Syndrome measurements are quantum operations that detect errors by
measuring the observable defined by the stabilizer generators of a
quantum error-correcting code. These measurements are typically realized
through circuits composed of CNOT and Hadamard gates, utilizing
ancillary qubits. This project focuses on designing shallow-depth
circuits capable of performing all necessary syndrome measurements
efficiently.
For example, in the Steane code, the syndrome measurements involve
the operators \(Z_1 Z_2 Z_3 Z_4\) and
\(X_1 X_2 X_3 X_4\). The corresponding
circuits are shown in the following figure.
Two syndrome measurements for Steane
code
Since both measurements act on qubits 1, 2, 3, and 4, and we need to
measure all these two measurements (and others), gate collisions may
occur if the operations are not properly scheduled.
Now we define the cost of a gate scheduling for a syndrome
measurement circuit more formally, following the basic rules:
For simplicity, we exclude single-qubit gates (primarily H gates)
and measurements from our analysis, as done in prior work
[TBLC22].
We discretize time into steps (TICKs), where each CNOT gate
operation consumes exactly one time unit.
Parallel execution is permitted for non-conflicting CNOT gates,
with the constraint that each qubit can participate in at most one CNOT
operation per time step.
2 Problems
Optimal Gate Scheduling: Determine the
minimum-depth circuit schedule for syndrome measurement in QLDPC codes
[BE21] under idealized hardware conditions (no topological constraints).
You may use SMT solvers for this task.
Hardware-Aware Scheduling (Research-level):
Extend this to realistic hardware with limited qubit connectivity. While
the problem was considered in [YZF+25] for superconducting architectures
(e.g., Google’s Sycamore) with the help of SWAP gates, we instead focus
on the Reconfigurable Atom Array (RAA) platform for
which there are custom-made compilers (OLSQ-RAA)
[TBLC22]. The challenge lies in co-optimizing syndrome measurement
circuits with RAA’s reconfigurable
connectivity.