For decades, the quantum algorithms conversation has been dominated by two landmark results: Shor's algorithm for integer factorisation and Grover's algorithm for unstructured search. Both offer provable quantum speedups over the best known classical algorithms, but both require fault-tolerant quantum computers with millions of qubits to run at practically useful scales.

A new generation of quantum algorithms is changing the conversation. Designed for the noisy intermediate-scale quantum (NISQ) devices available today — and the early fault-tolerant systems expected within the next five years — these algorithms are targeting specific, commercially valuable problems where quantum computers may offer a practical advantage sooner than previously expected.

Variational Quantum Algorithms

The most prominent class of near-term quantum algorithms is variational quantum algorithms (VQAs), which use a hybrid quantum-classical loop to optimise a parameterised quantum circuit. The quantum computer evaluates a cost function; a classical optimiser adjusts the circuit parameters; and the loop iterates until convergence.

The Variational Quantum Eigensolver (VQE) applies this approach to quantum chemistry, finding the ground-state energy of molecular Hamiltonians. The Quantum Approximate Optimisation Algorithm (QAOA) applies it to combinatorial optimisation problems. Both have been demonstrated on real quantum hardware, though the question of whether they offer a practical advantage over classical methods remains open.

Quantum Linear Systems

The HHL algorithm (Harrow, Hassidim, Lloyd) offers an exponential speedup for solving systems of linear equations — a fundamental operation in machine learning, fluid dynamics, and financial modelling. However, the speedup comes with significant caveats: the input must be efficiently loadable into quantum memory (the quantum RAM problem), and the output is a quantum state that must be sampled rather than read directly.

Recent work has focused on identifying specific problem instances where these caveats do not apply, and where the HHL speedup translates to a genuine practical advantage. Quantum linear systems algorithms for portfolio optimisation and credit risk modelling have shown promise in simulation studies.

"The question is not whether quantum algorithms are theoretically faster. The question is whether they are practically faster for the specific problems that matter to businesses. That is a much harder question to answer."
— Scott Aaronson, University of Texas at Austin

Quantum Simulation

The application where quantum computers have the clearest theoretical advantage — and where near-term devices are already producing useful results — is quantum simulation: using quantum computers to simulate quantum systems. This is the application Richard Feynman originally envisioned when he proposed the idea of quantum computing in 1982.

IBM, Google, and IonQ have all demonstrated quantum simulations of small molecular systems that match or exceed the accuracy of classical methods at comparable computational cost. As qubit counts and fidelities improve, quantum simulation is expected to become the first domain where quantum computers deliver unambiguous commercial value — with applications in drug discovery, materials science, and catalyst design.

Error Mitigation: Bridging NISQ and Fault Tolerance

A critical enabling technology for near-term quantum algorithms is quantum error mitigation — techniques that reduce the impact of noise on quantum computations without the overhead of full error correction. Methods such as zero-noise extrapolation, probabilistic error cancellation, and symmetry verification can significantly improve the accuracy of NISQ computations at the cost of additional circuit runs.

IBM's Qiskit Runtime and Quantinuum's TKET both include built-in error mitigation modules, making these techniques accessible to application developers without requiring deep expertise in quantum error theory. The combination of error mitigation and variational algorithms is enabling useful quantum computations on today's hardware that would not be possible without these techniques.