Quantum computers can solve certain kinds of problems far more efficiently than classical computers, including some that would take conventional machines an impractical amount of time to handle. Their most important potential applications include factoring large integers, simulating the behavior of molecules and materials, and accelerating particular mathematical and search-related tasks.
But the distinction is more subtle than saying quantum computers can solve problems that classical computers cannot. Classical computers can, in principle, solve every problem that a quantum computer can solve, provided they have sufficient time and memory. The difference is that quantum computers may solve some problems using dramatically fewer computational steps.
That advantage comes from quantum mechanics, the physical theory governing matter and energy at very small scales. Quantum computers use properties such as superposition, interference, and entanglement to process information in ways that conventional computers cannot directly reproduce efficiently.
Understanding what these machines can do requires separating three questions: which problems offer a genuine quantum advantage, why quantum mechanics helps with those problems, and which widely publicized applications remain beyond today’s practical capabilities.
How quantum computers differ from classical computers
Classical computers process information using bits, each of which represents either 0 or 1. Modern processors manipulate enormous numbers of these bits through logical operations, executing everything from arithmetic calculations to artificial intelligence workloads.
Quantum computers use quantum bits, or qubits. A qubit can be prepared in a quantum superposition of the states 0 and 1. When measured, it produces a conventional result, either 0 or 1, but before measurement, its quantum state can encode probability amplitudes associated with both possibilities.
These amplitudes are not simply ordinary probabilities. They can have positive or negative signs, or more generally, complex phases. Quantum operations can cause amplitudes to reinforce one another through constructive interference or cancel one another through destructive interference.
This interference is central to quantum computing. A successful quantum algorithm arranges its operations so that amplitudes associated with useful answers become more prominent, while those associated with less useful outcomes diminish.
Entanglement is another important feature. When qubits become entangled, their joint quantum state cannot be fully described by treating each qubit as an independent system. These correlations can support computational processes that are difficult to simulate efficiently using classical methods.
A register of nnn qubits has a quantum state described by 2n2^n2n complex amplitudes. That exponential growth is one reason quantum systems can become difficult for classical computers to simulate. However, it does not mean a quantum computer can read out 2n2^n2n independent answers from a single calculation. Measuring the register produces only one outcome at a time.
The real advantage comes from designing quantum operations that extract useful information from this complex state. More qubits alone do not guarantee better performance, and many quantum algorithms offer no meaningful advantage over their classical counterparts.
Factoring large numbers and breaking some encryption systems
One of the best-established theoretical applications of quantum computing is integer factorization: finding the prime numbers that multiply together to produce a given integer.
For example, factoring 15 means identifying its prime factors, 3 and 5. Small numbers are easy to factor, but the difficulty can increase substantially for carefully chosen integers with hundreds or thousands of digits.
This matters because widely used public-key cryptography systems, particularly RSA, rely on the difficulty of factoring sufficiently large products of prime numbers. Multiplying two large primes together is straightforward. Recovering those primes from their product is much harder for known classical algorithms at appropriately chosen sizes.
In 1994, mathematician Peter Shor developed a quantum algorithm that can factor integers in polynomial time, meaning its computational requirements grow much more favorably with input size than those of the best known general-purpose classical factoring algorithms.
Shor’s algorithm does not simply try every possible factor simultaneously and reveal the correct one. Instead, it transforms the factoring problem into a related mathematical problem called order finding. Quantum operations and interference then help extract the periodic structure needed to determine the factors.
With a sufficiently large, reliable quantum computer, Shor’s algorithm could make factoring integers used in vulnerable RSA implementations practical. A related quantum algorithm can solve the discrete logarithm problem, threatening other public-key cryptographic systems, including those based on elliptic curves.
The qualification is important: no existing quantum computer is known to be capable of breaking properly implemented, large-scale RSA encryption using Shor’s algorithm. The challenge is building a machine with enough high-quality qubits, sufficiently low error rates, and the error correction needed to run the algorithm at the required scale.
The prospect of future quantum attacks is already influencing cybersecurity. Organizations are developing and adopting post-quantum cryptography: cryptographic methods designed to resist attacks from both classical and quantum computers. These methods generally rely on mathematical problems other than the factoring and discrete logarithm problems that threaten conventional public-key systems.
Quantum computers would not automatically defeat every form of encryption. Properly implemented symmetric encryption and cryptographic hash functions face different threats. Grover’s quantum search algorithm can, under appropriate conditions, reduce the number of operations needed for an exhaustive search from roughly NNN to roughly N\sqrt{N}N, where NNN is the number of possibilities. For an idealized key search, this represents a quadratic speedup rather than the dramatic change associated with Shor’s algorithm. Increasing symmetric key sizes can help compensate for that advantage.
Simulating molecules and materials
Quantum simulation may ultimately be one of the most scientifically valuable uses of quantum computers.
Molecules, atoms, and materials obey quantum mechanics. Their electrons can occupy interacting quantum states, and the behavior of one electron may depend on the states of many others. Calculating these interactions accurately can become extremely difficult as a system grows more complex.
Classical computers already simulate many chemical systems successfully. Researchers use sophisticated approximations to predict molecular structures, reaction energies, electronic properties, and other quantities. These methods are powerful, but some systems are especially challenging because their behavior depends on complicated correlations between electrons.
The underlying difficulty is that an exact quantum description of a system can require an enormous amount of information. For a system of many interacting particles, the number of possible configurations can grow exponentially with system size. Classical algorithms can exploit mathematical structure to reduce this burden, but they cannot eliminate the difficulty in every case.
A quantum computer offers a different approach: use a controllable quantum system to represent and evolve another quantum system.
Rather than explicitly tracking every relevant amplitude on a conventional computer, a quantum processor can encode aspects of the target system in its own quantum state. Researchers can then perform operations designed to estimate properties such as the system’s energy or the behavior of its electrons.
This could improve calculations for chemical reactions, catalysts, superconducting materials, batteries, and other systems in which quantum interactions are important. Better predictions could help scientists identify promising compounds or materials before investing in expensive laboratory experiments.
For example, designing a more effective catalyst requires understanding how electrons are distributed and how chemical bonds change during a reaction. Small errors in calculated energies can sometimes lead to incorrect predictions about which reaction pathway is most favorable. More accurate calculations could help researchers distinguish between competing candidates.
However, quantum simulation is not automatically superior for every molecule. Many chemically important systems can already be modeled effectively using classical techniques, and classical simulation methods continue to improve. A useful quantum advantage would need to appear in a specific problem where a quantum algorithm achieves meaningful accuracy with fewer resources than the best practical classical alternatives.
The long-term promise is substantial, but it remains a research goal rather than a general capability of today’s quantum hardware.
Solving certain mathematical problems more efficiently
Beyond factoring, quantum computers offer advantages for several carefully defined mathematical problems. These examples illustrate why quantum computing is not simply a faster version of an ordinary processor.
One important category is the hidden structure of functions. In certain mathematical problems, an unknown pattern or periodic relationship is embedded in the outputs of a function. Quantum algorithms can use superposition and interference to identify aspects of that structure more efficiently than known classical algorithms.
Shor’s algorithm belongs to this broader family of techniques. Its factoring speedup depends on finding a period associated with modular arithmetic, rather than directly testing possible factors one by one.
Another example is the Bernstein-Vazirani problem, in which a hidden binary string determines how a particular function responds to binary inputs. Under the standard oracle model, a quantum algorithm can identify the string with one query, while a classical deterministic algorithm generally requires multiple queries proportional to the string’s length.
The example is mathematically clean, but its practical importance is limited. It assumes access to a specially defined function through a query interface. It does not establish that quantum computers can accelerate ordinary software or arbitrary calculations by the same amount.
This distinction matters throughout quantum computing research. An algorithm may demonstrate a rigorous speedup under a particular mathematical model without providing a useful advantage for a real-world task. Practical value depends on whether the problem occurs in an important application, whether the required operations can be implemented efficiently, and whether the quantum machine outperforms the best classical approach after accounting for overhead.
Searching large spaces of possibilities
Many computational problems involve finding a particular item among a large set of candidates. Examples include checking possible solutions to a constraint problem, identifying a matching record, or searching a database.
Grover’s algorithm provides a quadratic speedup for unstructured search. If there are NNN possible candidates and a procedure can identify whether a candidate is a valid solution, classical exhaustive search requires on the order of NNN evaluations in the worst case. Grover’s algorithm can find a solution using on the order of N\sqrt{N}N evaluations, assuming an appropriate quantum implementation and a suitable distribution of solutions.
For a search space of one million candidates, the theoretical comparison is roughly one million evaluations versus about one thousand quantum oracle queries, ignoring constants and implementation costs.
That sounds extraordinary, but the comparison needs context. A quantum oracle is a reversible quantum operation that encodes the test for a valid candidate. Building that operation may be expensive, particularly when the original task involves complex software, large datasets, or repeated interactions with classical hardware.
Furthermore, a quadratic speedup is not the same as an exponential one. As search spaces grow, reducing the number of evaluations from NNN to N\sqrt{N}N can be valuable, but it does not make every large search easy.
Grover’s algorithm is most relevant when the problem genuinely requires searching a large, unstructured space and the candidate-testing procedure can be implemented efficiently on a quantum computer. If a classical algorithm can exploit sorting, indexing, mathematical structure, or other shortcuts, the quantum advantage may be smaller or disappear.
Optimizing complex systems: a promising but uncertain application
Optimization involves finding the best solution among many possibilities, subject to an objective such as minimizing cost, reducing travel time, or maximizing efficiency.
Airlines must assign aircraft and crews to routes. Manufacturers must schedule production. Delivery companies must plan routes. Energy systems must balance supply and demand while respecting technical constraints. These problems can involve huge numbers of possible configurations.
Some optimization problems become computationally difficult as their size grows, particularly when they contain many interacting constraints. This has led to interest in quantum algorithms such as the quantum approximate optimization algorithm, or QAOA, and quantum annealing, which is designed to search for low-energy states of a mathematical model representing the problem.
These methods use quantum operations or quantum physical dynamics to explore candidate solutions. Their goal is often to find a good solution efficiently, not necessarily to prove that the absolute best solution has been found.
The important limitation is that there is no general proof that quantum computers will outperform classical computers on practical optimization problems across the board. Classical optimization methods are highly developed, and they often produce excellent solutions even for difficult problems.
Some quantum algorithms have theoretical advantages under particular assumptions, and researchers continue to investigate whether those advantages can survive the costs of encoding a real problem, running the computation, correcting errors, and extracting a useful answer.
Quantum optimization could eventually prove valuable for selected applications. For now, it is more accurate to describe it as an active area of research than as an established source of broad commercial advantage.
What quantum computers cannot do
The popular image of quantum computing sometimes suggests a machine that evaluates every possible answer at once and immediately selects the right one. That is not how quantum computation works.
Superposition allows a quantum state to encode many possible outcomes, but measurement does not expose the entire state. A quantum algorithm must use carefully chosen operations to make useful information more likely to appear in its output. Poorly designed operations can produce no advantage at all.
Nor can quantum computers solve mathematically impossible problems merely by being quantum. If a problem is undecidable, meaning no algorithm can correctly solve every possible instance, changing the physical hardware does not make a universal solution available. Quantum computing changes computational efficiency for certain tasks, not the fundamental limits of logic and computation.
Even for solvable problems, quantum computers are not necessarily faster. Ordinary arithmetic, document editing, web browsing, and many everyday computing tasks do not benefit from quantum algorithms. Classical processors are inexpensive, reliable, versatile, and extremely effective at the operations that dominate ordinary computing.
Quantum machines also face physical challenges. Qubits are sensitive to interactions with their environments, which can disturb their quantum states and introduce errors. Quantum error correction can protect information by encoding a logical qubit across multiple physical qubits, but doing so adds substantial hardware and computational overhead.
As a result, the number of physical qubits in a machine is not a sufficient measure of its useful computational power. The quality of those qubits, the available operations, the error-correction method, and the structure of the algorithm all matter.
A small quantum processor may demonstrate an interesting scientific result without being able to solve a commercially valuable problem better than a classical computer. Conversely, a future error-corrected machine could make a narrow but important class of previously impractical calculations feasible.
How scientists determine whether a quantum computer has an advantage
A quantum computer demonstrates a meaningful computational advantage when it can complete a task more effectively than the best relevant classical alternatives. The comparison must account for more than the number of operations performed inside the processor.
Researchers consider runtime, accuracy, memory requirements, energy use where relevant, and the resources needed to prepare inputs and interpret outputs. They also examine whether a classical method can exploit structure that an initial comparison overlooked.
One useful distinction is between quantum supremacy, often used to describe a demonstration in which a quantum device performs a particular computational task beyond the practical reach of available classical methods, and quantum advantage, which usually emphasizes a useful improvement for a meaningful task. Terminology varies, and neither label by itself establishes broad practical value.
A specialized quantum experiment might be difficult to reproduce classically without being useful for chemistry, logistics, or cryptography. Establishing a real-world advantage requires a relevant problem, a fair comparison, and an implementation that can operate at the necessary scale.
The most consequential future results are therefore unlikely to be defined by qubit counts alone. They will be defined by what the machines can reliably calculate, how much those calculations cost, and whether the results improve on what classical computers can achieve.
Quantum computers are not universal replacements for classical machines. Their promise is more specific: quantum mechanics gives them access to computational techniques that can dramatically reduce the effort required for certain problems, particularly integer factorization and some forms of quantum simulation and structured mathematical computation. Other potential applications, including optimization, remain less certain.
The central question is not whether quantum computers can do everything classical computers cannot. It is whether they can exploit the structure of a particular problem to deliver an answer that would otherwise be prohibitively expensive to obtain. For a limited but potentially important set of tasks, the theory says they can. The engineering challenge is to build machines that can realize those advantages reliably and at useful scale.