This document illustrates the content of the
IBM's basics of quantum information course
with Scala code.
This section deals with probabilistic information and quantum information.
The main difference between probabilistic information and quantum information is that the former involves non-negative real numbers and the latter involves complex numbers.
Real is a type alias for Double.
Complex is a case class.
reis the real part of a complex number,imis the imaginary part of a complex number.
Scalar is a trait that specifies a type class for type parameter S as Scalar[S].
A type class specification declares, and default defines common members of its type parameter.
realScalar is a given that implements type class Scalar by substituting type argument Real for type
parameter S as Scalar[Real].
A type class implementation defines the declared members of its type class specification.
isValidScalar is true if and only if a real number is positive or zero.
Not using the absolute value function for the scalar norm of a real number is justified by the fact that a real number is required to be positive or zero.
scalarNorm does not use the absolute value function because the real number is positive or zero.
complexScalar defines two extra members, r resp. i to construct real resp imaginary complex numbers.
NormedVector is a trait that specifies a generic value class with parameter S, required to be a Scalar, as
NormedVector[S: Scalar].
A value class declares, and default defines common members for values that are instances of it.
When constructing such an instance, all declared members need to be defined.
seq is the sequence of scalars of a normed vector.
norm is the Manhattan, L1-norm for vectors of non-negative real numbers resp. the square of the
Euclidean, L2-norm for vectors of complex numbers.
Not using the square root function for the norm of vectors of complex numbers is justified by the fact that this norm
is required to be equal to 1.0.
and
NormedColumnVector and NormedRowVector differ in the way they implement toString.
NormedColumnVectorSpace is a trait that specifies a generic value class with parameter S, required to be a
Scalar, as NormedColumnVectorSpace[S: Scalar].
dim, the dimension of the normed column vector space is the only declared member, all other ones are defined.
normedColumnVector constructs normed column vectors.
They are required to have norm equal to 1.0.
normedRowVector constructs normed row vectors.
They are required to have norm equal to 1.0.
indices and δ are auxiliary members that are used to define basis (normed) vectors.
basisColumnVector constructs basis (normed) column vectors.
basisRowVector constructs basis (normed) row vectors.
basisColumnVectors consists of all (normed) basis column vectors
isBasisColumnVector checks if a normed column vector is a basis (normed) column vector.
linearCombination constructs linear combinations of normed column vectors.
They are required to have norm equal to 1.0.
o defines the inner product of a normed row vector and a normed column vector.
asLinearBasisVectorCombination defines a normed column vector ncv as a linear combination of basis column vectors
with coefficients basisRowVector(i) o ncv for all indices i in 0 to dim - 1.
ProbabilisticStateVector.scala
ProbabilisticStateVector is a type alias for NormedColumnVector[Real].
Note: in IBM's basics of quantum information course probabilistic state vectors are called probability vectors.
QuantumStateVector is a type alias for NormedColumnVector[Complex].
ProbabilisticStateVectorSpace.scala
ProbabilisticStateVectorSpace is a type alias for NormedColumnVectorSpace[Real].
QuantumStateVectorSpace is a type alias for NormedColumnVectorSpace[Complex].
ProbabilisticStateVectorExamples.scala
QuantumStateVectorExamples.scala
ProbabilisticStateVectorSpaceSuite.scala
QuantumStateVectorSpaceSuite.scala
sbt:qc> test
qc.scalar.ProbabilisticStateVectorSpaceSuite:
+ norm == 1.0 0.005s
+ probabilisticStateVector == linearBasisVectorCombination 0.0s
qc.scalar.QuantumStateVectorSpaceSuite:
+ norm == 1.0 0.005s
+ quantumStateVector == linearBasisVectorCombination 0.0s
[info] Passed: Total 4, Failed 0, Errors 0, Passed 4
[success]