Automated Synthesis of Fault-Tolerant State Preparation Circuits for Quantum Error-Correction Codes

A central ingredient in fault-tolerant quantum algorithms is the initialization of a logical state for a given quantum error-correcting code from a set of noisy qubits. A scheme that has demonstrated promising results for small code instances that are realizable on currently available hardware composes a non-fault-tolerant state preparation circuit with a verification circuit that checks for spreading errors. Known circuit constructions of this scheme are mostly obtained manually, and no algorithmic techniques for constructing depth- or gate-optimal circuits exist. As a consequence, the current state-of-the-art exploits this scheme only for specific code instances and mostly for the special case of distance <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>d</a:mi><a:mo>=</a:mo><a:mn>3</a:mn></a:math> codes only. In this work, we propose an automated approach for synthesizing fault-tolerant state preparation circuits for arbitrary CSS codes. We utilize methods based on satisfiability solving (SAT) to construct fault-tolerant state preparation circuits consisting of depth- and gate-optimal preparation and verification circuits. We also provide heuristics that can synthesize fault-tolerant state preparation circuits for code instances where no optimal solution can be obtained in an adequate time. Moreover, we give a general construction for nondeterministic state preparation circuits for codes beyond distance 3. Numerical evaluations using <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>d</c:mi><c:mo>=</c:mo><c:mn>3</c:mn></c:math>, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>d</e:mi><e:mo>=</e:mo><e:mn>5</e:mn></e:math>, and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>d</g:mi><g:mo>=</g:mo><g:mn>7</g:mn></g:math> codes confirm that the generated circuits exhibit the desired scaling of the logical error rates. The resulting methods are publicly available as part of the (MQT) at . Such methods are an important step in providing fault-tolerant circuit constructions that can aid in near-term demonstrations of fault-tolerant quantum computing.