So I take efficient bureaucracy to mean how politics was efficiently conducted in Germany, which allowed them to resolve conflicts better and have a pinpoint focus, or vision, of what they wanted. The sources I have divide the political apparatus of Germany into pre-1848 and post-1848, and have some focus on Bismarck's interpretation of political bureaucracy of the two periods. If this is not what your question is asking, then please ignore this. However, just to start, Prussia's efficiency is, in my perspective, derived from the lack of power in the Prussian Assembly and the ability of the " top brass" to conduct business unimpeded, but I think it is explained below.

Let's start off with a direct quote from Otto von Bismarck: “Constitutional life is a series of compromises. If these are frustrated, conflicts arise. Conflicts are questions of power, and whoever has power to hand, can go his own way” (Steinberg, Bismarck: A Life, p472). This was the man that engineered politics in Prussia, and he embodied the mindset of his time.

Prior to 1848, the provincial diets that had been set up to appease liberals had little to no power; even their advisory capacity was ignored and they lacked the confidence of the public (Berdahl, The Politics of the Prussian Nobility, p186). Opinions on what to do swayed back and forth. The King was not decisive enough to formulate a plan on how to handle the liberal fervor that was so taking the country. The Frankfurt Assembly had gathered in 1849 to make offers to the Prussian king, giving him control of a unified Germany with liberal conditions (Kolkey, Germany on the March, p75).

So Otto von Bismarck, in a speech to the Prussian Assembly declared, “The crown of Frankfort may be very brilliant, but the gold which gives genuineness to its brilliance must first be obtained by melting down the Prussian crown; and I have no faith that a recasting in the mould of the Constitution will succeed” (Legge, Rhyme and revolution in Germany, p522). This was a speech that damned liberalism in Germany. However, Bismarck ultimately needed to secure Prussia as the sole ruler of the unified Germany, without any of the liberal fetter presented by the Frankfurt Assembly. He advised the king to reject the deal, and the Frankfurt Assembly failed (Kolkey, Germany on the March, p75).

With the threat of the Frankfurt Assembly over, Bismarck still had to deal with the spread of Prussian liberalism to bring public support behind the king. To do this, Bismarck needed to give a new Prussian Diet just enough power to appease them while giving them enough representative features so that lower-class Prussians would be appeased. What he devised was a brilliant formulation of a representative assembly that did exactly what it should do and no more: “By granting the liberals a few of their requests (such as a parliament), Bismarck minimized the revolution’s effect by creating a government with a parliament, but one which was so restricted that it was as if it did not exist at all. Even had the parliament not been so weak, the effect would have been nearly the same because the limited membership would have allowed betterment of only the upper classes” (Eyck, Bismarck and the German Empire, p60). Bismarck deftly put aside any meaningful impact the revolution might have had on Prussian politics, and at the same time put the Prussian public solely behind the crown.

Pre-and post 1848, the political apparatus was pretty weak, and allowed the Junkers to control and direct affairs. A top-heavy political system can be extremely efficient, and was a major factor in the German Unification.

Edit: Just wanted to say, some of these sources (well, really, all of them) are fascinating reads. I was most of the way through Kolkey's book before I even knew it! Him and Berdahl are the most interesting reads, since they (especially Berdahl) focus on topics that I hadn't seen much about before hand.

Edit2: See below about Beamten comment, should have written about that in main comment.

In short, the effectiveness of the Prussian bureaucracy was its coordination of the vast majority of society towards a collective goal of national strength.

By leveraging the small population into targeted positions (civil service, military) within a greater coherent strategy, the Prussians were able to create an incredibly efficient and coordinated state.

To borrow from Marx and de Tocqueville’s analysis of political economy, the coordination of the base (civil society) and the superstructure (political society) was both strong, as well as culturally celebrated in Prussia. The greatest honour was left to those who maintained their individual function to the greatest of their capacity in contributing to the strength of the Prussian state.

In Prussia, every individual member of society had a specific role that was celebrated as contributing to the strength of the collective.

For all of his dislike of Prussian elites, Hitler’s bureaucratisation of Germany into a coordinated whole was the why the Third Reich was able to do what Fredrick did centuries before. And in the end, the bureaucratisation of genocide is why the Holocaust was so frighteningly efficient.

I would assume this is all talking about the 19th century; is this a typo?

In short, the time from 1650 to 1750 was a time of bureaucratic consolidation in Prussia, when the needs of the state and the personality of the monarchs tuned the bureaucratic apparatus to be more effective in some areas than in most other countries.

This started with the growing demand of the standing army for money - the Great Elector expanded the standing army tenfold. To meet this, a ministry was created which superseded the traditional contributions of the estates to the financing of the army. In effect, this ministry (the Kriegskommisariat) began to take the adminstration of the land away from the locals. Combine that with the fact that the Elector's domains made up a third of Brandenburg and half of the duchy of Prussia, which he wanted to be better developed and created a ministry for (the Geheime Hofkammer) the centralization of the administration had started.

His grandson, also called Friedrich Wilhelm (the I. - as King) continued to strengthen a centralized administration, he merged those institutions into a super ministry (the General-Ober-Finanz-Kriegs- und Domainen-Directorium), which was responsible for nearly all of the financial - and economic politics and administration of the Kingdom of Prussia. He used this Generaldirectorium to tax and develop the country, to store grain and distribute when famine occured, to resettle devastated areas (after the Great Northern War, and several famines). Friedrich Wilhelm hated corruption (he had some corrupt bureaucrats executed) and was a stickler to details, which cultivated a sense of fearfull excellence in the bureaucrats of Prussia. His way of organizing the army (he demanded the sons of the nobility to be officers in his army) also lead the local autonomous administration (which partly existed into the days of Bismarck!) into (mostly) cooperating with the King.

Another important thing happened during F.W. reign: the university of Halle started to produce able administrators for the Prussian state. It was a centre of pietistic influence in Germany. In 1717, Friedrich Wilhelm made schooling compulsatory (this lead to Prussia becoming one of the most literate states in Europe during most of its existence). In the 1720s, it was made a mandatory prequesite for (higher) state service to have attended Halle university for two (later four) semesters.

After Friedrich Wilhelm's son, Friedrich II., succeeded to the throne, the university of Halle became a centre of the enlightenment, while still providing nearly all of the higher bureaucrats in Prussia. Friedrich Wilhelm and Friedrich II. spent a lot of effort in encouraging the growth of manufactories in their territories, through the Generaldirectorium. This interventionalist kind of economic planning became increasingly out of touch with contemporary economic thinking during Friedrich's later reign, however. Like his father, Friedrich was obsessed with details and a workaholic, so the working climate didn't really change for bureaucrats.

When Friedrich annexed Silesia, he had a competent bureaucracy which was adjusted to the demands of the army and the task of integrating Silesia into Prussia (and to rebuild it, if needed).

The quite excellent Iron Kingdom by Christopher Clark has more than plenty information on Prussian administrative history and was used as the main source for this post.