Andrew, a quantum information researcher at the University of Maryland? Andrew Childs said: "This is one of the reasons why the weather is unpredictable and the complicated fluid flow is difficult to understand. If we can understand these nonlinear dynamics, we can solve some difficult calculation problems. 」
This is not a fantasy, it may come true soon. In the independent research published in June 5438+065438+ 10, both the team led by Childs and the team of MIT described a powerful tool that can make quantum computers better simulate nonlinear dynamics.
Compared with traditional computers, quantum computers can make use of quantum phenomena to perform some specific calculations more effectively. It is precisely because of these functions that quantum computers can quickly overturn complex linear differential equations. For a long time, researchers have been hoping to solve nonlinear problems through clever quantum algorithms.
Although the specific methods used in these two studies are quite different, they both use a new method to disguise nonlinearity as a more understandable linear approximation set. So there are two different ways to solve nonlinear problems with quantum computers.
MáriaKieferová, a quantum computing researcher at the University of Technology in Sydney, said: "The interesting thing about these two papers is that they found a mechanism. Given some assumptions, they have an effective algorithm. This is really exciting, and both studies use very clever technology. 」
It's like teaching a car to fly.
For more than ten years, quantum information researchers have been trying to use linear equations as the key to solve nonlinear differential equations, but it has been difficult to make progress and finally made a breakthrough at 20 10. Dominic Berry of Macquarie University in Sydney established the first algorithm to solve exponential linear differential equations on quantum computers instead of traditional computers. Soon, Berry's attention turned to nonlinear differential equations. Berry said, "We have done some work before, but the efficiency is very low. 」
Andrew Childs of the University of Maryland led one of two research efforts to enable quantum computers to better simulate nonlinear dynamics. His team's algorithm uses a technique called "Kalman linearization" to transform these nonlinear systems into a series of more understandable linear equations.
The problem is that the physics on which quantum computers are based is linear in nature. "It's like teaching a car to fly," said Bobak Kiani, co-author of the MIT study. 」
Therefore, the trick is to find a way to transform a mathematical nonlinear system into a linear system. Childs said: "We want to have some linear systems, because this is the function of our toolbox. The two teams did this in two different ways.
In 1960s and 1930s, Childs' team used an outdated mathematical technique, namely Kalman linearization, to transform nonlinear problems into linear equations. Unfortunately, there are infinitely many equations in the system of equations. Researchers must figure out which equations they can delete from them to get a good enough approximation. "Stop at the equation 10? Or equation 20? " Nuno Loureiro, a plasma physicist at the Massachusetts Institute of Technology and co-author of the Maryland study, said. The team proved the nonlinear equations in a specific range, and they can truncate the infinite equations and solve the equations.
The MIT team's paper uses different methods to model the nonlinear problem as Bose-Einstein condensation. This is a state of matter, and the interaction within the particle group near absolute zero leads to the same behavior of each individual particle. Because particles are interrelated, the behavior of each particle will affect other particles and feed back to particles with nonlinear circulation characteristics.
The method of MIT is to use Bose-Einstein mathematical method to link nonlinearity with linearity, so as to simulate this nonlinear phenomenon on quantum computer. Therefore, by imagining each nonlinear problem as a different pseudobose-Einstein condensate, the algorithm deduces an effective linear approximation. "Give me your favorite nonlinear differential equation, and I will build a Bose-Einstein condensate for you to simulate it," said Tobias Osborne, a quantum information scientist at Leibniz University in Hanover who was not involved in the two studies. "This is an idea that I like very much. 」
The algorithm of the team led by MIT models any nonlinear problem as Bose-Einstein condensation, which is a strange state of matter in which interconnected particles behave the same.
Berry thinks these two papers are important in different ways (he is not involved in either of them). He said: "But in the end, their importance shows that it is possible to obtain nonlinear behavior by using these methods. 」
Know your limits
Although these achievements are important, they are still only the first step to solve nonlinear systems. Before the hardware needed to realize these methods becomes a reality, more research may focus on analyzing and perfecting each method. Kifrova said: "With these two algorithms, we can really look to the future. But if we want to use them to solve practical nonlinear problems, we need a quantum computer with thousands of qubits to minimize errors and noises, which is far beyond the existing possibilities.
At the same time, these two algorithms can only deal with mild nonlinear problems. Maryland's research accurately quantifies how many new nonlinear parameters R can be handled, and R represents the ratio of nonlinearity to linearity of the problem, that is, the trend of the problem becoming nonlinear and the friction of the system on the track.
"Childs' research is very strict in mathematics, including when it can be used and when it cannot be used. Osborne said, "I think this is really interesting. This is the core contribution. 」
According to Kiani, the research led by MIT did not strictly prove any theorem that restricted its algorithm. But the team plans to further understand the limitations of the algorithm by running small-scale tests on quantum computers, and then deal with more challenging problems.
The most important warning of these two techniques is that quantum solutions are fundamentally different from classical solutions. Quantum states correspond to probabilities, not absolute values. For example, you don't need to observe the airflow around various parts of the jet fuselage, but get the average speed or detect stagnant air. Kiani said: "The fact that the result belongs to quantum mechanics means that a lot of work still needs to be done to analyze this state. 」
In the next five to ten years, researchers are bound to test many successful quantum algorithms for practical problems, but it is important not to over-promise what quantum computers can do. Osborne said, "We will try all kinds of things. Moreover, if we consider the limitations, it may limit our creativity. 」
Link: https://www.quantazine.org/new-quantum-algorithms-finally-crack-nonlinear-equations-20210105/