Thumbnail made out of the Qiskit logo (no endorsement claimed), a code sample generated by Perplexity and its terminal output for the CNOT version of the Deutsch-Jozsa algorithm (simulating oracle that CNOT’s all |0> bits), the formula in Deutsch and Jozsa’s paper [11] and the circuit in a paper [7].

CODE #6: QUANTUM COMPUTING AND OPTIMIZATION THROUGH DEUTSCH’S & DEUTSCH-JOZSA ALGORITHMS (QISKIT/PYTHON)

I. Introduction from the previous article on Hadamard gate and on the use of Qiskit

Last time, we looked at the Hadamard gate, a fundamental building block of quantum computing circuits. This article builds on the previous, therefore it is expected of you to read it [2], or to have a gist of the basics of quantum computing. This is because we will study Deutsch’s algorithm, made out of the Pauli-X, Oracle, Hadamard and measurement channels [1]. Then, we may look at how the Deutsch-Jozsa algorithm expands on the former.

One more point: while we used QuTiP for the previous issue, we will this time use another Python library that deals with quantum computing circuits perhaps moreso than QuTiP, the Qiskit library. What’s interesting is that its development is profoundly tied to IBM, an industry “core” [4] also known to allow various audiences access to its quantum computers, along possessing one of the world’s most performant (1'000s of qubits for Condor) quantum computer in 2023 [3].

“We have introduced IBM Condor, a 1,121 superconducting qubit quantum processor based on our cross-resonance gate technology. Condor pushes the limits of scale and yield in chip design with a 50% increase in qubit density, advances in qubit fabrication and laminate size, and includes over a mile of high-density cryogenic flex IO wiring within a single dilution refrigerator. With performance comparable to our previous 433-qubit Osprey, it serves as an innovation milestone, solving scale and informing future hardware design.” [3]

II. Quantum computing… An industry of growth, or hype in tech?

Discussing the algorithms we plan to take on will lead us to tackle a classical example of optimization, thanks to quantum computing, of problems we face with classical computing. Said optimization could lead to more demand for the hardware of quantum computing, were we to find more applications, potentially similar to the rush for gold we find nowadays with LLMs and AI data centers — as it’s far from the dematerialized and unconsequential fable we’re sold — , in what is probably a financial bubble that will eventually crash, or self-correct by absorbing risk through individual bankruptcies or tragedies. If the right actors act economically, it may lead to a new crisis in a context of global recession that will make us regret 2008? But this was mere speculation.

What can be substantiated is the potential for growth that world-stage organizations perceive in the industry. Notably the OECD and EPO [4].

“Quantum technologies have the potential to enable powerful new forms of computing, safeguard critical communications and achieve levels of precision in sensing and measurement that enable groundbreaking advancements, ranging from medical imaging to navigation to environmental monitoring. […] Over 30 countries have formulated tailored policies to support the responsible development and adoption of quantum technologies, including 18 OECD countries that have comprehensive national quantum strategies [/] To support the design and implementation of these policies, this report provides a comprehensive overview of the ecosystems that sustain quantum technologies. It charts the development of quantum technologies through a dynamic network of actors: research institutions, innovative start-ups, established firms, investors, and public authorities.” [4]

[Figure 1] “Figure E11. Estimated annual quantum R&D size: funding and implied award size 2015–2023” [4], displays a sustained though volatile growth from 2015 to 2023 in terms of the size of annual R&D funding and implied award size. The unit of measure is the USD PPP, neutralizing local inflation rates and therefore being useful to uniformize the study.

“Figure E11 shows that the share of quantum R&D funding relative to total R&D funding in the OECD Fundstat database (which covers Government Budget Allocation for R&D) increased steadily over the last decade, from approximately 0.4% in 2015 to 1.1% by 2023, with a peak of 1.2% in 2022. In parallel, the share of quantum-related project awards grew proportionally, reaching nearly 0.8% of all funded projects by the end of the observed period. Notably, the implied award size for quantum R&D peaked in 2023 and has consistently exceeded that of non-quantum R&D projects since 2018 (except in 2020, potentially due to shifting priorities during the COVID-19 pandemic).” [4]

[Figure 2] “Figure E7. Highest educational achievement of quantum firms’ founders” [4], made in 2025, shows that as opposite to all firms, where 10% of founders possess a PhD (not including postgraduates, which count as Master’s), 40% of founders a Master’s degree and 50% a Bachelor’s degree, 58% of quantum founders possess a PhD, 30% a Master’s and 12% a Bachelor’s. This is a radical shift explained elsewhere.

“Employees of quantum firms also typically possess advanced scientific and engineering qualifications. Job postings reveal that demand for quantum-related skills is heavily concentrated in a small number of technical and research-oriented occupations such as computer science (26%), science and research (25%) and education and training (10%). By contrast, commercialisation-oriented occupations such as business management, marketing and sales account for less than 10% of vacancies altogether. [/] The high educational level and strong research orientation of quantum founders and employees are reflected in the scientific character of quantum patenting. As shown in Figure E8, a significantly larger share of quantum patents cite non-patent literature (NPL) compared with patents in other technology fields. This indicates a close proximity between quantum innovation and scientific research, as most NPL citations refer to academic journals and other outputs from basic research.” [4]

But the report is not strictly about quantum computing. There are other voices in the space, albeit perhaps less as formal. That discuss quantum computing, from viewpoints that seem more enlightened compared to the general public, as perhaps an area where investment, not just monetarily but also careerwise [6], may not be the most optimal due to… hype? A cryptographer [5] said the following in April 2025:

“To show to the students what is the level of the hype, I googled “quantum computer breakthrough” to make a slide with current developments. The result was unbelievable. In a matter of minutes I found five companies that claimed to have made a breakthrough in quantum computing only last week! You would think this is a case of major company espionage, or maybe an extreme coincidence. But I think it shows another effect: the need to publish. (More on this below.)” [5]

Redditor ShoshiOpti, in February 9, 2026, answered to a thread titled “Is quantum computing more than a hype?” [6], published the same day with this answer:

“I currently research quantum information and before doing my Phd I worked in software / cryptography. And I stay away from QC investments and even research grants despite the money it would offer. [/] We don’t really know how useful quantum computing will be. There is currently a narrow band of computational problems that it works amazing at, but their commercial / industrial use cases is speculative at best. We have no idea if better chemical simulation will directly translate into [better] results than our current AI / classical pipeline or if it will just be a significant but still marginal increase in output. The classical computing baseline is shifting so fast [it’s] impossible to get a read on the gap. [/] In computer science language we are looking at NP(ish) hard problems that have known quantum solutions for them, not all NP hard [refers to time complexity in classical programming] solutions do. Combinatorial optimization is the key subset […] People [overfocus] on what quantum computing can solve while ignoring the other bottlenecks it creates. State preparing, fault tolerance, I/O constraints, oracle assumptions, encoding problem Hamiltonian, [SpecTrap] gap scaling, verification, instance to instance variability […] all create major engineering bottlenecks to making commercially viable solutions beyond just having quantum computing available for experimental runs. [/] TLDR: quantum computing is likely over-hyped and its commercial viability is [overstated] to the public. People hype its potential to get research grants and investments, and while I’d love QC to be mature and available, it almost certainly is not the holy grail it is hyped up to be.” [6]

[Figure 6] Google Trends 2004–2026 graph.

It shows a reoccurring in the past of the query for “quantum stock”, much earlier than for “quantum ETF”, which seems to be a more recent financial product. Still it shows speculation or investment interests, driving traffic. News were peaking in December 2025, probably with the release of the industry-wide study by EPO and the OECD [4]. “Breakthrough” is much rarer and would come from people genuinely interested in the academic, or at least theoretical side, perhaps also practical from the viewpoint of innovation, maybe industry insiders. “News” is too generalistic to make sense of, everyone would query that. “Deutsch-Jozsa” is the rarest since it’s a single expansion for an algorithm, while “sensing” or “error correction” are concepts themselves more reaching.

Still we can see three orders of magnitude of SEO interest:

• the technical,
• the financial,
• the generalistic.

It may help give more substance to the anatomy of the discourse in this section, through a different lens.

Something else is interesting: a study by Olivier Ezratty worked on exactly this subject. This is just before (2022, a truly golden-age limit) the contrast to be made with the elephant in the room: LLMs, and the hype in technology they demonstrate.

“[Quantum hype] fails with exaggerated overpromises and underdeliveries that last too long. It could cut short research and innovation funding, creating some sort of quantum winter. After looking at the shape and form of technology and science hypes and driving some lessons from past hypes, we investigate the current quantum hype and its specifics. We find that, although there is some significant uncertainty on the potential to create real scalable quantum computers, the scientific and vendor fields are relatively sane and solid compared to other technology hypes.” [18]

III. Deutsch’s algorithm: an example of optimization from classical to quantum computing

“The idea is defined as follows; given a black box, known as the oracle, which consists of an unknown binary function of one bit f(x) : {0,1} → {0,1}, the goal is to decide in a deterministic way if the function is balanced or constant [using] the oracle only one time. The classical algorithms need two instances of application [sometimes called two queries] of the oracle to solve the [problem] in a deterministic way, while quantum Deutsch algorithm uses the oracle a single instance [again, or one query], as long as there are no [errors] in the calculation process. The Deutsch algorithm can be summarized in the following scheme shown in the Fig. 2.” [7]

[Figure 3] “FIGURE 2. The Deutsch Algorithm.” [7], CC BY-NC 4.0.

“In this circuit, both inputs x and y to the block Uf receive one qubit of data having a state neither pure state |0 nor pure state |1 but in between. This state is called “bell-state” [this seems false; commented right below, confusion!] and the characteristic of this state is that a qubit will generate 0 or 1 of equal probability after measurement. The final result measured at the output of the Hadamard gate at the output shows with certainty the nature of f(x). A 0 at the output will prove f(x) is of constant type while a 1, will show f(x) is of balanced type.” [14]

This quote denotes something interesting: maybe a common mistake people make is assuming there is a Bell state with the CNOT gates, not only in DA but also in DJA. Maybe arXiv is a red flag? [19]

“I’m as guilty as many other researchers in my field of the “put something on [arXiv] as soon as I submit it” (in fact for conferences like ICLR [International Conference on Learning Representations] this is a requirement), so take this with a grain of salt. […] On the other, it’s problematic because it encourages people to rush to put half-baked ideas on arXiv, or even full-baked(?) ideas, without the benefit of peer review and scientific approval of the community. This leads to a lot of hype-y papers with serious experimental flaws being pushed around, and people (even serious academics, sometimes) taking them seriously. The even bigger problem is that this has lead some of the authors of such papers (or their friends) to shut down competing work by claiming that they got there first (in reviews, public comments, etc), even if the competing work is going through the “proper” channels of peer review (and the earlier work has not been validated by the community yet).”[19]

Apparently, it’s even something used for generalization of DJA for QKD and other applications:

“We present a generalized Deutsch–Jozsa (DJ) quantum algorithm that not only determines both the global type of an unknown Boolean function (constant or balanced) but also determines explicit output values of the function in a single oracle query. Unlike the original DJ algorithm, which identifies only whether a function is constant or balanced, our generalization retrieves actual function output values at the same time with using a Bell state as ancilla. This makes a richer function characterization with minimal queries to have practical quantum advantages, e.g. data classification, logistic regression, and quantum cryptography.” [17]

To model this in Python using Qiskit, first notice that both basis states are at 0, the default basis state in Qiskit. Also, x being an identity in this oracle [Figure 3], then only y is important, that is our second qubit. If the function is balanced, then a control NOT gate suffices to simulate the oracle’s behavior, approximation of a XOR logic gate. If the function is a constant, then we need not use any control NOT gate. We will delineate this with a comment, but will still keep the CNOT version for balanced: “CNOT gate is an example that implements a balanced function for which f(0)=0 and f(1)=1” [8].

# LLMs consulted for this sample: Perplexity, ChatGPT
from qiskit import QuantumCircuit

qc = QuantumCircuit(2, 1)
qc.x(1)
qc.h([0, 1])

qc.cx(0, 1)
# Kept: simulates balanced function oracle behavior, with XOR. [8]
# Removed: simulates constant function oracle behavior, no XOR. [8]

qc.h(0)
qc.measure(0, 0)

print(qc.draw())

┌───┐          ┌───┐┌─┐
q_0: ┤ H ├───────■──┤ H ├┤M├
     ├───┤┌───┐┌─┴─┐└───┘└╥┘
q_1: ┤ X ├┤ H ├┤ X ├──────╫─
     └───┘└───┘└───┘      ║
c: 1/═════════════════════╩═
                          0

     ┌───┐┌───┐┌─┐
q_0: ┤ H ├┤ H ├┤M├
     ├───┤├───┤└╥┘
q_1: ┤ X ├┤ H ├─╫─
     └───┘└───┘ ║
c: 1/═══════════╩═
                0

“A CNOT gate takes two qubits, a control qubit and a target qubit, and flips the state of the target qubit if the control qubit is in the state |1>. But it does nothing to the target if the control qubit is in the state |0>. So Uf(f2) is just a CNOT on q0 and q1. The CNOT gate will do nothing to the first term since the control qbit is in the state 0, but on the second term, this 0 flips to 1 and this 1 flips to 0. And if we factor out a minus sign, we’ll see that the state of qubit 1 is unchanged. But this term has picked up a - sign. So the state of qubit 0 is actually the one that has changed. It went from being 0+1 to being 0–1. When we apply the final Hadamard to qubit 0, this transforms it to the state 1. So, if the function in the black-box is f2, we’ll measure a one instead of a zero. […] Intuitively, you may think that qubit 0 would remain unchanged. And yet somehow, our operations on qubit 1 actually change the state of qubit 0. By changing the relative phase between the two parts of its superposition. This phenomenon is known as the phase kickback mechanism.” [13]

On a tangent, it was advanced [7] that the classical Deutsch algorithm was twice as efficient as querying with a sequential program. This is because we absolutely need both answers to compare them and arrive to the correct conclusion of identifying whether a black-box function is constant or balanced. We could also compare both computational scenarios to decision trees, which help us understand better the efficiency with layers. This will absolutely be clearer for the rest of this piece.

[Figure 4] PlantUML representation of the decision tree.

Remark: if we use the classical decision tree [Figure 4], it seems to be a scenario where we can reasonably convoke Shannon entropy — disclaimer: I’m not sure about this — , since we have uncertainty and that would permit us to measure it. Indeed, admitting we expand this classical decision scenario to have three inputs and to want to check whether the final output is {0, 0, 0} or {1, 1, 1} or the 2^n-2 solutions (that makes 2 solutions for 8 outcomes for n = 3, because 2³-2 = 8–2 = 6; for n = 4 we find 2⁴-2 = 16–2 = 14), we have a measure of uncertainty that is coincidental with power relationship. Hence the logarithm in Shannon entropy. Its basis is 2.

[Figure 5] “6. CHOICE,UNCERTAINTY AND ENTROPY”, formalization of Shannon entropy equation. [9, p. 11]

One element matters, K is a positive constant so that there is a conservation of the negative sign in front of it. We see it too in the von Neumann entropy formula (S = -tr(p ln p), which we won’t use but it’s interesting to notice. In our case n is the number of inputs, K is arbitrary (“H = ∑ pi log pi (the constant K merely amounts to a choice of a unit of measure)” [9, p. 11]).

“In quantum computing, information theory takes on new dimensions due to the probabilistic nature of quantum mechanics. Quantum information theory extends classical concepts, addressing the unique properties of quantum states. This includes the study of quantum entropy, quantum channels, and quantum entanglement. [/] A central aspect of quantum information theory is how information is transmitted and processed using quantum states. Unlike classical systems, quantum systems can encode information in entangled qubits, allowing for phenomena like quantum teleportation, which allows the transmission of quantum states between distant systems without physically moving the qubits themselves.” [15]

IV. Deutsch-Jozsa, the generalization with exponential gain

“An early example of an algorithm for which quantum computers are expected to drastically outperform classical computers (cf. quantum algorithm).” [10]

But if we take a glance at the original paper:

“A class of problems is described which can be solved more efficiently by quantum computation than by any classical or stochastic method. The quantum computation solves the problem with certainty in exponentially less time than any classical deterministic computation. [/] If we wish, more realistically, to model the oracle’s size and running time, then we could assume the oracle size, for generalf, to be O(N), this being the size of the oracle which simply contains a ROM list of the function values. Also for any f there exists an oracle which operates in a time of O(lnN): e.g. to look up f(k) it could traverse a binary tree following the binary expansion of k. From these estimates, and (*), it follows that any classical computer requires at least polynomial time, whereas the quantum computer requires only logarithmic time, again providing an exponential saving. “ [11]

Figure 7. “Problem statement”, Wikipedia [12].

“The scaling is exponential, so even if we had just 10 qubits, they could be in a superposition state of 2 to the 10, which is 1024 possible combinations of zeroes and ones. The exponential scaling of the state space of qubits is called “quantum parallelism” and is partially responsible for the advantage of quantum computers over classical computers.” [13]

Figure 8. “Deutsch-Jozsa algorithm quantum circuit”, Peplm. [12]

Still, we’re only concerned with balanced or constant functions for Deutsch-Jozsa, a condition for the optimality of the function over classical computing, or even Deutsch’s algorithm (following the 1985 formalization, I presume) alternatives.

Figure 8 simplifies the different implementations to generalize to n-qubit queries. |1> is |0> facing a Pauli-X gate, doing a bitflip to |1>. The n/ in front of |0> signifies to do as many Hadamard operations on the different qubits, as well as input them all in the oracle Uf, which still does an identity of inputs |0> and not for the one in state |1>, which we never measure. We only put n bits in initial state |0>, prior to Hadamard that puts them in superposition |+>. The last qubit is called in the video “this last qubit for the output state” [13].

“If the function is balanced, then when we measure the qubits following the Hadamard, the phase kickback mechanism will cause one or more of those bits to be |1>. That might not be so easy to see this time.” [13]

To understand Deutsch-Jozsa, thus, we need to consider whether the |1> ancilla bit manages to do even a single CNOT, letting one or several of the qubits starting in |0> to finish in |1> despite two Hadamard gates putting them back to the original position [2].

Now, onto simulating it in Qiskit:

from qiskit import QuantumCircuit

qc = QuantumCircuit(3, 1)
qc.x(2)
qc.h([0, 1, 2])
qc.h([0, 1])
qc.measure(0, 0)
qc.measure(1, 0)

print(qc.draw()).draw())

This outputs a simple n=2 DJA-like circuit, approximating deterministically what the behavior of the oracle would be for a function that is constant, and thus not balanced. Otherwise, the ancilla qubit performs CNOT gate operations as the control bit.

“Ancilla bits are extra bits (units of information) used in computing paradigms requiring reversible operations, such as classical reversible computing and quantum computing. Unlike classical computing, where bits can be freely set to 0 or 1, reversible computation requires all operations on computer memory to be invertible. Ancilla bits, whose initial state is known, provide the necessary “workspace” for performing operations that would otherwise erase information. They play a crucial role in implementing complex logic gates and enabling universal computation within these reversible models.” [16]

     ┌───┐┌───┐┌─┐   
q_0: ┤ H ├┤ H ├┤M├───
     ├───┤├───┤└╥┘┌─┐
q_1: ┤ H ├┤ H ├─╫─┤M├
     ├───┤├───┤ ║ └╥┘
q_2: ┤ X ├┤ H ├─╫──╫─
     └───┘└───┘ ║  ║
c: 1/═══════════╩══╩═
                0  0

from qiskit import QuantumCircuit

qc = QuantumCircuit(3, 1)
qc.x(2)
qc.h([0, 1, 2])

# This line sets up 2 control channels thanks
# to an array in the positional argument for targets
qc.cx(2, [0, 1])

qc.h([0, 1])
qc.measure(0, 0)
qc.measure(1, 0)

print(qc.draw())

     ┌───┐     ┌───┐┌───┐     ┌─┐   
q_0: ┤ H ├─────┤ X ├┤ H ├─────┤M├───
     ├───┤     └─┬─┘├───┤┌───┐└╥┘┌─┐
q_1: ┤ H ├───────┼──┤ X ├┤ H ├─╫─┤M├
     ├───┤┌───┐  │  └─┬─┘└───┘ ║ └╥┘
q_2: ┤ X ├┤ H ├──■────■────────╫──╫─
     └───┘└───┘                ║  ║
c: 1/══════════════════════════╩══╩═
                               0  0    0  0

The latter performs the successive CNOT operations, just like in DA. This is the action simulating the oracle for a function that is balanced, and therefore not constant. We can generalize this to n bits. This is probably like a Toffoli gate.

# Generated by Perplexity
from qiskit import QuantumCircuit

n = 3
qc = QuantumCircuit(n + 1, n)

qc.x(n)
qc.h(range(n + 1))
targets = list(range(n))
qc.cx(n, targets)
qc.h(range(n))
qc.measure(list(range(n)), list(range(n)))

print(qc.draw())

     ┌───┐     ┌───┐┌───┐     ┌─┐
q_0: ┤ H ├─────┤ X ├┤ H ├─────┤M├───────────
     ├───┤     └─┬─┘├───┤┌───┐└╥┘     ┌─┐
q_1: ┤ H ├───────┼──┤ X ├┤ H ├─╫──────┤M├───
     ├───┤       │  └─┬─┘├───┤ ║ ┌───┐└╥┘┌─┐
q_2: ┤ H ├───────┼────┼──┤ X ├─╫─┤ H ├─╫─┤M├
     ├───┤┌───┐  │    │  └─┬─┘ ║ └───┘ ║ └╥┘
q_3: ┤ X ├┤ H ├──■────■────■───╫───────╫──╫─
     └───┘└───┘                ║       ║  ║
c: 3/══════════════════════════╩═══════╩══╩═
                               0       1  2

V. Conclusion

In part I, we introduced the subject by establishing a progressive relationship with the former issue [2], and some of the progressive elements (CNOT gates, measurement channels, Hadamard gates, oracle gates) contained therein. In part II, we searched for hints as to whether the field of quantum physics was blooming, as well as the more restricted field of quantum computing. We found interesting facts, such as the high level of education of founders in the quantum physics sector [4] as well as the sustained growth on the 2015–2023 period leading up to now [4]. But we also discussed several interpretations of quantum computing notably as possibly over-hyped [5, 6]. In part III, however, we have explored how the Deutsch algorithm worked as a classic example of the advantage of quantum computing, with superposition, as opposite to classical computing, or maybe we should say digital computing [14 presented some analog computing analogy to superposition]. Here, however, it was still in the domain of theory with the case of a black-box function that is promised either a constant or balanced distribution. Permutations of the function can perhaps lead to interesting findings later on [17]. We simulated DA with the Python library Qiskit. But this was also the occasion to expand on the interest of decision trees, in relation to Shannon entropy. In part IV, it is the generalization of the former, the Deutsch-Jozsa algorithm, with n qubits, that showed us the exponential power of quantum computing over classical computation. We simulated the constant and balanced cases, to show different oracle behaviors. Further experimentation could be done by exploring other permutations, or we could look at other algorithms with ancilla bits, notably to explore Toffoli gates in other algorithms.

References

[1] Thread. “Definition of a quantum gate”, Quantum Computing Stack Exchange, asked May 27, 2024, modified May 29, 2024. https://quantumcomputing.stackexchange.com/questions/38503/definition-of-a-quantum-gate

[2] Article. “CODE #5: HADAMARD GATE IN QUANTUM COMPUTING (PYTHON)”, Emilia Lilith-Lolita Hoarfrost, Medium, June 29, 2026. https://medium.com/@emiliahoarfrost/code-5-hadamard-gate-in-quantum-computing-python-dc587bef34e8

[3] Article. “The hardware and software for the era of quantum utility is here”, Jay Gambetta, IBM, December 4, 2023. https://www.ibm.com/quantum/blog/quantum-roadmap-2033

[4] Study. “Mapping the global quantum ecosystem”, OECD, EPO, December 2025. ISBN: 978–3–89605–408–1; DOI: doi.org/10.65216/20251217–0001. https://link.epo.org/web/publications/studies/en-mapping-the-global-quantum-ecosystem-executive-summary.pdf

[5] Article. “The quantum computer hype”, Jurjen Bos, Medium, April 4, 2025. https://medium.com/@jnebos/the-quantum-computer-hype-af460fb349de

[6] Thread. “Is quantum computing more than a hype?”, u/ShoshiOpti, r/Physics, Reddit, February 9, 2026. https://www.reddit.com/r/Physics/comments/1r00liq/comment/o4f47oq/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button

[7] Study. “Performance and error modeling of Deutsch’s algorithm in IBM Q”, Efrain Buksman, André Luiz Fonseca de Oliveira, Carolina Allende, Universidad ORT Uruguay, Revista Mexicana de Física 66(2 Mar-Apr):239–245, March 2020. DOI: 10.31349/RevMexFis.66.239. https://www.researchgate.net/publication/342979428_Performance_and_error_modeling_of_Deutsch's_algorithm_in_IBM_Q

[8] Thread. “Deutsch’s algorithm in Qiskit”, Quantum Computing Stack Exchange, asked May 4, 2020 by marissalianam, edited the same day by Martin Vesely, answered the same day by Davit Khachatryan. https://quantumcomputing.stackexchange.com/a/11852

[9] Paper. A Mathematical Theory of Communication, Shannon, C.E., Bell System Technical Journal, 27, 379–423, 1948. http://dx.doi.org/10.1002/j.1538-7305.1948.tb01338.x https://people.math.harvard.edu/~ctm/home/text/others/shannon/entropy/entropy.pdf

[10] Entry. “Deutsch-Jozsa algorithm”, Urs Schreiber, nLab, August 2022-February 2024. https://ncatlab.org/nlab/show/Deutsch-Jozsa+algorithm

[11] Paper. “Rapid solution of problems by quantum computation.” David Deutsch, Richard Jozsa, Proc. R. Soc. Lond. A 439 1907 (1992) 553–558 DOI:10.1098/rspa.1992.0167. https://www.isical.ac.in/~rcbose/internship/lectures2016/rt08deutschjozsa.pdf

[12] Entry. “Deutsch–Jozsa algorithm”, Wikipedia, 2007–2026. https://en.wikipedia.org/wiki/Deutsch%E2%80%93Jozsa_algorithm

[13] Video. “Quantum vs Classical: Deutsch & Deutsch-Jozsa Algorithms Explained”, Katie McCormick, Qiskit, YouTube, April 30, 2025. https://www.youtube.com/watch?v=QcK0GK7DUh8

[14] Paper. “Deutsch Algorithm on Classical Circuits”, Osman Kaan Erol, Istanbul Technical University, Electrical-Electronics Faculty, Computer Engineering Dept., arXiv, submitted on March 21, 2008. https://doi.org/10.48550/arXiv.0803.3183 https://arxiv.org/abs/0803.3183

[15] Article. “Entropy and Information in Quantum Computing”, quantumphysicsguide, Deepesh Jain, September 9, 2024. https://quantumphysicsguide.in/entropy-and-information-in-quantum-computing/

[16] Entry. “Ancilla bit”, Wikipedia, 2023–2025. https://en.wikipedia.org/wiki/Ancilla_bit

[17] Paper. “Generalized Deutsch-Jozsa Algorithm for Applications in Data Classification, Logistic Regression, and Quantum Key Distribution”, D. Oblak, M. Zomorodi, M. Ghadimi, S. Moradi, S. Bakrani, N. Gohari-Kamel, V. Salari, arXiv:2512.00715v1, arXiv, November 30, 2025. https://arxiv.org/html/2512.00715v1

[18] Paper. “Mitigating the quantum hype”, Olivier Ezratty, February 10, arXiv, 2022. DOI: https://doi.org/10.48550/arXiv.2202.01925 https://arxiv.org/abs/2202.01925

[19] Thread. “Let’s discuss: is arXiv always good?”, u/egrefen, r/MachineLearning, Reddit, November 23, 2015. https://www.reddit.com/r/MachineLearning/comments/3twwwh/comment/cxa0u53/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button

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