Quantum Physics

One hundred years after Heisenberg’s Uncertainty Principle, the question of how to make simultaneous measurements of noncommuting observables lingers. I will survey one hundred years of measurement theory, which brings us to the point where we can formulate how to measure any set observables weakly and simultaneously and then concatenate such measurements continuously to determine what is a strong measurement of the same observables. The description of the measurements is independent of quantum states---this we call instrument autonomy---and even independent of Hilbert space---this we call the universal Instrument Manifold Program. But what space, if not Hilbert space? It’s a whole new world: the Kraus operators of an instrument live in a Lie-group manifold generated by the measured observables themselves. I will describe measuring position and momentum and measuring the three components of angular momentum, special cases where the instrument approaches asymptotically a phase-space boundary of the instrumental Lie-group manifold populated by coherent states; these special universal instruments structure any Hilbert space in which they are represented. In contrast, for almost all sets of observables other than these special cases, the universal instrument descends into chaos ... literally. This work was done with Christopher S. Jackson, whose genius and vision inform every aspect.

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Zoom link https://pitp.zoom.us/j/94135518267?pwd=T2JOL21VaEcrY05KeG1SYTVYdHhxdz09

Finding ground states of quantum many-body systems is one of the most important---and one of the most notoriously difficult---problems in physics, both in scientific research and in practical applications.  Indeed, we know from a complexity-theoretic perspective that all methods  (quantum or classical) must necessarily fail to find the ground state efficiently in general. The ground state energy problem is already NP-hard even for classical, frustration-free, local Hamiltonians with constant spectral gap. For general quantum Hamiltonians, the problem becomes QMA-hard.

Nonetheless, as ground state problems are of such importance, and classical algorithms are often successful despite the theoretical exponential worst-case complexity, a number of quantum algorithms for the ground state problem have been proposed and studied. From quantum phase estimation-based methods, to adiabatic state preparation, to dissipative state engineering, to the variation quantum eigensolver (VQE), to quantum/probabilistic imaginary-time evolution (QITE/PITE).

Dissipative state engineering was first introduced in 2009 by Verstraete, Cirac and Wolf and by Kraus et al. However, it only works for the restricted class of frustration-free Hamiltonians.

In this talk, I will show how to construct a dissipative state preparation dynamics that provably produces the correct ground state for arbitrary Hamiltonians, including frustrated ones. This leads to a new quantum algorithm for preparing ground states: the Dissipative Quantum Eigensolver (DQE). DQE has a number of interesting advantages over previous ground state preparation algorithms:

• The entire algorithm consists simply of iterating the same set of   simple local measurements repeatedly.
• The expected overlap with the ground state increases monotonically with the length of time this process is allowed to run.
• It converges to the ground state subspace unconditionally, without any assumptions on or prior information about the Hamiltonian (such as spectral gap or ground state energy bound).
• The algorithm does not require any variational optimisation over parameters.
• It is often able to find the ground state in low circuit depth in practice.
• It has a simple implementation on certain types of quantum hardware, in particular photonic quantum computers.
• It is immune to errors in the initial state.
• It is inherently fault-resilient, without incurring any fault-tolerance overhead. I.e.\ not only is it resilient to errors on the quantum state, but also to faulty implementations of the algorithm itself; the overlap of the output with the ground state subspace degrades smoothly with the error rate, independent of the total run-time.

I give a mathematically rigorous analysis of the DQE algorithm and proofs of all the above properties, using non-commutative generalisations of methods from classical probability theory.

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Zoom link https://pitp.zoom.us/j/96022753460?pwd=SWlUVkVta1RyY3dsWUJWckRqOHdNdz09

Tsirelson’s problem involves two agents, Alice and Bob, who apply measurements on the same quantum system, K. It asks whether commutation, i.e., independence of whether Alice or Bob measures first, is sufficient to conclude that Alice and Bob’s measurements can be factorised so that they act non-trivially only on distinct subsystems of K. In this talk, I will present a “fully quantum generalisation” of this problem, where Alice and Bob’s measurements are replaced by operations on K that may depend on additional quantum inputs and produce quantum outputs. As for Tsirelson’s original problem, it turns out that commutation indeed implies factorisation, provided that all relevant systems are finite-dimensional.

This is joint work with Ramona Wolf; preprint available at arXiv:2308.05792.

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 Zoom link https://pitp.zoom.us/j/99031410183?pwd=MzVoQXpPSll6bFp1b1g3U2J4U21rZz09

Activation of Bell nonlocality refers to the phenomenon where some entangled mixed states that admit a local hidden variable model in the standard Bell scenario nevertheless reveal their nonlocal nature in more exotic measurement scenarios. It has recently been shown that by broadcasting the subsystems of a bipartite quantum state, one can activate Bell nonlocality and significantly improve noise tolerance bounds for device-independent and semi-device-independent entanglement certification, with a single copy of the state and local measurements. In this talk I will review the state of the art on activation of nonlocality and existence of local hidden variable models, introduce the broadcasting technique and outline several interesting results and research directions.

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Zoom link  https://pitp.zoom.us/j/91214624839?pwd=dXNUOERLMldScjczemlaSUhFbFNWQT09

A number of no-go theorems suggest that theories upholding both unitarity and relativity must deny that events are absolute. I'll show how quantum causal influences allow us to articulate an attractive conception of relational events. This will lead us towards a precise, observer-independent reformulation of quantum theory, in which relational events emerge from causation.

Zoom link: https://pitp.zoom.us/j/98649944363?pwd=Y25NOUROelY2ZmFpVWhTZFErV2MwUT09

In 2020 our Oxford-based Quantinuum team performed Quantum Natural Language Processing (QNLP) on IBM quantum hardware [1, 2].  Key to having been able to achieve what is conceived as a heavily data-driven task, is the observation that quantum theory and natural language are governed by much of the same compositional structure -- a.k.a. tensor structure.  

Hence our language model is in a sense quantum-native, and we provide an analogy with simulation of quantum systems in terms of algorithmic speed-up [forthcoming].  Meanwhile we have made all our software available open-source, and with support [github.com/CQCL/lambeq].    

The compositional match between natural language and quantum extends to other domains than language, and argue that a new generation of AI can emerge when fully pushing this analogy, while exploiting the completeness of categorical quantum mechanics / ZX-calculus [3, 4, 5] for novel reasoning purposes that go hand-in-hand with modern machine learning.

[1]     B. Coecke, G. De Felice, K. Meichanetzidis and A. Toumi (2020) Foundations for Near-Term Quantum Natural Language Processing.  https://arxiv.org/abs/2012.03755

[2]     R. Lorenz, A. Pearson, K. Meichanetzidis, D. Kartsaklis and B. Coecke (2020) QNLP in Practice: Running Compositional Models of Meaning on a Quantum Computer.  https://arxiv.org/abs/2102.12846

[3]     B. Coecke and A. Kissinger (2017) Picturing Quantum Processes. A first course on quantum theory and diagrammatic reasoning.  Cambridge University Press.

[4]     B. Coecke, D. Horsman, A. Kissinger and Q. Wang (2021) Kindergarten quantum mechanics graduates (...or how I learned to stop gluing LEGO together and love the ZX-calculus).  https://arxiv.org/abs/2102.10984

[5]     B. Coecke and S. Gogioso (2022) Quantum in Pictures.  Quantinuum, 2023.

Zoom Link: https://pitp.zoom.us/j/92333285960?pwd=MlpJSklmMlVlUlRTTWhsNjc2T2Y4QT09