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In this paper, we demonstrate the non-existence of a computational algorithm capable of determining whether the second homotopy group of any compact constructive topological space is trivial. This finding shows the inherent limitations of computer methods in resolving certain topological problems.

The mass production and the practical number of cryogenic quantum devices producible in a single chip are limited to the number of electrical contact pads and wiring of the cryostat or dilution refrigerator. It is, therefore, beneficial to contrast the measurements of hundreds of devices fabricated in a single chip in one cooldown process to promote the scalability, integrability, reliability, and reproducibility of quantum devices and to save evaluation time, cost and energy. Here, we use a cryogenic on-chip multiplexer architecture and investigate the statistics of the 0.7 anomaly observed on the first three plateaus of the quantized conductance of semiconductor quantum point contact (QPC) transistors. Our single chips contain 256 split gate field effect QPC transistors (QFET) each, with two 16-branch multiplexed source-drain and gate pads, allowing individual transistors to be selected, addressed and controlled through an electrostatic gate voltage process. A total of 1280 quantum transistors with nano-scale dimensions are patterned in 5 different chips of GaAs heterostructures. From the measurements of 571 functioning QPCs taken at temperatures T= 1.4 K and T= 40 mK, it is found that the spontaneous polarisation model and Kondo effect do not fit our results. Furthermore, some of the features in our data largely agreed with van Hove model with short-range interactions. Our approach provides further insight into the quantum mechanical properties and microscopic origin of the 0.7 anomaly in QPCs, paving the way for the development of semiconducting quantum circuits and integrated cryogenic electronics, for scalable quantum logic control, readout, synthesis, and processing applications.

Quantum point contact or QPC -- a constriction in a semiconducting two-dimensional (2D) electron system with a quantized conductance -- has been found as the building block of novel spintronic, and topological electronic circuits. They can also be used as readout electronic, charge sensor or switch in quantum nanocircuits. A short and impurity-free constriction with superconducting contacts is a Cooper pairs QPC analogue known as superconducting quantum point contact (SQPC). The technological development of such quantum devices has been prolonged due to the challenges of maintaining their geometrical requirement and near-unity superconductor-semiconductor interface transparency. Here, we develop advanced nanofabrication, material and device engineering techniques and report on an innovative realisation of nanoscale SQPC arrays with split gate technology in semiconducting 2D electron systems, exploiting the special gate tunability of the quantum wells, and report the first experimental observation of conductance quantization in hybrid InGaAs-Nb SQPCs. We observe reproducible quantized conductance at zero magnetic fields in multiple quantum nanodevices fabricated in a single chip and systematically investigate the quantum transport of SQPCs at low and high magnetic fields for their potential applications in quantum metrology, for extremely accurate voltage standards, and fault-tolerant quantum technologies.

While the application of out-of-plane magnetic fields was, so far, believed to be detrimental for the formation of Majorana phases in artificially engineered hybrid superconducting-semiconducting junctions, several recent theoretical studies have found it indeed useful in establishing such topological phases 1-5. Majorana phases emerge as quantized plateaus in the magnetoconductance of the hybrid junctions based on two-dimensional electron gases (2DEG) under fully out-of-plane magnetic fields. The large transverse Rashba spin-orbit interaction in 2DEG, together with a strong magneto-orbital effect, yield topological phase transitions to nontrivial phases hosting Majorana modes. Such Majorana modes are formed at the ends of 2DEG-based wires with a hybrid superconductor-semiconductor integrity. Here, we report on the experimental observation of such topological phases in Josephson junctions, based on In0.75Ga0.25As 2DEG, by sweeping out-of-plane magnetic fields of as small as 0 < B(mT) < 100 and probing the conductance to highlight the characteristic quantized magnetoconductance plateaus. Our approaches towards (i) creation and detection of topological phases in small out-of-plane magnetic fields, and (ii) integration of an array of topological Josephson junctions on a single chip pave the ways for the development of scalable quantum integrated circuits for their potential applications in fault-tolerant quantum processing and computing., Comment: 32 pages, 12 figures