The Kick-Off Event for the Thematic Einstein Semester “Mathematics for Quantum Technologies” (Summer 2024) is devoted to topics in quantum computing and quantum information theory from the perspective of academia and industry.

The event is funded by the Cluster of Excellence MATH+ and the Einstein Foundation Berlin.

Technical University Berlin

Main Building

Room H 3005

Straße des 17. Juni 135

10623 Berlin

**Registration has been closed prematurely as we have already received a large number of registrations.**

**Registered participants** are requested to **show up with sufficient time in advance** (at least 30 minutes before start). After 13.45, we will start to assign the remaining spots to **non-registered participants** on a first-come, first-served basis.

14:00 – 14:10 | Welcome Address: Michael Hintermüller (WIAS/ HU Berlin, Chair of MATH+) |

14:10 – 14:55 | Jens Eisert (FU Berlin)The mathematics of near-term quantum computing |

14:55 – 15:40 | Karl Jansen (Center for Quantum Technology and Applications, DESY Zeuthen)Quantum computing: a future perspective for scientific computing |

15:40 – 16:15 | Coffee Break |

16:15 – 16:45 | Almudena Carrera Vazquez (IBM Zürich)Scaling Hamiltonian simulation in the era of quantum utility |

16:45 – 17:15 | André Carvalho (Q-CTRL, Berlin)Performance enhancing infrastructure software for quantum technologies |

17:15 – 17:45 | Cristina Cirstoiu (Quantinuum, Cambridge)Quantum computing in noisy regimes: insights from group representation theory |

**Jens Eisert** (FU Berlin)

** The mathematics of near-term quantum computing**Quantum computers promise the efficient solution of some computational problems that are classically intractable. For many years, they have been primarily objects of theoretical study, as only in recent years, protagonists have set out to actually build intermediate-scale quantum computers. This creates an interesting state of affairs, but also begs for an answer to the question what such devices are possibly good for. In this talk, we discuss such questions from the perspective of mathematical physics. While we cannot provide a comprehensive answer, this talk will be dedicated to a number of results offering substantial progress along these lines. We will discuss rigorous quantum advantages in paradigmatic problems [1,2], and will explore the use of quantum computers in machine learning [3–5] and optimization [6]. We will also discuss limitations, by providing efficient classical algorithms for instances of quantum algorithms, hence “de-quantizing” them, and by identifying limitations to quantum error mitigation [9]. The talk will end with an invitation to view such near-term problems from the perspective of mathematics.

[1] Rev. Mod. Phys. 95, 035001 (2023)

[2] arXiv:2307.14424 (2023)

[3] Quantum 5, 417 (2021)

[4] Nature Comm. 15, 434 (2024)

[5] Nature Comm. 15, 2277 (2024)

[6] Science Adv. 10, eadj5170 (2024)

[7] arXiv:2309.11647 (2023)

[8] Phys. Rev. Lett. 131, 100803 (2023)

[9] Nature Phys., arXiv:2210.11505 (2024)

**Karl Jansen** (DESY Zeuthen)

**Quantum computing: a future perspective for scientific computing**

Quantum computing is rapidly emerging as a new method of scientific computing. It has the potential to solve problems much faster than it is possible with classical computers. Examples are applications in logistics, drug design, medicine, finances and many more. In addition, with quantum computers problems can be tackled that are very hard or even impossible to address with classical computers. In this talk, we will introduce the Center for Quantum Technology and Applications at DESY in Zeuthen and give real world examples of applications which can already now be computed on existing quantum computers.

**Almudena Carrera Vazquez** (IBM Zürich)

*Scaling Hamiltonian simulation in the era of quantum utility*

Simulating the time-evolution of a Hamiltonian stands out as a key application for quantum computing, promising significant breakthroughs across various fields. Yet, the capabilities of current quantum hardware are bounded by noise, short coherence times, and a limited number of qubits arranged in planar connectivity. In my talk, I will introduce three techniques that have been developed to extend the capabilities of quantum simulations to larger regimes. Firstly, multi-product formulas [1] offer a way to match the accuracy of higher order product formulas by combining the expectation values from shallow circuits classically [2]. Secondly, circuit cutting [3,4] is a technique for managing quantum circuits that exceed the qubit count or connectivity limits of available hardware [5]. Lastly, noise learning based error mitigation techniques [6] can push the boundaries of what can be achieved beyond classical brute-force computations [7].

[1] Quantum Info. Comput., 12(11–12):901–924 (2012)

[2] Quantum 7, 1067 (2023)

[3] New J. Phys. 23, 023021 (2021)

[4] IEEE Trans. Inf. Theory, 1 (2023)

[5] arXiv:2402.17833 (2024)

[6] Nature Physics 19, pp. 1116–1121 (2023)

[7] Nature 618, pp. 500–505 (2023)

**André Carvalho** (Q-CTRL, Berlin)

* Performance enhancing infrastructure software for quantum technologies*Excitement about the promise of quantum technologies is tempered by the reality that the hardware remains exceptionally fragile and error-prone, forming a bottleneck in the development of novel applications. In this talk we show how quantum control delivered by software could accelerate the adoption of quantum technologies by improving the performance of commercial quantum computers and quantum sensors. We will present the control methods underlying our performance enhancement solutions and discuss their deployment in real devices.

**Cristina Cirstoiu** (Quantinuum, Cambridge)

**Quantum computing in noisy regimes: insights from group representation theory**

Symmetries and structure lie at the heart of quantum mechanics. It is then not surprising that many quantum computing results rely on group-theoretic arguments. This talk will focus on how tools such as representation theory can give us insights into benchmarking the performance and limitations of noisy quantum computers. Specifically, when does noise facilitate an efficient classical simulation of a quantum computation?

I will discuss recent progress on efficient classical simulation algorithms of noisy quantum circuits at a fixed level of physical error per gate, with a focus on operator truncation methods. These techniques truncate the evolution of observables in the Heisenberg picture and enable rigorous trade-offs between complexity, average approximation error and physical noise. Representation theory and harmonic analysis on groups [1] facilitate a unifying framework of operator truncated classical simulations for noisy quantum tasks such as sampling and mean value estimation. Through this lens, we recover results for a series of applications in a noisy setting including variational algorithms [2], dynamical simulations and random circuit sampling [3].

[1] Phys. Rev. X 10, 041035

[2] https://arxiv.org/abs/2306.05400

[3] https://arxiv.org/abs/2211.03999

**Organizers**

Sven Burger (ZIB)

Patrick Gelß (ZIB)

Markus Kantner (WIAS)

Thomas Koprucki (WIAS)

Nathan Walk (FUB)

**Contact**

E-Mail: tes2024@mathplus.de