Current quantum computers are limited by noise, which severely affects their ability to run long quantum circuits. Thus, most known quantum algorithms are out of reach for such machines. This is even true for early-fault tolerant devices that may be available in a couple of years. Thus, if quantum computers are supposed to solve meaningful problems in the near future, we need to understand the power and capabilities of short quantum circuits. What kind of problems can be solved using short quantum circuits and which can't? Which currently known applications can be made more efficient such that they can be solved using short quantum circuits? What tools are needed to make progress in these questions?
To better comprehend the power of short quantum circuits, we try to answer how generic short quantum circuits can be distinguished from deep ones. More concretely: To which extent can a short quantum circuit look like a Haar-random unitary? We are tackling this questions using the theory of random quantum circuits and study connections to many-body physics, as well as to cryptographic and complexity-theoretic concepts. Beyond circuit depth, we are also interested in what other resources are needed to make a circuit look sufficiently random, for instance magic / non-stabilizerness.
In another line of research, we develop efficient methods for learning properties of quantum many-body states. This mainly concerns the theory of classical shadows, where random quantum circuits (RQCs) are extremely helpful. Here, fundamental insights can often be turned into drastical savings in the used quantum circuits and thus much more efficient protocols.
Randomness is a central ingredient in classical computing and information processing: It is used in probabilistic algorithms, in cryptography, and in complexity theory. It should thus come not as a surprise that quantum randomness also finds plenty of applications in the quantum world, which however stretch beyond the quantum counterparts of the classical usecases. It even seems that nature itself is a nearly perfect quantum scrambler: chaotic quantum systems have properties that are best describe through random quantum models and even black holes scramble information in a way that is best described by quantum randomness.
We study constructive ways of generating quantum randomness in terms of random quantum circuits. These are quantum circuits composed of randomly chosen local gates. With increasing circuit depth, these randomize more and more effectively. It was only recently established that already very short circuits randomize surprisingly well, which opens the possibility of constructing extremely efficient applications. Our research focuses on the question of how well short random quantum circuits randomize and how their efficiency depends on the circuit geometry, the used gate set, and other parameters. In this way, we lay the foundations of efficient and robust applications.
Quantum experiments are getting better and better in coherently manipulating many-body quantum systems. We can use this to study interesting many-body phenomena or manipulate quantum information in quantum computers. However, even with perfect quantum control, we are left with the highly non-trivial problem of extracting information from the prepared quantum states. An example for this would be the expectation value of a given Hamiltonian, because we are trying to find its ground state through a variational quantum algorithm.
Shadow estimation is a popular approach which utilizes randomized measurements to estimate expectation values of a quantum state using relatively few samples. For certain expectation values, such as state fidelities, a relativity high degree of randomization is necessary, typically involving deep circuits. We have worked on the performance of constant-depth random circuits in this context.