We report ultra-low temperature experiments on the obscure fractional quantum Hall effect (FQHE) at Landau level filling factor ν = 5/2 in a very high mobility specimen of µ = 1.7 × 10 7 cm 2 /Vs. We achieve an electron temperature as low as ∼ 4 mK, where we observe vanishing Rxx and, for the first time, a quantized Hall resistance, Rxy = h/(5/2e2 ) to within 2 ppm. Rxy at the neighboring odd-denominator states ν = 7/3 and 8/3 is also quantized. The temperature dependences of the Rxx-minima at these fractional fillings yield activation energy gaps ∆ 5/2 = 0.11 K, ∆ 7/3 = 0.10 K, and ∆ 8/3 = 0.055 K.PACS Numbers: 73.40Hm Electrons in two-dimensional systems at low temperatures and in the presence of an intense magnetic field condense into a sequence of incompressible quantum fluids with finite energy gaps for quasiparticle excitation, termed collectively the fractional quantum Hall effect (FQHE) [1]. These highly correlated electronic states occur at rational fractional filling ν = p/q of Landau levels. Their characteristic features in electronic transport experiments are vanishing resistance, R xx , and exact quantization of the concomitant Hall resistance, R xy , to h/(p/qe 2 ). Over the years, a multitude of FQHE states have been discovered -all q's being odd numbers. The only known exceptions are the states at half-filling of the second Landau level ν = 5/2 (=2+1/2) and ν = 7/2 (=3+1/2) [2-4]. Half-filled states in the lowest Landau level show no FQHE, whereas half-filled states in still higher Landau levels exhibit yet unresolved anisotropies [5]. Recent experiments in tilted magnetic field even seem to hint at a connection between the ν = 9/2 state and the state at ν = 5/2 [6].The origin of the ν = 5/2 and 7/2 states remains mysterious. Observation of odd-denominator FQHE states is intimately connected to the anti-symmetry requirement for the electronic wave function. An early, socalled hollow-core model [7] for the FQHE at ν = 5/2 and 7/2, which takes explicitly into account aspects of the modified single-particle wave functions of the second Landau level, arrived at a trial wave function. However, for a Coulomb Hamiltonian its applicability is problematic [8,9].With the advent of the composite fermion (CF) model [10] the existence of exclusively odd-denominator FQHE states is traced back to the formation of Landau levels of CFs emanating from even-denominator fillings, such as the sequence ν = p/(2p ± 1) from ν = 1/2. Evendenominator fillings themselves represent Fermi-liquid like states, resulting from the attachment of an even number of magnetic flux quanta to each electron. The obvious conflict between this theory and experiment at ν = 5/2 is resolved by invoking a CF-pairing mechanism [11][12][13][14]. In loose analogy to the formation of Cooper pairs in superconductivity such pairing creates a gapped, BCS-like ground state at ν = 5/2, called a "Pfaffian" state, which displays a FQHE. Indeed, an exact numerical diagonalization calculation by Morf [9] favors the Pfaffian state.The experimental si...
We describe the design, development and performance of a scanning probe microscopy (SPM) facility operating at a base temperature of 10 mK in magnetic fields up to 15 T. The microscope is cooled by a custom designed, fully ultra-high vacuum (UHV) compatible dilution refrigerator (DR) and is capable of in situ tip and sample exchange. Subpicometer stability at the tip-sample junction is achieved through three independent vibration isolation stages and careful design of the dilution refrigerator. The system can be connected to, or disconnected from, a network of interconnected auxiliary UHV chambers, which include growth chambers for metal and semiconductor samples, a field-ion microscope for tip characterization, and a fully independent additional quick access low temperature scanning tunneling microscope (STM) and atomic force microscope (AFM) system. To characterize the system, we present the cooling performance of the DR, vibrational, tunneling current, and tip-sample displacement noise measurements. In addition, we show the spectral resolution capabilities with tunneling spectroscopy results obtained on an epitaxial graphene sample resolving the quantum Landau levels in a magnetic field, including the sublevels corresponding to the lifting of the electron spin and valley degeneracies.
The discovery of topological order has revolutionized the understanding of quantum matter in modern physics and provided the theoretical foundation for many quantum error correcting codes. Realizing topologically ordered states has proven to be extremely challenging in both condensed matter and synthetic quantum systems. Here, we prepare the ground state of the toric code Hamiltonian using an efficient quantum circuit on a superconducting quantum processor. We measure a topological entanglement entropy near the expected value of ln 2, and simulate anyon interferometry to extract the braiding statistics of the emergent excitations. Furthermore, we investigate key aspects of the surface code, including logical state injection and the decay of the non-local order parameter. Our results demonstrate the potential for quantum processors to provide key insights into topological quantum matter and quantum error correction.
Realizing the potential of quantum computing requires sufficiently low logical error rates1. Many applications call for error rates as low as 10−15 (refs. 2–9), but state-of-the-art quantum platforms typically have physical error rates near 10−3 (refs. 10–14). Quantum error correction15–17 promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device18,19 and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits.
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