Quantum Computing
1. Give an introduction to quantum computing:
Answer: Quantum computing represents a paradigm shift in computation, moving
beyond the classical manipulation of bits to leveraging the principles of quantum
mechanics. This isn't merely about changing the physical hardware; it
fundamentally alters the very notion of computation. At its core, it replaces the
classical bit with the quantum bit, or qubit, unlocking new possibilities for
information processing.
2. What is the difference between a quantum and classical computer?
Answer: The fundamental distinction lies in how information is represented and
manipulated. Classical computers operate on bits, which exist in a definite state of
either 0 or 1. Quantum computers, however, utilize qubits, which can exist in a
superposition of both 0 and 1 simultaneously. This quantum nature of information
processing enables quantum computers to tackle certain problems with vastly
greater efficiency than their classical counterparts.
3. What is the unit complex number?
Answer: A unit complex number is a complex number with an absolute value (or
modulus) of 1. It can be expressed in the form eiθ=cos(θ)+isin(θ), where θ is a real
number representing the angle in the complex plane.
4. What is a general complex number?
Answer: A general complex number is a number of the form z=a+bi, where a and
b are real numbers, and i is the imaginary unit defined as −1. The part 'a' is the real
part, and 'b' is the imaginary part of the complex number.
5. What can the complex number a+bi be thought of as?
Answer: The complex number a+bi can be thought of as a point (a,b) in a two-
dimensional plane, where the horizontal axis represents the real part (a) and the
vertical axis represents the imaginary part (b). It can also be viewed as a vector
originating from the origin and terminating at the point (a,b).
6. What is the 2D plane called in complex numbers?
,Answer: The 2D plane in which complex numbers are geometrically represented is
called the complex plane or the Argand diagram.
7. What is the absolute value/modulus/magnitude of complex number a+bi?
Answer: The absolute value (or modulus or magnitude) of a complex number
z=a+bi is denoted by ∣z∣ or ∣a+bi∣ and is given by the formula: ∣a+bi∣=a2+b2
Geometrically, it represents the distance of the point (a,b) from the origin in the
complex plane.
8. What is the argument/phase of a+bi?
Answer: The argument (or phase) of a complex number z=a+bi, denoted by arg(z)
or ϕ, is the angle that the vector representing z in the complex plane makes with
the positive real axis. It can be found using the formula: tan(ϕ)=ab However, care
must be taken to determine the correct quadrant of ϕ based on the signs of a and b.
9. What is the complex conjugate of z=a+bi?
Answer: The complex conjugate of a complex number z=a+bi is denoted by z∗ or
zˉ and is given by: z∗=a−bi It is obtained by changing the sign of the imaginary
part of the complex number.
10. What are the properties of complex conjugates?
Answer: Some important properties of complex conjugates include:
(z1+z2)∗=z1∗+z2∗
(z1z2)∗=z1∗z2∗
(z∗)∗=z
z+z∗=2Re(z)=2a (twice the real part of z)
z−z∗=2iIm(z)=2bi (twice the imaginary part of z)
zz∗=∣z∣2=a2+b2
11. What is an n-dimensional complex vector space?
Answer: An n-dimensional complex vector space is a vector space where the
scalars are complex numbers, and it has a basis consisting of n linearly
independent vectors. Any vector in this space can be expressed as a linear
combination of these basis vectors with complex coefficients.
, 12. What is the basis of a complex vector space?
Answer: A basis of an n-dimensional complex vector space is a set of n linearly
independent vectors that span the entire space. This means that any vector in the
space can be uniquely expressed as a linear combination of the basis vectors using
complex scalar coefficients.
13. What is the space dimension?
Answer: The space dimension of a vector space is the number of vectors in any
basis of that vector space. It represents the minimum number of coordinates needed
to specify any point within the space.
14. What is the equation for the scalar product of two vectors?
Answer: For two vectors v=(v1,v2,...,vn) and w=(w1,w2,...,wn) in an n-
dimensional complex vector space, the scalar product (or dot product or inner
product) is often defined as: ⟨v,w⟩=v1∗w1+v2∗w2+...+vn∗wn=i=1∑nvi∗wi where
vi∗ denotes the complex conjugate of vi. In real vector spaces, the complex
conjugate is not needed.
15. What are 4 important properties of vector spaces?
Answer: Four important properties of vector spaces are closure under addition,
closure under scalar multiplication, the existence of a zero vector, and the
existence of an additive inverse for every vector. More formally:
Closure under addition: If u and v are vectors in the space, then their sum
u+v is also in the space.
Closure under scalar multiplication: If v is a vector in the space and c is a
scalar (complex in our case), then the product cv is also in the space.
Existence of a zero vector: There exists a vector 0 in the space such that for
any vector v in the space, v+0=v.
Existence of additive inverses: For every vector v in the space, there exists
a vector −v in the space such that v+(−v)=0.
16. Describe the Ket notation:
Answer: Ket notation, denoted by ∣ψ⟩, is a standard way to represent a column
vector (a state vector) in quantum mechanics and linear algebra. The symbol inside
1. Give an introduction to quantum computing:
Answer: Quantum computing represents a paradigm shift in computation, moving
beyond the classical manipulation of bits to leveraging the principles of quantum
mechanics. This isn't merely about changing the physical hardware; it
fundamentally alters the very notion of computation. At its core, it replaces the
classical bit with the quantum bit, or qubit, unlocking new possibilities for
information processing.
2. What is the difference between a quantum and classical computer?
Answer: The fundamental distinction lies in how information is represented and
manipulated. Classical computers operate on bits, which exist in a definite state of
either 0 or 1. Quantum computers, however, utilize qubits, which can exist in a
superposition of both 0 and 1 simultaneously. This quantum nature of information
processing enables quantum computers to tackle certain problems with vastly
greater efficiency than their classical counterparts.
3. What is the unit complex number?
Answer: A unit complex number is a complex number with an absolute value (or
modulus) of 1. It can be expressed in the form eiθ=cos(θ)+isin(θ), where θ is a real
number representing the angle in the complex plane.
4. What is a general complex number?
Answer: A general complex number is a number of the form z=a+bi, where a and
b are real numbers, and i is the imaginary unit defined as −1. The part 'a' is the real
part, and 'b' is the imaginary part of the complex number.
5. What can the complex number a+bi be thought of as?
Answer: The complex number a+bi can be thought of as a point (a,b) in a two-
dimensional plane, where the horizontal axis represents the real part (a) and the
vertical axis represents the imaginary part (b). It can also be viewed as a vector
originating from the origin and terminating at the point (a,b).
6. What is the 2D plane called in complex numbers?
,Answer: The 2D plane in which complex numbers are geometrically represented is
called the complex plane or the Argand diagram.
7. What is the absolute value/modulus/magnitude of complex number a+bi?
Answer: The absolute value (or modulus or magnitude) of a complex number
z=a+bi is denoted by ∣z∣ or ∣a+bi∣ and is given by the formula: ∣a+bi∣=a2+b2
Geometrically, it represents the distance of the point (a,b) from the origin in the
complex plane.
8. What is the argument/phase of a+bi?
Answer: The argument (or phase) of a complex number z=a+bi, denoted by arg(z)
or ϕ, is the angle that the vector representing z in the complex plane makes with
the positive real axis. It can be found using the formula: tan(ϕ)=ab However, care
must be taken to determine the correct quadrant of ϕ based on the signs of a and b.
9. What is the complex conjugate of z=a+bi?
Answer: The complex conjugate of a complex number z=a+bi is denoted by z∗ or
zˉ and is given by: z∗=a−bi It is obtained by changing the sign of the imaginary
part of the complex number.
10. What are the properties of complex conjugates?
Answer: Some important properties of complex conjugates include:
(z1+z2)∗=z1∗+z2∗
(z1z2)∗=z1∗z2∗
(z∗)∗=z
z+z∗=2Re(z)=2a (twice the real part of z)
z−z∗=2iIm(z)=2bi (twice the imaginary part of z)
zz∗=∣z∣2=a2+b2
11. What is an n-dimensional complex vector space?
Answer: An n-dimensional complex vector space is a vector space where the
scalars are complex numbers, and it has a basis consisting of n linearly
independent vectors. Any vector in this space can be expressed as a linear
combination of these basis vectors with complex coefficients.
, 12. What is the basis of a complex vector space?
Answer: A basis of an n-dimensional complex vector space is a set of n linearly
independent vectors that span the entire space. This means that any vector in the
space can be uniquely expressed as a linear combination of the basis vectors using
complex scalar coefficients.
13. What is the space dimension?
Answer: The space dimension of a vector space is the number of vectors in any
basis of that vector space. It represents the minimum number of coordinates needed
to specify any point within the space.
14. What is the equation for the scalar product of two vectors?
Answer: For two vectors v=(v1,v2,...,vn) and w=(w1,w2,...,wn) in an n-
dimensional complex vector space, the scalar product (or dot product or inner
product) is often defined as: ⟨v,w⟩=v1∗w1+v2∗w2+...+vn∗wn=i=1∑nvi∗wi where
vi∗ denotes the complex conjugate of vi. In real vector spaces, the complex
conjugate is not needed.
15. What are 4 important properties of vector spaces?
Answer: Four important properties of vector spaces are closure under addition,
closure under scalar multiplication, the existence of a zero vector, and the
existence of an additive inverse for every vector. More formally:
Closure under addition: If u and v are vectors in the space, then their sum
u+v is also in the space.
Closure under scalar multiplication: If v is a vector in the space and c is a
scalar (complex in our case), then the product cv is also in the space.
Existence of a zero vector: There exists a vector 0 in the space such that for
any vector v in the space, v+0=v.
Existence of additive inverses: For every vector v in the space, there exists
a vector −v in the space such that v+(−v)=0.
16. Describe the Ket notation:
Answer: Ket notation, denoted by ∣ψ⟩, is a standard way to represent a column
vector (a state vector) in quantum mechanics and linear algebra. The symbol inside
