Detection of weak periodical sound signal having negative SNR (will lock-in-amplifier be useful)
I do not think that you can detect a heartbeat using audio.  Let me explain why I think that:


Detecting a heartbeat from across a room would require a mic with a huge dynamic range.  By dynamic range, I am talking about the difference between the quietest sound and the loudest sound the mic can detect.  Just having a very sensitive mic won't help if the mic clips when the air conditioning kicks in, or someone clears their throat.  A good studio recording mic will have a dynamic range of around 130 dB.  My educated guess is that you will need at least 150 dB, or possibly even much higher.  I have never seen a mic that works at better than 150 dB.
You need to detect very faint signals.  How faint?  I regularly listen to digital systems with a noise floor of between -100 dBFS and -140 dBFS, and I frequently amplify the signal so all I am listening to is the noise floor and I've never heard a heartbeat.  I have heard basic breath noise from across a room, but never a heart beat.  So IF it is possible to hear a heartbeat from across the room then it is certainly way down there.  Way below what current professional audio systems are able to do.
Detecting a sine wave that is below the noise floor is possible.  Of course the lower down the sine wave is, the harder it is to detect it.  Most of the time, if the sine wave is 10 or 20 dB below the noise floor it is moderately difficult to detect.  In theory, you can detect sine wave any arbitrary level below the noise floor, but it requires more time to detect (more audio samples).  It might take many seconds of audio, possibly minutes, to detect a sine wave that is very low.
A heart beat is not a sine wave, and it does not qualify as a periodic wave because there are huge variations (from an audio processing point of view) in both pulse rate and audio frequency content.  While the variations are not huge over the span of several seconds, it can be huge over the many seconds to minutes required to pick out a sine wave-- and since it isn't a sine wave it gets several orders of magnitude harder.
The electronics for this will be amazingly difficult.  Designing a simple mic preamplifier that feeds an ADC with better than 130 dB SNR is beyond all but about 10 people in the United States.  You'll need a SNR of 150 to 190 dB, which is several orders of magnitude harder than that!  And there are no ADC's with that level of performance.


I suggest that you use an alternative method of detecting a pulse.

Here is a video of a system developed by MIT that magnifies motion, and does a very good job of detecting heartbeats. 

Also, there are many other visual systems that work more like a pulse oximeter but with a standard camera.  The iPhone App store has many of these.  There has also been some research into using something like the XBox Kinect, or similar systems, for detecting a pulse.  Any of these systems would be much easier (but still not easy) to implement than using an acoustic system.