What is the effect of Quantitative Easing on the US budget deficit? Is the deficit getting larger because of QE?
No, the deficit is not getting larger due to QE. Certainly not directly, because that's impossible, and also not indirectly, either.

Quantitative easing is a policy of purchasing government bonds with the intent of decreasing yields while injecting cash into the economy. It affects only the demand for government bonds, not the supply of bonds created by government borrowing. 

This shouldn't be a surprise, as the monetary authority (the central bank) and the fiscal authority (Congress) are independent of one another. Congress cannot (without rewriting the Federal Reserve Act) directly control monetary policy, nor can the Fed force the government to borrow more.

Indirectly, Congress could choose to borrow more as a result of QE, due to lower yields on government debt, as the lower debt payments make it cheaper to do so. However, there's not much evidence that this is the case. 

The deficit has fallen since 2009 (when it significantly increased due to passage of a large-scale stimulus program consisting primarily of tax cuts but also including direct federal spending, in response to the financial crisis and recession), both in nominal terms and as a percent of GDP. As a percentage of GDP (which is the appropriate long-term comparison, as taxable income is the best indication of a nation's ability to repay a debt), the deficit returned to a normal historical range by 2014.

So it's difficult to conclude even that QE encouraged spending, thereby increasing the deficit indirectly. Given two facts— that the Federal Reserve's QE programs started in earnest in 2009 and continued until late 2014, and that the federal deficit steadily fell during this time— it would be very strange indeed to conclude that QE had increased the deficit.