Qubit: The Unit of Quantum Computation

1 Introduction

The basic constituent of a quantum computer is a quantum bit or "qubit", which owes itsname to the more familiar classical "bit". The bit, itself an abbreviation of "binary digit",can take two values 0 and 1, which in a modern-day computer correspond to a transistorbeing in on or off states respectively. The quantum bit is a state vector, having two basisstates that are labelled |0〉 and |1〉. From our knowledge of quantum mechanics we knowthat a qubit can exist in a superposition of |0〉 and |1〉, quite unlike a classical bit. This andother quantum mechanical properties give quantum computers several unique featuresthat can be exploited to design novel algorithms and computational techniques.

2 The qubit: representation

2.1 The |0〉, |1〉 basis

The mathematical representation of a qubit is a point in two dimensional Hilbert space.A set of basis vectors in a 2D Hilbert space are given by,

|0〉 =(10

); |1〉 =

(01

)(1)

The most general qubit is represented as,

|ψ〉 = α|0〉+ β|1〉, (2)

where, α and β are in general complex numbers and are the probability amplitudes of thesystem to be in the two respective states. The basis state {|0〉, |1〉} is known as the standardbasis or the computational basis in 2D Hilbert space for the purpose of measurements.When we measure this qubit given in (2) in the standard basis, the probability of outcomecorresponding to |0〉 is |α|2 and the probability of outcome corresponding to |1〉 is |β|2.

2.2 The Bloch sphere

Since the net probability is one, |α|2 + |β|2 = 1 the qubit given in (2) can be represented inspherical polar coordinates as,

|ψ〉 = e−iφ2 cos

(θ

2

)|0〉+ e

iφ2 cos

(θ

2

)|1〉, (0 ≤ θ ≤ π; 0 ≤ φ ≤ 2π) (3)

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A qubit is thus a point on the surface of a sphere of unit radius, parametrized by thecoordinates (θ, φ) as represented in Fig. 1. The basis states |0〉 and |1〉 are then representedby the north and south poles of the Bloch sphere respectively. The convenience of theBloch sphere representation of a qubit will be evident when we study the evolution ofqubits and quantum gates.

Figure 1: Representation of a qubit in Bloch sphere.

Physical realisation of a qubit can be in many forms. Any two level quantum mechanicalsystem can be a qubit, Prominent examples being the up (| ↑〉 = |0〉) and down (| ↓〉 = |0〉)spin states of an electron, or vertical and horizontal polarization states of a photon, or theground and excited states of an atom.

3 Operators in H2

3.1 Pauli matrices

The three Pauli matrices σx, σy, σz are both hermitian and unitary matrices that provevery useful for studying two-level quantum systems. They satisfy σ2

x = σ2y = σ2

z = 1. It isstraight forward to check that σx, σy, and σz have two eigenvalues 1

2and −1

2, physically

corresponding to two measurable spin angular momentum values of the electron alongx, y and z directions respectively.For convenience, we have chosen the spin along z-direction to be the diagonal matrix σz.The eigenvectors of this matrix we are already familiar with, |0〉 (corresponding to up-spin) and |1〉 (corresponding to down-spin), with eigenvalues ±1 respectively.The eigenvectors of σx, on the other hand, are,

|±〉 = 1√2(|0〉 ± |1〉) (4)

2

Measuring spin: For appreciating the intricacies of quantum mechanics and its connec-tion with the classical world, let us consider an eigenstate of σx, |+〉 = 1√

2(|0〉+ |1〉). This

corresponds to a spin-12

particle with its spin oriented along the +x direction. On theother hand, this is also an equal superposition of the spin-up and spin-down states alongz-direction. Classical reasoning suggests that a measurement of spin along z-direction forthe above state should yield zero, since z and x directions are orthogonal to each other.But, as is evident, a measurement of spin along z actually yields ±1

2with equal probabil-

ities. In other words, the |+〉 collapses to either |0〉 or |1〉with probability 12.

The expected classical picture emerges if we perform a large number of measurement,where the average measurement of spin smooths out to 0. A single measurement, how-ever, always give either +1

2or −1

2.

3.1.1 Properties of Pauli matrices

The Pauli matrices are essential in studying quantum information theory and quantumcomputation. It is therefore worth examining their properties in some detail. It is easy tocheck that they obey the commutation relations,

σiσj − σjσi = [σi, σj] = 2εijkiσk (5)

where {i, j, k} ∈ {x, y, z}.This is structurally similar to the relation satisfied by the components of the orbital angu-lar momentum ~L = ~r × ~p.Further,

σiσj + σjσi = 2δij (6)

We can condense (5) and (6) into the useful relation:

σiσj = δijI+ iεijkσk (7)

It should also be added that the Pauli matrices with the 2 × 2 identity matrices forma complete basis in the space of 2 × 2 matrices. Any operator in H2 can therefore beexpanded as

A = a0I+ ~a.~σ (8)

3.2 Maximally unbiased basis

Let us start with the matrix σy =(0 −ii 0

).

We wish to diagonalize it. The eigenvalues equation reads, σyψ = λψ, where ψ =

(ψ1

ψ2

).

with ψ21 + ψ2

2 = 1. From the characteristic equation, Detσy − Iλ = 0 one gets

Det

(−λ −ii −λ

)=⇒ λ2 − 1 = 0 =⇒ λ = ±1 (9)

3

For λ = 1, ψ can be found from the equation(−1 −ii −1

)(ψ1

ψ2

)= 0 =⇒ −ψ1 − iψ2 = 0

iψ1 − ψ2 = 0=⇒ ψ2 = iψ1

The normalization condition yields,

(ψ?1 ψ?2

)(ψ1

ψ2

)= 1 =⇒ 2|ψ2

1| = 1

This gives ψ1 =1√2eiθ, with θ as an arbitrary phase.

An overall phase of a wave function is not a measurable quantity inquantum mechanicswhich allows us to write,

ψ+1 =1√2

(1i

), ψ−1 =

1√2

(1−i

)From the eigenvectors, a unitary matrix can be constructed,

U+1 =1√2

(1 1i −i

), such that, U−1

+1σyU+1 =

(1 00 −1

).

This physically amounts to rotating the spin operator σy along the y-direction to the z-direction with

σz =

(1 00 −1

)Analogously one finds that for Sy the eigenvectors are 1√

2

(11

), 1√

2

(1−1

)and as noted

earlier for σz, these two are(10

),(01

). To summarize we have obtained three sets of

orthogonal vectors, (10

),

(01

)1√2

(11

),

1√2

(1−1

)1√2

(1i

),

1√2

(1−i

)In the space of two dimensional column vectors there are two independent vectors andhence any two of the above set will supplies to describe a general vector, e.g., |ψ〉 =α |0〉 + β |1〉. It is worth emphasizing that a member of a given set is not orthogonal toanother member of a different set. In fact, it can be easily checked that the magnitude oftheir overlap is equal to 1√

2. For example,∣∣∣∣ 1√2

(1 1

) 1√2

(1i

)∣∣∣∣ = 1√2|1 + i| = 1√

2

4

This fact plays a crucial role in the theory of quantum measurement and quantum com-munication theory. These type of basis sets are known as maximally unbiased basis(MUB). In N dimension, we need to construct N + 1 sets , each set containing N orthog-onal vectors. The magnitude of the overlap from one set with any other from a differentset is 1√

N.

3.3 Rotation operators

The 2D operators,Rn(θ) = e

iθ2~σ·n (10)

where n is a unit vector, are known as rotation operators as their action on qubits is torotate them through an angle θ along an axis n on the Bloch Sphere.On expanding the exponential and using the identity (~σ · n)2 = I , we get the followingrelation

Rn(θ) = I cosθ

2+ n.~σ sin

θ

2(11)

These rotation operators prove useful for constructing different quantum gates.

4 Quantum gates

In classical computation, complex tasks are performed using algorithms that implementa sequence of simpler logic operations on bits. These operations are commonly referredto as gates, e.g., the NOT-gate that performs a bit-flip, i.e. changes 0 to 1 and 1 to 0, theOR- and AND-gates, etc. The quantum analogue of these logic gates are the 2D unitaryoperators that we have studied in the previous section.

4.1 Single-qubit gates

Let us now look at some important single-qubit gates.

4.1.1 Quantum NOT-gate

First, we construct a quantum version of a NOT-gate. The gate should change |0〉 to |1〉and |1〉 to |0〉. It is easy to see that the Pauli matrix σx fulfils the criteria:(

0 11 0

)(10

)=

(01

),

(0 11 0

)(01

)=

(10

)Thus, σx acts as the NOT gate on the computational basis states. However, note thatit need not act similarly on other basis; in particular, the eigenstates of σX remain un-changed under this operation.A quantum computer can go much beyond replicating the familiar logic operations ofclassical computers, as evident from our the following example - the uniquely quantum

5

√NOT-gate.

To construct this gate, we note that the NOT-gate can be expressed, up to a phase, as arotation of π about the x-axis,

σx =

(0 11 0

)= iRx(π) (12)

Since an overall phase is not important in quantum mechanical calculations, the√NOT-

gate is given (again up to an overall phase) by a rotation of π/2 about the x-axis,

√NOT = Rx(π/2) =

1√2

(1 −i−i 1

)(13)

4.1.2 Hadamard gate

The matrix H = 1√2

(1 11 −1

)diagonalizes σx:

H†σxH =

(1 00 −1

)(14)

which takes the state |0〉 to a linear superposition:

H |0〉 = |0〉+ |1〉√2

(15)

The above operator is called the Hadamard gate, another important single qubit gate. TheHadamard gate takes the computational basis states to the σx basis, |±〉.

Another operator H = 1√2

(1 1−1 1

)is a unitary but non-Hermitian matrix such that

H†(0 11 0

)H =

(−1 00 1

)Note that

H |0〉 = 1√2

(1 1−1 1

)(10

)=

1√2

(1−1

)=

1√2|0〉 − 1√

2|1〉

However, H12 = I , unlike H4 = I . The physical meaning of the above operations shouldbe explored by the enthusiastic reader.

4.1.3 Phase gate

The phase gate,

S =

(1 00 i

)(16)

6

introduces a relative phase of π in a superposition of computational basis states:

S|0〉+ |1〉√

2=|0〉+ i|1〉√

2(17)

The√NOT , the Hadamard and the phase gate are all examples of quantum gates that do

not have analogues in classical computation. Any 2 × 2 unitary operator can be used asan elementary logic operation as per our need.

4.2 Decomposition of single-qubit gates

While obtaining the matrix form of the√NOT gate, we expressed the NOT gate as a

rotation operator. More generally, any single qubit unitary operation can be written asU = eiαRn(θ), i.e., a rotation followed by a global phase shift.Any single qubit gate can be decomposed into various combinations of rotations andglobal phase shifts. A particularly important decomposition is the Z-Y decomposition:

U = eiαRz(β)Ry(γ)Rz(δ)

Such decompositions can in fact be carried out in terms of rotations along any two non-parallel axis n and m: U = eiαRn(β)Rm(γ)Rn(δ).Using the Z-Y decomposition, we can arrive at another useful representation of 2 × 2unitary operators:

U = eiαAσxBσxC (18)

where A, B, C are unitary and ABC = I.

5 Composite systems

Composite or multi-partite systems can be represented in quantum mechanics using thenotion of tensoor-product space. Suppose you have two systems A and B in states |ψ1〉Aand |ψ2〉b respectively. The state of the composite system A+B can be written in the form|ψ1〉A⊗|ψ2〉B where⊗ denotes the tensor product. For example, using qubit states |0〉 and|1〉 one can describe the two particle computational basis in the form

(10

)⊗(10

)= |0〉 ⊗ |0〉 = |00〉 =

1000

|01〉 =

0100

, |10〉 =

0010

, |11〉 =

0001

These are four orthogonal vectors describing four particles. They are independent too,therefore forming a basis in the composite four-dimensional Hilbert Space. The product

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state, along with superposition of probability amplitudes in quantum mechanics give riseto a bizarre quantum mechanical phenomenon called entanglement. Consider the state

1√2

1000

+1√2

0001

1√2

1001

produced by the superposition of two product states:

1√2

1001

=1

2|0 0〉+ 1√

2|1 1〉

This state can be written as a product of single-qubit states,

1√2

1001

6= |ψ1〉 ⊗ |ψ2〉

unlike say, 1000

= |0〉 ⊗ |0〉

=

(10

)⊗(10

)The former is a non-separable or entangled state, which plays a very important, almostmagical role in quantum computation.

6 Multi-qubit gates

Previously we have only looked at local operations, i.e., gates that act at the level of singlequbits. Quantum gates that act on multiple qubits are necessary for implementing severalquantum algorithms and can be easily constructed (at least theoretically).

6.1 CNOT gate

The classic example of such a gate is the two-qubit controlled-NOT or CNOT-gate. It takesas input a control and a target qubit and executes a NOT operation on the target only if thecontrol is in the state |1〉. If the control is set to |0〉, nothing happens to the target.

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6.1.1 Action of CNOT on H4 computational basis

|00〉 CNOT−−−−→ |00〉, |01〉 CNOT−−−−→ |01〉

|10〉 CNOT−−−−→ |11〉, |11〉 CNOT−−−−→ |10〉

6.1.2 Matrix representation

The matrix representation of CNOT is given by:

|0〉〈0| ⊗ I2 + |1〉〈1| ⊗ σx =

1 0 0 00 1 0 00 0 0 10 0 1 0

The CNOT gate belongs to a broad-class of controlled gates that execute some operationonly a control-qubit is set to the |1〉 state. An (n+1)-dimensional controlled-U -gate, whereU is an n-dimensional operator can be expressed as:

CU = |0〉〈0| ⊗ In + |1〉〈1| ⊗ U (19)

6.2 Universality of gates

Any arbitrary unitary operation, in any dimensions, can be decomposed into sequencesof single-qubit operations and the CNOT gate. These form, what is called, a universal setof quantum gates. Any operation conceivable in a quantum computer can be expressedin terms of such an universal set. In particular, as any single qubit operation can beexpressed as a finite sequence of rotations, the rotation operators along with CNOT aresufficient to construct all other unitary operations.Note that the universal set so obtained has and infinite number elements. What we wouldlike is to have a finite universal set but such a set is not sufficient to exactly reproduceany arbitrary unitary operation. However, we can think of finite sets of gates that canapproximate any unitary operator up to a desirable level of accuracy. One such set isformed by the Hadamard, the π

8or R(π

4) gate and the CNOT gate.

A few examples

Representing multi-qubit states and operators in matrix form can quite often prove con-fusing. The standard way of expressing two-qubit operators are illustrated with a fewexamples so that the reader gets familiarized.

1⊗ 1 =

1 0 0 00 1 0 00 0 1 00 0 0 1

9

Ax = 1⊗ σx =(1 00 1

)⊗(0 11 0

)=

0 1 0 01 0 0 00 0 0 10 0 1 0

Bx = σx ⊗ 1 =

(0 11 0

)⊗(1 00 1

)=

0 0 1 00 0 0 11 0 0 00 1 0 0

Ax, Bx =

0 1 0 01 0 0 00 0 0 10 0 1 0

0 0 1 00 0 0 11 0 0 00 1 0 0

−0 0 1 00 0 0 11 0 0 00 1 0 0

0 1 0 01 0 0 00 0 0 10 0 1 0

=

0 0 0 10 0 1 00 1 0 01 0 0 0

−0 0 0 10 0 1 00 1 0 01 0 0 0

= 0

Note that

Ax ⊗Bx =

0 0 0 10 0 1 00 1 0 01 0 0 0

= Cx

is a traceless matrix whose eigenstates are,1001

and

100−1

C2x =

0 0 0 10 0 1 00 1 0 01 0 0 0

0 0 0 10 0 1 00 1 0 01 0 0 0

=

1 0 0 00 1 0 00 0 1 00 0 0 1

Ax ×B†x = B†xA

†x

= BxAx = AxBx

A′x = 1⊗ σz =(1 00 1

)⊗(1 00 −1

)=

1 0 0 00 −1 0 00 0 −1 00 0 0 1

B′x = σz ⊗ 1 =

(1 00 −1

)⊗(1 00 1

)=

1 0 0 00 1 0 00 0 −1 00 0 0 −1

10

Again A′x and B′x are traceless matrices and are linearly independent.

A′xB′x =

1 0 0 00 −1 0 00 0 −1 00 0 0 1

1 0 0 00 1 0 00 0 −1 00 0 0 −1

=

1 0 0 00 −1 0 00 0 1 00 0 0 −1

= C ′x

B′xA′x =

1 0 0 00 1 0 00 0 −1 00 0 0 −1

1 0 0 00 −1 0 00 0 −1 00 0 0 1

=

1 0 0 00 −1 0 00 0 1 00 0 0 −1

= C ′x

σx ⊗ σx =(0 11 0

)⊗(0 11 0

)=

0 0 0 10 0 1 00 1 0 01 0 0 0

Similarly,

σz ⊗ σz =

1 0 0 00 −1 0 00 0 1 00 0 0 −1

σy ⊗ σy =

0 0 0 10 0 −1 00 −1 0 01 0 0 0

Also note that,

σx ⊗ σy =

0 0 0 10 0 −1 00 1 0 0−1 0 0 0

σy ⊗ σx =

0 0 0 10 0 1 00 −1 0 0−1 0 0 0

11

σy ⊗ σz =

0 0 1 00 0 0 −1−1 0 0 00 1 0 0

σz ⊗ σy =

0 1 0 0−1 0 0 00 0 0 −10 0 1 0

σx ⊗ σz =

0 0 1 00 0 0 −11 0 0 00 −1 0 0

σz ⊗ σx =

0 1 0 01 0 0 00 0 0 −10 0 −1 0

Problems

1. A = σx⊗σx andB = σy⊗σy. Find the eigenstates. Do these two operators commute?

2. Show that eiθσ/2 = cos(θ/2)I + isin(θ/2)(~σ · n).

3. Calculate Tr[eiσ.a eiσ.b].

4. Show that controlled NOT gate is Hermitian and Unitary.

5. Write down the matrix representation for the controlled-Z gate.

6. Compute the expectation values of the operators σi ⊗ σj on the state vector|ψ >= c|00 > +α|01 > +β|10 > +γ|11 > ,where the overall phase factor (up towhich |ψ > is defined) is chosen so that c is real (while α, β and γ are complexnumbers).

7. Consider the singlet state,

|ψ〉 = 1√2[| ↑〉1| ↓〉2 − | ↓〉1| ↑〉2], where | ↑〉=|0〉 =

(10

)and | ↓〉=|1〉 =

(01

)are two

orthogonal states.

(a) Show that the quantum mechanical expectation value,E(n,m) = 〈ψ|~σ.~n⊗ ~σ.~m|ψ〉, is given by E(n,m) = −cosφn,m,where, φn,m is the angle between unit vectors m and n. [2]

(b) The CHSH-inequality is given by,|E(n,m)− E(n,m′)|+ |E(n′,m′) + E(n′,m)| ≤ 2.Find the angles where the inequality is maximally violated. Interpret the result.

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8. Given the Pauli spin matrices σx, σy and σz, find out the eigenvalues and the eigen-vectors of the matrix e

−→σ .n, where n is the unit vector on the Bloch sphere, given byn = (sin θ cosφ, sin θ sinφ, cos θ) and −→σ = i σx + j σy + k σz.

(a) Express the eigenvectors in terms of the computational basis {|0〉, |1〉}

(b) Show explicitly using the different rotation matrices expressed in terms of theEuler angles, how you will arrive at these eigenstates from the state |0〉 in theBloch sphere.

9. Prove that (a.σ)(b.σ) = (a.b)I + iσ.(a× b).

10. Compute the expectation values of the operators σi ⊗ σj on the state vector |ψ〉 =c|00〉 + α|01〉 + β|10〉 + γ|11〉 ,where the overall phase factor (up to which |ψ〉 isdefined) is chosen so that c is real (while α, β and γ are complex numbers)

11. (a) Write down the truth table for the Toffoli Gate (a gate for reversible classicalcomputation, also known as the controlled-controlled-NOT gate) which actson three bits as follows: It flips the state of the third bit iff both the first andthe second bits are in the state 1.

(b) Suppose one represents the 8 possible states of the three bits by column vectorsas follows:

000 ≡

10000000

; 001 ≡

01000000

; 010 ≡

00100000

; 011 ≡

00010000

;

100 ≡

00001000

; 101 ≡

00000100

; 110 ≡

00000010

; 111 ≡

00000001

Construct the 8×8 matrix operatorGwhich corresponds to the action of ToffoliGate on the above vectors.

(c) Show thatG†G = I , whereG† is the adjoint operator ofG (transpose of complexconjugate ofG) and I is 8×8 the identity operator (this proves that G is a unitaryoperator, and thereby a valid quantum gate; use a clever method of identifyingknown matrices inside the gate before multiplying).

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References

[1] M.A. Nielsen & I.L. Chuang, Quantum Computation and Quantum Information, Cam-bridge University Press, ISBN 9781107002173

[2] John Preskill, Lecture Notes on Quantum Computation

[3] David Deutsch, Lectures on Quantum Computation

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