A primer for understanding quantum computing in 2025
In May 1981, MIT held a conference on the physics of computation. Organized in conjunction with IBM at MIT’s Endicott House in Dedham, Massachusetts, it brought together some of the world’s leading thinkers across the disciplines of physics and computer science. Attendees included Richard Feynman, arguably the preeminent scientist in the world at the time. While his best-known contributions are in the field of theoretical particle physics (for which he won the Nobel Prize in 1965), his understanding of general of physics at the deepest levels was unmatched.
At the conference, Feynman presented a talk called “Simulating physics with computers” [1] that would significantly challenge the foundational principles of classical computing as originally envisaged by Alan Turing in the mid-20th Century. Turing’s pioneering work had laid the groundwork for computation based on deterministic algorithms and binary logic, a paradigm that dominated thinking about computers for decades. Feynman, however, with his profound understanding of quantum mechanics, proposed that classical methods were insufficient to simulate quantum systems effectively. He argued that to truly capture the complexities of the quantum world, computation itself would need a radical transformation — one that harnessed the very principles of quantum physics.
This perspective was revolutionary. Feynman introduced the concept of quantum computing, where the states of bits (qubits) could exist simultaneously in superposition, and entanglement could be utilized to perform calculations impossible for classical systems. His vision laid the foundation not only for a new type of computing but also for a shift in how the scientific community understood problem-solving and computational limits.
While we have come a long way since that seminal talk, there are several challenges that still need to be tackled by the scientific community to make the dream of quantum computers a reality. In addition, there is a lot of confusion when topics related to quantum computing are discussed in the popular media. The aim of this article is twofold:
- Build a clear framework for understanding differences between quantum computing and traditional computing, both from a hardware and software perspective.
- Provide a brief overview of key results in the quantum computing field.
The quantum trifecta: Clarifying processes, algorithms, and equations
Along the way, the structure of this article attempts to clarify three terms that have the word “quantum” in them: Quantum processors, quantum algorithms, and quantum mechanics equations. Even though not all of them are directly related to quantum computing, an understanding of each one helps in forming an overall impression of the subject.
Quantum processors
Returning to Feynman’s talk in 1981, he argued that classical computers, despite their rapidly increasing power, are fundamentally unsuited to simulating the complexities of quantum mechanics. Why? Because classical computers work with bits — units of information that can be either 0 or 1. Every operation on a classical computer is a manipulation of these bits through well-defined logical gates. But the natural world, which is fundamentally described by quantum mechanics, doesn’t work that way.
Rather, if we want to compute the properties of the fundamental world, we must use qubits, which are the building blocks of quantum processors. Unlike classical bits, qubits can exist in a superposition of both 0 and 1 at the same time. While there is no intuitive way to think about superposition, it is the fundamental reality of how subatomic particles behave.
In addition to the basic property of superposition, there is an even weirder property of quantum mechanics called entanglement, which is needed for a quantum processor. Entanglement is a phenomenon that happens between two particles (or two qubits). If two qubits are entangled, measuring one of them instantly influences the other, even if they are light years away from each other.
Let’s pause here. If the previous sentence doesn’t cause confusion and surprise, read it again. This is the strangest phenomenon in all of physics and has caused over a century of debate, analysis, and confusion among the world’s leading minds about how to interpret it. Einstein famously called it “spooky action at a distance” and went to great lengths to try to explain it without having to rely on information being transmitted instantly (i.e., faster than the speed of light, thought to be the fastest anything can travel — this is often also called “non-locality”).
Models with so called “hidden variables” have aimed to explain away effects of non-locality, but these have been ruled out with experiments done over the past 50 years. In 2022, the minds behind these experiments were awarded the Nobel Prize in physics [2]. In these ways, it has been shown that entanglement effects through non-local behavior are a reality in our universe.
Now, what can be done with the principles of superposition and entanglement that isn’t feasible with the classical computers? In simple terms, a classical machine, constructed by bits, can be in only a single state of either 0 or 1 at any given time. This means that to process a set of inputs, you must process a single bit at a time as either a 0 or 1. In quantum machines, a qubit can exist as 0 or 1 or both at the same time (thanks to superposition). Someone having multiple qubits could represent all possible combinations of 0s and 1s at the same time.
Furthermore, using clever algorithms (explored further in the quantum algorithms section below), we can leverage the entanglement properties to process information at rates much faster than what classical machines can attain.
Unlike classical processors that manipulate bits using transistors, a quantum processor controls qubits through specialized quantum gates. These gates manipulate qubits to create and maintain superposition and entanglement — the core ingredients of quantum computation.
A range of techniques and physical systems can be used to build quantum computers that can maintain superposition and entanglement. A listing of some of the most promising approaches is included below:
1. Superconducting qubits
Superconducting qubits [3] are built using circuits made from superconducting materials cooled to near absolute zero. These circuits behave like artificial atoms that can exhibit quantum properties. This approach has gained significant attention due to its compatibility with existing semiconductor manufacturing processes. The advantage of superconducting qubits lies in their high gate speeds and scalability potential. However, they are also highly sensitive to noise and decoherence, requiring extremely low temperatures — below 20 degrees millikelvin — to operate effectively.
2. Trapped ions
The trapped ions approach [4] involves trapping individual ions (charged atoms) in place using electromagnetic fields within a vacuum chamber. Laser pulses are then used to manipulate their quantum states. Trapped ions boast high coherence times and precise control, making them highly reliable. However, the downside is that gate speeds are relatively slow compared to superconducting qubits, and scaling beyond a few dozen ions is challenging.
3. Topological qubits
Topological qubits [5] are being explored as a way to address the issue of error correction in quantum computing. They rely on manipulating quantum states in such a way that they are naturally resistant to certain types of errors. The idea is to make qubits less sensitive to disturbances from their environment, thus reducing the need for complex error correction mechanisms. While this approach is theoretically promising, practical implementation is still at a very early stage.
4. Photonic qubits
Photonic quantum computing [6] uses single photons — particles of light — as qubits. Manipulating them involves using beam splitters, phase shifters, and detectors. The major advantage of this approach is that it operates at room temperature, unlike most other technologies. Additionally, photons travel fast and are less susceptible to decoherence. However, reliably generating and detecting single photons remains a technical challenge, and using this technique to build scalable quantum gates is still difficult.
5. Neutral atoms
Neutral atoms [7] are trapped using optical tweezers and manipulated with laser pulses to perform quantum operations. This method offers high scalability due to the possibility of creating large arrays of atoms. Additionally, they have longer coherence times compared to superconducting qubits. However, achieving precise control over individual atoms and efficiently entangling them remains a work in progress.
6. Silicon spin qubits
This approach known as silicon spin qubits [8] uses the spin states of individual electrons or nuclei trapped in silicon-based quantum dots. Its main advantage is compatibility with existing semiconductor technology, which opens the door for mass production. The challenge, however, lies in achieving high fidelity operations and scaling beyond a few qubits.
7. Diamond NV centers
In this approach [9], defects in diamond crystals — where nitrogen atoms replace carbon atoms and create a vacancy — are used as qubits. These defects act as quantum bits that can be controlled and measured using lasers. Unlike most other approaches, diamond NV centers can operate at room temperature and exhibit long coherence times, making them robust to environmental noise. However, scalability remains a significant challenge.
8. Quantum annealing
Unlike general-purpose quantum computers, quantum annealers [10] are designed to solve optimization problems by exploiting quantum tunneling (the analog of sending a ball through a solid wall with a finite probability — something that can’t be done classically). D-Wave’s systems are currently the most commercially developed quantum devices, operating with thousands of qubits. However, their applicability is limited to specific types of problems and cannot provide the universal computational power of other quantum technologies.
Quantum algorithms
Now that we have discussed the types of hardware required, let’s discuss what can be done with that hardware. A qubit’s ability to exist in multiple states at once (superposition) allows quantum algorithms to process many possibilities simultaneously rather than one by one. In addition, using entangled qubits, we can construct operations, also called quantum gates, that can leverage interference to enhance the probability of correct answers while suppressing incorrect ones.
It’s easier to understand how algorithms are developed by introducing the following two objects:
- States: These are the objects that represent a bit or a qubit.
- Gates: These are objects that represent logical operations that act upon states.
Both these objects exist in classical and quantum computing, but are fundamentally different in that context. In classical computers, a bit can be in one of two states, 0 or 1. These states are definite and unambiguous at any given time. The gates that act upon the bits are the familiar logical gates that we learn about while studying mathematics or electronics (or even philosophy — see more about the philosophy of logic, for example, in George Boole’s book on “the laws of thought”). For instance, the AND gate takes in two bits as an input and outputs 1 if both bits are 1 and 0 otherwise.
In comparison, a single qubit can exist as a combination of 0 or 1 due to the principle of superposition. They can be written mathematically as:
|ψ⟩ = α|0⟩ + β|1⟩
Where alpha and beta are complex numbers that, when squared, represent probabilities. The |0⟩ and |1⟩ symbols are called basis states. These are vectors and in general anything written inside a |⟩ symbol is a vector (to be even more specific, this is often called a “ket” vector in quantum mechanics and it has a complex conjugate written as ⟨| called a “bra” vector — collectively combined to get a “braket” — terminology used in quantum mechanics). A properly defined state in quantum mechanics must have the probabilities of individual states add up to 1 and thus in the state above we must also ensure that alpha (modulus) squared, which is the probability of state |0⟩ and beta (modulus) squared, which is the probability of state |1⟩ must add to 1.
Already we see that a classical state is very different from quantum states. Now let’s look at the operations for qubits. One of the most important gates in quantum computing is one that creates a superposition state from a single state. This is called the Hadamard gate, H, and it works as follows:
H|0⟩ = (1/√2)(|0⟩ + |1⟩)
H|1⟩ = (1/√2)(|0⟩ – |1⟩)
Immediately you can see that there is no equivalent operation in classical computing that can create a superposition of bits, as it wouldn’t make sense physically.
There are many more gates in classical and quantum computers, but hopefully these examples give an idea of how different they are. Quantum algorithms are constructed by applying sequences of these gates to qubits.
To better appreciate this, let’s look at a “toy” problem of doing a search.
Lost password problem: Grover’s algorithm
Suppose you must find a two-letter password and you know that the two letters will come from the following four-letter set: A, B, C and D. In total, there are 16 possible combinations of passwords from these letters: AA, AB, AC, AD…. DD. If we want to search for the right password on a classical computer, the most efficient way to do it is just to try each of them one at a time and see whether it’s right. On average, this will take eight tries.
In contrast, on a quantum computer, there are five steps we can follow to get the right password.
Step 1: Create a superposition state
Construct a superposition of four qubits that gives 16 states, each representing one of the possible password combinations. For instance:
|ψ⟩ = (1/4)(|0000⟩ + |0001⟩ + |0010⟩ + … |1111⟩)
Where we encode |0000⟩ = AA, |0001⟩ = AB, and so on.
Step 2: Create an “Oracle” circuit
This is a quantum circuit made of a collection of quantum gates that are defined by the problem. The objective of this circuit is to tag the right answer. In this analogy, think of this step as checking each password (i.e., the 16 possible options) by typing them into a computer like you would do with a classical computer. The difference here is that the Oracle circuit can process all 16 possibilities at the same time (as opposed to 16 separately timed steps that would be required in classical computers). The specific implementation of these Oracle circuits is quite technical and would require a series of articles by themselves, thus we don’t attempt to explain it further here, but the key concept is exactly what we explain above, namely that they can try all of the states at once and identify the right one.
The tagged password cannot be obtained by a person and yet it has just been tagged as the right one in the overall quantum state. The “tag” is usually a phase shift (technically, this means it changes the imagery part of the state, which when squared gives the same number but when combined with other states can affect the overall probabilities — going beyond this explanation would require an article of its own) of the quantum state, which doesn’t change its probability. For instance, if the correct state is |0001⟩ = AB, then the Oracle might change the sign in front of this state in the overall state, which may now look like:
|ψ⟩ = (1/4)(|0000⟩ + |0001⟩ + |0010⟩ + … |1111⟩)
|ψₒ⟩ = (1/4)(|0000⟩ – |0001⟩ + |0010⟩ + … |1111⟩)
Where, if O is the Oracle circuit (operator), the following operation is used to construct this state:
O |ψ⟩ = |ψₒ⟩
Step 3: Amplify the amplitude
Even if the correct state has been “tagged,” the probability of getting the right state when a measurement is made is quite small (in the state above, each state has an equal probability of 1/16 of being observed if a measurement is made). Thus, the amplitude of the correct state needs to be amplified. This is done through a so called “diffusion operator/circuit,” which basically increases the amplitude of the desired state while reducing the amplitude of the other states. The effect of the amplitude amplification is that the “tagged” correct state is often lost (for instance, if it was tagged by changing the sign of the correct state, the diffusion operator might flip the sign to a positive number again).
Again, we don’t discuss the details of how this operator is constructed but it is conceptually still made of the same quantum gates that we discussed above.
Step 4: Repeat
The Oracle operator is applied again followed by the diffusion operator. This needs to be done about √N times, where N is the number of possible states, in this case 16, to get a high probability of measuring the correct state.
Step 5: Measure
Measure the quantum state and mathematically the most probable output will be the correct answer; in our example it will be |0001⟩ = AB.
The algorithm we described above is called Grover’s algorithm [11] and it is one of the most fundamental algorithms that can be used by quantum computers for many practical applications.
There are several other key algorithms that can be used in quantum computers. A brief, non-exhaustive, list of them is described below:
1. Quantum Fourier transform (QFT)
This approach [12] is the analog of the classical Fourier transform, which decomposes a function or quantum state into a sum of periodic components. Unlike its classical counterpart, the QFT can be implemented with a complexity of:
O(Nlog(N))
offering an exponential speed up over the classical algorithms that require:
O(N²)
It achieves this efficiency using controlled rotation gates applied sequentially on qubits.
2. Shor’s algorithm
Shor’s algorithm [13] is probably the most famous of all quantum algorithms. It provides an exponential speed up over classical algorithms for integer factorization, which is what most of the cybersecurity on the internet relies on (including public key cryptography — if you want to see the consequences of breaking this, a recent TV series called “” is an interesting watch). This algorithm runs in polynomial time (where N is the integer we are trying to factor):
O((logN)³)
compared to the best classical algorithms that scale exponentially with N.
3. Quantum phase estimation (QPE)
This is one of the most fundamental algorithms [14] in quantum computing as it performs the most universal quantum computation — finding the eigenvalue of a unitary operator. Almost any calculation in quantum mechanics relies on this type of computation, thus doing this rapidly can influence all types of physics and chemistry problems. The QPE allows for the extraction of information about the phase of a quantum state and it has a complexity of:
O(logN)
which is exponentially faster than any classical computation.
4. Quantum annealing (Adiabatic quantum computing)
This is a novel algorithm [15] designed to solve optimization problems by slowly evolving the quantum state of a system. It is also the most challenging to explain as it relies on the most fundamental technical details of quantum mechanics. Nevertheless, let’s try to explain it at high level. The idea is to start with a simple (energy function) whose ground state is easy to prepare (like the harmonic oscillator, as an example) and then gradually modify it to a more complex Hamiltonian whose ground state encodes the solution to the problem of interest (as an analogy, it’s a little bit like doing ). If the evolution is “slow enough,” the system remains in its ground state throughout the process, effectively finding the optimal solution. While this is not universal for all optimization problems, there are many interesting problems that can be reframed as quantum Hamiltonian optimization problems and that’s where the power of this can really be leveraged. Some examples where this has been thought of include logistics and Machine Learning, but this is very much an evolving field and so the potential applications maybe much larger.
Quantum mechanics equations
Quantum mechanics is the best theory we have developed so far that describes our world at the most fundamental level. The main equation that describes the world in quantum mechanics is the Schrödinger equation. This is a differential equation that describes how a quantum state (which is a vector) evolves. It looks like this:
|ψ(x,t)⟩ is the state of the quantum system, the function V(x,t) is called the potential energy and is defined by the system we are trying to model.
Solving the Schrödinger equation to understand how quantum states evolve is a fundamental task in modeling physical systems, particularly in chemistry and materials science. Depending on the complexity of the potential energy function, solving this equation often requires numerical integration through various computational techniques, such as Hartree–Fock, Density Functional Theory, and Quantum Monte Carlo methods [16,17,18]. While this is an important computational field, it is not intrinsically linked to quantum computing. Classical computers are well-equipped to solve the Schrödinger equation using approximate numerical methods, and advancements in this area primarily involve improving classical algorithms. Although quantum computers may eventually assist in solving certain instances of the Schrödinger equation more efficiently — potentially accelerating discoveries in chemistry and materials science — solving the equation itself is not inherently a quantum computing problem. Even though Feynman’s inspiration for quantum computing in his 1981 lecture came from simulating quantum systems, it is a common misconception to conflate classical techniques for solving the Schrödinger equation with quantum algorithms. The former runs entirely on classical computers, while the latter leverages fundamentally different principles.
Summary and future outlook
Quantum computing represents one of the most transformative frontiers of science and technology. By harnessing the unique principles of quantum mechanics, these advanced machines promise to solve problems that are currently beyond the capabilities of classical computers. From optimizing complex systems in logistics to accelerating breakthroughs in materials science and medicine, the potential applications are vast and awe-inspiring.
In this article, we have explored some of the foundational algorithms, such as Quantum Phase Estimation, which offers exponential speed-ups, and Quantum Annealing, which tackles intricate optimization challenges through the elegant manipulation of energy states. These techniques, though grounded in deep quantum principles, illustrate the sheer innovation driving the field forward.
While significant strides are being made in quantum hardware development with scalable and modular designs [19,20,21], many challenges remain. Issues such as error correction, qubit stability, and operational scalability demand continued collaboration among physicists, engineers, and computer scientists. The journey to a commercially viable quantum computer is as intricate and challenging as it is exciting, offering valuable opportunities for discovery at every step [22,23,24].
To provide a concise yet informative perspective, the aim of this article has been to serve as both an educational guide and a call to action. As we stand on the brink of a potential quantum revolution, the need for interdisciplinary expertise has never been more crucial. The path to realizing the full potential of quantum computing is not just about overcoming technical barriers — it is about fostering imagination, resilience, and a pioneering spirit that will drive the next wave of scientific and technological marvels.
To summarize the concepts presented in this article and provide a brief on how to think about quantum computing, I have made this summary slide, shown in Figure 2, which provides an overview and brief of different aspects relating to quantum computing and the key areas of development within them.
Darsh Kodwani is on .
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