I was watching Samurai Swordmaking, which was uploaded in 2013, where an old man forges a sword by hand using fire and hammer. Has anybody seen a modern technology for making a sword that is as effective as one produced using a traditional method?
This 2007 interview with Dr. Rick Vinci, a Stanford-trained materials scientist and engineer at Lehigh University, by NOVA does a good job of describing both the materials science and engineering of the samurai sword.
The sword is an excellent example of people, first of all, understanding the requirements of the particular application. Understanding that the cutting edge of the blade must have a certain sort of characteristic—hard so that it retains its sharpness and its ability to penetrate—while the back edge must have a different sort of characteristic—it must be tough in order to withstand damage and fracture. And then the ability to understand how to achieve these properties through the right choice of materials and the right processing.
In the interview, the question of whether industrial processes can reproduce the sword comes up directly:
Can we make a sword today that's as good as the samurai sword of yore?
There is no mystery at this point about exactly what the metals are, so yes. It is understood what makes up a samurai sword, and there is no inherent problem with reproducing it using industrial techniques.
It is unclear from this resource whether anyone actually uses industrial processes to make high-quality swords, or whether this is a simply a theoretical statement.
This is more of an aside than a full answer (but too long and with links not well handled by comments), but it is worth noting that the traditional Katanas evolved in a region where the naturally available iron was both rarer than in many areas in Europe and of lower quality. While the famed folding process does have other effects, one of the most significant was that it helped remove impurities and helped more evenly distribute carbon. In other words, they were essentially making steel.
The difference between steel and iron in terms of producing a weapon is enormous. For just one thing, iron tends to be more brittle. This means that it fails, and does so catastrophically, more easily than steel does.
But of course, there are numerous different grades of steel. It is too simplistic to say that one grade of steel is simply better than another, but some are clearly more suited to certain purposes than others.
To make a long story short, we can make vastly superior steel now with modern processes (and do so more consistently) than traditional Japanese sword-smithing did.
So, where my answer is incomplete is that I do not have a full answer as to whether there is a fully modern process that will produce one that was as effective, but I have a few comments. First, let me point out that you haven't defined "effective for what?" (And neither saying "killing" nor "warfare" is precise enough). Weapons, including swords, are products of their environment. You cannot readily say that a rapier is more effective than a greatsword or vice versa without precisely specifying the environment you are talking about. A greatsword is much more effective at fighting knights in full-plate than a rapier is, especially in a many-on-many scenario. On the other hand, if you are talking about a one-on-one duel between opponent wearing no armor, I would rather have the rapier. Even going back to feudal japan as our context, it is worth remember that Samurai often treated the katana as a sidearm and dueling weapon and often carried either a Yumi (bow) or a Yari (spear) as a primary weapon into warfare.
Now, almost without concern for the exact definition of "effective", superior metallurgy would give the modern swordmaker with modern techniques an enormous advantage. This is such a huge advantage that even if the modern technique were otherwise inferior to the traditional way that I suspect the answer would be "Yes, modern techniques would be more effective". But I do not have any specific citations for that and any more comprehensive examination would require defining precisely "effective for what" or perhaps "effective against what".
Not only can it be done, but it has been done and you can watch it:
In the TV show "Man at arms, reforged", when they do katanas, they do the folding and tempering but with power tools instead of hand hammers, recently designed clays, and modern furnaces and chemical treatments (in this case, they show that they cut some corners, because the objective here is not a weapons-grade quality, but all the ingredients are present).
Wired magazine published this article, about the advances of modern science in metallurgy, and how a professor and his team demonstrated this by forging a (better) sword then by traditional methods. They named this project the 'Dragon Slayer' - i.e.; blade to slay dragons!
Very interesting read. The short answer is yes, modern science can do a better and more consistent job.
For a more nuanced answer, a lot of it gets into what you consider to be a katana. Presently, it is extremely difficult to make a katana in Japan, as swordmaking was highly regulated post-WW2. During WW2, katana were machine-forged or sometimes even stamped, but they were for display, and were indeed inferior. Tool steel and high speed steel are equivalent or superior to the layered steel of traditional katana which, as per Timothy's answer, was more a matter of Japan having a shortage of good steel than it was of any truly special technique.