  A central goal in quantum error correction is to reduce the overhead of
fault-tolerant quantum computing by increasing noise thresholds and reducing
the number of physical qubits required to sustain a logical qubit. We introduce
a potential path towards this goal based on a family of dynamically generated
quantum error correcting codes that we call "hyperbolic Floquet codes.'' These
codes are defined by a specific sequence of non-commuting two-body measurements
arranged periodically in time that stabilize a topological code on a hyperbolic
manifold with negative curvature. We focus on a family of lattices for $n$
qubits that, according to our prescription that defines the code, provably
achieve a finite encoding rate $(1/8+2/n)$ and have a depth-3 syndrome
extraction circuit. Similar to hyperbolic surface codes, the distance of the
code at each time-step scales at most logarithmically in $n$. The family of
lattices we choose indicates that this scaling is achievable in practice. We
develop and benchmark an efficient matching-based decoder that provides
evidence of a threshold near 0.1% in a phenomenological noise model and 0.25%
in an entangling measurements noise model. Utilizing weight-two check operators
and a qubit connectivity of 3, one of our hyperbolic Floquet codes uses 400
physical qubits to encode 52 logical qubits with a code distance of 8, i.e., it
is a $[[400,52,8]]$ code. At small error rates, comparable logical error
suppression to this code requires 5x as many physical qubits (1924) when using
the honeycomb Floquet code with the same noise model and decoder.
