# Feynman's; Path Integral formalism on Quantum Computation

#### sardar

I have a nutty feeling that I can apply the Feynman Path Integral approach to calculate the efficient Quantum Computation paths on a Poincare or a Bloch sphere, can anyone help me with a formal introduction to teh path integral approach of QM, so that I can see ways I can apply it to Quantum Computation.

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#### StatMechGuy

Schulman's "Techniques and Applications of Path Integration" is a good one, although I'm not sure how you would use the path integral in quantum computation...

#### quetzalcoatl9

i think one way is this: if you work out the density matrix, what you get is something that looks like a classical polymer of masses connected by a harmonic potential that is temperature dependent. in the high temperature limit the bead polymer collapses to a single point particle.

i dont know a whole lot about this, but i think you simulate things classically (i.e. Metropolis monte carlo with a classical force field) but when you calculate the interactions you include all possible point distributions between the masses. your average observable are calculated over this distribution. obviously the more you have (large Trotter number) the more accurate your representation will be, but more expensive computationally.

example: http://cat.inist.fr/?aModele=afficheN&cpsidt=1354259

#### Rach3

Well, functional path integrals are for continuous systems, while the standard "unitary gate" formalism of quantum computing is 'discrete' (finite-dim. Hilbert space). The integrals dx_n would simply reduce to discrete sums, which doesn't give you much of an edge. Maybe some other computing system could use them - maybe one of the topolgical ones? (TQFT was shown to be equivalent to a universal quantum computer, and it's a QFT. The knot theoretic Jones Polynomial algorithms are supposed to be related.) I haven't a clue.

#### Rach3

sardar - The path integral formalism is introduced in any textbook on QFT (or to a lesser degree, QM).

#### quetzalcoatl9

it is one thing for the path integral formalism to give you nice and neat analytical results, but considering "all possible paths" in the same sense numerically is not feasible since a) you will never be able to consider each path in a finite representation and b) it is numerically unstable since the whole benefit hinges upon getting most of those paths to cancel - but taking the difference of two very large numbers on a computer tends to make errors accumulate. in other words, you are never going to actually calculate the quantum action integral - or worse - a quantum partition function.

look at the reference i gave above on bead polymers, that is what everyone practically uses in the context of a Monte Carlo method

#### sardar

Rach3 said:
Well, functional path integrals are for continuous systems, while the standard "unitary gate" formalism of quantum computing is 'discrete' (finite-dim. Hilbert space). The integrals dx_n would simply reduce to discrete sums, which doesn't give you much of an edge. Maybe some other computing system could use them - maybe one of the topolgical ones? (TQFT was shown to be equivalent to a universal quantum computer, and it's a QFT. The knot theoretic Jones Polynomial algorithms are supposed to be related.) I haven't a clue.

I was thinking to apply path integrals over a Bloch Sphere where an efficient quantum computation is a path from the initial to the final state, so if we can mathematically describe a way to find say a minimal distance path(e.g a geodesic over the sphere) then may be we can define the efficient ways to do quantum computation over the sphere.

Double posting

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#### Epicurus

The connection for path integrals to statistical systems is the density matrix as quetzalcoatl9 mentioned. I am involved in path integral monte carlo techniques and the path integral measure is only really meaningful for continuous systems. The discrete case for which you are talking is an NP problem analogous to the travelling salesman problem, the only way in which you could approximately solve is assuming that for a large number of qubits the path can be approximated as continuous. There is a rev. mod.phys. by D.M. Ceperley which outlines the path integral method in statisitical mechanics, and the best book I have seen on path integrals and statisitical mechanics is by the man himself, R. Feynman's Statistical mechanics.

it is one thing for the path integral formalism to give you nice and neat analytical results, but considering "all possible paths" in the same sense numerically is not feasible since a) you will never be able to consider each path in a finite representation and b) it is numerically unstable since the whole benefit hinges upon getting most of those paths to cancel - but taking the difference of two very large numbers on a computer tends to make errors accumulate. in other words, you are never going to actually calculate the quantum action integral - or worse - a quantum partition function.
Yes we can, that is what I do!

#### quetzalcoatl9

Epicurus said:
Yes we can, that is what I do!
Epicurus, correct me if I'm wrong (since I am not an expert in this kind of calculation) but I thought that by using a Metropolis monte carlo scheme you get around having to calculate the actual partition function since it is only the energy change that is considered, i.e.:

$$\frac{Pr(i)}{Pr(j)} = \frac{\frac{e^{-\beta E_i}}{Z}}{\frac{e^{-\beta E_j}}{Z}} = \frac{e^{-\beta E_i}}{e^{-\beta E_j}} = e^{-\beta(E_i - E_j)} = e^{-\beta \delta E}$$

in which case the paritition function drops out and you don't have to worry about it - just try a configuration and accept it with the above probabilty. The actual transition probability from i -> j can be shown from the above along with the condition of detailed balance.

At what point in time do you actually calculate the paritition function? Any averages that you need you can get from the already boltzman-weighted configurations, no?

#### Epicurus

quetzalcoatl9, you are right. There is no need to calculate Z as we are only interested in observables. The partition function in itself is a physically meaningless quantity.

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