  The density and temperature properties of the intergalactic medium (IGM)
reflect the heating and ionization history during cosmological structure
formation, and are primarily probed by the Lyman-alpha forest of neutral
hydrogen absorption features in the observed spectra of background sources
(Gunn & Peterson 1965). We present the methodology and initial results from the
Cholla IGM Photoheating Simulation (CHIPS) suite performed with the Graphics
Process Unit-accelerated Cholla code to study the IGM at high, uniform spatial
resolution maintained over large volumes. In this first paper, we examine the
IGM structure in CHIPS cosmological simulations that include IGM uniform
photoheating and photoionization models where hydrogen reionization completes
early (Haardt & Madau 2012) or by redshift z~6 (Puchwein et al. 2019).
Comparing with observations of the large- and small-scale Lyman-alpha
transmitted flux power spectra P(k) at redshifts 2 <~ z <~ 5.5, the relative
agreement of the models depends on scale, with the self-consistent Puchwein et
al. (2019) IGM photoheating and photoionization model in good agreement with
the flux P(k) at k >~ 0.01 s/km at redshifts 2 <~ z <~ 3.5. On larger scales
the P(k) measurements increase in amplitude from z~4.6 to z~2.2 faster than the
models, and lie in between the model predictions at 2.2 <~ z <~ 4.6 for k~=
0.002-0.01 s/km. We argue the models could improve by changing the HeII
photoheating rate associated with active galactic nuclei to reduce the IGM
temperature at z~3. At higher redshifts z>~4.5 the observed flux P(k) amplitude
increases at a rate intermediate between the models, and we argue that for
models where hydrogen reionization completes late (z~5.5 - 6) resolving this
disagreement will require inhomogeneous or 'patchy' reionization. (Abridged)
