How can I decrease noise while maintaining bandwidth and gain in my Transimpedance amplifier
First, lets not worry about the gain, but lets get the delta-charge from the photodiode converted into a healthy, buffered delta-voltage on output of a first stage. Assume the 20pF of PhotoDiode sets the total (sqrt(kt/c) noise floor, which is 7microvolts RMS. Your signal is 50uVpeak, or 25uV above or below some detection threshold. What can we do with that?

Let's bias a CE NPN bipolar at 26mA, providing GM of 1amp/volt or 50uA/50uV.
Assuming the collector resistor is 100 ohms (low in value, so we have some legit hope in reaching 2GHz BW), our output delta-voltage is 50uA * 100 ohms, or 5 milliVolts. [by the way, 1pF and 100 ohms is 100pS, or 1.6GHz F3dB]

Put an emitter follower on that, running at 10mA so Rout is 2.6 ohms, and AC couple THAT into a copy of the first stage, producing 500 millivolt output.

With nary a TIA in sight. 
The 2SC5646A has 12.5GHz Ftau at 3 volts and 15mA, with Cob of 0.5pF typical.







simulate this circuit &ndash; Schematic created using CircuitLab

Caveats: (1) the photodiode capacitance may cause oscillation of the CommonBase Q1. If so, solder SurfaceMount 10Ohms directly under the transistor base. Literally lift up the SOT-23 base lead and slip a 10 Ohm under that lead.
(2) for more speed, boost VDD; or reduce R1 to 50 Ohms; R1 has 0.5 + 0.5pF in parallel (COb Q1, Cob Q2), or 100picoSec tau, only 1.6GHz.
(3) reduce the transconductance_setting resistor [we want gm of 1/1_ohm] to 33 or 24 ohms, and reduce the pot base voltage; this provides more Vce on Q1 for higher bandwidth. (4) circuit has little headroom on Q2, so may saturate; consider reducing the lower collector resistor for more headroom.

What do we (think we) have here? Collector tau of 100 ohm * 0.5pF (only 1 of the Cob loads) or 50 picosecond Tau, supporting 15GB/sec datarates. With 20uA input, we get 100 ohm * 20uA = 2 milliVolts output (from Q1 collector, Q2 emitter) at 2.6 ohms Rout, or ~~ 50x the current as the PD provided.

We can trade off gain for bandwidth, increasing  R1 to 200 ohms with 100pS tau. That requires using 6 volts VDD and reducing Remitter of Q1.