Quantum Information

Assuming the polynomial hierarchy is infinite, we prove a sufficient condition for determining if uniform and polynomial size quantum circuits over a non-universal gate set are not efficiently classically simulable in the weak multiplicative sense. Our criterion exploits the fact that subgroups of SL(2; C) are essentially either discrete or dense in SL(2; C). Using our criterion, we give a new proof that both instantaneous quantum polynomial (IQP) circuits and conjugated Clifford circuits (CCCs) afford a quantum advantage. We also prove that both commuting CCCs and CCCs over various fragments of the Clifford group afford a quantum advantage, which settles two questions of Bouland, Fitzsimons, and Koh. Our results imply that circuits made of just U \otimes U-conjugated CZ gates afford a quantum advantage for almost all single-qubit unitaries U.

I will present a unified framework for understanding the statistics and anomalies of excitations—ranging from particles to higher-dimensional objects—in quantum lattice systems. We introduce a general method to compute the quantized statistics of Abelian excitations in arbitrary dimensions via Berry phases of locality-preserving symmetry operations, uncovering novel statistics for membrane excitations. These statistics correspond to quantum anomalies of generalized global symmetries and imply obstructions to gauging, enforcing long-range entanglement. In particular, we show that anomalous higher-form symmetries enforce intrinsic long-range entanglement, meaning that fidelity with any SRE states must exhibit exponential decay, unlike ordinary (0-form) symmetry anomalies. As an application, we identify a new example of (3+1)D mixed-state topological order with fermionic loop excitations, characterized by a breakdown of remote detectability linked to higher-form symmetry anomalies.

We present a tensor-network-based classical algorithm (equipped with guarantees) for simulating $n$-qubit quantum circuits with arbitrary single-qubit noise. Our algorithm represents the state of a noisy quantum system by a particular ensemble of matrix product states from which we stochastically sample a pure quantum state. Each single qubit noise process acting on a pure state is then represented by the ensemble of states that achieve the minimal average entanglement (the entanglement of formation) between the noisy qubit and the rest of the system. This approach provides a connection between the entanglement of formation and the accuracy of the simulation algorithm. For a given maximum bond dimension $\chi$ and circuit, our algorithm comes with an upper bound on the simulation error (in total variation distance), runs in $poly(n,\chi)$-time and improves upon related prior work (1) in scope: by extending from the three commonly considered noise models to general single qubit noise (2) in performance: by employing a state-dependent locally-entanglement-optimal unraveling and (3) in conceptual contribution: by showing that the fixed unraveling used in prior work becomes equivalent to our choice of unraveling in the special case of depolarizing and dephasing noise acting on a maximally entangled state. This is joint work with Simon Cichy, Paul K. Faehrmann, Lennart Bittel and Jens Eisert.

In this talk, I will discuss the complexity of a fermionic analogue of Quantum k-SAT. In this Fermionic k-SAT problem, one is given the task to decide whether there is a fermionic state in the null-space of a collection of fermionic, parity-conserving, projectors on n fermionic modes, where each fermionic projector involves at most k fermionic modes. We prove that this problem can be solved efficiently classically for k = 2. In addition, we show that deciding whether there exists a satisfying assignment with a given fixed particle number parity can also be done efficiently classically for Fermionic 2-SAT: this problem is a quantum-fermionic extension of asking whether a classical 2-SAT problem has a solution with a given Hamming weight parity. We also prove that deciding whether there exists a satisfying assignment for particle-number-conserving Fermionic 2-SAT for some given particle number is NP-complete. Complementary to this, we show that Fermionic 9-SAT is QMA_1-hard. 

High-rate quantum low-density parity-check (qLDPC) codes offer significantly lower encoding overhead compared to their topological counterparts by relaxing locality constraints. However, achieving full-fledged logical computation with these codes in physical systems with low space-time costs remains a formidable challenge. In the first part of this talk, I will provide an overview of recent advancements in implementing qLDPC codes as quantum memories on realistic platforms, such as reconfigurable atom arrays. Next, I will present a new scheme for performing parallelizable and locally addressable logical operations on homological product codes. This scheme extends the transversal CNOT gate from two identical CSS codes to two distinct, yet structurally similar, qLDPC codes, enabling efficient local addressing of collectively encoded information. We demonstrate that this approach achieves lower overhead in not only the space- but also the overall space-time overhead compared to surface-code-based computations. Finally, I will discuss new strategies for achieving highly space-time-efficient computations with qLDPC codes by leveraging algorithm-specific fault tolerance, designing tailored protocols for structured quantum algorithms.