# A Many-Method Attack on the Many Electron Problem

## Flatiron Institute

Uniting our everyday world with the quantum realm requires tackling titanic numbers. A single penny contains 2 billion trillion atoms with a total of 70 billion trillion electrons whizzing around them. The behavior of those electrons produces many of the penny’s properties, such as its conductivity and even its shininess.

Taming these electrons would yield society-changing benefits, such as enabling the design and control of materials with desirable properties, such as high-transition-temperature superconductivity. The goal of the Center for Computational Quantum Physics (CCQ) is to help make that future a reality.

The CCQ faces mind-boggling numbers beyond just the plethora of electrons. Particles in a quantum system can exist in many different configurations, called states. Each electron, for instance, can have an upward or downward spin. Completely understanding a quantum system requires calculating the system’s wave function, which describes how particles are distributed over all possible states.

For example, even if we consider spin alone, a group of 10 electrons can be in any one of 1,024 states. For a penny’s worth of electrons, the number of states would be billions of trillions of digits long. And calculating a material’s overall wave function is further complicated by quantum mechanical entanglement, which means that electrons influence one another so strongly that they can’t be treated individually.

The CCQ, launched in September 2017, has already established itself as an international leader in developing the computational methods needed to solve the so-called ‘many electron problem.’ “The center has several angles of attack on this problem,” says center director Antoine Georges, who leads the CCQ along with co-director Andrew Millis. “We’re bringing together the best methods and people, and I think after a year and a half of existence, we have an impressive set of methods and software.” Each method has strengths and weaknesses as well as synergies with other approaches.

CCQ project leader Olivier Parcollet says the center’s scientists pursue so many methods “because we don’t know which one is best, and we don’t even know if any one of them will be the best.” For example, Parcollet’s work focuses on a method of tackling the many electron problem known as quantum embedding, which originated from research pioneered by Georges in the 1990s. The method leverages the fact that physicists are often only interested in the behavior of a small part of a quantum system. Instead of performing a detailed calculation across the whole system, quantum embedding performs high-level calculation on only the area of interest. The rest of the system is treated more simply, drastically streamlining many quantum problems.

“We’re bringing together the best methods and people, and I think after a year and a half of existence, we have an impressive set of methods and software.”

Quantum embedding has limitations, though. The calculations often require experimental validation to ensure that the problem wasn’t oversimplified. Also, even a small chunk of a quantum system can be too computationally taxing to compute using conventional methods, requiring the additional use of other methods.

Another approach avoids deterministic computations altogether. Dubbed ‘the Monte Carlo method’ after the Mediterranean casino, the approach uses random sampling to compute the answer to a problem. A famous example involves randomly throwing stones into a square with a circle inscribed inside. The fraction of rocks that fall within the circle provides a rough estimation of pi divided by four: The more stones thrown, the more accurate the estimate.

Conventional methods such as integration can compute pi more quickly. However, for more complex tasks, such as integrating mathematical equations with many variables, or dimensions, Monte Carlo often wins. “If we do that integration in high dimensions, the clever, faster things suddenly become super slow, and this ‘dumb’ way wins out,” says CCQ senior scientist and group leader Shiwei Zhang, who develops algorithms for Monte Carlo methods.

Quantum physicists run Monte Carlo calculations until they reach a desired accuracy. The method isn’t always viable, though. In most quantum systems, the solution requires computing an answer that is the slight difference between large positive and large negative contributions from a quantum system’s wave function. In these cases, the time required to compute an accurate solution becomes extremely large.

Whereas Monte Carlo leverages randomness, a relative newcomer directly computes solutions using novel mathematics called tensor networks. The approach compacts quantum problems by bundling information about a system into multidimensional arrays called tensors. Quantum entanglement links these tensors into a network.

Similar to quantum embedding, tensor networks take advantage of the fact that only a small fraction of the states in a large quantum system are relevant to any particular physical situation. The organization of information about the system, therefore, can be streamlined.

As an analogy, one might treat a proton and an electron orbiting it as a single atom rather than tracking the details of each particle’s motion separately. Tensor-network code leverages patterns in the structure of the wave functions to produce a compact representation of the most important states in a quantum system. This approach makes problems smaller and more manageable, similar to the way streaming websites compress video files.

This approach outperforms other methods in certain situations, such as when the system is large only along one dimension or has relatively simple interactions. In other cases, though, the computational requirements balloon too high to be worthwhile. “Even though we have issues, some of the problems we’re tackling can’t be touched at all by other methods without making heavy approximations,” says CCQ research scientist Miles Stoudenmire, who develops and deploys tensor-network methods.

Tensor networks often take advantage of patterns in a quantum system that are too convoluted and complex for a person to uncover. Artificial intelligence techniques such as machine learning can help uncover such patterns through a complementary set of techniques.

“Sharing code helps the whole field advance … We need to minimize the costs of testing new ideas. If a student needs to reinvent the wheel for every new project, the field will slow.”

Computer programs that best world champions at board games and train self-driving cars inspired CCQ associate research scientist Giuseppe Carleo to explore artificial intelligence approaches. Similar to those applications, artificial intelligence in quantum physics improves by ingesting information. Over time, the code learns how to impersonate a quantum system. Similar to tensor networks, Carleo’s methods create a compact representation of the important states of a quantum system. In February 2018, Carleo and his colleagues published a paper in *Nature Physics* that demonstrated that machine-learning techniques could drastically reduce the time needed to reconstruct a wave function based on experimental results. Systems that would typically require thousands of years to be reconstructed could be thoroughly analyzed in hours.

Sometimes, however, machine-learning techniques take more time than they save. Carleo and others are trying to make the code more efficient. “We’re the newborn in the field,” Carleo says. “There’s an explosion of things going on, but we have a lot left to do.”

A crucial part of the CCQ’s mission is making its code available to the public via open-source libraries. Carleo shares machine-learning code through NetKet, Stoudenmire maintains a library called ITensor for tensor networks, Parcollet leads the TRIQS project for interacting quantum systems, and Zhang heads the AFQMC library for Monte Carlo codes.

Sharing code helps the whole field advance, Parcollet says. “The complexity of the methods is growing,” he says. “We need to minimize the costs of testing new ideas. If a student needs to reinvent the wheel for every new project, the field will slow.”