Quantum Simulation and Chemistry

Simulating quantum systems, like molecules, is the job quantum computers were born for. The memory a normal computer needs grows exponentially with the size of the system. A quantum computer's does not. Small molecules have been demonstrated on real hardware. Chemistry that is hard for normal computers has not. This chapter shows exactly what a chemistry circuit measures, and where today's honest frontier sits.

Why is simulating nature the natural job for a quantum computer?

In 1982 Richard Feynman made a simple point. Nature is quantum. So copying it faithfully on normal hardware carries an exponential cost. He proposed building quantum hardware to do the job directly.

The math behind his point is short. Picture a quantum system made of n parts, each with two states — electron spins, for example. It is described by amplitudes: one number for each possible arrangement of the parts. (Square an amplitude's size and you get a chance.) There are 2ⁿ arrangements, because each part doubles the count.

Worked example, step by step. Take n = 50 spins.

The state needs 2⁵⁰ = 1,125,899,906,842,624 amplitudes. That is about 1.1 × 10¹⁵.
Each amplitude is a complex number. Stored in "double precision," it takes 16 bytes (two 8-byte numbers).
Total memory: 1,125,899,906,842,624 × 16 = 18,014,398,509,481,984 bytes ≈ 18 petabytes.

That is beyond any single machine on Earth, for just 50 spins. At n = 60 it is a thousand times more. A quantum computer holds the same state with 50 qubits (quantum bits). That is because the qubits simply are a quantum system of the same kind.

An everyday example: to predict how a flag waves, you could write down the position of every thread. Or you could just hang a flag in the wind and watch. The quantum computer is like the flag in the wind. Where the picture breaks: watching a flag shows you everything at once. Reading a quantum computer only gives you one sample per run, so you must repeat it many times.

This is the one application where the data loading problem from chapter 1 mostly disappears. The "input" is a short description of a molecule — which atoms, and where. It is not a billion-record database. That is why chemistry and materials pass filter question 2, while search and machine learning struggle with it.