  Quantum computers are growing in size, and design decisions are being made
now that attempt to squeeze more computation out of these machines. In this
spirit, we design a method to boost the computational power of near-term
quantum computers by adapting protocols used in quantum error correction to
implement "Approximate Quantum Error Correction (AQEC)." By approximating
fully-fledged error correction mechanisms, we can increase the compute volume
(qubits $\times$ gates, or "Simple Quantum Volume (SQV)") of near-term
machines. The crux of our design is a fast hardware decoder that can
approximately decode detected error syndromes rapidly. Specifically, we
demonstrate a proof-of-concept that approximate error decoding can be
accomplished online in near-term quantum systems by designing and implementing
a novel algorithm in Single-Flux Quantum (SFQ) superconducting logic
technology. This avoids a critical decoding backlog, hidden in all offline
decoding schemes, that leads to idle time exponential in the number of T gates
in a program.
  Our design utilizes one SFQ processing module per physical qubit. Employing
state-of-the-art SFQ synthesis tools, we show that the circuit area, power, and
latency are within the constraints of contemporary quantum system designs.
Under pure dephasing error models, the proposed accelerator and AQEC solution
is able to expand SQV by factors between 3,402 and 11,163 on expected near-term
machines. The decoder achieves a $5\%$ accuracy-threshold and pseudo-thresholds
of $\sim$ $5\%, 4.75\%, 4.5\%,$ and $3.5\%$ physical error-rates for code
distances $3, 5, 7,$ and $9$. Decoding solutions are achieved in a maximum of
$\sim 20$ nanoseconds on the largest code distances studied. By avoiding the
exponential idle time in offline decoders, we achieve a $10$x reduction in
required code distances to achieve the same logical performance as alternative
designs.
