NISQ Algorithms and VQE

John Preskill coined "NISQ" in 2018 — Noisy Intermediate-Scale Quantum — to name the era we are in and, just as importantly, to set expectations about it. Intermediate-scale means roughly 50 to a few thousand qubits: past the point where a laptop can simulate the state vector, short of the millions error correction demands. Noisy means no error correction, so errors accumulate through the circuit unchecked.

The second word is the one that binds. With gate error rates around 10⁻³ for two-qubit gates on good hardware, a circuit of depth D on a handful of qubits accumulates total error roughly D × (number of gates per layer) × 10⁻³. Once that product approaches 1, the output is noise. In practice that caps useful circuits at hundreds of two-qubit gates, and it is a hard cap: no amount of clever compilation gets you past it, because the limit is physical.

Note what this does to the algorithms in the rest of this track. Shor's algorithm on a cryptographically interesting number needs circuit depth in the billions. Grover's needs √N sequential iterations, each containing an oracle circuit, so any instance large enough to be interesting needs depth far beyond NISQ reach — and worse, Grover's speed-up is quadratic, so noise-induced slowdown eats it easily. Neither is a NISQ algorithm, not by a factor of a thousand. Depth, not qubit count, is the wall.

So the NISQ question is: what can you compute with a wide but very shallow circuit? The dominant answer has been the variational family — VQE for chemistry and materials, QAOA for combinatorial optimisation, and quantum machine-learning variants. The shared idea is to make the quantum circuit as shallow as possible and hand everything else to a classical computer, running the two in a loop. It is a genuinely reasonable response to the constraint. Whether it delivers a practical advantage is still, honestly, unresolved.