  In existing general-purpose architectures for surface-code-based
fault-tolerant quantum computers, the cost of a quantum computation is
determined by the circuit volume, i.e., the number of qubits multiplied by the
number of non-Clifford gates. We introduce an architecture using non-2D-local
connections in which the cost does not scale with the number of qubits, and
instead only with the number of logical operations. Each logical operation has
an associated active volume, such that the cost of a quantum computation can be
quantified as a sum of active volumes of all operations. For quantum
computations with thousands of logical qubits, the active volume can be orders
of magnitude lower than the circuit volume. Importantly, the architecture does
not require all-to-all connectivity between N logical qubits. Instead, each
logical qubit is connected to O(log N) other sites. As an example, we show
that, using the same number of logical qubits, a 2048-bit factoring algorithm
can be executed 44 times faster than on a general-purpose architecture without
non-local connections. With photonic qubits, long-range connections are
available and we show how photonic components can be used to construct a
fusion-based active-volume quantum computer.
