# Why was Feynman hesitant about simulating fermions with a quantum computer?

Richard Feynman has a number of foundational publications from the early-mid 80's on quantum computing that I continue to read with awe and inspiration. As earlier discussed, the 1985 article "Quantum Mechanical Computers" is concerned with dissipative properties of quantum computers, and leans heavily into Bennett, Toffoli, Fredkin, and friends while using ladder operators to construct reversible circuits to mitigate the limitations estopped by Landauer.

However in his earlier 1982 talk on "Simulating Physics with Computers" (the one ending with the rallying cry that "Nature isn't classical, dammit!") Feynman appears more motivated to compare and contrast the capabilities of classical computers with those afforded by quantum mechanics. In the first half of the paper he also introduces ladder operators, and somehow envisions a quantum computer or quantum simulator as a spin-lattice with sites that are either occupied or not, with the ladder operators creating and annihilating therebetween.

While considering the capabilities of such a quantum computer operating on such a lattice, Feynman conjectured:

The question is, if we wrote a Hamiltonian which involved only these operators, locally coupled to corresponding operators on the other space-time points, could we imitate every quantum mechanical system which is discrete and has a finite number of degrees of freedom? I know, almost certainly, that we could do that for any quantum mechanical system which involves Bose particles. I'm not sure whether Fermi particles could be described by such a system. So I leave that open. Well, that's an example of what I mean by a general quantum mechanical simulator. I'm not sure that it's sufficient, because I'm not sure that it takes care of Fermi particles. (Emphasis added).

This hesitancy on Feynman's part about fermionic simulation confused David Deutsch, who in another landmark paper "Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer" stated of Feynman's quantum simulator "I do not understand why Feynman doubts that it can simulate fermion systems".

Then to my question, has anyone offered a reason:

• Why was Feynman hedging his guess about fermionic Hamiltonian simulation with a quantum computer?

Was it because he limited his perspective to such ladder operators (as fermions and bosons have different commutation relationships between creation and annihilation)?

Read very broadly, and using modern language, the former 1982 talk of Feynman appears to conjecture that (local) Hamiltonian simulation is in BQP, at least for bosons, while the latter 1985 paper anticipates a proof that Hamiltonian simulation is BQP-hard. We might also generously read the 1985 paper as outlining a proof at least that BPP$$\subseteq$$BQP, with the 1982 talk as conjecturing that BPP $$\subsetneq$$ BQP, both of which were also conjectured/explored by others, such as Benioff and Manin, at roughly the same time.

This might be ascribing too much to Feynman and the pioneers of the early '80s, but these comments about fermionic simulation are at least a curiosity.

I don't know how to reconcile Feynman's hesitancy about fermions with his description of a universal quantum simulator as spin-lattices of particles with spin 1/2 (e.g. of fermions). That is, he appears to appreciate that qubits could be fermions, but then doubts whether fermionic simulation is possible. But that's a misunderstanding I think on my part, as he also appears comfortable describing Bell's inequality using photons (bosons).

My guess is that this has to do with worrying about correctly capturing the antisymmetric properties of fermions (that when you swap two fermions, the wave function acquires a $$-1$$ phase). There is an approach to classically simulating quantum systems, called quantum monte carlo. In this approach sometimes one runs up against the problem that it looks like you need to have negative probabilities. This is called the sign problem. While bosonic system can have sign problems, for fermions the anti-symmetry directly leads to lots of negative sign probabilities. So likely Feynman was worried that a similar problem would arise in his simulation of fermions.