Further improvements to stabilizer simulation theory: classical rewriting of CSS-preserving stabilizer circuits, quadratic form expansions of stabilizer operations, and framed hidden variable models

TL;DR

The paper addresses the efficient simulation of stabilizer circuits by focusing on CSS-preserving operations. It proves that CSS-preserving stabilizer circuits can be exactly rewritten as $2n$-bit classical circuits, enabling exact reproduction of measurement statistics with no overhead, and provides two independent correctness proofs. It introduces a robust algebraic framework based on $\mathbb{Z}_4$-valued quadratic forms to represent Clifford channels, including a standard form that supports fast composition and Heisenberg evolution, and shows CSS-preserving channels reduce to linear forms. A time-evolving reference-frame formalism encoded by quadratic forms yields a contextual hidden-variable picture for general stabilizer circuits and frames that capture non-CSS resources, with practical implications for simulating near-stabilizer dynamics and magic states. The work connects classical rewriting, quadratic-form expansions, and frame-based HV models to offer new perspectives and potential performance gains for stabilizer and near-stabilizer simulators, with clear pathways to qudit generalization and higher Clifford-level extensions.

Abstract

Simulation of stabilizer circuits is a well-studied problem in quantum information processing, with a number of highly optimized algorithms available. Yet, we argue that further improvements can arise from the theoretical structure of stabilizer operations themselves. We focus on the subclass of stabilizer circuits composed of Calderbank-Shor-Steane (CSS)-preserving stabilizer operations, which naturally appear in fault-tolerant computations over CSS stabilizer codes. Using elementary circuit-transformation techniques, we show that such circuits can be exactly rewritten as classical probabilistic circuits that reproduce measurement statistics. This rewriting introduces no computational overhead, in contrast to the general case of stabilizer circuits. To clarify the origin of this simplification, we introduce the standard quadratic-form representation of general stabilizer operations (Clifford channels). It provides an efficient way to describe compositions of stabilizer operations and thus to simulate stabilizer circuits. CSS-preserving operations correspond to purely linear forms, which under a Walsh-Hadamard-Fourier transform yield a noncontextual hidden variable model, providing an alternative proof of the introduced rewriting. Finally, we develop a theory of reference frames for multiqubit systems, where frames are encoded by quadratic forms. This allows us to express stabilizer operations as probabilistic maps for proper reference frames. Non-CSS-preserving stabilizer circuits require dynamical modifications of reference frames, embodying a contextuality resource that leads to the computational overhead. This framework provides a new perspective on simulating stabilizer and near-stabilizer circuits within dynamically evolving quasiprobability models.

Figures (6)

• Figure 1: Schematic representation of the main setting. A quantum stabilizer circuit $\mathsf{QC}$ interacts with a classical controller through rounds of communication. Each round consists of sending measurement outcomes from the quantum circuit to the classical computer, which processes this information and sends back control parameters for conditional gates; in general, there may be multiple (or no) communication rounds. We rewrite $\mathsf{QC}$ to a classical circuit $\mathsf{CC}$. If the stabilizer circuit $\mathsf{QC}$ is CSS-preserving, then $\mathsf{CC}$ exactly reproduces its output statistics, and the interaction with the controller remains unchanged after the rewriting. If $\mathsf{QC}$ includes non-CSS-preserving operations, the controller must additionally compute information about the reference frames to ensure correctness. These reference frame updates are determined by the quadratic form expansions of the non-CSS-preserving gates. The example circuits used in the figure are taken from \ref{['subsec:non-CSS_incorrectness']} and further discussed in \ref{['subsubsec:frames_as_contexts']}.
• Figure 2: Venn diagram showing some classes of quantum operations that are relevant to our discussion.
• Figure 3: Simple examples of rewriting quantum circuits to classical circuits. (a) Reduction of the superdense coding protocol Bennett_1991 to classical circuit. (b) Reduction of the quantum teleportation circuit Bennett_1993 to classical circuit.
• Figure 4: The running times of four described methods depending on the number of drawn samples. The data was obtained by generating some random affine Boolean circuit and simulating it. This plot only illustrates the usual behaviour of the methods and may be inadequate for concrete circuits.
• Figure 5: Venn diagram showing subclasses of stabilizer operations. Adding the resource of graphness to CSS-preserving operations results in the class of real stabilizer operations, adding imaginarity results in the class of general stabilizer operations.