For Fixed Control Parameters the Quantum Approximate Optimization Algorithm's Objective Function Value Concentrates for Typical Instances

The Quantum Approximate Optimization Algorithm, QAOA, uses a shallow depth quantum circuit to produce a parameter dependent state. For a given combinatorial optimization problem instance, the quantum expectation of the associated cost function is the parameter dependent objective function of the QAOA. We demonstrate that if the parameters are fixed and the instance comes from a reasonable distribution then the objective function value is concentrated in the sense that typical instances have (nearly) the same value of the objective function. This applies not just for optimal parameters as the whole landscape is instance independent. We can prove this is true for low depth quantum circuits for instances of MaxCut on large 3-regular graphs. Our results generalize beyond this example. We support the arguments with numerical examples that show remarkable concentration. For higher depth circuits the numerics also show concentration and we argue for this using the Law of Large Numbers. We also observe by simulation that if we find parameters which result in good performance at say 10 bits these same parameters result in good performance at say 24 bits. These findings suggest ways to run the QAOA that reduce or eliminate the use of the outer loop optimization and may allow us to find good solutions with fewer calls to the quantum computer.Comment: 16 pages, 1 figur

Indoor Navigation with MEMS sensors

AbstractAccurate positioning becomes extremely important for modern application like indoor navigation and location-based services. Standalone GPS cannot meet this accuracy. In this paper a method to couple GPS and a high resolution MEMS pressure sensor is presented to improve vertical as well as horizontal (in urban canyon environment) positioning. Further, a step counter based on an accelerometer is improved with an altimeter for stair detection and automatic step length adaptation for dead reckoning inside buildings. Finally, a stand-alone system accurately tracks floor levels inside buildings, using only a pressure sensor

Differences in daptomycin and vancomycin ex vivo behaviour can lead to false interpretation of negative blood cultures

AbstractIn clinical studies on bacteraemia, the negativity of blood cultures is an important endpoint for comparing the efficacy of different therapeutic regimens. In FAN° anaerobic blood culture medium (BacT/ALERT system), daptomycin displayed increased MIC against Staphylococcus aureus and improved abolishment of its carryover effect in charcoal when compared with vancomycin. Differences between these two drugs can lead to a false interpretation of negative blood cultures. To compare different antibiotic regimens for the treatment of bacteraemia, preliminary studies are mandatory to ensure that ex vivo antibiotic behaviour is similar in the blood-culture system used

Approximating Fractional Time Quantum Evolution

An algorithm is presented for approximating arbitrary powers of a black box unitary operation, $\mathcal{U}^t$, where $t$ is a real number, and $\mathcal{U}$ is a black box implementing an unknown unitary. The complexity of this algorithm is calculated in terms of the number of calls to the black box, the errors in the approximation, and a certain `gap' parameter. For general $\mathcal{U}$ and large $t$, one should apply $\mathcal{U}$ a total of $\lfloor t \rfloor$ times followed by our procedure for approximating the fractional power $\mathcal{U}^{t-\lfloor t \rfloor}$. An example is also given where for large integers $t$ this method is more efficient than direct application of $t$ copies of $\mathcal{U}$. Further applications and related algorithms are also discussed.Comment: 13 pages, 2 figure