Microscopic Era makes progress in quantum coding and chemistry

(hereinafter referred to as "MicroEra") has used quantum coding technology to reveal the simple and beautiful quantum entanglement structure in chemical systems, and based on this, the approximate energy of the ground state of the system can be solved quickly. More details of the paper can be found at the end of the article. This breakthrough is expected to cause significant changes in both computational chemistry and quantum algorithms.

01Solving Schrödinger's equation

The establishment of quantum mechanics, the Schrödinger equation, provides the basic principles for describing the electronic structure in chemical systems. However, the exact solution of Schrödinger's equation is extremely difficult. For hydrogen atoms and hydrogen-like ions with only one electron, we can obtain analytical solutions. But for even a two-electron hydrogen molecule H2, we have so far had to rely on extremely tedious numerical methods.

In desperation, physicists/chemists have come up with an approximate solution: the Hatree-Fock approximation. In simple terms, the electrons are isolated one by one and solved individually, and then put together in an appropriate way. If the electrons are relatively independent of each other, the results of doing so are not bad; however, if there is a strong correlation between the electrons, or quantum entanglement, the results tend to be poor. As shown in the figure below, the Hartree-Fock approximation gives H2 energies that are closer to the exact value when the bond length is small, but far apart when the bond length is large. Various improvements have been made to the Hatree-Fock approximation, but they have not been able to solve the problem at all.

There is a view that the fermionic properties of electrons are at the root of this conundrum. One of the bizarre properties of fermions is that exchanging two identical fermions brings a negative sign that has to be exchanged again to be restored. In contrast, particles that are more consistent with our intuition are called bosons, such as photons. As early as 1928, Jordan and Wigner found a way to convert ordered fermionic systems into bosonic systems. After the rise of quantum computing, two other transformations were invented to convert fermions into bosons because the actual quantum bits constructed were all bosons. However, after doing these transformations, the expressions for the Hamiltonian quantities of electron interactions (which can be simply understood as energy) seem to be more complicated, and the corresponding Schrödinger equation is also more complicated. This makes the search for the electron entanglement structure even more confusing.

02Using quantum coding techniques

The key to the initial solution of this problem was the use of quantum coding techniques developed by quantum informaticists at the end of the last century, in particular, the stabilizer formulation. For physicists, there is a very straightforward way to understand this formulation: a stabilizer can be roughly understood as "a set of mutually permissive mechanical quantities"; and a stabilizer state is their "common eigenstate" with an eigenvalue of 1. Only for quantum bits, the stabilizer formulation has a richer and more elegant mathematical structure. With the "eye of the beholder", the entanglement structure of the electronic system can be understood at a glance. Here we still use the ground state of H2 molecule to illustrate briefly.

We know that the H2 molecule has two atomic orbitals, and considering the electron spin, there will be four spin orbitals. Accordingly, we can describe it with four quantum bits. The number of quantum bits can be cut (tapering) to two by using symmetry. This step is not necessary, but it makes the result more obvious. This gives us the following Hamiltonian (bond length of 0.74 Å).

H = -1.0534210769165204 * II

+ 0.39484436335590356 * IZ

- 0.39484436335590367 * ZI

+ 0.1812104620151969 * XX

- 0.011246157150821112 * ZZ.

where I is a 2*2 unit matrix and X,Y,Z are Pauli matrices. When the bond length is large, say at 2.8 Å, the Hamiltonian quantities are as follows.

H = -0.8284676561247681 * II

+ 0.016170000066607376 * IZ

- 0.016170000066607328 * ZI

+ 0.2930431286727852 * XX

- 0.0001469354633982234 * ZZ.

Make sure you pay attention to the change in the magnitude of the coefficients! Now I will tell you that for the former, the stabilizers are -IZ and ZI and the stabilizer state is |01>, i.e. the direct product state/Hatree-Fock state; for the latter, the stabilizers are -XX and ZZ and the stabilizer state is a two-bit entangled state: the

If you didn't see it at a glance, it means you still need to go back to Tai Shang Lao Jun's Bagua Furnace and practice for a while. Of course maybe the problem is not as simple as I make it out to be. For chemists have derived similar expressions for Hamiltonian quantities more than 20 years ago, but they don't seem to have been able to see the entanglement behavior of electrons from them.

Acting the Hamiltonian at each distance on each of these two states and taking the smaller of them gives the following curve.

It can be seen that by using a simple two-bit entangled state, almost all of the electron correlation energy has been recovered! One has to wonder how simple it really is!

You may suspect that the simple entanglement behavior is due to the simplicity of the H2 molecule. The MicroEra team further explored the LiH and BeH2 systems and found a very similar entanglement structure. In simple terms, it means that these molecules will first fill the extra electrons into the orbitals according to the Hartree-Fock method, and then entangle the last two active electrons in a similar way to H2. The team's next step is to study more complex molecules in order to find even more complex forms of entanglement.

You might also ask, you didn't get the exact ground state energy in the diagram using the entangled structure! That's true. Quantum entanglement is only the skeleton of quantum computing, and requires some more delicate manipulation to flesh and blood to bring out its full power. The same is true for nature. In fact, for the H2 molecule, we only need to do some perturbations on top of that to easily get the exact ground state: the

In turn, we can say that the formation process of entangled structure is actually embedded in the ground state.

Finally, to conclude, we borrow some ideas from the famous Chinese physicist Mr. Xiaogang Wen. According to him, nature is actually a sea of intertwined quantum bits, and the so-called fermionic properties are just a superficial phenomenon presented by the collective motion of these quantum bits. In the chemical system here, nature first entangled quantum bits two by two and then disguised them as fermions, and blinded us with this double encoding for almost a century. Now we are finally beginning to gradually unravel the mysteries of nature using quantum computing and quantum information technology, how could this not be exciting?

Author Bio.

Zuo Fen, Director of Algorithm, Shanghai MicroEra Digital Technology Co. D. in theoretical physics from the University of Science and Technology of China (USTC). She received her undergraduate degree from the Junior Class of USTC and worked as a postdoctoral fellow at the Institute of Theoretical Physics, the Institute of High Energy Physics, and the National Institute of Nuclear Physics of the Chinese Academy of Sciences. His main research interests are particle physics, string theory, quantum computing and related algebraic structures.

Thesis information：

Stabilizer Approximation, Xinying Li, Jianan Wang, Chuixiong Wu, Fen Zuo,

https://arxiv.org/abs/2209.09564