  Quantum computers promise to solve problems that are intractable for
classical computers, but qubits are vulnerable to many sources of error,
limiting the depth of the circuits that can be reliably executed on today's
quantum hardware. Quantum error correction has been proposed as a solution to
this problem, whereby quantum information is protected by encoding it into a
quantum error-correcting code. But protecting quantum information is not
enough, we must also process the information using logic gates that are robust
to faults that occur during their execution. One method for processing
information fault-tolerantly is to use quantum error-correcting codes that have
logical gates with a tensor product structure (transversal gates), making them
naturally fault-tolerant. Here, we test the performance of a code with such
transversal gates, the [[8,3,2]] color code, using trapped-ion and
superconducting hardware. We observe improved performance (compared to no
encoding) for encoded circuits implementing non-Clifford gates, a class of
gates that are essential for achieving universal quantum computing. In
particular, we find improved performance for an encoded circuit implementing
the control-control $Z$ gate, a key gate in Shor's algorithm. Our results
illustrate the potential of using codes with transversal gates to implement
non-trivial algorithms on near-term quantum hardware.
