# Introducing QCSimulator: A 5-qubit quantum computing simulator in R

## Introduction

My 5-qubit Quantum Computing Simulator,QCSimulator, is finally ready, and here it is! I have been able to successfully complete this simulator by working through a fair amount of material. To a large extent, the simulator is easy, if one understands how to solve the quantum circuit. However the theory behind quantum computing itself, is quite formidable, and I hope to scale this mountain over a period of time.

QCSimulator is now on CRAN!!!

The code for the QCSimulator package is also available at Github QCSimulator. This post has also been published at Rpubs as QCSimulator and can be downloaded as a PDF document at QCSimulator.pdf

Disclaimer: This article represents the author’s viewpoint only and doesn’t necessarily represent IBM’s positions, strategies or opinions

install.packages("QCSimulator")
library(QCSimulator)
library(ggplot2)

### 1. Initialize the environment and set global variables

Here I initialize the environment with global variables and then display a few of them.

rm(list=ls())
#Call the init function to initialize the environment and create global variables
init()

# Display some of global variables in environment
ls()
##  [1] "I16"     "I2"      "I4"      "I8"      "q0_"     "q00_"    "q000_"
##  [8] "q0000_"  "q00000_" "q00001_" "q0001_"  "q00010_" "q00011_" "q001_"
## [15] "q0010_"  "q00100_" "q00101_" "q0011_"  "q00110_" "q00111_" "q01_"
## [22] "q010_"   "q0100_"  "q01000_" "q01001_" "q0101_"  "q01010_" "q01011_"
## [29] "q011_"   "q0110_"  "q01100_" "q01101_" "q0111_"  "q01111_" "q1_"
## [36] "q10_"    "q100_"   "q1000_"  "q10000_" "q10001_" "q1001_"  "q10010_"
## [43] "q10011_" "q101_"   "q1010_"  "q10100_" "q10101_" "q1011_"  "q10110_"
## [50] "q10111_" "q11_"    "q110_"   "q1100_"  "q11000_" "q11001_" "q1101_"
## [57] "q11010_" "q11011_" "q111_"   "q1110_"  "q11100_" "q11101_" "q1111_"
## [64] "q11110_" "q11111_"
#1. 2 x 2 Identity matrix
I2
##      [,1] [,2]
## [1,]    1    0
## [2,]    0    1
#2. 8 x 8 Identity matrix
I8
##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
## [1,]    1    0    0    0    0    0    0    0
## [2,]    0    1    0    0    0    0    0    0
## [3,]    0    0    1    0    0    0    0    0
## [4,]    0    0    0    1    0    0    0    0
## [5,]    0    0    0    0    1    0    0    0
## [6,]    0    0    0    0    0    1    0    0
## [7,]    0    0    0    0    0    0    1    0
## [8,]    0    0    0    0    0    0    0    1
#3. Qubit |00>
q00_
##      [,1]
## [1,]    1
## [2,]    0
## [3,]    0
## [4,]    0
#4. Qubit |010>
q010_
##      [,1]
## [1,]    0
## [2,]    0
## [3,]    1
## [4,]    0
## [5,]    0
## [6,]    0
## [7,]    0
## [8,]    0
#5. Qubit |0100>
q0100_
##       [,1]
##  [1,]    0
##  [2,]    0
##  [3,]    0
##  [4,]    0
##  [5,]    1
##  [6,]    0
##  [7,]    0
##  [8,]    0
##  [9,]    0
## [10,]    0
## [11,]    0
## [12,]    0
## [13,]    0
## [14,]    0
## [15,]    0
## [16,]    0
#6. Qubit 10010
q10010_
##       [,1]
##  [1,]    0
##  [2,]    0
##  [3,]    0
##  [4,]    0
##  [5,]    0
##  [6,]    0
##  [7,]    0
##  [8,]    0
##  [9,]    0
## [10,]    0
## [11,]    0
## [12,]    0
## [13,]    0
## [14,]    0
## [15,]    0
## [16,]    0
## [17,]    0
## [18,]    0
## [19,]    1
## [20,]    0
## [21,]    0
## [22,]    0
## [23,]    0
## [24,]    0
## [25,]    0
## [26,]    0
## [27,]    0
## [28,]    0
## [29,]    0
## [30,]    0
## [31,]    0
## [32,]    0

The QCSimulator implements the following gates

1. Pauli X,Y,Z, S,S’, T, T’ gates
3. 2,3,4,5 qubit CNOT gates e.g CNOT2_01,CNOT3_20,CNOT4_13 etc
4. Toffoli State,SWAPQ0Q1

### 2. To display the unitary matrix of gates

To check the unitary matrix of gates, we need to pass the appropriate identity matrix as an argument. Hence below the qubit gates require a 2 x 2 unitary matrix and the 2 & 3 qubit CNOT gates require a 4 x 4 and 8 x 8 identity matrix respectively

PauliX(I2)
##      [,1] [,2]
## [1,]    0    1
## [2,]    1    0
##           [,1]       [,2]
## [1,] 0.7071068  0.7071068
## [2,] 0.7071068 -0.7071068
S1Gate(I2)
##      [,1] [,2]
## [1,] 1+0i 0+0i
## [2,] 0+0i 0-1i
CNOT2_10(I4)
##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    0    0    0    1
## [3,]    0    0    1    0
## [4,]    0    1    0    0
CNOT3_20(I8)
##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
## [1,]    1    0    0    0    0    0    0    0
## [2,]    0    0    0    0    0    1    0    0
## [3,]    0    0    1    0    0    0    0    0
## [4,]    0    0    0    0    0    0    0    1
## [5,]    0    0    0    0    1    0    0    0
## [6,]    0    1    0    0    0    0    0    0
## [7,]    0    0    0    0    0    0    1    0
## [8,]    0    0    0    1    0    0    0    0

### 3. Compute the inner product of vectors

For example of phi = 1/2|0> + sqrt(3)/2|1> and si= 1/sqrt(2)(10> + |1>) then the inner product is the dot product of the vectors

phi = matrix(c(1/2,sqrt(3)/2),nrow=2,ncol=1)
si = matrix(c(1/sqrt(2),1/sqrt(2)),nrow=2,ncol=1)
angle= innerProduct(phi,si)
cat("Angle between vectors is:",angle)
## Angle between vectors is: 15

### 4. Compute the dagger function for a gate

The gate dagger computes and displays the transpose of the complex conjugate of the matrix

TGate(I2)
##      [,1]                 [,2]
## [1,] 1+0i 0.0000000+0.0000000i
## [2,] 0+0i 0.7071068+0.7071068i
GateDagger(TGate(I2))
##      [,1]                 [,2]
## [1,] 1+0i 0.0000000+0.0000000i
## [2,] 0+0i 0.7071068-0.7071068i

### 5. Invoking gates in series

The Quantum gates can be chained by passing each preceding Quantum gate as the argument. The final gate in the chain will have the qubit or the identity matrix passed to it.

# Call in reverse order
# Superposition states
# |+> state
##           [,1]
## [1,] 0.7071068
## [2,] 0.7071068
# |-> ==> H x Z
##            [,1]
## [1,]  0.7071068
## [2,] -0.7071068
# (+i) Y ==> H x  S
##                      [,1]
## [1,] 0.7071068+0.0000000i
## [2,] 0.0000000+0.7071068i
# (-i)Y ==> H x S1
##                      [,1]
## [1,] 0.7071068+0.0000000i
## [2,] 0.0000000-0.7071068i

### 6. More gates in series

#### TGate of depth 2

The Quantum circuit for a TGate of Depth 2 is

Implementing the quantum gates in series in reverse order we have

# Invoking this in reverse order we get
result=measurement(a)

plotMeasurement(result)

### 7. Invoking gates in parallel

To obtain the results of gates in parallel we have to take the Tensor Product Note:In the TensorProduct invocation the Identity matrix is passed as an argument to get the unitary matrix of the gate. Q0 – Hadamard-Measurement Q1 – Identity- Measurement

#
b = DotProduct(a,q00_)
plotMeasurement(measurement(b))

b = DotProduct(a,q00_)
plotMeasurement(measurement(b))

### 8. Measurement

The measurement of a Quantum circuit can be obtained using the measurement function. Consider the following Quantum circuit
Q0 – H-T-H-T-S-H-T-H-T-H-T-H-S-Measurement where H – Hadamard gate, T – T Gate and S- S Gate

measurement(a)
##          0        1
## v 0.890165 0.109835

### 9. Plot measurement

Using the same example as above Q0 – H-T-H-T-S-H-T-H-T-H-T-H-S-Measurement where H – Hadamard gate, T – T Gate and S- S Gate we can plot the measurement

result = measurement(a)
plotMeasurement(result)

### 10. Evaluating a Quantum Circuit

The above procedures for evaluating a quantum gates in series and parallel can be used to evalute more complex quantum circuits where the quantum gates are in series and in parallel.

Here is an evaluation of one such circuit, the Bell ZQ state using the QCSimulator (from IBM’s Quantum Experience)

# 1st composite
# Output of CNOT
b = CNOT2_01(a)
# 2nd series
#3rd composite
d= TensorProd(I2,c)
# Output of 2nd composite
e = DotProduct(b,d)
#Action of quantum circuit on |00>
f = DotProduct(e,q00_)
result= measurement(f)
plotMeasurement(result)

### 11. Toffoli State

This circuit for this comes from IBM’s Quantum Experience. This circuit is available in the package. This is how the state was constructed. This circuit is shown below

The implementation of the above circuit in QCSimulator is as below

# Computation of the Toffoli State
H=1/sqrt(2) * matrix(c(1,1,1,-1),nrow=2,ncol=2)
I=matrix(c(1,0,0,1),nrow=2,ncol=2)

# 1st composite
# H x H x H
a = TensorProd(TensorProd(H,H),H)
# 1st CNOT
a1= CNOT3_12(a)

# 2nd composite
# I x I x T1Gate
b = TensorProd(TensorProd(I,I),T1Gate(I))
b1 = DotProduct(b,a1)
c = CNOT3_02(b1)

# 3rd composite
# I x I x TGate
d = TensorProd(TensorProd(I,I),TGate(I))
d1 = DotProduct(d,c)
e = CNOT3_12(d1)

# 4th composite
# I x I x T1Gate
f = TensorProd(TensorProd(I,I),T1Gate(I))
f1 = DotProduct(f,e)
g = CNOT3_02(f1)

#5th composite
# I x T x T
h = TensorProd(TensorProd(I,TGate(I)),TGate(I))
h1 = DotProduct(h,g)
i = CNOT3_12(h1)

#6th composite
# I x H x H
j1 = DotProduct(j,i)
k = CNOT3_12(j1)

# 7th composite
# I x H x H
l1 = DotProduct(l,k)
m = CNOT3_12(l1)
n = CNOT3_02(m)

#8th composite
# T x H x T1
o1 = DotProduct(o,n)
p = CNOT3_02(o1)
result = measurement(p)
plotMeasurement(result)

### 12. GHZ YYX measurement

Here is another Quantum circuit, namely the entangled GHZ YYX state. This is

and is implemented in QCSimulator as

# Composite 1
b= CNOT3_12(a)
c= CNOT3_02(b)
# Composite 2
e= DotProduct(d,c)
#Composite 3
g= DotProduct(f,e)
#Composite 4
j = DotProduct(i,g)
result=measurement(j)
plotMeasurement(result)

## Conclusion

The 5 qubit Quantum Computing Simulator is now fully functional. I hope to add more gates and functionality in the months to come.

Feel free to install the package from Github and give it a try.

Disclaimer: This article represents the author’s viewpoint only and doesn’t necessarily represent IBM’s positions, strategies or opinions

References

My other posts on Quantum Computing

You may also like
For more posts on other topics like Cloud Computing, IBM Bluemix, Distributed Computing, OpenCV, R, cricket please check my Index of posts

# Taking a closer look at Quantum gates and their operations

This post is a continuation of my earlier post ‘Exploring Quantum gate operations with QCSimulator’. Here I take a closer look at more quantum gates and their operations, besides implementing these new gates in my Quantum Computing simulator, the  QCSimulator in R.

Disclaimer: This article represents the author’s viewpoint only and doesn’t necessarily represent IBM’s positions, strategies or opinions

In  quantum circuits, gates  are unitary matrices which operate on 1,2 or 3 qubit systems which are represented as below

1 qubit
|0> = $\begin{pmatrix}1\\0\end{pmatrix}$ and |1> = $\begin{pmatrix}0\\1\end{pmatrix}$

2 qubits
|0> $\otimes$ |0> = $\begin{pmatrix}1\\ 0\\ 0\\0\end{pmatrix}$
|0> $\otimes$ |1> = $\begin{pmatrix}0\\ 1\\ 0\\0\end{pmatrix}$
|1> $\otimes$ |o> = $\begin{pmatrix}0\\ 0\\ 1\\0\end{pmatrix}$
|1> $\otimes$ |1> = $\begin{pmatrix}0\\ 0\\ 0\\1\end{pmatrix}$

3 qubits
|0> $\otimes$ |0> $\otimes$ |0> = $\begin{pmatrix}1\\ 0\\0\\ 0\\ 0\\0\\ 0\\0\end{pmatrix}$
|0> $\otimes$ |0> $\otimes$ |1> = $\begin{pmatrix}0\\ 1\\0\\ 0\\ 0\\0\\ 0\\0\end{pmatrix}$
|0> $\otimes$ |1> $\otimes$ |0> = $\begin{pmatrix}0\\ 0\\1\\ 0\\ 0\\0\\ 0\\0\end{pmatrix}$

|1> $\otimes$ |1> $\otimes$ |1> = $\begin{pmatrix}0\\ 0\\0\\ 0\\ 0\\0\\ 0\\1\end{pmatrix}$
Hence single qubit is represented as 2 x 1 matrix, 2 qubit as 4 x 1 matrix and 3 qubit as 8 x 1 matrix

1) Composing Quantum gates in series
When quantum gates are connected in a series. The overall effect of the these quantum gates in series is obtained my taking the dot product of the unitary gates in reverse. For e.g.

In the following picture the effect of the quantum gates A,B,C is the dot product of the gates taken reverse order
result = C . B . A

This overall action of the 3 quantum gates can be represented by a single ‘transfer’ matrix which is the dot product of the gates

If we had a Pauli X followed by a Hadamard gate the combined effect of these gates on the inputs can be deduced by constructing a truth table

 Input Pauli X – Output A’ Hadamard – Output B |0> |1> 1/√2(|0>  – |1>) |1> |0> 1/√2(|0>  + |1>)

Or

|0> -> 1/√2(|0>  – |1>)
|1> -> 1/√2(|0>  + |1>)
which is
$\begin{pmatrix}1\\0\end{pmatrix}$  ->1/√2 $\begin{pmatrix}1\\0\end{pmatrix}$$\begin{pmatrix}0\\1\end{pmatrix}$ = 1/√2  $\begin{pmatrix}1\\-1\end{pmatrix}$
$\begin{pmatrix}0\\1\end{pmatrix}$  ->1/√2 $\begin{pmatrix}1\\0\end{pmatrix}$ + $\begin{pmatrix}0\\1\end{pmatrix}$ = 1/√2  $\begin{pmatrix}1\\1\end{pmatrix}$
Therefore the ‘transfer’ matrix can be written as
T = 1/√2 $\begin{pmatrix}1 & 1\\ -1 & 1\end{pmatrix}$

2)Quantum gates in parallel
When quantum gates are in parallel then the composite effect of the gates can be obtained by taking the tensor product of the quantum gates.

If we consider the combined action of a Pauli X gate and a Hadamard gate in parallel

 A B A’ B’ |0> |0> |1> 1/√2(|0>  + |1>) |0> |1> |1> 1/√2(|0>  – |1>) |1> |0> |0> 1/√2(|0>  + |1>) |1> |1> |0> 1/√2(|0>  – |1>)

Or

|00> => |1> $\otimes$ 1/√2(|0>  + |1>) = 1/√2 (|10> + |11>)
|01> => |1> $\otimes$ 1/√2(|0>  – |1>) = 1/√2 (|10> – |11>)
|10> => |0> $\otimes$ 1/√2(|0>  + |1>) = 1/√2 (|00> + |01>)
|11> => |0> $\otimes$ 1/√2(|0>  – |1>) = 1/√2 (|10> – |11>)

|00> = $\begin{pmatrix}1\\ 0\\ 0\\0\end{pmatrix}$ =>1/√2$\begin{pmatrix} 0\\ 0\\ 1\\ 1\end{pmatrix}$
|01> = $\begin{pmatrix}0\\ 1\\ 0\\0\end{pmatrix}$ =>1/√2$\begin{pmatrix} 0\\ 0\\ 1\\ -1\end{pmatrix}$
|10> = $\begin{pmatrix}0\\ 0\\ 1\\0\end{pmatrix}$ =>1/√2$\begin{pmatrix} 1\\ 0\\ 1\\ -1\end{pmatrix}$
|11> = $\begin{pmatrix}0\\ 0\\ 0\\1\end{pmatrix}$ =>1/√2$\begin{pmatrix} 1\\ 0\\ -1\\ -1\end{pmatrix}$

Here are more Quantum gates
a) Rotation gate
$U = \begin{pmatrix}cos\theta & -sin\theta\\ sin\theta & cos\theta\end{pmatrix}$

b) Toffoli gate
The Toffoli gate flips the 3rd qubit if the 1st and 2nd qubit are |1>

 Toffoli gate Input Output |000> |000> |001> |001> |010> |010> |011> |011> |100> |100> |101> |101> |110> |111> |111> |110>

c) Fredkin gate
The Fredkin gate swaps the 2nd and 3rd qubits if the 1st qubit is |1>

 Fredkin gate Input Output |000> |000> |001> |001> |010> |010> |011> |011> |100> |100> |101> |110> |110> |101> |111> |111>

d) Controlled U gate
A controlled U gate can be represented as
controlled U = $\begin{pmatrix}1 & 0 & 0 & 0\\ 0 &1 &0 & 0\\ 0 &0 &u11 &u12 \\ 0 & 0 &u21 &u22 \end{pmatrix}$   – (A)
where U = $\begin{pmatrix}u11 &u12 \\ u21 & u22\end{pmatrix}$

e) Controlled Pauli gates
Controlled Pauli gates are created based on the following identities. The CNOT gate is a controlled Pauli X gate where controlled U is a Pauli X gate and matches the CNOT unitary matrix. Pauli gates can be constructed using

a) H x X x H = Z    &
H x H = I

b) S x X x S1
S x S1 = I

the controlled Pauli X, Y , Z are contructed using the CNOT for the controlled X in the above identities
In general a controlled Pauli gate can be created as below

f) CPauliX
Here C is the 2 x2  Identity matrix. Simulating this in my QCSimulator
CPauliX I=matrix(c(1,0,0,1),nrow=2,ncol=2)
# Compute 1st composite
a = TensorProd(I,I)
b = CNOT2_01(a)
# Compute 1st composite
c = TensorProd(I,I)
#Take dot product
d = DotProduct(c,b)
#Take dot product with qubit
e = DotProduct(d,q)
e
}

Implementing the above with I, S, H gives Pauli X, Y and Z as seen below

library(QCSimulator)
I4=matrix(c(1,0,0,0,
0,1,0,0,
0,0,1,0,
0,0,0,1),nrow=4,ncol=4)

#Controlled Pauli X
CPauliX(I4)
##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    0    1    0    0
## [3,]    0    0    0    1
## [4,]    0    0    1    0
#Controlled Pauli Y
CPauliY(I4)
##      [,1] [,2] [,3] [,4]
## [1,] 1+0i 0+0i 0+0i 0+0i
## [2,] 0+0i 1+0i 0+0i 0+0i
## [3,] 0+0i 0+0i 0+0i 0-1i
## [4,] 0+0i 0+0i 0+1i 0+0i
#Controlled Pauli Z
CPauliZ(I4)
##      [,1] [,2] [,3] [,4]
## [1,]    1    0    0    0
## [2,]    0    1    0    0
## [3,]    0    0    1    0
## [4,]    0    0    0   -1

g) CSWAP gate

q00=matrix(c(1,0,0,0),nrow=4,ncol=1)
q01=matrix(c(0,1,0,0),nrow=4,ncol=1)
q10=matrix(c(0,0,1,0),nrow=4,ncol=1)
q11=matrix(c(0,0,0,1),nrow=4,ncol=1)
CSWAP(q00)
##      [,1]
## [1,]    1
## [2,]    0
## [3,]    0
## [4,]    0
#Swap qubits
CSWAP(q01)
##      [,1]
## [1,]    0
## [2,]    0
## [3,]    1
## [4,]    0
#Swap qubits
CSWAP(q10)
##      [,1]
## [1,]    0
## [2,]    1
## [3,]    0
## [4,]    0
CSWAP(q11)
##      [,1]
## [1,]    0
## [2,]    0
## [3,]    0
## [4,]    1

h) Toffoli state
The Toffoli state creates a 3 qubit entangled state 1/2(|000> + |001> + |100> + |111>)

Simulating the Toffoli state in IBM Quantum Experience we get

h) Implementation of Toffoli state in QCSimulator

#ToffoliState
# Computation of the Toffoli State
H=1/sqrt(2) * matrix(c(1,1,1,-1),nrow=2,ncol=2)
I=matrix(c(1,0,0,1),nrow=2,ncol=2)

# 1st composite
# H x H x H
a = TensorProd(TensorProd(H,H),H)
# 1st CNOT
a1= CNOT3_12(a)

# 2nd composite
# I x I x T1Gate
b = TensorProd(TensorProd(I,I),T1Gate(I))
b1 = DotProduct(b,a1)
c = CNOT3_02(b1)

# 3rd composite
# I x I x TGate
d = TensorProd(TensorProd(I,I),TGate(I))
d1 = DotProduct(d,c)
e = CNOT3_12(d1)

# 4th composite
# I x I x T1Gate
f = TensorProd(TensorProd(I,I),T1Gate(I))
f1 = DotProduct(f,e)
g = CNOT3_02(f1)

#5th composite
# I x T x T
h = TensorProd(TensorProd(I,TGate(I)),TGate(I))
h1 = DotProduct(h,g)
i = CNOT3_12(h1)

#6th composite
# I x H x H
j1 = DotProduct(j,i)
k = CNOT3_12(j1)

# 7th composite
# I x H x H
l1 = DotProduct(l,k)
m = CNOT3_12(l1)
n = CNOT3_02(m)

#8th composite
# T x H x T1
o1 = DotProduct(o,n)
p = CNOT3_02(o1)
result = measurement(p)
plotMeasurement(result)

The measurement is identical to the that of IBM Quantum Experience

Conclusion:  This post looked at more Quantum gates. I have implemented all the gates in my QCSimulator which I hope to release in a couple of months.

Disclaimer: This article represents the author’s viewpoint only and doesn’t necessarily represent IBM’s positions, strategies or opinions

References
1. http://www1.gantep.edu.tr/~koc/qc/chapter4.pdf
3. https://quantumexperience.ng.bluemix.net/

Take a look at my other posts at
1. Index of posts

# Exploring Quantum Gate operations with QCSimulator

Introduction: Ever since I was initiated into Quantum Computing, through IBM’s Quantum Experience I have been hooked. My initial encounter with domain made me very excited and all wound up. The reason behind this, I think, is because there is an air of mystery around ‘Quantum’ anything.  After my early rush with the Quantum Experience, I have deliberately slowed down to digest the heady stuff.

This post also includes my early prototype of a Quantum Computing Simulator( QCSimulator) which I am creating in R. I hope to have a decent Quantum Computing simulator available, in the next couple of months. The ideas for this simulator are based on IBM’s Quantum Experience and the lectures on Quantum Mechanics and Quantum Computation by Prof Umesh Vazirani from University of California at Berkeley at edX. This calls to this simulator have been included in R Markdown file and has been published at RPubs as Quantum Computing Simulator

Disclaimer: This article represents the author’s viewpoint only and doesn’t necessarily represent IBM’s positions, strategies or opinions

In this post I explore quantum gate operations

A) Quantum Gates
Quantum gates are represented as a n x n unitary matrix. In mathematics, a complex square matrix U is unitary if its conjugate transpose Uǂ is also its inverse – that is, if
U ǂU =U U ǂ=I

a) Clifford Gates
The following gates are known as Clifford Gates and are represented as the unitary matrix below

1. Pauli X
$\begin{pmatrix}0&1\\1&0\end{pmatrix}$

2.Pauli Y
$\begin{pmatrix}0&-i\\i&0\end{pmatrix}$

3. Pauli Z
$\begin{pmatrix}1&0\\0&-1\end{pmatrix}$

1/√2 $\begin{pmatrix}1 & 1\\ 1 & -1\end{pmatrix}$

5. S Gate
$\begin{pmatrix}1 & 0\\ 0 & i\end{pmatrix}$

6. S1 Gate
$\begin{pmatrix}1 & 0\\ 0 & -i\end{pmatrix}$

7. CNOT
$\begin{pmatrix}1 & 0 & 0 &0 \\ 0 & 1 & 0 &0 \\ 0& 0 & 0&1 \\ 0 & 0 & 1 & 0\end{pmatrix}$

b) Non-Clifford Gate
The following are the non-Clifford gates
1. Toffoli Gate
T = $\begin{pmatrix}1 & 0\\ 0 & e^{^{i\prod /4}}\end{pmatrix}$

2. Toffoli1 Gate
T1 =$\begin{pmatrix}1 & 0\\ 0 & e^{^{-i\prod /4}}\end{pmatrix}$

B) Evolution of a 1 qubit Quantum System
The diagram below shows how a 1 qubit system evolves on the application of Quantum Gates.

C) Evolution of a 2 Qubit  System
The following diagram depicts the evolution of a 2 qubit system. The 4 different maximally entangled states can be obtained by using a Hadamard and a CNOT gate to |00>, |01>, |10> & |11> resulting in the entangled states  Φ+, Ψ+, Φ, Ψrespectively

D) Verifying Unitary’ness
XXǂ = XǂX= I
TTǂ = TǂT=I
SSǂ = SǂS=I
The Uǂ  function in the simulator is
Uǂ = GateDagger(U)

E) A look at some Simulator functions
The unitary functions for the Clifford and non-Clifford gates have been implemented functions. The unitary functions can be chained together by invoking each successive Gate as argument to the function.

1. Creating the dagger function
TDagger = GateDagger(TGate)
TDagger x TGate

H
##           [,1]       [,2]
## [1,] 0.7071068  0.7071068
## [2,] 0.7071068 -0.7071068
HDagger = GateDagger(H)
HDagger
##           [,1]       [,2]
## [1,] 0.7071068  0.7071068
## [2,] 0.7071068 -0.7071068
HDagger %*% H
##      [,1] [,2]
## [1,]    1    0
## [2,]    0    1
T
##      [,1]                 [,2]
## [1,] 1+0i 0.0000000+0.0000000i
## [2,] 0+0i 0.7071068+0.7071068i
TDagger = GateDagger(T)
TDagger
##      [,1]                 [,2]
## [1,] 1+0i 0.0000000+0.0000000i
## [2,] 0+0i 0.7071068-0.7071068i
TDagger %*% T
##      [,1] [,2]
## [1,] 1+0i 0+0i
## [2,] 0+0i 1+0i

2. Angle between 2 vectors – Inner product
The angle between 2 vectors can be obtained by taking the inner product between the vectors

#1. a is the diagonal vector 1/2 |0> + 1/2 |1> and b = q0 = |0>
diagonal <-  matrix(c(1/sqrt(2),1/sqrt(2)),nrow=2,ncol=1)
q0=matrix(c(1,0),nrow=2,ncol=1)
innerProduct(diagonal,q0)
##      [,1]
## [1,]   45
#2. a = 1/2|0> + sqrt(3)/2|1> and  b= 1/sqrt(2) |0> + 1/sqrt(2) |1>
a = matrix(c(1/2,sqrt(3)/2),nrow=2,ncol=1)
b = matrix(c(1/sqrt(2),1/sqrt(2)),nrow=2,ncol=1)
innerProduct(a,b)
##      [,1]
## [1,]   15

3. Chaining Quantum Gates
For e.g.
H x q0
S x H x q0 == > SGate(Hadamard(q0))

Or
H x S x S x H x q0 == > Hadamard(SGate(SGate(Hadamard))))

# H x q0
##           [,1]
## [1,] 0.7071068
## [2,] 0.7071068
# S x H x q0
##                      [,1]
## [1,] 0.7071068+0.0000000i
## [2,] 0.0000000+0.7071068i
# H x S x S x H x q0
##      [,1]
## [1,] 0+0i
## [2,] 1+0i
# S x T x H x T x H x q0
##                      [,1]
## [1,] 0.8535534+0.3535534i
## [2,] 0.1464466+0.3535534i

4. Measurement
The output of Quantum Gate operations can be measured with
measurement(a)
measurement(q0)
measurement(a)

measurement(q0)
##   0 1
## v 1 0
##     0   1
## v 0.5 0.5
measurement(a)
##           0         1
## v 0.8535534 0.1464466

5. Plot the measurements

plotMeasurement(q1)

plotMeasurement(a)

6. Using the QCSimulator for one of the Bell tests
Here I compute the following measurement of  Bell state ZW  with the QCSimulator

When this is simulated on IBM’s Quantum Experience the result is

Below I simulate the same on my R based QCSimulator

# Compute the effect of the Composite H gate with the Identity matrix (I)
a=kronecker(H,I,"*")
a
##           [,1]      [,2]       [,3]       [,4]
## [1,] 0.7071068 0.0000000  0.7071068  0.0000000
## [2,] 0.0000000 0.7071068  0.0000000  0.7071068
## [3,] 0.7071068 0.0000000 -0.7071068  0.0000000
## [4,] 0.0000000 0.7071068  0.0000000 -0.7071068
# Compute the applcation of CNOT on this result
b = CNOT(a)
b
##           [,1]      [,2]       [,3]       [,4]
## [1,] 0.7071068 0.0000000  0.7071068  0.0000000
## [2,] 0.0000000 0.7071068  0.0000000  0.7071068
## [3,] 0.0000000 0.7071068  0.0000000 -0.7071068
## [4,] 0.7071068 0.0000000 -0.7071068  0.0000000
# Obtain the result of CNOT on q00
c = b %*% q00
c
##           [,1]
## [1,] 0.7071068
## [2,] 0.0000000
## [3,] 0.0000000
## [4,] 0.7071068
# Compute the effect of the composite HxTxHxS gates and the Identity matrix(I) for measurement
e=kronecker(I, d,"*")

# Applying  the composite gate on the output 'c'
f = e %*% c
# Measure the output
g <- measurement(f)
g
##          00        01        10        11
## v 0.4267767 0.0732233 0.0732233 0.4267767
#Plot the measurement
plotMeasurement(g)

aa
which is exactly the result obtained with IBM’s Quantum Experience!

Conclusion : In  this post I dwell on 1 and 2-qubit quantum gates and explore their operation. I have started to construct a  R based Quantum Computing Simulator. I hope to have a reasonable simulator in the next couple of months. Let’s see.

Watch this space!

Disclaimer: This article represents the author’s viewpoint only and doesn’t necessarily represent IBM’s positions, strategies or opinions

Index of posts

# A primer on Qubits, Quantum gates and Quantum Operations

Introduction: After my initial encounter with IBM’s Quantum Experience, and my playing around with qubits, quantum gates and trying out the Bell experiment I can now say that I am fairly hooked to quantum computing.

So, I decided that  before going any  further,  that I  needed to  spend a little time more, getting to know more about the basics of Dirac’s bra-ket notation, qubits and ensuring that my knowledge is properly “chunked” ( See Learning to Learn: Powerful mental tools to master tough subjects, a really good course!!)

So, I started to look around for material on Quantum Computing, and finally landed on the classic course “Quantum Mechanics and Quantum Computing”, from University of California, Berkeley by Prof Umesh V Vazirani at edX. I have started to audit the course (listen in, without doing the assignments). The Prof is unbelievably good, and makes the topic both interesting and absorbing. This post is based on my notes of week 1 & 2 lectures. I have tried to articulate as best as I can, what I have understood of the lectures, though I would strongly recommend  you to, at least audit the archived course. By the way, I also had to refresh my knowledge of basic trigonometry and linear algebra. My knowledge of the basics of matrix manipulation, vectors etc. were buried deep within the sands of time. Luckily for me, they were reasonably intact.

A) Quantum states
A hydrogen atom has 1 electron in orbit. The electron can be either in the idle state or in the excited state. We can represent the idle state with |0> and the excited state with |1>, which is Dirac’s ‘ket’ notation.

This electron will be in a superposition state which is represented by
Ψ = α |0> + β |1> where α & β are complex numbers and obey | α|2 + | β|2 = 1
For e.g. we could have the superposition state
$\varphi= \frac{1}{2} + \frac{1i}{2}|0> + \frac{1}{\sqrt{2}}|1>$
It can be seen that | α|2 = $\frac{1}{2}$ and | β|2 = = $\frac{1}{2}$
For a complex number α = a+ bi  == > | α| = $\sqrt{a^{2} + b^{2}}$

B) Measurement
However, when the electron or the qubit is measured,  the state of superposition collapses to either |0> or |1> with the following probability’s

The resulting state is
|0> with probability | α|2
|1> with probability | β |2

And | α|2 + | β|2 = 1 because the sum of the probabilities must add up to 1  i.e. $\sum p_{i} = 1$

C) Geometric interpretation
Let us consider a qubit that is in a superposition state
Ψ = α |0> + β |1> where α & β are complex numbers and obey | α|2 + | β|2 = 1

We can write this as a vector  $\begin{pmatrix} \alpha\\ \beta \end{pmatrix}$ representing the state of the electron

It can be seen that if
$\alpha=1$ and $\beta=0$ then |0> = $\begin{pmatrix}1\\ 0\end{pmatrix}$
and
$\alpha=0$ and $\beta=1$  then |1> = $\begin{pmatrix}0\\ 1\end{pmatrix}$

Measuring a qubit in standard basis
If we represent the qubit geometrically then the superposition can be represented as a vector which makes an angle $theta$ with |0>
$\varphi = cos\theta|0> + sin\theta|1>$

Measuring this  $\varphi$ is the projection of on one of the standard basis |0> or |1>
The output is is |0> with probability $cos^{2}\theta$ and
|1> with probability $sin^{2}\theta$

D) Measuring a qubit in any basis

The qubit can be measured in any arbitrary basis. For e.g.  if
Ψ = α |0> + β |1>   and we have the diagonal basis
|u>| and |u’> as shown and Ψ makes an angle $\theta$ with |u> then we can write

|u> with probability cos2Θ
And |u’> with probability sin2Θ

E) K Qubit system
Let us assume that we have a Quantum System with k qubits
|0>, |1>, |2>… |k-1>

The qubit will be in a superimposed state
Ψ = α0 |0>+ α1 |1> + α2|2> + … + αk-1 |k-1>
Where αj is a complex vector with the property ∑ αj = 1

Here Ψ is a unit vector in a K dimensional complex vector space, known as Hilbert Space
For e.g. a 3 qubit quantum system
$\varphi = \frac{1}{2}+\frac{i}{2}|0> - \frac{1}{2} |1> + \frac{1}{2}|2>$
Then P(0) = ½ P(1) = ¼ and P(2) = ¼   ∑Pj = 1

We could also write
$\varphi = \begin{pmatrix} \alpha_{0}\\ \alpha_{1}\\ .. \\\alpha_{k-1}\\\end{pmatrix}$
Or
Ψ = α0 |0>+ α1 |1> + α2|2> + … + αk-1 |k-1>

F) Measuring the angle between 2 complex vectors
To measure the angle between 2 complex vectors
$\varphi = \begin{pmatrix} \alpha_{0}\\ \alpha_{1}\\\alpha_{2}\\\end{pmatrix}$
And
$\phi = \begin{pmatrix} \beta_{0}\\ \beta_{1}\\\beta_{2}\\\end{pmatrix}$
we need to take the inner product of the complex conjugate of the 1st vector and the 2nd
cosΘ = inner product  => $\bar{\varphi} . \phi$
$cos\theta = \bar\alpha_{0}\beta_{0} + \bar\alpha_{1}\beta_{1} + \bar\alpha_{2}\beta_{2}$

For e.g.
If $\varphi = \frac{1}{2} |0> + \frac{\sqrt 3}{2}|1>$ which makes 60 degrees with the |0> basis
And
$\phi = \frac{1}{\sqrt 2} |0> + \frac{1}{\sqrt 2}|1>$ which is the diagonal |+> basis
Then the angle between these 2 vectors are obtained by taking the inner product
cosΘ =  ½ * 1/√2 + √3/2 * ½

G) Measuring Ψ in |+> or |-> basis
For e.g. if
$\varphi = \frac{1}{2} |0> + \frac{\sqrt 3}{2}|1>$   – (A)

Then we can specify/measure Ψ in |+> or |-> basis as follow
|+> = 1/√2(|0> + |1>) and |-> = 1/√2(|0> – |1>)
Or |0> = 1/√2(|+> + |->)   and |1> = 1/√2(|+> – |->)

We can write
Ψ= α|+> + β|->

Substituting for |0> and |1>  in (A) we get
$\varphi = \frac{1+ \sqrt 3}{2 \sqrt 2} |+> \frac{1 - \sqrt 3 }{2 \sqrt 2} |->$

H) Quantum gates
I) Clifford gates
Pauli gates
a) Pauli X
The Pauli X gate does a bit flip
|0> ==>  X|0> ==> |1>
|1> ==>  X|1> ==> |0>
and is represented
$\begin{pmatrix}0&1\\1&0\end{pmatrix}$

b) Pauli Z
This gates does a phase flip and is represented as the 2 x 2 unitary matrix
$\begin{pmatrix}1&0\\0&-1\end{pmatrix}$

c) Pauli Y
The Pauli operator Y  does both  a bit and a phase flip. The Y operator is represented as
$\begin{pmatrix}0&-i\\i&0\end{pmatrix}$

K) Superposition gates
Superposition is the concept that adding quantum states together results in a new quantum state. There are 3 gates that perform superposition of qubits the H, S and S’ gate.
The H gate, also known as the Hadamard Gate when applied |0> state results in the qubit being half the time in  |0> and the other half in |1>
The H gate can be represented as
1/√2 $\begin{pmatrix}1 & 1\\ 1 & -1\end{pmatrix}$

b) S gate
The S gate can be represented as
$\begin{pmatrix}1 & 0\\ 0 & i\end{pmatrix}$

c) S’ gate
And the S’ gate is
$\begin{pmatrix}1 & 0\\ 0 & -i\end{pmatrix}$

L) Non-Clifford Gates
The quantum gates discussed in my earlier post Pauli X, Y, Z, H, S and S1 are members of a special group of gates known as the ‘Clifford group’.
The non-Clifford gates, discussed are the T and  Tǂ gates

These are given by
T = $\begin{pmatrix}1 & 0\\ 0 & e^{^{i\prod /4}}\end{pmatrix}$
Tǂ =$\begin{pmatrix}1 & 0\\ 0 & e^{^{-i\prod /4}}\end{pmatrix}$

M) 2 qubit system
A 2 qubit system

A 2 qubit system is a superposition of all possible 2 qubit states. A 2 qubit system and can be represented as

Ψ = α00 |00> + α01 |01> + α10 |10> + α11 |11>

Measuring the 2 qubit system, as earlier, results in the collapse of the superposition and the result is one of 4 qubit states. The probability of the measure state is the square of the amplitude | αij|2

N) Entanglement
A  2 qubit system in which we have
Ψ = α0 |0> + α1 |1> and Φ= β0 |0> + β1 |1> the superimposed state is obtained by taking the tensor product of the 2 qubits
$\chi = (\alpha_{0} |0> + \alpha_{1}|1>)\otimes (\beta_{0} |0> + \beta_{1} |1>)$
Where $\otimes$ is the tensor product

The state
Ψ = 1/√2|00> + 1/√2|11>
Is called an ‘entangled’ state because it cannot  be reduced to a product of 2 vectors

N) 2 qubit gates

A 2 qubit system is a superposition of all possible 2 qubit states. A 2 qubit system and can be represented as
Ψ = α00 |00> + α01 |01> + α10 |10> + α11 |11>

Measuring the 2 qubit system, as earlier, results in the collapse of the superposition and the result is one of 4 qubit states. The probability of the measure state is the square of the amplitude | αij|2
More specifically a 2 qubit system in which we have
Ψ = α0 |0> + α1 |1> and Φ= β0 |0> + β1 |1> the superimposed state is obtained by taking the tensor product of the 2 qubits
$\chi = (\alpha_{0} |0> + \alpha_{1}|1>)\otimes (\beta_{0} |0> + \beta_{1} |1>)$

Where $\otimes$ is the tensor product
The state
Ψ = 1/√2|00> + 1/√2|11>
Is called an ‘entangled’ state because it cannot  be reduced to a product of 2 vectors
A quantum gate is a 2 x 2 unitary matrix U such that
α0 |0> + α1 |1>    == > Quantum Gate == > β0|0> + β1 |1>

Unitary functions: In mathematics, a complex square matrix U is unitary if its conjugate transpose U* is also its inverse
If
U=$\begin{pmatrix}a & c \\b & d\end{pmatrix}$
And U* = $\begin{pmatrix}\bar a & \bar b \\ \bar c & \bar d\end{pmatrix}$
Then
UU* = I where I is the Identity matrix

2 qubit gates is  4 x 4 unitary matrix
For a 2 qubit that is in the superposition state
Ψ = α00 |00> + α01 |01> + α10 |10> + α11 |11>

A 2 qubit gate’s operation on Ψ is
$\begin{pmatrix}a & e & i & m \\ b & f & j & n\\ c & g & k & o\\ d & h & l & p\end{pmatrix}$ * $\begin{pmatrix}\alpha_{00}\\ \alpha_{01}\\ \alpha_{10}\\\alpha_{11}\end{pmatrix}$

One important 2 qubit gate is the CNOT gate which is shown  below

The CNOT gate is represented by the following unitary matrix
$\begin{pmatrix}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\end{pmatrix}$

If 2  2×2 qubit gates were applied to 2 qubits the composite gate would be Tensor product of the 2 matrices

u1 = $\begin{pmatrix}a & c\\ b & d\end{pmatrix}$
u2 = $\begin{pmatrix}e & g\\ f & h\end{pmatrix}$
then
U= $u1 \otimes u2$
U = $\begin{pmatrix}a\begin{pmatrix}e & g\\ f & h\end{pmatrix} & c\begin{pmatrix}a & c\\ b & d\end{pmatrix}\\ b \begin{pmatrix}a & c \\ b & d\end{pmatrix}& d\begin{pmatrix}a & c\\ b & d\end{pmatrix}\end{pmatrix}$
The above product is also known as the Kronecker product

O) Tensor product of 2 qubits
Ψ = α0 |0> + α1 |1> and Φ= β0 |0> + β1 |1>
\$latex \varphi \otimes \phi= α0 β0|0>|0> + α0 β1|0>|1> + α1 β0|1>|0> + α1 β1|1>|1>
= α0 β0|00> + α0 β1|01> + α1 β0|10> + α1 β1|11>

If a Z gate and a Hadamard gate H were applied on 2 qubits, it is interest to know what the resulting composite gate would be.

Z = $\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}$ and H = $\begin{pmatrix}\frac{1}{\sqrt 2} & \frac{1}{\sqrt 2} \\ \frac{1}{\sqrt 2} & \frac{-1}{\sqrt 2}\end{pmatrix}$
The composite gate is obtained by the tensor product of
$Z \otimes H$
Hence the result of the composite gate is
=$\begin{pmatrix}1\begin{pmatrix}\frac{1}{\sqrt 2} & \frac{1}{\sqrt 2}\\ \frac{1}{\sqrt 2}& \frac{-1}{\sqrt 2}\end{pmatrix} & 0\\ 0 & -1\begin{pmatrix}\frac{1}{\sqrt 2} & \frac{1}{\sqrt 2}\\ \frac{1}{\sqrt 2} &\frac{-1}{\sqrt 2} \end{pmatrix}\end{pmatrix}$
=$\begin{pmatrix}\begin{pmatrix}\frac{1}{\sqrt 2} & \frac{1}{\sqrt 2}\\ \frac{1}{\sqrt 2}& \frac{-1}{\sqrt 2}\end{pmatrix} & 0\\ 0 & -\begin{pmatrix}\frac{-1}{\sqrt 2} & \frac{-1}{\sqrt 2}\\ \frac{-1}{\sqrt 2} &\frac{1}{\sqrt 2} \end{pmatrix}\end{pmatrix}$

which is the entangled state.

Conclusion: This post includes most of the required basics to get started on Quantum Computing. I will probably add another post detailing the operations of the Quantum Gates on qubits.

Note:
1.The equations and matrices have been created using LaTeX notation using the online LaTex equation creator
2. The figures have been created using the app Bamboo Paper, which I think is cooler than creating in Powerpoint