Current quantum computers are limited by noise, which severely affects their ability to run long quantum circuits. Most known quantum algorithms are out of reach for such machines — and this remains true for the early-fault-tolerant devices expected in the coming years. So if quantum computers are 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 so that they fit into the short-circuit regime? What mathematical tools are needed to make progress on these questions?
To get a handle on what short quantum circuits can do, we ask how generic short circuits can be distinguished from deep ones — or, equivalently, to which extent a short circuit can look like a Haar-random unitary. We tackle this through the theory of random quantum circuits, with connections to many-body physics and 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 — most importantly magic / non-stabilizerness.
A central place in which these ideas fundamentally matter is the extraction of information from quantum systems. A quantum state cannot be read out directly and properties of interest such as fidelities, expecatation values, entanglement measures, correlation functions need to be estimated from measurements, ideally with minimal overhead. This particularly relevant for quantum simulation: For a long time to come, quantum computational resources will remain scarce, and the cost of the information-extraction step threatens to dominate the overall running time of a quantum application. Understanding it is therefore essential for the practical utility of quantum computers.
Randomness sits at the foundation of classical computing — in probabilistic algorithms, cryptography, and complexity theory — and quantum randomness plays a similarly central role in the quantum world. Many quantum experiments rely on it: classical shadow protocols, benchmarking schemes, and tests of quantum advantage all draw on the statistical properties of random unitaries. Nature itself appears to be a remarkably efficient quantum scrambler, with chaotic systems and even black holes modeled by quantum random matrix ensembles.
We study how quantum randomness emerges in short quantum circuits. Until recently it was believed that genuine randomness only appears at linear circuit depth, but a series of breakthroughs has shown that already logarithmic depth can suffice for many purposes. We have contributed to this picture by showing that the emergence of 2-designs in random circuits is essentially controlled by a single, easier-to-establish property — anti-concentration — making low-depth designs much more universal than previously thought (Heinrich, Haferkamp, Roth & Helsen, arXiv:2510.23719). A complementary direction connects random matrix product unitaries to free probability (Dowling et al., arXiv:2508.00051).
An equally important question is where this picture breaks down. Random circuits over subgroups of the unitary group — the Clifford, orthogonal, symplectic, and matchgate groups — behave very differently from the full unitary case. We have proven linear-depth lower bounds for designs over each of these groups (Grevink et al., arXiv:2506.23925), ruling out a naive transfer of the qubit techniques to fermionic and other settings. Yet weaker forms of randomness, such as additive-error state designs and anti-concentration, often still emerge at logarithmic depth. The onset of randomness in shallow circuits is a widespread but subtle phenomenon, and pinning down exactly which randomness emerges when — and at what cost in other resources — is one of our central goals.
Even with perfect quantum control, extracting information from a quantum many-body state is highly non-trivial. Properties of interest — expectation values, fidelities, entanglement measures — must be inferred from measurement statistics, and doing this efficiently is a central question for the practical use of quantum computers.
Shadow estimation is a particularly powerful approach: a random unitary is applied to the unknown state, the state is measured, and the resulting data is post-processed into estimators of the desired property. The efficiency depends on a delicate interplay between the quantum state, the observable, and the chosen random ensemble. For certain expectation values, such as state fidelities, a high degree of randomization is needed and typically involves deep circuits — but the recent progress on low-depth designs (see above) opens the door to substantial reductions. We have, for example, derived closed-form expressions for shadow estimation with brickwork circuits that interpolate cleanly between local and global measurement regimes (Arienzo et al., Quantum Inf. Comput. 23, 961).
Beyond the noiseless case, we ask how robust these protocols are under realistic conditions. Classical shadows are surprisingly stable under generic noise: we have shown that the noise-induced estimation bias is governed by the non-stabilizerness of the target observable (Brieger, Heinrich, Roth & Kliesch, PRL 134, 090801) — providing the first quantitative link between magic and the robustness of property estimation, and pointing towards general principles we are now exploring.
Not all quantum gates are equal. The Clifford gates (Hadamard, phase, CNOT) are simple to implement on fault-tolerant quantum computers and can be efficiently simulated classically — but they are not universal. Achieving the full computational power of quantum mechanics requires non-Clifford gates such as the T or Toffoli gate, which carry a substantial overhead on real hardware. Quantum states and circuits that lie beyond the reach of Clifford operations are called non-stabilizer, and their non-stabilizerness — also known as magic — is the quantum-computational resource that genuinely separates quantum from classical computing.
Magic plays a double role in our research. On the one hand, it is a resource whose mathematical structure we want to understand: in earlier work we mapped out the symmetries of the stabilizer polytope (Heinrich & Gross, Quantum 3, 132) and clarified the resource-theoretic foundations of magic, including a separation between its axiomatic and operational formulations (Heimendahl, Heinrich & Gross, J. Math. Phys. 63, 112201). On the other hand, magic is a cost — and one whose interaction with short quantum circuits is increasingly central to our work.
A particularly clean way to study this interaction is through doped Clifford circuits, in which Clifford layers are interleaved with a controlled number of non-Clifford gates. We introduced these as a way to construct higher-order unitary designs at a precisely tunable non-Clifford cost (Haferkamp et al., Commun. Math. Phys. 397, 995), and they have since become a standard model in the study of near-stabilizer quantum states and circuit complexity. More recently, we have studied doped real Clifford circuits and the closely related question of anti-concentration (Magni, Heinrich, Leone & Turkeshi, arXiv:2512.15880). The broader picture we are after is: how much magic does a given task — randomization, property estimation, classical simulation — actually require, and how does this trade against circuit depth and other resources?
Many quantum protocols — and most property-estimation protocols in particular — rely on a substantial amount of classical post-processing. For classical shadows, for instance, one must classically simulate the inverse of the random measurement circuit and then evaluate the target observable. The cost of this step can easily dominate the cost of the entire protocol, especially when the circuit contains non-Clifford gates or the observable is highly non-stabilizer.
This makes the classical simulation of Clifford-dominated circuits a central computational primitive for the near-term era of quantum computing — and an interesting mathematical problem in its own right. We develop techniques based on stabilizer decompositions and the stabilizer rank that exploit the structure of Clifford-dominated circuits to achieve practical speed-ups well beyond what worst-case asymptotic bounds suggest. A recent line of work uses symmetries of the stabilizer extent to accelerate simulation, with concrete gains on benchmark instances (Camillo, Peres, Heinrich & Bermejo-Vega, arXiv:2510.18977).
On the theoretical side, we are interested in the fundamental limits of stabilizer-based simulation. The best known lower bounds on the stabilizer rank of magic states are merely quadratic, while the best upper bounds are exponential — a striking gap that we hope to close. Beyond their intrinsic interest, sharper bounds would tighten the resource accounting in property-estimation protocols and clarify the boundary between classical and quantum computational power.
Before a quantum computer can run useful algorithms, we need to know how well it works. Benchmarking and characterization protocols quantify the noise and imperfections of quantum hardware — ideally without requiring expensive full tomography of the device.
A workhorse of this field is randomized benchmarking (RB), in which random sequences of gates are applied and the decay of the resulting signal reveals average gate fidelities. Classical RB, however, relies on inverting the gate sequence, requires structured Clifford gate sets, and is hard to interpret for the native gates of real hardware. We have developed a flexible framework — filtered randomized benchmarking — that lifts these restrictions: arbitrary compact groups, no inverse gate needed, native gates handled directly, and rigorous guarantees under realistic gate-dependent noise (Heinrich, Kliesch & Roth, arXiv:2212.06181). The framework extends to settings well beyond the qubit case — for instance, the first rigorous benchmarking protocol for passive bosonic transformations (Arienzo, Grinko, Kliesch & Heinrich, PRX Quantum 6, 020305).
Benchmarking is closely related to property estimation: both rely on randomized measurements, both exploit unitary designs, and both face the same underlying challenge of extracting clean information from noisy quantum operations. A key technical insight from our work — that gate noise closes the spectral gap of the moment operator of random quantum circuits — has since been demonstrated experimentally by others. Understanding this interplay between scrambling and noise feeds directly into protocols for gate calibration and Hamiltonian engineering, and into the noise robustness of estimation more broadly. Much of this work is carried out in close collaboration with experimental partners developing next-generation trapped-ion devices.