Quantum Physics

The formalism of quantum theory over discrete systems is extended in two significant ways. First, tensors and traceouts are generalized, so that systems can be partitioned according to almost arbitrary logical predicates. Second, quantum evolutions are generalized to act over network configurations, in such a way that nodes be allowed to merge, split and reconnect coherently in a superposition. The hereby presented mathematical framework is anchored on solid grounds through numerous lemmas. Indeed, one might have feared that the familiar interrelations between the notions of unitarity, complete positivity, trace-preservation, non-signalling causality, locality and localizability that are standard in quantum theory be jeopardized as the partitioning of systems becomes both logical and dynamical. Such interrelations in fact carry through, albeit two new notions become instrumental: consistency and comprehension.

Joint work with Amélia Durbec and Matt Wilson

Reference: https://arxiv.org/abs/2110.10587

Zoom Link: https://pitp.zoom.us/j/97185954578?pwd=OC9mUzl4L3V4WDZzVEZoekpOS24wQT09

Bell’s theorem is typically understood as proof that quantum theory is incompatible with local hidden variable (LHV) models. In recent years, however, LHV models have been recognized as a particular and simple case of much more general causal networks that can give rise to new and stronger forms of nonclassicality. And, since nonlocality is a resource in a variety of applications, it is thus natural to ask whether these novel forms of nonclassical behavior can also be put to use in information processing. In this seminar, I will present recent results exploring quantum correlations in several such causal scenarios, ranging from foundational questions such as freedom of choice in Bell experiments to more applied situations covering cryptography, distributed computing, and game theory.

Zoom Link: TBD

General relativity does not distinguish a preferred reference frame, and conservatively one ought to expect that its quantization does not necessitate such background structure. However, this desire stands in contrast to orthodox formulations of quantum theory which rely on a background time parameter external to the theory, and in the case of quantum field theory a spacetime foliation. Such considerations have led to the development of the Page-Wootters formalism, which seeks to describe motion relative to a reference frame internal to a quantum theory that encompasses both the system of interest and employed reference frame. I will begin by reviewing a modern formulation of the Page-Wootters formalism in terms of Hamiltonian constraints, generalized coherent states, and covariant time observables. I will then present Kuchar’s criticisms of the Page-Wootters formalism, and discuss their resolution by showing the equivalence between the formalism and relational Dirac observables. These Dirac observables will then be used to introduce a gauge-invariant, relational notion of subsystems and entanglement. Finally, a field-theoretic extension of the Page-Wootters formalism will be introduced and used to recover the Schwinger-Tomonaga equation.

Zoom Link: https://pitp.zoom.us/j/91420728439?pwd=cXlGZ21tTGZEUjFDVjRKMWxaVFlVZz09

Aside from quantum mechanics, other areas in physics constrain information processing. From relativity, we know information cannot move faster than the speed of light. In quantum gravity, we expect but don't fully understand additional constraints, for instance on how densely information can be stored. Can we develop an understanding of information processing in the context of quantum gravity? Towards doing so, we consider quantum information within ``holographic'' spacetimes, which have an alternative, non-gravitational, description. In that context questions about information processing in the presence of gravity can be translated to different questions about quantum information without gravity. As a specific example, we study constraints on computation within a gravitating region, and use the holographic description to argue that gravity constrains the complexity of operations that can happen inside the region. Along the way, we are led to new connections relating quantum gravity, position-based cryptography, quantum secret sharing and complexity theory. 

Topological quantum codes illustrate a variety of concepts in quantum many-body physics. They also provide a realistic and resource-efficient approach to building scalable quantum computers. In this talk, I will focus on fault-tolerant quantum computing with a new topological code, the 3D subsystem toric code (STC). The 3D STC can be realized on the cubic lattice by measuring small-weight local parity checks. The 3D STC is capable of handling measurement errors while performing reliable quantum error correction and implementing logical gates in constant time. This dramatically reduces the computational time overhead and gives the 3D STC a resilience to time-correlated noise.

This talk consists of two parts. At first, I will focus on geometric obstructions to decoding Hawking radiation (python’s lunch). Harlow and Hayden argued that distilling information out of Hawking radiation is computationally hard despite the fact that the quantum state of the black hole and its radiation is relatively un-complex. I will trace this computational difficulty to a geometric obstruction in the Einstein-Rosen bridge connecting the black hole and its radiation. Inspired by tensor network models, I will present a conjecture that relates the computational hardness of distilling information to geometric features of the wormhole.

Then, with the long-term goal of studying quantum gravity in the lab, I will discuss a proposal for a holographic teleportation protocol that can be readily executed in table-top experiments. This protocol exhibits similar behavior to that seen in recent traversable wormhole constructions. I will introduce the concept of "teleportation by size" to capture how the physics of operator-size growth naturally leads to information transmission. The transmission of a signal through a semi-classical holographic wormhole corresponds to a rather special property of the operator-size distribution we call "size winding". 

Zoom Link: https://pitp.zoom.us/j/93957279481?pwd=eGVTU1MwOGNWNkMyYlRiWGo0QnFldz09

The mixing time of Markovian dissipative evolutions of open quantum many-body systems can be bounded using optimal constants of certain quantum functional inequalities, such as the logarithmic Sobolev constant, which is equivalent to some form of entropy decay. For classical spin systems, the positivity of such constants follows from a mixing condition on the Gibbs measure, via quasi-factorization results for the entropy. Inspired by the classical case, we present a strategy to derive the positivity of the logarithmic Sobolev constant associated to the dynamics of certain quantum systems from some clustering conditions on the Gibbs state of a local, commuting Hamiltonian. Subsequently, we apply it to show that for a finite-range, translation-invariant commuting Hamiltonian on a spin chain, the Davies semigroup describing the reduced dynamics resulting from the joint Hamiltonian evolution of a spin chain weakly coupled to a large heat bath thermalizes rapidly at any temperature. This, in particular, rigorously establishes the absence of dissipative phase transition for Davies evolutions over translation-invariant spin chains.
 

A quantum error-correcting code depends on a classical decoding algorithm that uses the outcomes of stabilizer measurements to determine the error that needs to be repaired. Likewise, the design of a decoding algorithm depends on the underlying physics of the quantum error-correcting code that it needs to decode. The surface code, for instance, can make use of the minimum-weight perfect-matching decoding algorithm to pair the defects that are measured by its stabilizers due to its underlying charge parity conservation symmetry. In this talk I will argue that this perspective on decoding gives us a unifying principle to design decoding algorithms for exotic codes, as well as new decoding algorithms that are specialised to the noise that a code will experience. I will describe new decoders for exotic fracton codes we have designed using these principles. I will also discuss how the symmetries of a code change if we focus on restricted noise models, and how we have leveraged this observation to design high-threshold decoders for biased noise models. In addition to these examples, this talk will focus on recent work on decoding the color code, where we found a high-performance decoder by investigating the defect conservation laws at the boundaries of the color code. Remarkably, our results show that we obtain an advantage by decoding this planar quantum error-correcting code by matching defects on a manifold that has the topology of a Moebius strip.

Zoom Link: https://pitp.zoom.us/j/91540245974?pwd=RDkzaVJZZ2tkTldxM2pkdXU5VHlIZz09

Quantifying quantum states' complexity is a key problem in various subfields of science, from quantum computing to black-hole physics. Motivated by the expected behavior of wormholes in quantum gravity, Brown and Susskind conjectured that the quantum complexity of the state output by a random circuit on n qubits grows linearly as more and more random gates are applied, until saturating after a number of gates exponential in n. We prove this conjecture by studying the dimension of the set of all unitaries that can be accessed with a given arrangement of two-qubit gates. Our core technical contribution is a lower bound on this dimension, using techniques from algebraic geometry and considerations based on Clifford circuits. In the second part of my talk, I'll discuss some thermodynamic and effective information-theoretic aspects of the complexity of quantum states and its growth in quantum many-body systems, establishing a resource theory to capture a notion of quantum complexity and drawing a connection between the concepts of complexity and entropy.

Joint work with: Jonas Haferkamp, Teja Naga Bhavia Kothakonda, Anthony Munson, Jens Eisert, Nicole Yunger Halpern

Zoom Link: https://pitp.zoom.us/j/94288479163?pwd=Nm8wOUdReGhreDErdUpJTzFETlBUUT09

Quantum metrology, which studies parameter estimation in quantum systems, has many important applications in science and technology, ranging from frequency spectroscopy to gravitational wave detection. Quantum mechanics imposes a fundamental limit on the estimation precision, called the Heisenberg limit, which is achievable in noiseless quantum systems, but is in general not achievable in noisy systems. This talk is a summary of some recent works by the speaker and collaborators on error-corrected quantum metrology. Specifically, we present a necessary and sufficient condition for achieving the Heisenberg limit in noisy quantum systems. When the condition is satisfied, the Heisenberg limit is recovered by a quantum error correction protocol which corrects all noises while maintaining the signal; when it is violated, we show the estimation limit still in general has a constant factor improvement over classical strategies, and is achievable using approximate quantum error correction. Both error correction protocols can be optimized using semidefinite programs. Examples in some typical noisy systems will be provided.

Zoom Link: https://pitp.zoom.us/j/95698542740?pwd=OWJtcTViKzZqNDg1bWk4cDFtaTRxZz09