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Probing Hilbert space fragmentation and the block Inverse Participation Ratio

We consider a family of quantum many-body Hamiltonians that show exact Hilbert space fragmentation in certain limits. The question arises whether fragmentation has implications for Hamiltonians in the vicinity of the subset defined by these exactly fragmented models, in particular in the thermodynamic limit. We attempt to illuminate this issue by considering distinguishable classes of transitional behaviour between fragmented and non-fragmented regimes and employing a set of numerical observables that indicate this transition. As one of these observables we present a modified inverse participation ratio (IPR) that is designed to capture the emergence of fragmented block structures. We compare this block IPR to other definitions of inverse participation ratios, as well as to the more traditional measures of level spacing statistics and entanglement entropy. In order to resolve subtleties that arise in the numerics, we use perturbation theory around the fragmented limit as a basis for defining an effective block structure. We find that our block IPR predicts a boundary between fragmented and non-fragmented regimes that is compatible with results based on level statistics and bipartite entanglement. A scaling analysis indicates that a finite region around the exactly fragmented limit is dominated by effects of approximate fragmentation, even in the thermodynamic limit, and suggests that fragmentation constitutes a phase. We provide evidence for the universality of our approach by applying it to a different family of Hamiltonians, that features a fragmented limit due to emergent dipole-conservation.

What Does ‘(Non)-Absoluteness of Observed Events’ Mean?

Recently there have emerged an assortment of theorems relating to the ‘absoluteness of emerged events,’ and these results have sometimes been used to argue that quantum mechanics may involve some kind of metaphysically radical non-absoluteness, such as relationalism or perspectivalism. However, in our view a close examination of these theorems fails to convincingly support such possibilities. In this paper we argue that the Wigner’s friend paradox, the theorem of Bong et al and the theorem of Lawrence et al are all best understood as demonstrating that if quantum mechanics is universal, and if certain auxiliary assumptions hold, then the world inevitably includes various forms of ‘disaccord,’ but this need not be interpreted in a metaphysically radical way; meanwhile, the theorem of Ormrod and Barrett is best understood either as an argument for an interpretation allowing multiple outcomes per observer, such as the Everett approach, or as a proof that quantum mechanics cannot be universal in the sense relevant for this theorem. We also argue that these theorems taken together suggest interesting possibilities for a different kind of relational approach in which dynamical states are relativized whilst observed events are absolute, and we show that although something like ‘retrocausality’ might be needed to make such an approach work, this would be a very special kind of retrocausality which would evade a number of common objections against retrocausality. We conclude that the non-absoluteness theorems may have a significant role to play in helping converge towards an acceptable solution to the measurement problem.

General gravitational charges on null hypersurfaces

We perform a detailed study of the covariance properties of the gravitational symplectic potential on a null hypersurface, and of the different polarizations that can be used to study conservative as well as leaky boundary conditions. We study the symmetry groups that arise with different %boundary conditions in the phase space prescriptions, and determine the fields that have anomalous transformations. This allows us to identify a one-parameter family of covariant symplectic potentials. Imposing stationarity as in the original Wald-Zoupas prescription, one recovers the unique symplectic potential of Chandrasekaran, Flanagan and Prabhu. The associated charges are all conserved on non-expanding horizons, but not on flat spacetime. We show that it is possible to demand a weaker notion of stationarity which selects another symplectic potential, again in a unique way, and whose charges are conserved on both non-expanding horizons and flat light-cones. Furthermore, the flux of future-pointing diffeomorphisms at leading-order around an outgoing flat light-cone is positive and reproduces the tidal heating term plus a memory-like term. Our results have applications for dynamical notions of entropy, and are useful to clarify the interplay between different boundary conditions, charge prescriptions, and symmetry groups that can be associated with a null boundary. We also study the conformal conservative boundary conditions suggested by the alternative polarization and identify under which conditions they define a non-ambiguous variational principle.

Null Raychaudhuri: Canonical Structure and the Dressing Time

We initiate a study of gravity focusing on generic null hypersurfaces, non-perturbatively in the Newton coupling. We present an off-shell account of the extended phase space of the theory, which includes the expected spin-2 data as well as spin-0, spin-1 and arbitrary matter degrees of freedom. We construct the charges and the corresponding kinematic Poisson brackets, employing a Beltrami parameterization of the spin-2 modes. We explicitly show that the constraint algebra closes, the details of which depend on the non-perturbative mixing between spin-0 and spin-2 modes. Finally we show that the spin zero sector encodes a notion of a clock, called dressing time, which is dynamical and conjugate to the constraint. It is well-known that the null Raychaudhuri equation describes how the geometric data of a null hypersurface evolve in null time in response to gravitational radiation and external matter. Our analysis leads to three complementary viewpoints on this equation. First, it can be understood as a Carrollian stress tensor conservation equation. Second, we construct spin-$0$, spin-$2$ and matter stress tensors that act as generators of null time reparametrizations for each sector. This leads to the perspective that the null Raychaudhuri equation can be understood as imposing that the sum of CFT-like stress tensors vanishes. Third, we solve the Raychaudhuri constraint non-perturbatively. The solution relates the dressing time to the spin-$2$ and matter boost charge operators. Finally we establish that the corner charge corresponding to the boost operator in the dressing time frame is concave. These results show that the notion of an observer can be thought of as emerging from the gravitational degrees of freedom themselves. We briefly mention that the construction offers new insights into focusing conjectures.

A trace inequality for Euclidean gravitational path integrals (and a new positive action conjecture)

The AdS/CFT correspondence states that certain conformal field theories are equivalent to string theories in a higher-dimensional anti-de Sitter space. One aspect of the correspondence is an equivalence of density matrices or, if one ignores normalizations, of positive operators. On the CFT side of the correspondence, any two positive operators $A,B$ will satisfy the trace inequality $operatorname{Tr}(AB) leq operatorname{Tr}(A) operatorname{Tr}(B)$. This relation holds on any Hilbert space ${cal H}$ and is deeply associated with the fact that the algebra $B({cal H})$ of bounded operators on ${cal H}$ is a type I von Neumann factor. Holographic bulk theories must thus satisfy a corresponding condition, which we investigate below. In particular, we argue that the Euclidean gravitational path integral respects this inequality at all orders in the semi-classical expansion and with arbitrary higher-derivative corrections. The argument relies on a conjectured property of the classical gravitational action, which in particular implies a positive action conjecture for quantum gravity wavefunctions. We prove this conjecture for Jackiw-Teitelboim gravity and we also motivate it for more general theories.

Global Synthesis of CNOT Circuits with Holes

A common approach to quantum circuit transformation is to use the properties of a specific gate set to create an efficient representation of a given circuit’s unitary, such as a parity matrix or stabiliser tableau, and then resynthesise an improved circuit, e.g. with fewer gates or respecting connectivity constraints. Since these methods rely on a restricted gate set, generalisation to arbitrary circuits usually involves slicing the circuit into pieces that can be resynthesised and working with these separately. The choices made about what gates should go into each slice can have a major effect on the performance of the resynthesis. In this paper we propose an alternative approach to generalising these resynthesis algorithms to general quantum circuits. Instead of cutting the circuit into slices, we “cut out” the gates we can’t resynthesise leaving holes in our quantum circuit. The result is a second-order process called a quantum comb, which can be resynthesised directly. We apply this idea to the RowCol algorithm, which resynthesises CNOT circuits for topologically constrained hardware, explaining how we were able to extend it to work for quantum combs. We then compare the generalisation of RowCol using our method to the naive “slice and build” method empirically on a variety of circuit sizes and hardware topologies. Finally, we outline how quantum combs could be used to help generalise other resynthesis algorithms.

Matter relative to quantum hypersurfaces

We explore the canonical description of a scalar field as a parameterized field theory on an extended phase space that includes additional embedding fields that characterize spacetime hypersurfaces $mathsf{X}$ relative to which the scalar field is described. This theory is quantized via the Dirac prescription and physical states of the theory are used to define conditional wave functionals $|psi_phi[mathsf{X}]rangle$ interpreted as the state of the field relative to the hypersurface $mathsf{X}$, thereby extending the Page-Wootters formalism to quantum field theory. It is shown that this conditional wave functional satisfies the Tomonaga-Schwinger equation, thus demonstrating the formal equivalence between this extended Page-Wootters formalism and standard quantum field theory. We also construct relational Dirac observables and define a quantum deparameterization of the physical Hilbert space leading to a relational Heisenberg picture, which are both shown to be unitarily equivalent to the Page-Wootters formalism. Moreover, by treating hypersurfaces as quantum reference frames, we extend recently developed quantum frame transformations to changes between classical and nonclassical hypersurfaces. This allows us to exhibit the transformation properties of a quantum field under a larger class of transformations, which leads to a frame-dependent particle creation effect.

Predictive Modelling of Quantum Process with Neural Networks

Complete characterization of an unknown quantum process can be achieved by process tomography, or, for continuous time processes, by Hamiltonian learning. However, such a characterization becomes unfeasible for high dimensional quantum systems. In this paper, we develop the first neural network algorithm for predicting the behavior of an unknown quantum process when applied on a given ensemble of input states. The network is trained with classical data obtained from measurements on a few pairs of input/output quantum states. After training, it can be used to predict the measurement statistics of a set of measurements of interest performed on the output state corresponding to any input in the state ensemble. Besides learning a quantum gate or quantum circuit, our model can also be applied to the task of learning a noisy quantum evolution and predicting the measurement statistics on a time-evolving quantum state. We show numerical results using our neural network model for various relevant processes in quantum computing, quantum many-body physics, and quantum optics.

Sequence Processing with Quantum Tensor Networks

We introduce complex-valued tensor network models for sequence processing motivated by correspondence to probabilistic graphical models, interpretability and resource compression. Inductive bias is introduced to our models via network architecture, and is motivated by the correlation structure inherent in the data, as well as any relevant compositional structure, resulting in tree-like connectivity. Our models are specifically constructed using parameterised quantum circuits, widely used in quantum machine learning, effectively using Hilbert space as a feature space. Furthermore, they are efficiently trainable due to their tree-like structure. We demonstrate experimental results for the task of binary classification of sequences from real-world datasets relevant to natural language and bioinformatics, characterised by long-range correlations and often equipped with syntactic information. Since our models have a valid operational interpretation as quantum processes, we also demonstrate their implementation on Quantinuum’s H2-1 trapped-ion quantum processor, demonstrating the possibility of efficient sequence processing on near-term quantum devices. This work constitutes the first scalable implementation of near-term quantum language processing, providing the tools for large-scale experimentation on the role of tensor structure and syntactic priors. Finally, this work lays the groundwork for generative sequence modelling in a hybrid pipeline where the training may be conducted efficiently in simulation, while sampling from learned probability distributions may be done with polynomial speed-up on quantum devices.

A clarification on prevailing misconceptions in unimodular gravity

The traditional presentation of Unimodular Gravity (UG) consists on indicating that it is an alternative theory of gravity that restricts the generic diffeomorphism invariance of General Relativity. In particular, as often encountered in the literature, unlike General Relativity, Unimodular Gravity is invariant solely under volume-preserving diffeomorphisms. That characterization of UG has led to some confusion and incorrect statements in various treatments on the subject. For instance, sometimes it is claimed (mistakenly) that only spacetime metrics such that $|$det $g_{mu nu}| = 1$ can be considered as valid solutions of the theory. Additionally, that same (incorrect) statement is often invoked to argue that some particular gauges (e.g. the Newtonian or synchronous gauge) are not allowed when dealing with cosmological perturbation theory in UG. The present article is devoted to clarify those and other misconceptions regarding the notion of diffeomorphism invariance, in general, and its usage in the context of UG, in particular.