Short-time Quantum Dynamics: Classical Simulation, Quantum Advantage, and Characterization
Candidate: Yinchen Liu
Supervisor: David Gosset
Location: QNC 2101
Date/Time: July 29, 2026, 1:00 p.m.
Abstract
This thesis presents several results on short-time quantum dynamics, with themes spanning classical simulation, quantum advantage, and characterization. A prime example of short-time quantum dynamics is that of a shallow quantum circuit, whose circuit depth is a constant independent of the number of qubits. Another example is the dynamics generated by a single 2-qubit gate, even the identity gate, on a noisy quantum computer.
In Chapter 2, we give classical simulation algorithms for peaked shallow quantum circuits. We say an n-qubit quantum circuit is peaked if it has an output probability that is at least inverse-polynomially large as a function of n. We describe a classical algorithm with quasipolynomial n^O(log n) runtime that approximately samples from the output distribution of a peaked shallow quantum circuit. We give even faster algorithms for circuits composed of nearest-neighbour gates on a D-dimensional grid of qubits, with polynomial n^O(1) runtime if D = 2 and almost-polynomial n^O(log log n) runtime for D > 2. Our sampling algorithms can be used to estimate the output probabilities of shallow quantum circuits to within a given inverse-polynomial additive error, and for cases other than D = 2, our algorithms achieve improved runtime scalings compared to prior state-of-the-art algorithms.
In Chapters 3 and 4, we study shallow 1D and 2D random quantum circuits. The output states of these circuits only exhibit entanglement among spatially neighbouring qubits due to locally constrained lightcones. However, entanglement between distant qubits could be generated if we were to measure all other qubits of an output state in the computational basis. In 2D, it is conjectured that this long-range measurement-induced entanglement (MIE) proliferates when the circuit depth exceeds some constant critical value d^*. For 2D circuits composed of Haar random 2-qubit gates, it is also conjectured that this coincides with a quantum advantage phase transition in the classical hardness of sampling from their output distributions.
In Chapter 3, we first demonstrate numerically and rigorously the emergence of long-range MIE in the output states of 1D and 2D random Clifford circuits in various settings. Then, we prove an unconditional quantum advantage for any distribution of random Clifford circuits satisfying a long-range GHZ-type MIE property. This could be viewed as establishing a scaled-down version of the quantum advantage phase transition originally conjectured for shallow 2D Haar random quantum circuits.
In Chapter 4, we concentrate on the computational-basis output distributions of shallow 2D random Clifford circuits with depth d>d^*. We propose and support with numerical evidence a conjecture called the Scrooge hypothesis that relates the distribution of pure post-measurement states on a subset of qubits, generated by measuring all other qubits of an output state in the computational basis, to stabilizer Scrooge ensembles, which are natural generalizations of the distribution of uniformly random stabilizer states. We show that the Scrooge hypothesis implies a detailed characterization of the way local and global correlations coexist in the output distribution of a typical shallow 2D random Clifford circuit, corroborating numerically observed behaviours. Building on the structural characterization, we show that a column-by-column algorithm, which only ever keeps track of O(log n) columns of qubits at a time, is able to correctly simulate the marginal output distribution if a contiguous region of O(log n) qubits is traced out. The correctness of the column-by-column simulation algorithm implies that any global correlation residing in the output distribution is fragile against depolarizing noise.
In Chapter 5, we tackle the problem of characterizing the short-time dynamics of a single 2-qubit gate implemented on current quantum computing hardware. Building on prior work, we devise a suite of algorithms utilizing alternating optimization techniques to simultaneously and self-consistently estimate state preparation and measurement (SPAM) errors and best-fit Lindbladian models for a set of gates. We benchmark our algorithms extensively using synthetic 2-qubit gate data under a range of realistic error models and apply them to detect cross-talk on real superconducting-qubit hardware.