2. Circuit Decomposition
• A set of quantum gates is said to be universal if any unitary
transformation of the quantum data can be efficiently
approximated arbitrarily well as a sequence of gates in the
set.
• These gates can easily be transpiled by using C-NOT (Cx)
gates and hadamard gates with single qubit rotations.
• These forms universal gate set as well as any unitary can be
implemented using them.
• Our task is to form an efficient unitary matrix that can be
decomposed into lesser number of known quantum gates.
3. Unitary
• Unitarmy atrices are the complex analog of real orthogonal
matrices.
• A unitary matrix is a matrix whose inverse equals it conjugate
transpose.
• U is unitary, the it holds following properties:
• U is invertible and U− 1 = U*.
• The columns of U form an orthonormal basis.
• The rows of U form an orthonormal basis.
• U is an isometry with respect to the inner product determined
by U.
• U is a normal matrix with eigenvalues lying on the unit circle. .
4. Isometry
• an isometry is a transformation which maps elements to the
same or another metric space such that the distance
between the image elements in the new metric space is
equal to the distance between the elements in the original
metric space.
• An isometric surjective linear operator on a Hilbert space is
called a unitary operator.
5. Transpilation
• It is defined as source-to-source compilation. Tools have
been written to do this and they are called transpilers.
Transpilers take the source code and convert it into another
language.
• The ouput is generally understandable by a human.
• This output still has to go through a compiler or interpreter
to be able to run on the machine.
6. Compilation Vs Transpilation
• It is an umbrella term to
describe a program that
takes source code written in
one language and produce a
(or many) output file in
some other language.
• compiler such as gcc which
takes in C code as input and
produces a binary
executable (machine code)
as output.
• They are a subset of
compilers which take in a
source code file and convert
it to another source code
file in some other language
or a different version of the
same language.
• The output is generally
understandable by a human.
This output still has to go
through a compiler or
interpreter.
7. Cost
• Cost is the most crucial factor for designing or
implementation of any circuit.
• when it comes to quantum circuits it becomes the most
unavoidable part of design technique.
• The hadamard-unitary sandwich technique has an amazing
cost reduction potential in quantum circuits.
8. Circuit Cost
• Here we will look at the number of CNOT-gates, and the
number of u3-gates, to determine the cost of our
decomposition.
• We used only two gates Hadamard (u3-gate) and Controlled-
Not (Cx) gate in the circuit.
• We determined the cost as:
Cost = 10. ncx + nu3
