Manuscript in preparation · Mingyuan Wang and Stefan Krastanov
Abstract
The order of noisy checks in partial graph-state distillation determines how effectively they suppress errors, even when the target, auxiliaries and hardware are fixed. Exploiting this control freedom is computationally difficult: explicit state propagation is exponential, and the space of complete check orders is factorial. We make sequence selection tractable as an approximate optimization problem by combining an exact characterization of ordering structure with a reduced model of conditional dynamics. The structural analysis identifies equivalent orders and explains how local faults couple several precedence relations. The reduced model, PairMoment, replaces the exponentially large target distribution with a quadratic number of single-label and pair moments. Coupling this compact conditional evolution to bounded search avoids exhaustive enumeration while preserving information needed to compare noisy sequences. Exact-reference benchmarks show shortlists containing near-optimal orders on the tested systems, and information-order comparisons show that better state prediction need not give better selection. The resulting approach makes sequence control computationally accessible within partial graph-state distillation.
Research overview
In partial graph-state distillation, the order of local purification steps can change the final error distribution. My work studies this sequence-selection problem under correlated noise and finite resources.
Why does the order of purification steps matter?
A purification step changes errors in its target region, but it also changes correlations elsewhere in the state. These correlations influence the effectiveness of later operations. The cost of a step therefore depends on the history of the sequence, and choosing a good next operation requires information about more than its isolated effect.
How can the sequence search be organized?
I formulated sequence selection as a combinatorial optimization problem. The framework identifies structural equivalences among purification sequences and organizes their history-dependent costs through predecessor sets. It also captures higher-order ordering effects that cannot be described by simple pairwise preferences between operations.
What does PairMoment track?
PairMoment is a reduced dynamical model that tracks single-vertex error probabilities and pair correlations. It approximates the full graph-basis probability distribution with an O(n²) state representation, where n is the number of vertices. The objective is to retain correlations relevant to sequence-dependent performance while making repeated evaluation practical for optimization.
How does this connect to my earlier work?
The GHZ and graph-preserving frameworks exploit symmetry and topology known before a circuit is applied. Sequence control adds a changing structure: the correlations relevant to a later step evolve during purification. The same modeling principle applies, but the reduced representation must track the information produced by earlier operations.
Manuscript
Mingyuan Wang and Stefan Krastanov. Sequence Control for Partial Distillation of Graph States with Pair-Moment Propagation. Manuscript in preparation.
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