How Much Is the Cost of Implementing Arithmetic on a Quantum Computer?

Abstract

This book chapter explores the transition from classical to quantum computing, emphasizing the capabilities and challenges associated with quantum bits (qubits). Unlike classical computing, where the information is represented as binary digits, known as bits, quantum bits, or qubits, although they share similarities with classical bits, such as 0 and 1 states, they operate in superposition, simultaneously encompassing both 0 and 1 states. This unique property allows quantum computers to perform certain computations, traditionally done sequentially on classical computers, more efficiently, in a single operation with a qubit. Through comparative analysis, we investigate the feasibility and costs of integrating quantum computing into daily applications, examining the potential for personal quantum computing devices. Our findings highlight significant advancements in computational speed of routine mathematical operations, although not yet economically viable or competitive in the current market. Future directions question the market readiness for quantum computing, suggesting a pivotal shift towards cloud-based quantum computing resources.

1. Introduction

“To Rachele and Mattia, the next users of this amazing technology.”

Quantum mechanics came as a thunderstorm to the fundamental pillars in our classical understanding of the physical realm. Since its inception in the 1920s, it has enriched our comprehension significantly, marked by pioneering contributions from scientists like Niels Bohr, Werner Heisenberg, and Erwin Schrödinger, and has led to technological innovations that have become cornerstones of modern civilization. The theory has revolutionized the electronics and the information science with the transistor, the backbone of contemporary electronics [1, 2], and the advent of optical communication through lasers [3] are among the myriad contributions of quantum mechanics to technology and information science. Furthermore, recent advancements in quantum chemistry have facilitated the creation of novel materials [4] and pharmaceuticals [5], underscoring the theory’s broad impact. Today, quantum technologies contribute significantly to the Gross Domestic Product (GDP) of numerous countries, embodying a substantial sector within the global economy [6]. Nonetheless, similar to the slow assimilation of Maxwell’s equations in the nineteenth century, which required a century to fully grasp and catalyze technological advancements, it is to expect that the recent developments in experimental techniques for generating and controlling quantum states will lay the foundations for a potential second quantum revolution. This resurgence holds the promise of reshaping societal and economic landscapes once more. Quantum mechanics, initially a purely theoretical framework, underwent a transformative journey spanning over half a century. It transitioned from a mere physical theory to a cornerstone of our understanding and manipulation of information—the very foundation of modern information technology. The old Shannon’s paradigm according to which the purely mathematical bit can be implemented in the physical word has now been replaced by Landauer’s view, in which the information cannot be detached from its physical embodiment [7]. In 1982, the so-called “no-cloning theorem” confirmed new rules must be considered when information was quantically encoded [8]. In the same year, R. Feynman clearly predicted the quantum computer supremacy on the classical machines [9], which highlights the unique and transformative nature of quantum information processing. Subsequent milestones, including the first establishment of a quantum cryptography protocol published by Bennett and Brassard [10] in 1984, which allowed the creation of symmetric keys with perfect secrecy between the end points of a channel able to transmit quantum correlations, and the first, albeit primitive, implementation in 1989 [11] over a 25-cm free space, have added credibility to its claim of being a viable technology. This milestone effectively inaugurated the domain of quantum key distribution (QKD), marking a pivotal moment in its practical application, further exemplifying quantum mechanics’ profound impact on communication security and computational theory. Nevertheless, the no-cloning principle, which gives the security of quantum key distribution (QKD) and quantum cryptography, also imposes restrictions on the faithful amplification of quantum signals. This limitation ultimately constrains the maximum achievable distance of QKD transmissions. It’s worth noting that while the no-cloning principle prohibits the direct copying of quantum states, it does not rule out the potential for quantum repeaters. Quantum repeaters are devices capable of establishing quantum correlations over vast distances without resorting to state replication. Therefore, they offer a promising solution to overcome the inherent limitations of QKD, enabling it to transcend physical constraints such as distance and losses [12], overcoming its inherent limitations. The advent of quantum cryptography gained significant traction in 1994, notably propelled by Peter Shor’s groundbreaking quantum algorithm [13]. This pivotal moment made computer scientists and cryptographers worldwide realize the potential threat posed by quantum computers to existing public key ciphers and cryptographic systems. It became evident that if quantum computers were successfully developed, they could potentially compromise the security of current cryptographic protocols. The main question became, after the development of quantum computers, how long it would take before the rise of cryptographic vulnerabilities. While a quantum computer boasting 53 qubits has, in mere minutes, tackled computations that traditional computers would need millennia to solve [14], quantum key distribution systems are already in a commercial stage and many companies and governments have invested heavily in researching quantum information technologies, which reflect the rapid progression and application potential of quantum technologies. Moreover, as we stand on the brink of a new technological era, the contributions of quantum mechanics to sensing, metrology, and beyond continue to expand the horizons of what is scientifically and practically achievable. Particularly, integrating quantum computers into daily applications notably involves leveraging quantum and reversible arithmetic techniques, advancing beyond traditional methods. This encompasses employing Draper’s pioneering work on Quantum Fourier Transform (QFT)-based arithmetic [15], a critical foundation for quantum computation, alongside exploring modern advancements like ripple-array approaches [16]. These methodologies are essential for harnessing the unique computational capabilities of quantum systems, enabling more efficient processing and problem-solving strategies that significantly outperform classical computing paradigms in specific tasks. Despite existing challenges that must be addressed for its broader application, the impact of quantum technologies on our future technological landscape is undeniable. These advancements are poised to significantly influence various sectors, underscoring their emerging critical role in shaping the next technological era [17, 18]. While the potential for considerably faster computations is evident [19], the practical implications of integrating quantum hardware into everyday computing pose critical questions regarding cost-effectiveness and energy efficiency. This requires a shift in inquiry towards a more targeted analysis: “What is the comparative cost analysis between implementing quantum arithmetic operations versus classical operations in terms of computational resources and energy consumption?” This refined question underscores the need to evaluate the practical implications of quantum computing, focusing on the balance between its revolutionary computational speed and the logistical and economic challenges of adopting quantum hardware for everyday computing.

2. Classical and quantum arithmetic

2.1 Adding binary numbers

Binary addition operates on a straightforward principle, mirroring the conventional arithmetic process but within the binary numeral system. This approach ensures that adding two binary numbers adheres to specific rules, yielding results comparable to those of traditional arithmetic, albeit expressed in binary form. This foundational concept underscores the logical consistency between binary and standard arithmetic operations. By doing binary addition, two numbers in their binary representation are added to get the resultant number. This operation is typically performed using an adder, a fundamental component within the arithmetic logic unit. In the arithmetic logic unit, the adder serves as the foundational element for various arithmetic functions, including subtraction, multiplication, and division [20]. Algorithms for these operations are intricately linked to the principles of binary addition. The rules dictating binary addition are as follows.

2.1.1 The rules

We want to add two four bit numbers $A_3{A}_2{A}_1{A}_0$ and $B_3{B}_2{B}_1{B}_0$. We have to work through each column from the right to the left. In the following example, the first number is 1001 $(A_3=1$, $A_2=0$, $A_1=0$, $A_0=1)$ and the second one is 1111 $(B_3=1$, $B_2=1$, $B_1=1$, $B_0=1)$. Starting with the rightmost bit, we first have to calculate 1 $\left(A_0\right)$ plus 1 $\left(B_0\right)$, and 1 $\left(C_1\right)$ carries out to the next column.

Step 1 (S1):

$$\begin{array}{r}\\ 1001\\ \underset{¯}{+1111}\\ 0\end{array}\begin{array}{r}1\\ 1001\\ \underset{¯}{+1111}\\ 0\end{array}$$

Now, the second step consists in calculating the second column: 0 $\left(A_1\right)$ plus 1 $\left(B_1\right)$ and adding finally the carry 1 $\left(C_1\right)$. Again 1 carries out to the next column $\left(C_2\right)$. We advance further, in the third column we have 0 $\left(A_2\right)$ plus 1 $\left(B_2\right)$ and the carry 1 $\left(C_2\right)$. Again 0 is the result with the carry 1 $\left(C_3\right)$.

Step 2 (S2):

$$\begin{array}{r}11\\ 1001\\ \underset{¯}{+1111}\\ 00\end{array}\begin{array}{r}111\\ 1001\\ \underset{¯}{+1111}\\ 000\end{array}$$

In the last step, we have to sum 1 $\left(A_3\right)$ and 1 $\left(B_3\right)$ adding the carry 1 $\left(C_3\right)$. The result is 1000 with one more carry 1 $\left(C_4\right)$ for the next column.

Step 3 (S3):

$$\begin{array}{r}1111\\ 1001\\ \underset{¯}{+1111}\\ 1000\end{array}\begin{array}{r}1111\\ 1001\\ +1111\\ \overline{1\leftarrowtail 1000}\end{array}$$

Formally, we can summarize the sum protocol in the following way:

$$\begin{array}{r}{C}_4{C}_3{C}_2{C}_1\\ A_3{A}_2{A}_1{A}_0\\ \underset{¯}{+B_3{B}_2{B}_1{B}_0}\\ C_4\leftarrowtail S_3{S}_2{S}_1{S}_0\end{array}$$

2.1.2 The half adder: the circuit

We begin building a logical circuit that adds two bits following the added rules. We have four situations to face:

$$\begin{array}{r}0\\ +0\\ \overline{\mathit{Sum}0}\\ \mathit{\text{Carry}}0\end{array}\begin{array}{r}1\\ +0\\ \overline{\mathit{Sum}1}\\ \mathit{\text{Carry}}0\end{array}$$

$$\begin{array}{r}0\\ +1\\ \overline{\mathit{Sum}1}\\ \mathit{\text{Carry}}0\end{array}\begin{array}{r}1\\ +1\\ \overline{\mathit{Sum}0}\\ \mathit{\text{Carry}}1\end{array}$$

We name the input bits A and B, the sum S, and the carry bit C. The four situations can be summarized in a so-called truth table (see Table 1). The sum can be implemented with a logical XOR. The carry can be expressed with a logical AND.

A	B	S	C
0	0	0	0
1	0	1	0
0	1	1	0
1	1	0	1

Table 1.

Truth table for the logical half adder.

Figure 1 shows how the circuit, so-called half adder, works in practice.

Figure 1.

Logical circuit for a half adder (electronic circuit simulation by Brian studio): (a) A = 0, B = 0; (b) A = 0, B = 1; (c) A = 1, B = 0; (d) A = 1, B = 1.

2.1.3 The full adder: the circuit

For the next bits, the logical circuit is more complex, it is not enough for a half adder. We have to add them along with whatever was carried in. Since this allows for a carry, it is called full adder. We call the two input bits A and B. The bit we carry is $C_{\mathit{\text{in}}}$. We build again the truth table for understanding the gates we need to implement the logical operation:

From the truth table (Table 2), we can see that the carry out is 1 when A and B are 1, or when $C_{\mathit{\text{in}}}=1$ and $\mathrm{A}\oplus \mathrm{B}=1$:

A	B	$C_{\mathit{\text{in}}}$	S	$C_{\mathit{\text{out}}}$
0	0	0	0	0
0	0	1	1	0
1	0	0	1	0
1	0	1	0	1
0	1	0	1	0
0	1	1	0	1
1	1	0	0	1
1	1	1	1	1

Table 2.

Truth table for the logical full adder.

$$C_{\mathit{\text{out}}}=\mathit{AB}+C_{\mathit{\text{in}}}\left(A\oplus B\right)$$E1

From the truth table, we also notice that the sum S is 1, whenever one of the inputs (Figure 2) A, B, or $C_{\mathit{\text{in}}}$ is 1, or when all three are 1. This is equivalent to XOR of all the three bits:

Figure 2.

Full adder circuit diagram.

$$S=A\oplus B\oplus C_{\mathit{\text{in}}}$$E2

It is now evident that to sum two four bit numbers, the circuit must have one half adder and three full adder. Figure 3 shows how the circuit works in practice.

Figure 3.

Logical circuit for a full adder (electronic circuit simulation by Brian studio): (a) A = 0, B = 0; (b) A = 0, B = 1; (c) A = 1, B = 0; (d) A = 1, B = 1.

3. Reversible and irreversible processes in physics

In the Resnick, Halliday, and Krane book [21], we can read this definition:

…In a reversible process, we make a small change in a system and its environment; by reversing that change, the system and its environment will return to their original conditions.

The definitions in other introductory books are not much different or clearer [22, 23]. In thermodynamics, the principle of reversibility encompasses the concept that certain processes are capable of being inverted, ensuring that a system can revert to its original state without exerting any lasting influence on other systems that have interacted with it (an ideal example of this process would be the singular swing of a frictionless pendulum). Irreversibility emerges because of the intricate interactions between a given system and other concurrent processes. Irreversible processes inherently involve dissipative phenomena, such as heat transfer across a finite temperature gradient or the manifestation of irreversible chemical reactions. In essence, the dichotomy between reversible and irreversible processes encapsulates the intricate dance of thermodynamic principles within the intricate choreography of the physical world. On the contrary, quantum mechanics and quantum circuits inherently embody reversibility, a consequence of the unitarity principle intrinsic to quantum mechanics. Based on the strict reversibility of the laws of microphysics, Landauer [24], Bennett [25], Priese [26], Fredkin and Toffoli [27], Feynman [28], and others envisioned a reversible computer that cannot allow any ambiguity in backward steps of a calculation. In this perspective, operations described by unitary operators, preserving purity and information, define the reversible transformations in quantum circuits.

3.1 Reversible gate

A reversible gate is a type of logic gate that allows the original input to be derived from its output. For instance, the NOT gate is an example of a reversible gate. Based on the truth table, the outputs of a NOT gate are distinct,

enabling the operation to be reversed. Hence, if the output from the gate is 1, the input must have been 0, and the reverse is also true. The reversibility of the NOT gate stems from the fact that the input can be always unequivocally determined (Table 3).

A	$\overline{A}$
0	1
1	0

Table 3.

Truth table for the NOT gate.

3.2 Irreversible gate

An irreversible gate is the opposite of a reversible gate. Given the output of the gate, we cannot determine unequivocally what the input is. An example is the AND gate (Table 4).

A	B	AB
0	0	0
1	0	0
0	1	0
1	1	1

Table 4.

Truth table for the AND gate.

From the truth table, the output of the AND gate being 1 clearly indicates that both inputs are 1. However, if the output is 0, it does not uniquely specify the inputs, which could be 00, 01, or 10. Thus, from the output alone, the inputs to the AND gate cannot typically be determined, making it an irreversible gate. Note that the AND gate accepts two input bits, allowing for four possible combinations (00, 10, 01, and 11), but produces only one output bit, which can be either 0 or 1. The reduction in possible output states compared to input states means some information is lost, making it impossible to reliably deduce the inputs from the output. Moreover, in situations where there are more input bits than output bits, the circuit remains irreversible because some output scenarios do not correspond uniquely to specific inputs. Consider, for example, the following truth table (Table 5).

$A_{\mathit{\text{in}}}$	$B_{\mathit{\text{out}}}$	$C_{\mathit{\text{out}}}$
0	0	1
1	1	0

Table 5.

Truth table for an irreversible gate.

If both outputs are 0, the input remains unspecified according to the truth table; hence, its inverse is not fully determined. Therefore, a logic gate must have at least as many input bits as output bits to be reversible [29]. This is a necessary condition but not always sufficient. For instance, the gate described below is reversible (Table 6).

A	B	C	D
0	0	1	1
1	0	1	0
0	1	0	1
1	1	0	0

Table 6.

Truth table for reversible gate.

The gate is reversible because, given the output, it is always possible to determine the input. No information is lost since we can always recover the inputs from the outputs. Moreover, we have just seen, the AND gate is not reversible.

3.2.1 Making reversible gates irreversible

We consider a gate with inputs A and B and outputs f(A, B), which is a function that output 0 or 1 depends on the inputs A and B. For example, for AND gate, the function would map $f\left(0,0\right)=0$, $f\left(0,1\right)=0$, $f\left(1,0\right)=0$, and $f\left(1,1\right)=1$. We can make it reversible with XORing output with a third input C (see Table 7).

A	B	C	A	B	$f\left(A,B\right)\oplus C$
0	0	0	0	0	$f\left(0,0\right)\Rightarrow 0$
0	0	1	0	0	$\overline{f}\left(0,0\right)\Rightarrow 1$
0	1	0	0	1	$f\left(0,1\right)\Rightarrow 0$
0	1	1	0	1	$\overline{f}\left(0,1\right)\Rightarrow 1$
1	0	0	1	0	$f\left(1,0\right)\Rightarrow 0$
1	0	1	1	0	$\overline{f}\left(1,0\right)\Rightarrow 1$
1	1	0	1	1	$f\left(1,1\right)\Rightarrow 1$
1	1	1	1	1	$\overline{f}\left(1,1\right)\Rightarrow 0$

Table 7.

Truth table for the XORing irreversible gate.

In the rightmost column of the truth table, we used $f\left(A,B\right)\oplus 0=f\left(A,B\right)$ and $f\left(A,B\right)\oplus 1=\overline{f}\left(A,B\right)$. So this implements the original gate when C = 0. But the overall circuit is reversible. In the output, every permutation of A and B shows up twice, once with f(A, B) and once with $\overline{f}\left(A,B\right)$, ensuring that the outputs are unique [30]. The technique can be generalized several ways. Figure 4 shows how to implement the logical circuit for a generic two input ports/one output logical gate.

Figure 4.

XORing the GATE.

4. Quantum adder

4.1 Qubit

A quantum bit, often referred to as a qubit, shares similarities with and differs from classical bits. Like its classical counterpart, a qubit can assume either of the two values, 0 or 1. In the language of quantum physics, using Dirac notation, these states are represented by the kets:

$$\left|0\right.⟩=\left[\begin{array}{c}1\\ 0\end{array}\right]\left|1\right.⟩=\left[\begin{array}{c}0\\ 1\end{array}\right]$$E3

A qubit possesses the ability to store both classical and quantum information concurrently, a phenomenon called quantum superposition. Analogous to classical bits, where the number of states increases exponentially with the number of bits (e.g., 8 bits allowing representation of integers from 0 to $2^8-1$), qubits follow a similar pattern. Still, with 8 qubits, it is possible to simultaneously encode all integers within the range of 0 to $2^8-1$. Consequently, the memory capacity of a quantum computer grows exponentially with the number of qubits. This exponential growth implies that the number of bits required to represent a set of qubits also increases exponentially with the set length. A qubit in superposition can be expressed as a linear combination of computational basis states, where an arbitrary qubit $\left|\psi \right.⟩$ is defined as ${\left|\alpha \right|}^2+{\left|\beta \right|}^2=1$. The numbers $\alpha$ and $\beta$, known as the probability amplitudes, weigh the probability of measuring a basis state. r. Those are complex numbers with the restriction of ${\left|\alpha \right|}^2+{\left|\beta \right|}^2=1$.

4.2 Quantum gate: CNOT and Toffoli gate

Quantum gates act on qubits like logic gates act on bits. The controlled-NOT gate or CNOT inverts the right qubit if the qubit is 1:

$$\mathit{\text{CNOT}}\left|00\right.⟩=\left|00\right.⟩$$E4

$$\mathit{\text{CNOT}}\left|01\right.⟩=\left|01\right.⟩$$E5

$$\mathit{\text{CNOT}}\left|10\right.⟩=\left|11\right.⟩$$E6

$$\mathit{\text{CNOT}}\left|11\right.⟩=\left|10\right.⟩$$E7

In this arrangement, the qubit on the left is known as the control qubit, and the qubit on the right is termed the target qubit. It is important to note that the control qubit remains unchanged by the CNOT operation, while the target qubit is transformed into the XOR of the input values:

$$\mathit{\text{CNOT}}\left|a\right.⟩\left|b\right.⟩=\left|a\right.⟩\left|a\oplus b\right.⟩$$E8

Thus, the CNOT gate is a quantum XOR gate. We can implement the sum using two control not, the second CNOT turns the previous in:

$$\mathit{\text{CNOT}}\left|c\right.⟩\left(\mathit{\text{CNOT}}\left|a\right.⟩\left|b\right.⟩\right)=\left|c\right.⟩\left|a\right.⟩\left|a\oplus b\oplus c\right.⟩$$E9

This quantum circuit can implement the quantum sum [30]. It is clear that due to the reversibility in quantum computing, we have to pay for the quantization by including one more qubit at the input stage (the ancilla qubit) in comparison to the bits in the classical circuit.

The Toffoli gate is a three-qubit gate in quantum computing, also called Controlled-Controlled-NOT gate.

$$\mathit{\text{CCNOT}}\left|000\right.⟩=\left|001\right.⟩$$E10

$$\mathit{\text{CCNOT}}\left|001\right.⟩=\left|001\right.⟩$$E11

$$\mathit{\text{CCNOT}}\left|010\right.⟩=\left|010\right.⟩$$E12

$$\mathit{\text{CCNOT}}\left|011\right.⟩=\left|011\right.⟩$$E13

$$\mathit{\text{CCNOT}}\left|100\right.⟩=\left|100\right.⟩$$E14

$$\mathit{\text{CCNOT}}\left|101\right.⟩=\left|101\right.⟩$$E15

$$\mathit{\text{CCNOT}}\left|110\right.⟩=\left|111\right.⟩$$E16

$$\mathit{\text{CCNOT}}\left|111\right.⟩=\left|110\right.⟩$$E17

$$\mathit{\text{CCNOT}}\left|c\right.⟩\left|b\right.⟩\left|a\right.⟩=\left|a\right.⟩\left|b\right.⟩\left|\mathit{ab}\oplus c\right.⟩$$E18

Since the Toffoli gate is a quantum universal gate, quantum computers can efficiently convert classical algorithm utilizing Toffoli gates. This allows to create an AND of A and B, XORed with C:

$$\mathit{\text{CCNOT}}\left|c\right.⟩\left|a\right.⟩\left|b\right.⟩=\left|a\right.⟩\left|b\right.⟩\left|\mathit{ab}\oplus c\right.⟩$$E19

$$\mathit{\text{CNOT}}\left|a\right.⟩\left|b\right.⟩=\left|a\right.⟩\left|a\oplus b\right.⟩$$E20

4.3 Quantum adder implemented on the IBM platform

Quantum computing is emerging from the research in the laboratory onto the marketplace. Numerous enterprises are in the process of crafting prototype quantum processors. These early stage quantum processors, termed noise-intermediate-scale quantum (NISQ) devices, are characterized by their susceptibility to decoherence, making them not fault-tolerant devices, and they are of intermediate-scale nature since they feature a moderate quantity of qubits. Several companies have opened access to their preliminary quantum processors for experimental purposes. Pioneering this accessibility, IBM was the foremost to provide cloud-based access to their quantum processors via the internet. IBM’s quantum computing resources, including their smaller quantum processors, are accessible to the public via their platform (https://quantum-computing.ibm.com), while larger and more advanced processors are offered through commercial access. This arrangement facilitates a range of engagement with quantum computing, from educational to research-oriented applications. The Circuit Composer on the website provides a drag-and-drop interface for the programming quantum circuits. Using the Circuit Composer, it is possible to create the circuits presented in the previous subsection, as shown in Figure 5. We note that q[0], q[1], and q[2] are the three qubit to sum, q[0] is the carry in. The three not gates $\oplus$ initialize to 1 the input values. q[3] and q[4] are the ancilla qubits for making reversible, respectively, q[1] $\oplus$ q[2] and q[0] $\oplus$ q[1] $\oplus$ q[2]. The two Toffoli gates implement the logical AND between q[1], q[2] (q[1]q[2]) and between q[0], q[1] $\oplus$ q[2] (q[0][q[1] $\oplus$ q[2]]) with the support of the two ancilla qubits q[5] and q[6]. The last qubit q[7] produces ${\mathrm{C}}_{\mathit{\text{out}}}$ through an anti-Toffoli gate, implementing an OR gate between q[1]q[2] and $\mathrm{q}\left[0\right]\left(\mathrm{q}\left[1\right]\oplus \mathrm{q}\left[2\right]\right)={\mathrm{C}}_{\mathit{\text{out}}}$.

Figure 5.

Quantum adder on the IBM platform.

5. Results and discussion

Using the Circuit Composer, shown in Figure 5, it is possible to create the circuits presented in the previous subsection. Where q[0], q[1], and q[2] are the three qubits to sum, q[0] is the carry in. The three NOT gates initialize the input values to 1. On January 4th, 2024, we ran the quantum circuit 1024 times on IBM’s IBM_Brisbane 13 processor, with 127 qubits available. The following table (see Table 8) summarizes the quasi-probability distribution for 12 outputs after 1024 trials in 17 seconds (in queue: 2 h 26 m 30.9 s) when q[0] = 1, q[1] = 1, q[2] = 1. Quasi-probability distributions are distinct from standard probability distributions as they permit negative values, yet their total must still equate to 1, akin to regular probabilities. This characteristic often emerges in the context of error mitigation strategies, illustrating the unique aspects of quantum computing calculations. Negative quasi-probabilities may appear when using error mitigation techniques [31]. The implemented protocol [32] excels in capturing correlated noise efficiently. It is designed to scale effectively for deployment in large quantum devices and serves as a demonstration of the concept of probabilistic error cancelation (PEC) on a superconducting quantum processor. PEC, as showcased in this implementation, is a technique adept at reversing well-characterized noise channels, leading to accurate and unbiased estimates of physical observables. The table shows the result q[4] = S = 1 and $\mathrm{q}\left[7\right]=C_{\mathit{\text{out}}}=1$ with 65% confidence, while q[4] = S = 1 and $\mathrm{q}\left[7\right]=C_{\mathit{\text{out}}}=0$ shows a quasi-probability next to the 18%. The IBM platform offers the possibility to run quantum circuit on an advanced quantum cloud-based computer simulator. We ran the same circuit on the ibmq_qsam_simulator, performing the simulation on a classic computer that resides on the cloud and simulates a 32-qubit computer. Figure 6 indicates the result q[4] = S = 1 and $\mathrm{q}\left[7\right]={\mathrm{C}}_{\mathit{\text{out}}}=1$ with 100% confidence. The quantum adder circuit exhibits consistent results between classical and quantum hardware; however, the intricacy of the circuit renders qubit $\mathrm{q}\left[7\right]=C_{\mathit{\text{out}}}$ susceptible to noise, with a frequency of 18%. In a subsequent experiment on the same day, the circuit was rerun with altered input values (q[0] = 1, q[1] = 0, q[2] = 1) on the identical platform and quantum hardware. Following 1024 trials and 18 seconds of utilizing the ibm_brisbane device, the outcomes were as follows: q[4] = S = 0 and $\mathrm{q}\left[7\right]=C_{\mathit{\text{out}}}=1$ with a quasi-probability of 0.426898811824693, and q[4] = S = 1 and $\mathrm{q}\left[7\right]=C_{\mathit{\text{out}}}=1$ with a quasi-probability of 0.341177111084729. This last result underlines the importance of noise reduction for the next generation of quantum computers and the need for the introduction of error-correction protocols in famous complex circuits like those employed in Shor’s algorithm or Grover’s algorithm [33]. Notably, the simulation using the ibmq_qsam_simulator classical cloud computer yielded q[4] = S = 0 and $\mathrm{q}\left[7\right]=C_{\mathit{\text{out}}}=1$ with a quasi-probability of 100% (see Figure 6). We executed a for loop 1024 times to implement the same logical classical operation consistently on a laptop with Intel(R) Core(TM) i7-8650U CPU@ 1.90 GHz-2.11 GHz and 16.0 GB RAM. The program in the MATLAB environment (@Matworks) when ${\mathrm{A}}_1=1$${\mathrm{A}}_0=1$ and ${\mathrm{C}}_{\mathit{\text{in}}}=0$ outputs the correct result with probability 100% in less than 1 millisecond (in average). In present-day classical computers, the probability of encountering errors is exceedingly low, often reported at the scale of $p\approx {10}^{-18}$. In contrast, for cutting-edge quantum computers, the error probability is notably higher, typically ranging around ${10}^{-2}$ or, under optimal conditions, ${10}^{-3}$ [34]. The process of addressing errors in classical bit addition revolves around encoding redundant information from one bit across multiple bits. This ensures that if an error occurs in one of the redundant bits, the encoded data remain intact. A basic illustration of this concept involves a three-bit code employing a majority voting mechanism. To retrieve the encoded data, all three bits are measured, and the result is determined by the majority vote among them. Encoded information is determined through a majority vote among three bits. For instance, if a distortion changes the third bit in a sequence, transforming 111 to 110, the predominance of 1 s in the sequence 110 suggests the original encoded information was a 1. This method underscores the resilience of error-correction codes in maintaining data integrity despite errors (Table 8) [35].

Figure 6.

Quantum adder quasi-probability histogram on the IBM QSAM (classical quantum) platform when q[0] = 1, q[1] = 1, q[2] = 1.

q[7]	q[5]	q[2]	q[1]	q[0]	Frequency
1	0	0	1	1	0.005963903511760564
1	0	1	0	1	0.005789115501746171
1	0	1	1	0	0.007055794279213567
1	0	1	1	1	0.13005895397294628
1	1	1	0	1	−0.003797515073072619
1	1	1	1	0	−0.01562920614909066
1	1	1	1	1	0.6528051438929247
0	0	0	1	1	0.0004680269228594772
0	0	1	1	0	−0.0007243258136245517
0	0	1	1	1	0.037259824909123135
0	1	1	1	0	−0.0024218131094713726
0	1	1	1	1	0.18317209715468513

Table 8.

Quasi-probability for the quantum adder implemented on the IBM platform.

5.1 Error correction

Correcting errors on quantum bits is indeed a more intricate challenge. This complexity posed a significant challenge for researchers in the early stages of quantum computing. The nuanced nature of quantum information and the susceptibility of qubits to environmental factors make error correction a substantial and ongoing focus of research in the field, especially for large and complex circuits. Overcoming these challenges is crucial for the development of reliable and scalable quantum computing technologies. The major obstacles thought to prevent quantum error are:

1. the no-cloning theorem [8];

2. measuring a quantum state causes it to collapse into an eigenstate of the measured observable [36];

3. within classical computing, a singular error type exists, the bit flipping, which toggles the bit between 0 and 1. In contrast, a quantum bit faces a multitude of potential errors, primarily due to various phase shifts, highlighting the complexity of quantum error correction [37].

In a bit flip channel, with three qubits it is possible to encode and protect the information, but it increases the hardware resources necessary for implementing the quantum circuit. To protect the information from the flip errors in a bit flip channel (the X gate), it is possible to encode the qubit using three qubits, not cloning, but applying the CNOT gate:

$$\mid \psi ⟩=\alpha \mid 0⟩+\beta \mid 1⟩\to \alpha \mid 000⟩+\beta \mid 111⟩=\mid {\psi }_3⟩$$E21

If we perform the parity measurement both on qubits 1,2 and 2,3, i.e., measure the product ${\sigma }_{z_1}\otimes {\sigma }_{z_2}$ and the product ${\sigma }_{z_1}\otimes {\sigma }_{z_2}$ (where ${\sigma }_{z_i}$ is the Z 2 x 2 Pauli matrix acting on the qubit i):

$${\sigma }_{z_1}\otimes {\sigma }_{z_2}\mid {\psi }_3⟩,$$E22

$${\sigma }_{z_2}\otimes {\sigma }_{z_3}\mid {\psi }_3⟩,$$E23

the carried information is not changed, the quantum state $\mid {\psi }_3⟩$ just gains a global phase factor marking a possible flip. In this case, applying an X gate to this qubit (${\sigma }_{x_3}$ 2 x 2 Pauli matrix), it flips back to the original state. The protocol works also when the rotation is not a flip (X gate), but an arbitrary rotation $\vartheta$ (RX gate on the IBM platform). In quantum calculations (or computing), the role of stabilizers in error correction, as above, can be elucidated by examining how a quantum state $\mid \psi ⟩$ interacts with a unitary operator $\mathit{U}$. Specifically, if the application of $\mathit{U}$ to $\mid \psi ⟩$ results in no change to the state $\left(\mathit{U}|\psi ⟩=|\psi ⟩\right)$, then $\mid \psi ⟩$ is considered to be stabilized by $\mathit{U}$, indicating a foundational aspect of understanding error-correction codes through stabilizers. Implementing then is crucial in minimizing the likelihood of a single-qubit error propagating into a genuine error within an encoded single-logical-qubit state, which is composed of multiple physical qubits. However, this initial step towards achieving error resilience in quantum computing comes at a cost. The incorporation of error correction inevitably escalates circuit complexity and adds to the computational overhead. Moving beyond this challenge, the realization of fault-tolerant quantum computing necessitates the development of robust schemes and allocation of resources for executing logical operations and measurements on logical qubits in a manner that is inherently resilient to errors. While delving into the intricacies of achieving fault tolerance is beyond the scope of this chapter, it is indeed possible and represents a pivotal aspect of advancing quantum computing capabilities.

6. Conclusions

Quantum computation stands as a groundbreaking paradigm with the potential to significantly reduce complexity in mathematical problem-solving. Contemporary accessibility to quantum hardware through cloud architectures, as exemplified by platforms such as Quantum IBM on IBM and AWS on Amazon, presents varied pricing plans in terms of U.S. dollars per second. However, adhering to fundamental principles in quantum mechanics, quantum circuits inherently necessitate reversibility, leading to an augmented demand for basic resources (qubits). In this investigation, we explored the hardware cost implications of implementing a classical logical circuit, specifically the full adder, on an IBM quantum computer. In contrast to its classical counterpart, the quantum circuit demands six input qubits (three input values q[0],q[1],q[2] and three ancillas q[3],q[5],q[6]) and two output qubits (q[4] = S and $\mathrm{q}\left[7\right]={\mathrm{C}}_{\mathit{\text{out}}}$), despite both circuits addressing a binary logic operation with three input bits (${\mathrm{A}}_0{\mathrm{A}}_1$ and ${\mathrm{C}}_{\mathit{\text{in}}}$) and two output bits (S and ${\mathrm{C}}_{\mathit{\text{out}}}$). The classical logic circuit can be effortlessly executed on commonplace calculators or personal laptops, offering straightforward implementation, reliable results, and swift outputs within milliseconds for a thousand trials. While the classical logic circuit seamlessly executes on standard calculators or personal laptops, featuring a straightforward implementation, dependable results, and rapid outputs within milliseconds for a thousand trials, the quantum counterpart introduces additional complexity. The quantum circuit, despite its higher hardware demands, is subject to the distinctive principles of quantum mechanics, and its execution involves the utilization of ancillary qubits for specific functionalities. This disparity highlights the inherent distinctions between classical and quantum computing paradigms, underscoring the need for a nuanced consideration of resource requirements when transitioning from classical to quantum implementations of logical circuits. Nowadays, the most common and efficient business solution on the market is shared quantum hardware in cloud architecture, but subject to queuing delays. Results are obtained within seconds for a thousand trials, yet they are influenced by noise and temporal stability issues. In this perspective, for routine day-to-day mathematical operations, quantum calculators or laptops do not presently emerge as economically viable, reliable, scalable, user-friendly, or competitive products in the market. Future prospects may witness alternative platforms based on diverse quantum technology approaches, such as photonic or ion trap-based circuits, presenting compelling business models for personalized quantum products. At present, the considerable expense associated with superconducting quantum bits (at approximately $10,000 [38]) necessitates the introduction of quantum facilities into the market primarily in the form of shared resources accessible via the internet, adopting time-of-use plans priced in U.S. dollars per second.

Acknowledgments

The authors would like to gratefully acknowledge the financial support from FAPESP, Brazil (São Paulo Research Foundation – Processes number: CEPID 2013/00793-6, 2018/11283-7 and AUXPE 2013/21569-1), CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior), CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) Brazil, Universidade Federal de São Carlos (UFSCar), and Instituto de Física (IF) at the University of Brasilia (UnB).

Conflict of interest

The authors declare no conflict of interest.