Seminar

In this talk I will explain how some techniques coming from the local theory of Banach spaces can be used to obtain claims about the security of protocols for Position Based Cryptography. In particular, I will show how the knowledge about certain geometrical properties of particular Banach spaces (tensor norms on tensor products of Hilbert spaces) can be translated into lower bounds on the resources needed for cheating in this cryptographic task. I will finish pointing out some open problems and future directions suggested by our work.

Gauge theory is a powerful theoretical framework for understanding the fundamental forces in the standard model. Simulating the real time dynamics of gauge theory, especially in the strong coupling regime, is a challenging classical problem.  Quantum computers offer a solution to this problem by taking advantage of the intrinsic quantum nature of the systems. The Schwinger model, that is the 1+1 dimensional U(1) gauge theory coupled to fermions, has served as a testbed for different methods of quantum simulation.

Quantum computers are expected to dramatically outperform classical computers for certain computational problems. Originally developed for simulating quantum physics, quantum algorithms have been subsequently developed to address diverse computational challenges.

We examine the distribution over measurement outcomes of noisy random quantum circuits in the low-fidelity regime. We will show that, for local noise that is sufficiently weak and unital, the output distribution p_noisy of typical circuits can be approximated by F*p_ideal + (1−F)*p_unif, where F is the probability that no local errors occur, p_ideal is the distribution that would arise if there were no errors, and p_unif is the uniform distribution.

Monitored random unitary circuits with intermittent measurements can host a phase transition between a pure and a mixed phase with different entanglement entropy behaviors with the system size. Recently, it was demonstrated that these phase transitions can be locally probed via entangling reference qubits to the quantum circuit and studying the purification dynamics of the reference qubits. After disentangling from the circuit, the state of the reference qubit can be determined according to the measurement outcomes of the qubits in the circuit.

Nuclear magnetic resonance (NMR) spectroscopy is a useful tool in understanding molecular composition and dynamics, but simulating NMR spectra of large molecules becomes intractable on classical computers as the spin correlations in these systems can grow exponentially with molecule size. In contrast, quantum computers are well suited to simulate NMR spectra of molecules, particularly zero- to ultralow field (ZULF) NMR where the spin-spin interactions in the molecules dominate.

Quantum error correction is expected to play an important role in the realization of large-scale quantum computers. At the lowest level, it takes advantage of embedding qubits in a larger Hilbert space, giving redundancy which allows measurements which preserve logical information while revealing the presence of errors. While many codes rely on multiple physical systems, Bosonic codes make use of the higher dimensional Hilbert space of a single harmonic oscillator mode.

Lieb-Robinson bounds are powerful analytical tools for constraining the dynamic and static properties of non-relativistic quantum systems. Recently, a complete picture for closed systems that evolve unitarily in time has been achieved. In experimental systems, however, interactions with the environment cannot generally be ignored, and the extension of Lieb-Robinson bounds to dissipative systems which evolve non-unitarily in time remains an open challenge.

Quantum circuits have been widely used as a platform to explore universal properties of generic quantum many-body systems. In this talk, I will present our work in which we construct a field theoretical approach to study quantum circuits. We reformulate the sigma model for time periodic Floquet systems using the replica method, and apply it to the study of spectral statistics of the evolution operator of quantum circuits.

To harness the potential of noisy intermediate-scale quantum devices, it is paramount to find the best type of circuits to run hybrid quantum-classical algorithms. Key candidates are parametrized quantum circuits that can be effectively implemented on current devices. Here, we evaluate the capacity and trainability of these circuits using the geometric structure of the parameter space via the effective quantum dimension [1]. We find that the capacity exhibits scaling laws and striking differences depending on the type of entangling gates used.