# Idea¶

The original idea of quantum computing was motivated by the challenge of simulating quantum systems, as the requirements in computational power and in memory scale exponentially with the system size.

In 1982, Richard Feynman explained his vision of a quantum computer that would be able to simulate quantum systems in a far more efficient way that classical computers, because it would be of the same nature that the system to simulate: “Let the computer it-self be built of quantum mechanical elements which obey quantum mechanical laws.” [39]

The kind of computer that Feynman envisaged was a universal quantum simulator,
made of a circuit of universal quantum gates,
and is currently refered to as a **digital quantum simulator**.
The Hamiltonian of the system to be simulated is approximated by a circuit,
with the essential condition that for the simulation to be efficient,
this approximation, including the preparation of the initial state,
must be done with polynomial resources.

A current “standard” threshold is considered to be at about \(N = 40\) : storing \(2^{40} \approx 10^{12}\) numbers as 32-bit floats takes about 4 TB of memory 1. Approximate statistical methods, namely Monte Carlo algorithms, can be used for many problems, but not for all quantum systems, especially for fermionic systems, as for example the electronic structure of a molecule.

But the power of quantum computing is not restricted to these simulations.

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- 1
Preskill mentions a “milestone” of 50 qubits [108], that is “beyond what can be simulated by brute force using the most powerful existing digital supercomputers”. This estimation relies on classical simulations of typically a lattice of 7 x 7 qubits [19]. It is practically the same then the 40 qubits named above.

In Google’s 2019 quantum supremacy demonstration [7] it is also a number of 43 qubits that is given as the current maximum number of qubits to be simulated on the Jülich supercomputer: “Above this size, there is not enough random access memory (RAM) to store the quantum state”.