Synthesis and Optimization of Encoding Circuits for Fault-Tolerant Quantum Computation

TL;DR

This paper introduces new methods for synthesizing efficient encoding circuits for arbitrary stabilizer codes, significantly reducing two-qubit gate count and circuit depth for fault-tolerant quantum computation.

Contribution

It develops greedy, rollout-based, and SMT-based algorithms for stabilizer code encoder synthesis, improving efficiency and optimality over previous methods.

Findings

Up to 43% reduction in two-qubit gate count.

Up to 70% reduction in circuit depth.

Effective for a broad set of stabilizer codes, including holographic and qLDPC codes.

Abstract

Preparing arbitrary logical states is a central primitive for universal fault-tolerant quantum computation and the cost of encoded-state preparation contributes directly to the overall resource overhead. This makes the synthesis of efficient general-state encoding circuits an important problem, particularly with respect to two-qubit gate count and circuit depth. Yet the synthesis of such encoders has been studied less extensively than general Clifford circuit synthesis or the preparation of specific logical Pauli-eigenstates. In this work, we develop methods for synthesizing efficient encoders for arbitrary stabilizer codes. We formulate encoder synthesis as a search over stabilizer tableaus and introduce greedy and rollout-based algorithms that exploit the freedom among stabilizer-equivalent realizations of the same encoding isometry. For code families with a modular structure, such as generalized concatenated and holographic codes, we show how large encoders can be assembled from optimized local constituent encoders, and we use SMT-based exact synthesis to obtain optimal local circuits for small instances. We further evaluate the proposed methods on a broad set of stabilizer codes, including holographic and quantum low-density parity-check (qLDPC) codes, and compare them against recent encoder-synthesis methods and existing constructions from the literature, obtaining improvements of up to 43% in two-qubit gate count and up to 70% in depth. Our results support the optimization of encoded-state preparation in several fault-tolerant quantum-computing schemes, and all methods are openly available as part of the Munich Quantum Toolkit.