Quantum mechanics as an operationally time symmetric probabilistic theory
Ognyan Oreshkov Université Libre de Bruxelles
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A universal Hamiltonian simulator: the full characterization
Gemma De Las Cuevas Universität Innsbruck
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Asymptotically Optimal Topological Quantum Compiling
Vadym Kliuchnikov University of Waterloo
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Simulating Hamiltonian evolution with quantum computers
Richard Cleve Institute for Quantum Computing (IQC)
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Quantum States are Consistent Probability Distributions
Nadish de Silva University of Oxford
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A Non-Local Reality: Is there a Phase Uncertainty in Quantum Mechanics?
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Niayesh Afshordi University of Waterloo
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Elizabeth Gould Arthur B. McDonald Canadian Astroparticle Physics Research Institute
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Quantum mechanics as an operationally time symmetric probabilistic theory
Ognyan Oreshkov Université Libre de Bruxelles
The standard formulation of quantum mechanics is operationally asymmetric with respect to time reversal---in the language of compositions of tests, tests in the past can influence the outcomes of test in the future but not the other way around. The question of whether this represents a fundamental asymmetry or it is an artifact of the formulation is not a new one, but even though various arguments in favor of an inherent symmetry have been made, no complete time-symmetric formulation expressed in rigorous operational terms has been proposed. Here, we discuss such a possible formulation based on a generalization of the usual notion of test. We propose to regard as a test any set of events between an input and an output system which can be obtained by an autonomously defined laboratory procedure. This includes standard tests, as well as proper subsets of the complete set of outcomes of standard tests, whose realization may require post-selection in addition to pre-selection. In this approach, tests are not expected to be operations that are up to the choices of agents---the theory simply says what circuits of tests may occur and what the probabilities for their outcomes would be, given that they occur. By virtue of the definition of test, the probabilities for the outcomes of past tests can depend on tests that take place in the future. Such theories have been previously called non-causal, but here we revisit that notion of causality. Using the Choi-Jamiolkowski isomorphism, every test in that formulation, commonly regarded as inducing transformations from an input to an output system, becomes equivalent to a passive detection measurement applied jointly on two input systems---one from the past and one from the future. This is closely related to the two-state vector formalism, but it comes with a conceptual revision: every measurement is a joint measurement on two separate systems and not on one system described by states in the usual Hilbert space and its dual. We thus obtain a static picture of quantum mechanics in space-time or more general structures, in which every experiment is a local measurement on a global quantum state that generalizes the recently proposed quantum process matrix. The existence of two types of systems in the proposed formalism allows us to define causation in terms of correlations without invoking the idea of intervention, offering a possible answer to the problem of the meaning of causation. The framework is naturally compatible with closed time-like curves and other exotic causal structures. -
A universal Hamiltonian simulator: the full characterization
Gemma De Las Cuevas Universität Innsbruck
We show that if the ground state energy problem of a classical spin model is NP-hard, then there exists a choice parameters of the model such that its low energy spectrum coincides with the spectrum of \emph{any} other model, and, furthermore, the corresponding eigenstates match on a subset of its spins. This implies that all spin physics, for example all possible universality classes, arise in a single model. The latter property was recently introduced and called ``Hamiltonian completeness'', and it was shown that several different models had this property. We thus show that Hamiltonian completeness is essentially equivalent to the more familiar complexity-theoretic notion of NP-completeness. Additionally, we also show that Hamiltonian completeness implies that the partition functions are the same. These results allow us to prove that the 2D Ising model with fields is Hamiltonian complete, which is substantially simpler than the previous examples of complete Hamiltonians. Joint work with Toby Cubitt. -
A note on emergent classical behavior and approximations to decoherence functionals
Henrique Gomes University of Oxford
Although it can only be argued to have become consequential in the study of quantum cosmology, the question ``Why do we observe a classical world? " has been one of the biggest preoccupations of quantum foundations. In the consistent histories formalism, the question is shifted to an analysis of the telltale sign of quantum mechanics: superposition of states. In the consistent histories formalism, histories of the system which ``decohere", i.e. fall out of superposition or have negligible interference can be subjected to a notion of classical probability. In this paper we use an extension of Kirchoff's diffraction formula for wave functions on configuration spaces to give a different analysis and an approximation of decoherence. The Kirchoff diffraction formula lies conveniently at the midway between path integrals, wave equations, and classical behavior. By using it, we formulate an approximate dampening of the amplitude of superposition of histories. The dampening acts on each middle element of the fine-grained history {c_\alpha}, and is a function of the angle formed between {c_{n-1},c_n} and {c_n,c_{n+1}}, as classical trajectories in configuration space. As an example we apply the formalism to a modified gravity theory in the ADM gravitational conformal superspace. -
Asymptotically Optimal Topological Quantum Compiling
Vadym Kliuchnikov University of Waterloo
In a topological quantum computer, universality is achieved by braiding and quantum information is natively protected from small local errors. We address the problem of compiling single-qubit quantum operations into braid representations for non-abelian quasiparticles described by the Fibonacci anyon model. We develop a probabilistically polynomial algorithm that outputs a braid pattern to approximate a given single-qubit unitary to a desired precision. We also classify the single-qubit unitaries that can be implemented exactly by a Fibonacci anyon braid pattern and present an efficient algorithm to produce their braid patterns. Our techniques produce braid patterns that meet the uniform asymptotic lower bound on the compiled circuit depth and thus are depth-optimal asymptotically. Our compiled circuits are significantly shorter than those output by prior state-of-the-art methods, resulting in improvements in depth by factors ranging from 20 to 1000 for precisions ranging between 10^{−10} and 10^{−30}. -
Simulating Hamiltonian evolution with quantum computers
Richard Cleve Institute for Quantum Computing (IQC)
In 1982, Richard Feynman proposed the concept of a quantum computer as a means of simulating physical systems that evolve according to the Schrödinger equation. I will explain various quantum algorithms that have been proposed for this simulation problem, including my recent work (jointly with Dominic Berry and Rolando Somma) that significantly improves the running time as a function of the precision of the output data. -
Can ‘sub-quantum’ theories based on a background field escape Bell’s no-go theorem ?
Louis Vervoort Université de Montréal
In systems described by Ising-like Hamiltonians, such as spin-lattices, the Bell Inequality can be strongly violated. Surprisingly, these systems are both local and non-superdeterministic. They are local, because 1) they include only local, near-neighbor interaction, 2) they satisfy, accordingly, the Clauser-Horne factorability condition, and 3) they can violate the Bell Inequality also in dynamic Bell experiments. Starting from this result we construct an elementary hidden-variable model, based on a generalized Ising Hamiltonian, describing the interaction of the Bell-particles with a stochastic ‘background’ medium. We suggest that such a model is a simple version of a variety of recently developed ‘sub-quantum’ theories, by authors as Nelson, Adler, De la Pena, Cetto, Groessing, Khrennikov, all based on a background field. We investigate how the model might be turned into a realistic theory. Finally, it appears that background-based models can be tested and discriminated from quantum mechanics by a straightforward extension of existing experiments. -
Universal fault-tolerant quantum computation with only transversal gates and error correction
Transversal implementations of encoded unitary gates are highly desirable for fault-tolerant quantum computation. It is known, however, that transversal gates alone cannot be computationally universal. I will show that the limitation on universality can be circumvented using only fault-tolerant error correction, which is already required anyway. This result applies to ``triorthogonal'' stabilizer codes, which were recently introduced by Bravyi and Haah for state distillation. I will show that triothogonal codes admit transversal implementation of the controlled-controlled-Z gate, and then demonstrate a transversal Hadamard construction which uses error correction to preserve the codespace. I will also discuss how to adapt the distillation procedure of Bravyi and Haah to Toffoli gates, improving on existing Toffoli distillation schemes. -
Coalgebras, Models and Logics for Quantum Systems
Coalgebras are a flexible tool commonly used in computer science to model abstract devices and systems. Coalgebraic models also come with a natural notion of logics for the systems being modelled. In this talk we will introduce coalgebras and aim to illustrate their usefulness for modelling physical systems. Extending earlier work of Abramsky, we will show how a weakening of the usual morphisms for coalgebras provides the flexibility to model quantum systems in an easy to motivate manner. We will then investigate how a natural extension to the usual notion of coalgebraic logic can be used to produce logics for reasoning about quantum systems and protocols. No prior knowledge of coalgebras will be assumed for this talk, and the emphasis throughout will be on examples rather than technical details. -
Quantum States are Consistent Probability Distributions
Nadish de Silva University of Oxford
We describe a notion of state for a quantum system which is given in terms of a collection of empirically realizable probability distributions and is formally analogous to the familiar concept of state from classical statistical mechanics. We first demonstrate the mathematical equivalence of this new notion to the standard quantum notion of density matrix. We identify the simple logical consistency condition (a generalization of the familiar no-signalling condition) which a collection of distributions must obey in order to reconstruct the unique quantum state from which they arise. In this way, we achieve a formal expression of the common intuition of a quantum state as being classical distributions on compatible observables. -
Quantum Observables as Real-valued Functions and Quantum Probability
Quantum observables are commonly described by self-adjoint operators on a Hilbert space H. I will show that one can equivalently describe observables by real-valued functions on the set P(H) of projections, which we call q-observable functions. If one regards a quantum observable as a random variable, the corresponding q-observable function can be understood as a quantum quantile function, generalising the classical notion. I will briefly sketch how q-observable functions relate to the topos approach to quantum theory and the process called daseinisation. The topos approach provides a generalised state space for quantum systems that serves as a joint sample space for all quantum observables. This is joint work with Barry Dewitt. -
A Non-Local Reality: Is there a Phase Uncertainty in Quantum Mechanics?
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Niayesh Afshordi University of Waterloo
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Elizabeth Gould Arthur B. McDonald Canadian Astroparticle Physics Research Institute
A century after the advent of Quantum Mechanics and General Relativity, both theories enjoy incredible empirical success, constituting the cornerstones of modern physics. Yet, paradoxically, they suffer from deep-rooted, so-far intractable, conflicts. Motivations for violations of the notion of
relativistic locality include the Bell's inequalities for hidden variable theories, the cosmological horizon problem, and Lorentz-violating approaches to quantum geometrodynamics, such as Horava-Lifshitz gravity. Here, we explore a recent proposal for a ``real ensemble'' non-local description of quantum mechanics, in which ``particles'' can copy each others' observables AND phases, independent of their spatial separation. We first specify the exact theory, ensuring that it is consistent and has (ordinary) quantum mechanics as a fixed point, where all particles with the same observables have the same phases. We then study the stability of this fixed point numerically, and analytically, for simple models. We provide evidence that most systems (in our study) are locally stable to small deviations from quantum mechanics, and furthermore, the phase variance per observable, as well as systematic deviations from quantum mechanics, decay as ~ (EnergyXTime)^{-n}, where n > 2. Interestingly, this convergence is controlled by the absolute value of energy (and not energy difference). Finally, we discuss different issues related to this theory, as well as potential implications for early universe, and the cosmological constant problem. -
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From empirical practice to observables and the action principle
Physical theories ought to be built up from colloquial notions such as ’long bodies’, ’energetic sources’ etc. in terms of which one can define pre-theoretic ordering relations such as ’longer than’, ’more energetic than’. One of the questions addressed in previous work is how to make the transition from these pre-theoretic notions to quantification, such as making the transition from the ordering relation of ’longer than’ (if one body covers the other) to the notion of how much longer. In similar way we introduce dynamical notions ’more impulse’ (if in a collision one object overruns the other) and ’more energetic’ (if the effect of one source exceeds the effect of the other). In a physical model - built by coupling congruent standard actions - those basic pre-theoretic notions become measurable. We derive all (classical and relativistic) equations between basic physical quantities of Energy, Momentum and Inertial Mass and ultimately the principle of least action.