I've been thinking a bit lately about pump jets and whether new Chinese SSNs will feature them. I want to dive into (no pun) how they work and what would limit China from being able to use them. To see pump jets' benefit we first have to examine the physics of cavitation. Let's first consider the relationship between the speed of a flow and its pressure given by a simplified form of the Bernoulli equation:
P + cV^2 = constant (when omitting changes in height of the flow).
This is the conservation of energy applied to fluid flows. We see that pressure and flow speed are "inversely" proportional (not exactly, but close enough for our purposes). Let's apply this equation to a submarine's propeller - as the propeller turns in the water, the speed of the water relative to the propeller increases, which causes the pressure of the water near the propeller to drop. This is the start of our problems.
A liquid boils when its vapour pressure (a function of temperature) exceeds the ambient pressure. If you took a glass of water at room temperature and pulled a sufficient vacuum around it, it would boil. Similarly at the interface between the propeller and the surrounding water - the pressure in that region is low enough that the water begins to boil. It forms small steam bubbles that travel along the propeller's flow into the surrounding water. The pressure there is much higher than the bubble's pressure which causes the bubble to implode, and the shockwave from that sudden implosion is what's heard as cavitation.
How to address this? The Bernoulli equation suggests two ways not to cross that critical pressure threshold - spin the propeller slower or raise the pressure of the surrounding water. Spinning the propeller slower is certainly a possibility, which is why submarines rarely travel at flank speed unless absolutely necessary. One way to "raise" the pressure is to dive deeper; however, the bathymetry might not permit that (which is the case in the SCS and is part of the reason why it's such a challenging environment for submarines) or the depth necessary might exceed what the submarine's structure can tolerate.
Pump jets offer a way of increasing the pressure of the surrounding water without counting on gravity to do it for you. The exact mechanics aren't necessary to get into, the only thing to note for our purposes is that pump jets can increase the pressure of a flow at the same speed relative to a bare propeller. Great, right? Not so fast. Let's look at the Bernoulli equation again for two flows at the same speed.
P_1 + cV^2 = c_1
P_2 + cV^2 = c_2
Since P_1 (the pump jet flow) > P_2 (the conventional propeller flow), by necessity c_1 > c_2. This means that the energy of the first flow is greater than the energy of the second, which means the work the submarine must do to create the first flow exceeds the work it needs for the second, and the higher the desired pressure and turning speed (a product of the propeller's rotational speed and radius), the more the work.
From the point of view of propelling the submarine, pressurizing its outward flow is entirely wasted energy. The only thing that matters is the speed of the flow. Therefore, to pressurize the flow the submarine equipped with a pump jet requires a more powerful prime mover than the same submarine equipped with a propeller. This is why we don't see pump jets equipped on SSKs, whose power output is already limited. It's also why we don't see them on earlier generations of SSNs as their reactors did not have the power density to support this "waste".
The conclusion from all this is that whether we'll see pump jets on Type 09-IIIB/V submarines doesn't depend on the complexities of the pump jet itself, rather on how successfully China has been able to miniaturize nuclear reactors of sufficiently high output to achieve the power density required to support a pump jet.