Grover’s quantum search algorithm, proposed by Lov Grover in 1996 [18] , is a quantum algorithm for searching specific items in unsorted databases. It uses quantum operations, such as the Hadamard transform and phase inversion, to amplify the target item’s amplitude. The algorithm runtime is proportional to the square root of the database size, faster than classical algorithms. However, it has limitations, such as a quadratic speedup and requires a quantum computer.

First, we define the initial state of the quantum system as:

$|{\psi }_0\rangle =\frac{1}{\sqrt{N}}\underset{x=0}{\overset{N-1}{{\displaystyle \sum }}}|x\rangle$(1)

where N is the size of the database and $|x\rangle$ represents the state of the quantum system corresponding to the item $x$ in the database.

Next, we define the quantum oracle that marks the target item as:

$U_{\omega }|x\rangle ={\left(-1\right)}^{f\left(x\right)}|x\rangle$(2)

where $f\left(x\right)=1$ if $x$ is the target item and $f\left(x\right)=0$ otherwise.

We then apply the Grover iteration, which consists of two steps:

Apply the quantum oracle:

$|{\psi }_1\rangle =U_{\omega }|{\psi }_0\rangle$(3)

Apply the Grover diffusion operator:

$|{\psi }_2\rangle =U_s|{\psi }_1\rangle$(4)

where the Grover diffusion operator is defined as:

$U_s=2|{\psi }_0\rangle \langle {\psi }_0|-I$(5)

and I is the identity operator.

The Grover iteration is repeated k times, where k is approximately equal to:

$k\approx \frac{\pi }{4}\sqrt{N}$(6)

After k iterations, the quantum system is measured, and the target item is found with high probability.

Shor’s quantum algorithm, proposed by Peter Shor in 1994 [19] , efficiently factored large composite numbers into prime factors using quantum modular exponentiation. The algorithm uses a random number a between 1 and N − 1, and a quantum circuit constructed using quantum Fourier transform and modular exponentiation. The runtime is $O\left({\left(\log N\right)}^3\right)$, faster than classical algorithms but requires a large number of qubits.

Here are some of the key equations involved in the algorithm:

The quantum Fourier transform is used to convert the period finding problem into a phase estimation problem. The quantum Fourier transform of a state $|x\rangle$ is defined as:

$ℱ|x\rangle =\frac{1}{\sqrt{N}}\underset{y=0}{\overset{N-1}{{\displaystyle \sum }}}{\text{e}}^{2\pi ixy/N}|y\rangle$(7)

where i is the imaginary unit and N is the size of the quantum register.

The quantum modular exponentiation is used to compute $a^x\mathrm{mod}N$ efficiently using a quantum computer. It is defined as:

$U_f|x\rangle |y\rangle =|x\rangle |y\oplus f\left(x\right)\rangle$(8)

where $f\left(x\right)=a^x\mathrm{mod}N$ and $\oplus$ represents bitwise addition modulo 2.

The phase estimation algorithm is used to estimate the phase of the eigenvalue of the unitary operator $U_f$. It is based on the application of the quantum Fourier transform to a superposition of eigenstates of $U_f$:

$|\psi \rangle =\frac{1}{\sqrt{r}}\underset{j=0}{\overset{r-1}{{\displaystyle \sum }}}|u^j\rangle$(9)

where u is the eigenvector of $U_f$ corresponding to the eigenvalue ${\text{e}}^{2\pi i\theta }$.

The period r of the function $f\left(x\right)=a^x\mathrm{mod}N$ can be obtained from the phase estimate using continued fractions:

$\theta \approx \frac{x}{q}=\left[0;a_1,a_2,\cdots ,a_k\right]$(10)

where q is a power of 2 and the coefficients $a_i$ are obtained from the convergents of the continued fraction expansion of x/q.

Once the period r is obtained, N can be factored using classical methods. The factors are given by:

$N=\text{gcd}\left(a^{r/2}\pm 1,N\right)\times \text{gcd}\left(a^{r/2}\mp 1,N\right)$(11)

where gcd denotes the greatest common divisor.

To prove that the set of quantum gates, denoted by $Q$, is effectively denumerable, we need to show that there exists a bijective mapping from the set of quantum gates to the set of natural numbers. In other words, we need to show that we can assign a unique natural number to each quantum gate. The set of quantum gates typically consist of a finite number of gates, such as H, X, Y, Z, CNOT, etc. Each gate can be uniquely identified by its name and the qubits it operates on. To establish a bijective mapping, we can assign a unique natural number to each possible combination of gate name and qubits.

For numbering the X gate:

$\beta \left(X\left(n\right)\right)=19\left(n-1\right)+0$(12)

For numbering the Y gate:

$\beta \left(Y\left(n\right)\right)=19\left(n-1\right)+1$(13)

For numbering the Z gate:

$\beta \left(Z\left(n\right)\right)=19\left(n-1\right)+2$(14)

For numbering the H gate:

$\beta \left(H\left(n\right)\right)=19\left(n-1\right)+3$(15)

For numbering the S gate:

$\beta \left(S\left(n\right)\right)=19\left(n-1\right)+4$(16)

For numbering the T gate:

$\beta \left(T\left(n\right)\right)=19\left(n-1\right)+5$(17)

For numbering the R4 gate:

$\beta \left(R4\left(n\right)\right)=19\left(n-1\right)+6$(18)

For numbering the Measure gate:

$\beta \left(Measure\left(n\right)\right)=19\left(n-1\right)+7$(19)

For numbering the $R_x\left(\theta \right)$ gate:

$\beta \left(R_x\left(n,\theta \right)\right)=19\text{Π}\left(n-1,\theta \right)+8$(20)

For numbering the $R_y\left(\theta \right)$ gate:

$\beta \left(R_y\left(n,\theta \right)\right)=19\text{Π}\left(n-1,\theta \right)+9$(21)

For numbering the $R_z\left(\theta \right)$ gate:

$\beta \left(R_z\left(n,\theta \right)\right)=19\text{Π}\left(n-1,\theta \right)+10$(22)

For numbering the CNOT gate:

$\beta \left(\text{CNOT}\left(m,n\right)\right)=19\text{Π}\left(m-1,n-1\right)+11$(23)

For numbering the SWAP gate:

$\beta \left(\text{SWAP}\left(m,n\right)\right)=19\text{Π}\left(m-1,n-1\right)+12$(24)

For numbering the Controlled-Y gate:

$\beta \left(\text{CY}\left(m,n\right)\right)=19\text{Π}\left(m-1,n-1\right)+13$(25)

For numbering the Controlled-Z gate:

$\beta \left(\text{CZ}\left(m,n\right)\right)=19\text{Π}\left(m-1,n-1\right)+14$(26)

For numbering the Controlled-S gate:

$\beta \left(\text{CS}\left(m,n\right)\right)=19\text{Π}\left(m-1,n-1\right)+15$(27)

For numbering the Controlled-T gate:

$\beta \left(\text{CT}\left(m,n\right)\right)=19\text{Π}\left(m-1,n-1\right)+16$(28)

For numbering the Toffoli gate:

$\beta \left(\text{Toffoli}\left(k,m,n\right)\right)=19\zeta \left(k,m,n\right)+17$(29)

where

$\text{Π}\left(m,n\right)=2^m\left(2n+1\right)-1$(30)

and

$\zeta \left(k,m,n\right)=\text{Π}\left(\text{Π}\left(k-1,m-1\right),n-1\right)$(31)

To find ${\beta }^{-1}\left(q\right)$, find u and r such that $q=19u+r$ with $0\le r<19$. This value of r indicates which kind of gate ${\beta }^{-1}$ is.

$\text{if}r=0,\text{then}{\beta }^{-1}\left(q\right)=X\left(u+1\right)$(32)

$\text{if}r=1,\text{then}{\beta }^{-1}\left(q\right)=Y\left(u+1\right)$(33)

$⋮$

$\text{if}r=17,\text{then}{\beta }^{-1}\left(q\right)=\text{Toffoli}\left(k,m,n\right)$(34)

where

$\left(k,m,n\right)={\zeta }^{-1}\left(u\right)$(35)

and

${\zeta }^{-1}\left(q\right)=\left({\text{Π}}_1\left({\text{Π}}_1\left(q\right)\right)+1,{\text{Π}}_2\left({\text{Π}}_1\left(q\right)\right)+1,{\text{Π}}_2\left(q\right)+1\right)$(36)

where ${\text{Π}}_1$ and ${\text{Π}}_2$ are computable functions defined as follows.

${\text{Π}}_1\left(q\right)={\left(q+1\right)}_1$(37)

and

${\text{Π}}_2\left(q\right)=\frac{1}{2}\left(\frac{q+1}{2^{{\text{Π}}_1\left(q\right)}}-1\right)$(38)

Note that at r = 18 is kept unused for future usages.

To prove that the set C of all quantum circuits is effectively denumerable, we need to show that there exists a bijective mapping from the set of quantum circuits to the set of natural numbers. In other words, we need to show that we can assign a unique natural number to each quantum circuit. A quantum circuit consists of a sequence of quantum gates. To establish a bijective mapping, we can use a standard encoding scheme, such as the Gödel numbering. If circuit C is a sequence of gates: $G_1,G_2,\cdots ,G_s$ then

$\gamma \left(C\right)=\tau \left(\beta \left(G_1\right),\beta \left(G_2\right),\cdots ,\beta \left(G_s\right)\right)$(39)

where

$\tau \left(v_1,\cdots ,v_k\right)=2^{v_1}+2^{v_1+v_2+1}+\cdots +2^{v_1+v_2+\cdots +k-1}-1$(40)

Calculating ${\tau }^{-1}\left(c\right)$, relies on the fact that every natural number can be expressed as a binary number. Given c, we can find a unique numbers k and $0\le u_1<u_2<\cdots <u_k$ such that

$c+1=2^{u_1}+2^{u_2}+\cdots +2^{u_k}$(41)

So, ${\tau }^{-1}\left(c\right)=\left(v_1,v_2,\cdots ,v_k\right)$ where $v_1=u_1$ and

$v_{i+1}=u_{i+1}-u_i-1$(42)

where $1\le i\le k$

For a circuit C, the number $\gamma \left(C\right)$ is called the code or the number of that circuit. We can say that $C_n$ = the circuit with code number n = ${\gamma }^{-1}\left(n\right)$. We say that $C_n$ is the nth circuit.

Note that if $m\ne n$, then $C_m$ is different from $C_n$ even that they may compute the same operation.

Figure 1 shows the circuit of EPR creation. This circuit consists of Hadamard and CNOT gates. Let us calculate the number or the code of this circuit.

It can be listed as:

= $2^2+2^{2+49+1}-1$

= $4+2^{52}-1$

= 4,503,599,627,370,499

Figure 2 shows the circuit of SWAP operation. This circuit consists of three CNOT gates. Let us calculate the number or the code of this circuit.

It can be listed as:

= 562,949,953,421,312

+ 1,208,925,819,614,629,174,706,176

+ 1,361,129,467,683,753,853,853,498,429,727,072,845,824

− 1

= 1,645,504,558,087,453,812,587,913,611,736,524,018,557,

890,457,442,949,424,440,410,111

Note that this code for numbering the SWAP gate is the equivalent to

$\beta \left(\text{SWAP}\left(1,2\right)\right)=19\text{Π}\left(0,1\right)+12=50$

based on $\text{Π}$

$\text{Π}\left(0,1\right)=2^0\left(2\times 1+1\right)-1=2$

and $\tau \left(50\right)=2^{50}-1=1125899906842623$

Figure 3 shows the teleportation circuit. This circuit consists of Hadamard, CNOT, Measure, Controlled-Z, and Controlled- X gates. Let us calculate the number or the code of this circuit.

It can be listed as:

= τ(22, 182, 49, 3, 7, 26, 185, 87)

= $2^{22}+2^{22+182+1}+2^{22+182+49+2}+2^{22+182+49+3+3}$

+ $2^{22+182+49+3+7+4}+2^{22+182+49+3+7+26+5}$

+ $2^{22+182+49+3+7+26+185+6}+2^{22+182+49+3+7+26+185+87+7}$

− 1

= $2^{22}+2^{205}+2^{255}+2^{259}+2^{267}+2^{294}+2^{480}+2^{568}-1$

= 4,194,304

+ 51,422,017,416,287,688,817,342,786,954,917,203,

280,710,495,801,049,370,729,644,032

+ 57,896,044,618,658,097,711,785,492,504,343,953,

926,634,992,332,820,282,019,728,792,003,956,564,819,

968

+ 926,336,713,898,529,563,388,567,880,069,503,262,

826,159,877,325,124,512,315,660,672,063,305,037,119,

488

+ 237,142,198,758,023,568,227,473,377,297,792,835,

283,496,928,595,231,875,152,809,132,048,206,089,502,

588,928

+ 31,828,687,130,226,345,097,944,463,881,396,533,

766,429,193,651,030,253,916,189,694,521,162,207,

808,802,136,034,115,584

+ 3,121,748,550,315,992,231,381,597,229,793,166,

305,748,598,142,664,971,150,859,156,959,625,371,

738,819,765,620,120,306,103,063,491,971,159,826,

931,121,406,622,895,447,975,679,288,285,306,290,

176

+ 966,134,380,754,314,586,173,837,972,732,996,836,

074,731,832,426,608,749,664,308,812,862,879,785,

572,390,106,134,048,441,645,480,644,490,615,904,

007,875,544,294,341,269,665,260,746,913,935,727,

168,366,770,187,174,245,203,705,856

− 1

= 966,134,380,754,314,586,173,837,975,854,745,386,

390,724,063,808,205,979,457,475,118,611,477,928,237,

361,257,025,034,088,639,205,159,924,866,743,949,573,

929,210,916,805,793,710,442,371,339,067,812,831,970,

988,587,388,382,478,335

Figure 4 shows the circuit of Quantum Fourier Transform (QFT). This circuit consists of Hadamard, Controlled-S, Controlled-T, and SWAP gates. Let us calculate the number or the code of this circuit.

It can be listed as:

= τ(3, 34, 73, 22, 224, 41, 88)

= $2^3+2^{3+34+1}+2^{3+34+73+2}+2^{3+34+73+22+3}$

+ $2^{3+34+73+22+224+4}+2^{3+34+73+22+224+41+5}$

+ $2^{3+34+73+22+224+41+88+6}$

- 1

= $2^3+2^{38}+2^{112}+2^{135}+2^{360}+2^{402}+2^{491}-1$

= 8 + 274,877,906,944

+ 5,192,296,858,534,827,628,530,496,329,220,096

+ 43,556,142,965,880,123,323,311,949,751,266,331,

066,368

+ 2,348,542,582,773,833,227,889,480,596,789,337,

027,375,682,548,908,319,870,707,290,971,532,209,

025,114,608,443,463,698,998,384,768,703,031,934,976

+ 10,328,999,512,347,634,358,623,676,688,012,047,497,

318,823,171,316,894,051,322,637,426,162,590,488,067,

364,778,518,581,413,120,551,325,743,612,687,890,989,

973,504

+ 6,393,341,031,047,152,089,869,511,126,616,404,594,173,

128,996,177,860,916,959,553,453,312,761,321,102,879,990,

006,386,899,074,031,556,935,325,554,936,640,763,689,877,

454,191,182,408,307,282,280,448

− 1

= 6,393,341,031,047,152,089,869,511,136,945,404,106,

523,111,897,384,311,438,199,490,431,228,373,829,447,

149,723,877,932,645,107,329,335,974,222,586,036,484,

943,430,874,337,072,758,146,938,842,382,343

The backward way of exploring quantum algorithms is shown in Figure 5 . The algorithm takes the initial and last values of numbering. It starts with setting empty set to Cs which is the list of valid quantum circuits. It loops until it reaches a stopping condition or the loop iterator reaches the last value of numbering. At each iteration, it calculates the valid quantum circuit that is associated with a number equals to the iterator. Validity of a quantum circuit means that it performs a desirable operation based on evaluation of an expert. The author played the role of expert for recognizing useful quantum circuits.

One of useful applications of quantum circuits is Quantum Key Distribution (QKD) BB84 protocol [20] which is proved to be secure [21] . Quantum Key

Distribution (QKD) is a cryptographic protocol that uses quantum mechanics to establish a shared key between Alice and Bob. The BB84 protocol, named after its inventors Charles Bennett and Gilles Brassard, involves initialization, transmission, basis selection, measurement, public announcement, key sifting, error estimation, privacy amplification, and key generation [22] . A full software stack to integrate QKD in a cloud context was proposed in a recent publication [23] . Another recent paper presented a simulation of the QKD BB84 protocol using IBM’s Qiskit quantum computing platform [24] . The simulation involves implementing the protocol with and without an eavesdropper Eve.

Table 4 shows the quantum part of the channel between Alice and Bob implementing BB84 Protocol on 5 Qubits. There is also a classical channel between them. The proposed approach can generate the circuits of BB84 protocol. But it depends on the ability of the implementation runner to recognize the circuits.