Current graduate students

IQC Seminar - Ian George, UIUC

As quantum networks approach real life implementation, a theoretical understanding of their limitations becomes practically important. In this talk, I will discuss recent works with my collaborators where we characterize the ability to prepare a target quantum state over simple networks using local operations (LO) and limited correlated resources in the one-shot setting.

Quantum Nano Centre (QNC) Room 0101, 200 University Avenue West, Waterloo, ON

IQC Colloquium, Harry Buhrman - QuSoft

One of the major challenges in computer science is to establish lower bounds on the resources, usually time, that are needed to solve computational problems. This holds in particular for computational problems that appear in practise. One way towards dealing with this situation is the study of fine- grained complexity where we use special reductions to prove time lower bounds for many diverse problems based on the conjectured hardness of some key problems.

Critical Phase and Spin Sharpening in SU(2)-Symmetric Monitored Quantum Circuits

Monitored quantum circuits exhibit entanglement transitions at certain measurement rates. Such a transition separates phases characterized by how much information an observer can learn from the measurement outcomes. We study SU(2)-symmetric monitored quantum circuits, using exact numerics and a mapping onto an effective statistical-mechanics model. Due to the symmetry's non-Abelian nature, measuring qubit pairs allows for nontrivial entanglement scaling even in the measurement-only limit. We find a transition between a volume-law entangled phase and a critical phase whose diffusive purification dynamics emerge from the non-Abelian symmetry. Additionally, we identify a “spin-sharpening transition.” Across the transition, the rate at which measurements reveal information about the total spin quantum number changes parametrically with system size.

Reference https://journals.aps.org/prb/abstract/10.1103/PhysRevB.108.054307

Analog Quantum Simulation via Parametric Interactions in Superconducting Circuits

While universal quantum computers are still years away from being used for simulating complicated quantum systems, analog quantum simulators have become an increasingly attractive approach to studying classically intractable quantum systems in condensed matter physics, chemistry, and high-energy physics. In this dissertation, we utilize superconducting cavities and qubits to establish analog quantum simulation (AQS) platforms to study systems of interest. 

An approach of AQS that has gained interest lately is the use of photonic lattices to simulate popular lattice models. These systems consist of an array of cavities or resonators arranged on a lattice with some couplings graph between modes. We propose an in situ programmable platform based on a superconducting multimode cavity. The unique design of the cavity allows us to program arbitrarily connected lattices where the coupling strength and phase of each individual coupling are highly programmable via parametrically activated interactions. Virtually any quadratic bosonic Hamiltonian can be realized in our platform with a straightforward pumping scheme.

The effectiveness of the cavity-based AQS platform was demonstrated by the experimental simulation of two interesting models. First, we simulated the effect of a fictitious magnetic field on a 4-site plaquette of a bosonic Creutz ladder, a paradigmatic topological model from high-energy physics.  Under the right magnetic field conditions, we observed topological features such as emergent edge states and localized soliton states. The platform's ability is further explored by introducing pairing (downconversion) terms to simulate the Bosonic Kitaev chain (BKC), the bosonic version of the famous Fermionic Kitaev chain that hosts Majorana fermions. We observe interesting properties of BKC, such as chiral transport and sensitivity to boundary conditions.  

In the final part of the dissertation, we propose and implement a parametrically activated 3-qubit interaction in a circuit QED architecture as the simplest building block to simulate lattice gauge theories (LGT). LGT is a framework for studying gauge theories in discretized space-time, often used when perturbative methods fail.   The gauge symmetries lead to conservation laws, such as Gauss's law in electrodynamics, which impose constraints tying the configuration of the gauge field to the configuration of ''matter'' sites.  Therefore, any quantum simulation approach for LGTs must maintain these conservation laws, with one strategy in AQS being to build them in at the hardware level.  Here, the gauge constraints are explicitly included using a higher-order parametric process between three qubits. The simplest 2-site U(1) LGT building block is realized with two qubits as matter sites and a third qubit as the gauge field mediating the matter-matter interaction, which is crucial to maintain the symmetry of U(1) LGTs.  

Modeling and managing noise in quantum error correction 

Simulating a quantum system to full accuracy is very costly and often impossible as we do not know the exact dynamics of a given system. In particular, the dynamics of measurement noise are not well understood. For this reason, and especially in the context of quantum error correction, where we are studying a larger system with branching outcomes due to syndrome measurement, studies often assume a probabilistic Pauli (or Weyl) noise model on the system with probabilistically misreported outcomes for the measurements. In this thesis, we explore methods to decrease the computational complexity of simulating encoded memory channels by deriving conditions under which effective channels are equivalent up to logical operations. Leveraging this method allows for a significant reduction in computational complexity when simulating quantum error correcting codes. We then propose methods to enforce a model consistent with the typical assumptions of stochastic Pauli (or Weyl) noise with probabilistically misreported measurement outcomes. First, via a new protocol we call measurement randomized compiling, which enforces an average noise on measurements wherein measure- ment outcomes are probabilistically misreported. Then, by another new protocol we call logical randomized compiling, which enforces the same model on syndrome measurements and a probabilistic Pauli (or Weyl) noise model on all other operations (including idling). Together, these results enable more efficient simulation of quantum error correction systems by enforcing effective noise of a form which is easier to model and by reducing the simulation overhead further via symmetries. The enforced effective noise model is additionally consistent with standard error correction procedures and enables techniques founded upon the standard assumptions to be applied in any setting where our protocols are simultaneously applied. 

An estimation theoretic approach to quantum realizability problems

This thesis seeks to develop a general method for solving so-called quantum realizability problems, which are questions of the following form under which conditions does there exists a quantum state exhibiting a given collection of properties? The approach adopted by this thesis is to utilize mathematical techniques previously developed for the related problem of property estimation which is concerned with learning or estimating the properties of an unknown quantum state. Our primary result is to recognize a correspondence between (i) property values which are realized by some quantum state, and (ii) property values which are occasionally produced as estimates of a generic quantum state. In Chapter 3, we review the concepts of stability and norm minimization from geometric invariant theory and non-commutative optimization theory for the purposes of characterizing the flow of a quantum state under the action of a reductive group.

In particular, we discover that most properties of quantum states are related to the gradient of this flow, also known as the moment map. Afterwards, Chapter 4 demonstrates how to estimate the value of the moment map of a quantum state by performing a covariant quantum measurement on a large number of identical copies of the quantum state. These measurement schemes for estimating the moment map of a quantum state arise naturally from the decomposition of a large tensor-power representation into its irreducible sub-representations.

Then, in Chapter 5, we prove an exact correspondence between the realizability of a moment map value on one hand and the asymptotic likelihood it is produced as an estimate on the other hand. In particular, by composing these estimation schemes, we derive necessary and sufficient conditions for the existence of a quantum state jointly realizing any finite collection of moment maps. Finally, in Chapter 6 we apply these techniques to the quantum marginals problem which aims to characterize precisely the relationships between the marginal density operators describing the various subsystems of composite quantum state. We make progress toward an analytic solution to the quantum marginals problem by deriving a complete hierarchy of necessary inequality constraints.