert space H is a d × d matrix U such that
UU † = I d , where U † represents the conjugate transpose. A quantum algorithm operates on a
product space H in ⊗ H out ⊗ H work and consists of n unitary transformations U 1 , ..., U n in this space.
H in represents the input to the algorithm, H out the output, and H work the work space. A classical
input x to the quantum algorithm is converted to the quantum state |x, 0, 0〉. Then, the unitary
transformations are applied one-by-one, resulting in the final state
|ψ x 〉 = U n ...U 1 |x, 0, 0〉 .
The final state is measured, obtaining (a, b, c) with probability |〈a, b, c|ψ x 〉| 2 . The output of the
algorithm is b.
We will implement an oracle O : X → Y by a unitary transfor-
Quantum-accessible Oracles.
mation O where
O|x, y, z〉 = |x, y + O(x), z〉
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8. Quantum-Secure Message Authentication Codes