Even though quantum computers are still in their crawling phase, computer scientists continue to push their limits. Recently, a group of scientists used a two-qubit quantum system to model the energies of a hydrogen molecule and found that using an iterative algorithm to calculate each digit of the phase shift gave very accurate results. Their system, while not directly extensible, has the potential to help map the energies of more complex molecules and could result in significant time and power savings compared to classical computers.

There are some situations, like quantum states of particles, that classical computers can only approximate, and they often do so quite poorly, with high degrees of uncertainty despite extensive computing time. For modeling quantum situations, there's no better tool than another quantum system that can be used to store and process the relevant data, as quantum computers can explore many possible states at once (though only one state can be measured as the outcome).

First, a quick rundown on quantum computers: while a regular computer processes and uses bits comprised of zeroes and ones, a quantum computer processes qubits, or bits that can store a superposition of both zero and one that will be in only one of these states when read out. In other words, when a qubit is measured, its superposition collapses to one of its available states (in this case, zero or one).

Modeling the energy levels of molecular hydrogen requires calculating the distance between the two atoms and the effect of different levels of excitation. A group of scientists designed a three-step method to handle this: encode the wave function of the molecule using qubits, simulate time evolution with quantum logic gates, and then use an iterative phase estimation algorithm to reduce the error, using the output of each trial as the input for the next. To get the energy, they calculated the phase shift of the molecule's wave function as a series of bits, and calculated one bit at a time with the qubit system.

To model a hydrogen molecule (two bonded hydrogen atoms), scientists injected two photons into an optical circuit, with each photon's polarization representing the encoding for a "control" qubit and a "register" qubit. The register represents an eigenstate, or one accepted energy configuration of the hydrogen molecule, and the control is in an equal superposition of a vertical and horizontal polarization.

The photons are then passed through a logic gate that represents an evolution of the wave function over time. The gate polarizes the control photon, forcing it to collapse into either a vertical or horizontal state. It also performs an operation on the register photon if the control comes out of the gate horizontally polarized, or leaves the register photon alone if the control becomes vertically polarized. The position of the control photon is measured and converted to a bit—0 for horizontal, and 1 for vertical. This represents one pass through the optical circuit.

The algorithm used 31 samples, or photon pairs, for each bit, and a "majority vote" was taken using all the samples—the resulting number is used as one digit in the binary expansion of the phase shift. The next iteration put all the photon pairs through the same circuit again, using the output of the first iteration as input for the second. With each new time around, the output was used to simulate a different time point in the evolution of the hydrogen molecule.

The least significant digit was always calculated first, then the next most significant, as this order allows for the best estimation of the most significant digits. The finished product looks something like 0.01001011101011100000, and varies depending on the excitation of the atoms and their distance from each other. Researchers found that they could calculate the phase shift out to 20 significant digits before the least significant digit stopped strongly favoring one value over another.

The results of the experiment mirrored very closely the energy curves of a hydrogen molecule as a function of the atomic separation, indicating that this is an excellent method for studying the energies of molecules. While the general approach to the problem, in particular the use of an iterative algorithm to estimate the phase of the wave function, proved accurate, the system that was used is only applicable to the hydrogen molecule. Simulating larger molecules requires more qubits and logic gates, which decrease the accuracy of measurements.

The precision in the hydrogen molecule system is high because the error introduced by one gate is always a constant, and can be corrected for by the classical method of the majority vote. If it were possible to look at the system after each gate, correct the error, and continue on, large systems requiring multiple logic gates would work pretty well. However, a quantum system can only be observed once it has run its course, and each logic gate roughly doubles the error each time the photons pass through it. Therefore, some new quantum correction techniques will have to be introduced before quantum computers can take on larger molecules. Despite these limitations, the work is an important demonstration of the promise of quantum computing, and shows that the techniques we already have can be put to practical use.

Nature, 2010. DOI: 10.1038/NCHEM.483