How Many Qubits Are Needed to Accurately Simulate Hydrogen Molecules?

[jimgraber]

Recently the NIST group demonstrated a two qubit programmable quantum computer,
http://arxiv.org/abs/0908.3031
And another consortium used a two qubit computer in an iterative fashion to compute some properties of the hydrogen molecule.
http://arxiv.org/abs/0905.0887

How many qubits are required in principle to fully describe a hydrogen atom? A hydrogen molecule?
TIA
Jim Graber

 

[SpectraCat]

Recently the NIST group demonstrated a two qubit programmable quantum computer,
http://arxiv.org/abs/0908.3031
And another consortium used a two qubit computer in an iterative fashion to compute some properties of the hydrogen molecule.
http://arxiv.org/abs/0905.0887

How many qubits are required in principle to fully describe a hydrogen atom? A hydrogen molecule?
TIA
Jim Graber

I guess it depends what you mean by "fully describe". As I understand it, you need 1 qubit for each independent particle that you want to simulate quantum mechanically. So, if you treat nuclei classically (a la BO separation), then you need 1 qubit per electron. I suppose if you wanted to include quantum nuclear effects, you could add a qubit for each nucleus that you want to simulate quantum mechanically .. say to study proton tunneling. Of course, I am not sure they know how to implement more than two qubits yet, although I have seen Martin Gruebele talk about the theoretical possibility of as many as 8 laser-based qubits before.

 

[f95toli]

8 qubits have already been implemented using NMR, about 7 or 8 years ago. There have also been a few demonstrations of Shor's algorithm using the same technique.
However, the problem with NMR is that it does not scale very well, meaning it is extremely unlikely that one would be able to build a practical QC using this technique.

Also, it is not correct to say that one need one qubit per electron. Normally when people talk about QC (algorithms etc) we refer to systems where the qubits are really bits, i.e. two state systems (that can be put in a superposition, but they are still bits). Hence, they are not really different from classical computers in that respect. This means that you would need a QC with at least a few hundred qubits before it would be more useful than a classical computer.

However, there are other approaches. Most of them involves using one quantum system to "emulate" another, this is what e.g. the Brisbane group did recently and it is also -roughly- the approach used by D-Wave Systems with their adiabatic QC.
But neither of these techniques are what one would normally refer to as quantum computing as such (there are no proper gate operations etc)

 

[SpectraCat]

8 qubits have already been implemented using NMR, about 7 or 8 years ago. There have also been a few demonstrations of Shor's algorithm using the same technique.
However, the problem with NMR is that it does not scale very well, meaning it is extremely unlikely that one would be able to build a practical QC using this technique.

Also, it is not correct to say that one need one qubit per electron. Normally when people talk about QC (algorithms etc) we refer to systems where the qubits are really bits, i.e. two state systems (that can be put in a superposition, but they are still bits). Hence, they are not really different from classical computers in that respect. This means that you would need a QC with at least a few hundred qubits before it would be more useful than a classical computer.

However, there are other approaches. Most of them involves using one quantum system to "emulate" another, this is what e.g. the Brisbane group did recently and it is also -roughly- the approach used by D-Wave Systems with their adiabatic QC.
But neither of these techniques are what one would normally refer to as quantum computing as such (there are no proper gate operations etc)

Sorry, I should have made it clear that my response was confined to laser-optical quantum computers, as they were implemented by the Brisbane group for quantum simulations. In that case, it really is one qubit per electron, but as you say, it is not really a digital computer implementation, but rather more like an analog one, where the design of the entire computer must be changed for each different calculation.

 

[jimgraber]

My question is really aimed at how many qubits are needed, not how or if it can be done. But I want to provide a complete description of the system, not just solve for the energy levels. Also I am thinking in terms of non-relativistic quantum mechanics (NRQM), not quantum field theory (QFT). In other words, I am thinking of the hydrogen atom as a two body problem, (which could equally well be positronium) and the hydrogen molecule as a four body problem.

I have another question: In the case of the hydrogen atom, can you reduce the dimensions of the problem by exploiting spherical symmetry?
Does this reduce the number of qubits required?
Thanks again,
Jim Graber

 

[f95toli]

My question is really aimed at how many qubits are needed, not how or if it can be done. But I want to provide a complete description of the system, not just solve for the energy levels. Also I am thinking in terms of non-relativistic quantum mechanics (NRQM), not quantum field theory (QFT). In other words, I am thinking of the hydrogen atom as a two body problem, (which could equally well be positronium) and the hydrogen molecule as a four body problem.

But that is not a well defined question. "Conventional" quantum computers are digital, so there really is no such thing as "minimum" number of bits for a given problem (beyond the fact that you have to be able to represent all the numbers etc, e.g. you need at least 8 qubits in order to use Shor's algorithm to factorize the number 15). A practical quantum computer would just act like a very fast digital signal processor for certain problems(sorting, factorization etc). Hence, there is -from a "problem solving" point of view- no fundamental difference between a QC and conventional computer, except for the speed of execution.

Also, even if we extend the concept of a QC to the work done in Brisbane etc we are still relying on finding systems that are described by similar equations; and the only system that is decribed by exactly the same equations as a hydrogen atom is another hydrogen atom.

It is a bit like calculating the properties of a shock absorbed for a car by using an analogue electronic circuit (op-amps, resistors etc); if you build the right circuit the differential equations will be identical (this is the whole idea behind analogue computers) but that does not mean that an electronics circuit could ever be used to "calculate" all the properties of a real mechanical system.

 

[jimgraber]

Oh yes, I see I have not phrased my question correctly.
I guess I may be really looking for something like the number of qubit registers required, or the dimensions of the problem. Maybe I should withdraw my question and try to rephrase it. I am now more than ever puzzled by the use of iterative techniques to compute multiple bits with only a two qubit computer. Maybe you can shed some light on this. I understand that they moved other components around and sort of re-tuned the machine in between iterations. But I don't understand how it works overall. Is there an analog-like convergence involved?