Quantum Computing Applications: What Problems Can It Solve?

Quantum computing is often described as a revolutionary technology that will solve previously impossible problems. The reality is more nuanced. Quantum computers offer genuine advantages for specific problem types, while offering little or no advantage for others. This tutorial surveys the most important application areas and explains the current state of each.

How to Think About Quantum Advantage

A quantum computer has “quantum advantage” for a problem when it can solve that problem faster than any classical computer. This is not a binary property: advantages can be quadratic, polynomial, or exponential, and they depend on problem size, hardware quality, and available classical algorithms.

There are three categories of quantum application worth distinguishing:

1. Near-term (NISQ era): algorithms that might run on current noisy hardware with a few hundred to a few thousand imperfect qubits
2. Fault-tolerant era: algorithms that require error-corrected logical qubits, likely years away at scale
3. Unclear timeline: applications where quantum advantage has been conjectured but not proven

Chemistry and Materials Science

Chemistry simulation is widely considered the most promising near-to-medium-term application. The core problem is computing the ground state energy of molecules, which determines their chemical properties and reaction behavior.

Classical computers simulate quantum chemistry by approximating the wavefunction. These approximations break down for strongly correlated systems. Quantum computers can represent molecular wavefunctions exactly (using qubits to encode electron states), at a cost that scales polynomially rather than exponentially.

Current status: Small molecules (H2, LiH, BeH2) have been simulated on quantum hardware using the Variational Quantum Eigensolver (VQE). Industrially relevant molecules require error-corrected hardware.

Key target applications:

• Drug discovery: simulating protein-ligand binding to find candidate drugs
• Materials design: discovering new catalysts, superconductors, and battery materials
• Fertilizer production: the Haber-Bosch process uses 1-2% of global energy; a better catalyst could save enormously

# Example: computing H2 ground state energy with PennyLane
import pennylane as qml
from pennylane import qchem

# Build the H2 Hamiltonian
import numpy as np
symbols = ["H", "H"]
coordinates = np.array([0.0, 0.0, 0.0, 0.0, 0.0, 1.4])  # flat array, Bohr
H, n_qubits = qchem.molecular_hamiltonian(symbols, coordinates)

print(f"H2 Hamiltonian has {n_qubits} qubits and {len(H.operands)} terms")

Optimisation

Many real-world problems are combinatorial optimisation problems: finding the best solution out of an astronomically large set of possibilities. Logistics routing, financial portfolio construction, manufacturing scheduling, and supply chain management all involve this type of problem.

Quantum annealing (offered by D-Wave) is designed specifically for optimisation. Gate-based quantum computers can use QAOA (Quantum Approximate Optimisation Algorithm) for similar problems.

Current status: Quantum advantage for optimisation has not been conclusively demonstrated. Classical heuristics (simulated annealing, genetic algorithms) are highly competitive. Quantum optimisation may show advantage for specific problem structures.

Key target applications:

• Vehicle routing for logistics companies
• Portfolio optimisation in finance
• Traffic flow optimisation in cities
• Drug candidate screening (combinatorial chemistry)

# QAOA for MaxCut (partition graph to maximize crossing edges)
from qiskit_optimization.applications import Maxcut
import networkx as nx

# Simple graph: triangle
graph = nx.Graph()
graph.add_edges_from([(0, 1), (1, 2), (0, 2)])

maxcut = Maxcut(graph)
problem = maxcut.to_quadratic_program()
print(problem.export_as_lp_string())

Cryptography

Quantum computers threaten some widely used cryptographic systems. Shor’s algorithm can factor large integers and compute discrete logarithms in polynomial time; tasks that underlie RSA, elliptic-curve cryptography, and Diffie-Hellman key exchange.

Current status: Breaking real RSA-2048 keys would require millions of error-corrected qubits. Current gate-based quantum computers have at most a few thousand physical qubits (IBM’s largest processors are ~1,000-2,000 qubits; Google Willow used 105 qubits for its benchmarking demonstration), and error correction requires roughly 1,000 physical qubits per logical qubit. Cryptographically relevant quantum attacks are estimated to be at least 10-20 years away.

Post-quantum cryptography: NIST standardized the first set of quantum-resistant algorithms in 2024 (CRYSTALS-Kyber, CRYSTALS-Dilithium, and others). Organizations should begin transitioning to these algorithms now, regardless of when quantum computers become cryptographically relevant.

Quantum key distribution (QKD) uses quantum mechanics (not quantum computing) to distribute cryptographic keys with information-theoretic security. QKD is already deployed in some networks.

Machine Learning

Quantum machine learning (QML) is an active research area exploring whether quantum computers can speed up machine learning algorithms. Claims range from exponential speedups to modest advantages for specific data structures.

Current status: Most proposed quantum speedups in machine learning come with significant caveats: they require quantum RAM (not yet practical), assume specific data access models, or apply only to narrow problem classes. Demonstrating genuine QML advantage over classical ML on practical tasks remains an open problem.

Near-term approaches:

• Quantum kernel methods: use quantum circuits to compute kernel functions for SVMs
• Variational quantum circuits as neural network layers
• Quantum generative models (QGANs, quantum Boltzmann machines)

import pennylane as qml
import numpy as np

# A simple quantum kernel function
dev = qml.device("default.qubit", wires=2)

@qml.qnode(dev)
def quantum_kernel(x1, x2):
    """Compute <x1|x2> in feature Hilbert space."""
    qml.AngleEmbedding(x1, wires=[0, 1])
    qml.adjoint(qml.AngleEmbedding)(x2, wires=[0, 1])
    return qml.probs(wires=[0, 1])

x1 = np.array([0.5, 1.2])
x2 = np.array([0.3, 0.8])
similarity = quantum_kernel(x1, x2)[0]
print(f"Quantum kernel value: {similarity:.4f}")

Financial Modelling

Quantum computing has attracted interest in finance for Monte Carlo simulation, risk analysis, and option pricing. Quantum amplitude estimation can provide a quadratic speedup over classical Monte Carlo sampling.

Current status: Quadratic speedup for Monte Carlo is theoretically established. Whether this is practically useful before fault-tolerant hardware arrives is uncertain; the constant factors and overhead of quantum hardware are large.

Target applications:

• Option pricing and derivative valuation
• Value-at-risk calculations
• Portfolio optimisation (overlaps with optimisation category above)
• Fraud detection

Sensing and Metrology

Quantum sensing is separate from quantum computing but uses similar underlying physics. Quantum sensors use entanglement and superposition to measure physical quantities (magnetic fields, gravity, time) with precision beyond classical limits.

Current status: This is perhaps the most mature quantum technology category. Quantum sensors are already deployed in research settings and are closer to commercialization than quantum computers.

Applications:

• Navigation (quantum gyroscopes, accelerometers)
• Medical imaging (quantum MRI, magnetoencephalography)
• Defense (quantum radar, submarine detection)
• Resource exploration (detecting underground mineral deposits)

What Quantum Computers Are Not Good For

It is equally important to know where quantum advantage does not exist:

• General data processing, databases, and business software
• Searching sorted data (binary search is already optimal)
• Training large language models or classical deep learning
• Any problem that is already efficiently solvable classically

The common thread among problems with quantum advantage is that they involve quantum systems directly (chemistry), have specific mathematical structures (factoring), or allow interference to be exploited (search, phase estimation).

Timeline Summary

Application	Estimated Timeline	Current Readiness
Small molecule simulation	3-7 years (fault-tolerant)	Demonstrated on hardware (small scale)
Cryptography break	15-25 years	Early demonstrations only
Optimisation advantage	Uncertain	No conclusive demonstration
QML advantage	Uncertain	Active research, no clear win
Quantum sensing	Now	Deployed in research settings

Next Steps

• VQE Explained (the leading near-term algorithm for chemistry)
• Quantum Key Distribution (quantum-secure communication)
• Quantum Machine Learning Intro (getting started with QML)
• Quantum Computing vs. Classical (a detailed comparison of capabilities)