Study Protocol for a Randomised Controlled Trial on Pulmonary Metastasectomy vs. Standard of Care in Colorectal Cancer Patients With ≥ 3 Lung Metastases (PUCC-Trial).

This is a multicentre prospective randomised controlled trial for patients with 3 or more resectable pulmonary metastases from colorectal carcinoma. The study investigates the effects of pulmonary metastasectomy in addition to standard medical treatment in comparison to standard medical treatment plus possible local ablative measures such as SBRT. This trial is intended to demonstrate an overall survival difference in the group undergoing pulmonary metastasectomy. Further secondary and exploratory endpoints include quality of life (EORTC QLQ-C30, QLQ-CR29 and QLQ-LC29 questionnaires), progression-free survival and impact of mutational status. Due to the heterogeneity and complexity of the disease and treatment trajectories in metastasised colorectal cancer, well powered trials have been very challenging to design and execute. The goal of this study is to create a setting which allows treatment as close to the real life conditions as possible but under well standardised conditions. Based on previous trials, in which patient recruitment in the given setting hindered successful study completion, we decided to (1) restrict inclusion to patients with 3 or more metastases (since in case of lesser, surgery will probably be the preferred option) and (2) allow for real world standard of care (SOC) treatment options before and after randomisation including watchful waiting (as opposed to a predefined treatment protocol) and (3) possibility that patient can receive SOC externally (to reduce patient burden). Moreover, we chose to stipulate 12 weeks of systemic treatment prior to possible resection to further standardize treatment response and disease course over a certain period of time. Hence, included patients will be in the disease state of oligopersistence rather than primary oligometastatic. The trial was registered in the German Clinical Trials Register (DRKS-No.: DRKS00024727).

Hamiltonian simulation algorithms for near-term quantum hardware.

The quantum circuit model is the de-facto way of designing quantum algorithms. Yet any level of abstraction away from the underlying hardware incurs overhead. In this work, we develop quantum algorithms for Hamiltonian simulation "one level below" the circuit model, exploiting the underlying control over qubit interactions available in most quantum hardware and deriving analytic circuit identities for synthesising multi-qubit evolutions from two-qubit interactions. We then analyse the impact of these techniques under the standard error model where errors occur per gate, and an error model with a constant error rate per unit time. To quantify the benefits of this approach, we apply it to time-dynamics simulation of the 2D spin Fermi-Hubbard model. Combined with new error bounds for Trotter product formulas tailored to the non-asymptotic regime and an analysis of error propagation, we find that e.g. for a 5 × 5 Fermi-Hubbard lattice we reduce the circuit depth from 1, 243, 586 using the best previous fermion encoding and error bounds in the literature, to 3, 209 in the per-gate error model, or the circuit-depth-equivalent to 259 in the per-time error model. This brings Hamiltonian simulation, previously beyond reach of current hardware for non-trivial examples, significantly closer to being feasible in the NISQ era.

Uncomputability of phase diagrams.

The phase diagram of a material is of central importance in describing the properties and behaviour of a condensed matter system. In this work, we prove that the task of determining the phase diagram of a many-body Hamiltonian is in general uncomputable, by explicitly constructing a continuous one-parameter family of Hamiltonians H(φ), where [Formula: see text], for which this is the case. The H(φ) are translationally-invariant, with nearest-neighbour couplings on a 2D spin lattice. As well as implying uncomputablity of phase diagrams, our result also proves that undecidability can hold for a set of positive measure of a Hamiltonian's parameter space, whereas previous results only implied undecidability on a zero measure set. This brings the spectral gap undecidability results a step closer to standard condensed matter problems, where one typically studies phase diagrams of many-body models as a function of one or more continuously varying real parameters, such as magnetic field strength or pressure.

Size-driven quantum phase transitions.

Can the properties of the thermodynamic limit of a many-body quantum system be extrapolated by analyzing a sequence of finite-size cases? We present models for which such an approach gives completely misleading results: translationally invariant, local Hamiltonians on a square lattice with open boundary conditions and constant spectral gap, which have a classical product ground state for all system sizes smaller than a particular threshold size, but a ground state with topological degeneracy for all system sizes larger than this threshold. Starting from a minimal case with spins of dimension 6 and threshold lattice size [Formula: see text], we show that the latter grows faster than any computable function with increasing local spin dimension. The resulting effect may be viewed as a unique type of quantum phase transition that is driven by the size of the system rather than by an external field or coupling strength. We prove that the construction is thermally robust, showing that these effects are in principle accessible to experimental observation.

The complexity of divisibility.

We address two sets of long-standing open questions in linear algebra and probability theory, from a computational complexity perspective: stochastic matrix divisibility, and divisibility and decomposability of probability distributions. We prove that finite divisibility of stochastic matrices is an NP-complete problem, and extend this result to nonnegative matrices, and completely-positive trace-preserving maps, i.e. the quantum analogue of stochastic matrices. We further prove a complexity hierarchy for the divisibility and decomposability of probability distributions, showing that finite distribution divisibility is in P, but decomposability is NP-hard. For the former, we give an explicit polynomial-time algorithm. All results on distributions extend to weak-membership formulations, proving that the complexity of these problems is robust to perturbations.