xii Notation
set of all finite words in the alphabet A
group of characters on the Abelian group E,
i.e., Hom(E, U(1))
complex conjugate of z
space of linear functionals on the space M
ξ| bra-vector (p. 56)
ket-vector (p. 56)
ξ|η inner product
Hermitian adjoint operator
G unitary operator corresponding to the
permutation G (p. 61)
IL identity operator on the space L
ΠM projection (the operator of projecting
onto the subspace M)
TrF A partial trace of the operator A over the
space (tensor factor) F (p. 96)
A · B superoperator ρ AρB (p. 108)
n-th tensor degree of M
C(a,b,... ) space generated by the vectors a, b, . . .
Λ(U) operator U with quantum control (p. 65)
U[A] application of the operator U to a quantum
register (set of qubits) A (p. 58)
E[A], E(n, k) error spaces (p. 156)
σ (α1,β1,...,αn,βn) basis operators on the space
(p. 162)
SympCode(F, μ) symplectic code (p. 168)
| · | cardinality of a set or modulus of a number
· norm of a vector (p. 71)
or operator norm (p. 71)
trace norm (p. 98)

superoperator norm (p. 110)
Pr[A] probability of the event A
P (·|·) conditional probability (in various contexts)
P (ρ, M) quantum probability (p. 95)
f(n) = O(g(n)) there exist numbers C and n0
such that f(n) Cg(n) for all n n0
f(n) = Ω(g(n)) there exist numbers C and n0
such that f(n) Cg(n) for all n n0
f(n) = Θ(g(n)) f(n) = O(g(n)) and f(n) = Ω(g(n)) at the
same time
f(n) = poly(n) means the same as f(n) =
poly(n, m) abbreviation for poly(n + m)
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