Further improvements to stabilizer simulation theory: classical rewriting of CSS-preserving stabilizer circuits, quadratic form expansions of stabilizer operations, and framed hidden variable models
Vsevolod I. Yashin, Evgeniy O. Kiktenko, Vladimir V. Yatsulevich, Aleksey K. Fedorov
quant-phSubmitted Nov 7, 2025• Announced Nov 10, 2025• 0 citations
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.
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35 pages, 6 figures, 2 tables. Comments are welcome!
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.
Generic quantum-circuit simulation appears intractable for conventional computers and may be unnecessary because useful quantum circuits exhibit significant structure that can be exploited during simulation. For example, Gottesman and Knill identified an important subclass, called stabilizer circuits, which can be simulated efficiently using group-theory techniques and insights from quantum physics. Realistic circuits enriched with quantum error-correcting codes and fault-tolerant procedures are dominated by stabilizer subcircuits and contain a relatively small number of non-Clifford components. Therefore, we develop new data structures and algorithms that facilitate parallel simulation of such circuits. Stabilizer frames offer more compact storage than previous approaches but require more sophisticated bookkeeping. Our implementation, called Quipu, simulates certain quantum arithmetic circuits (e.g., reversible ripple-carry adders) in polynomial time and space for equal superpositions of n-qubits. On such instances, known linear-algebraic simulation techniques, such as the (state-of-the-art) BDD-based simulator QuIDDPro, take exponential time. We simulate quantum Fourier transform and quantum fault-tolerant circuits using Quipu, and the results demonstrate that our stabilizer-based technique empirically outperforms QuIDDPro in all cases. While previous high-performance, structure-aware simulations of quantum circuits were difficult to parallellize, we demonstrate that Quipu can be parallelized with a nontrivial computational speedup.
As quantum information processing gains traction, its simulation becomes increasingly significant for engineering purposes - evaluation, testing and optimization - as well as for theoretical research. Generic quantum-circuit simulation appears intractable for conventional computers. However, Gottesman and Knill identified an important subclass, called stabilizer circuits, which can be simulated efficiently using group-theory techniques. Practical circuits enriched with quantum error-correcting codes and fault-tolerant procedures are dominated by stabilizer subcircuits and contain a relatively small number of non-stabilizer components. Therefore, we develop new group-theory data structures and algorithms to simulate such circuits. Stabilizer frames offer more compact storage than previous approaches but requires more sophisticated bookkeeping. Our implementation, called Quipu, simulates certain quantum arithmetic circuits (e.g., ripple-carry adders) in polynomial time and space for equal superpositions of n-qubits. On such instances, known linear-algebraic simulation techniques, such as the (state-of-the-art) BDD-based simulator QuIDDPro, take exponential time. We simulate various quantum Fourier transform and quantum fault-tolerant circuits with Quipu, and the results demonstrate that our stabilizer-based technique outperforms QuIDDPro in all cases.
Simulation of quantum information processing remains a major challenge with important applications in quantum computer science and engineering. Generic quantum-circuit simulation appears intractable for conventional computers and may be unnecessary because useful quantum circuits exhibit significant structure that can be exploited during simulation. For example, Gottesman and Knill identified an important subclass, called stabilizer circuits, which can be simulated efficiently using the Heisenberg representation for quantum computers. Stabilizer circuits are exclusively composed of stabilizer gates -- Hadamard, Phase and CNOT -- followed by one-qubit measurements in the computational basis. Such circuits are applied to a computational-basis state and produce so-called stabilizer states. Aaronson and Gottesman generalized stabilizer-circuit simulation to additionally handle a small number of non-stabilizer gates. We design new, more efficient data structures and algorithms for such beyond-stabilizer simulation using superpositions of stabilizer states. One such data structure, a stabilizer frame, offers more compact storage than previous approaches but require additional algorithms to maintain the global phases of each state in the superposition. To explore the advantages and limitations of our technique, we analyze the geometric structure of stabilizer states and their embedding in Hilbert space. Our analysis includes results on the computational geometry of stabilizer states such as efficient algorithms for computing distances, angles and volumes between them. The main advantages of using stabilizer-state superpositions to simulate quantum circuits are: (i) stabilizer subcircuits are simulated with high efficiency, (ii) superpositions can be restructured and compressed on the fly during simulation to reduce resource requirements, and (iii) operations performed on such superpositions lend themselves to distributed or asynchronous processing. Our software implementation, called Quipu, simulates certain quantum arithmetic circuits (e.g., reversible ripple-carry adders) and quantum Fourier transform circuits in polynomial time and space for specific input states. On such instances, known linear-algebraic simulation techniques, such as the (state-of-the-art) BDD-based simulator QuIDDPro, take exponential time. We simulate quantum fault-tolerant circuits using Quipu, and the results indicate that our stabilizer-based technique empirically outperforms QuIDDPro in all cases. While previous structure-aware simulations of quantum circuits were difficult to parallellize, we demonstrate a parallel version of Quipu that achieves a nontrivial speedup.
Modeling and simulation are essential for predicting and verifying the behavior of fabricated quantum circuits, but existing simulation methods are either impractically costly or require an unrealistic simplification of error processes. We present a method of simulating noisy Clifford circuits that is both accurate and practical in experimentally relevant regimes. In particular, the cost is weakly exponential in the size and the degree of non-Cliffordness of the circuit. Our approach is based on the construction of exact representations of quantum channels as quasiprobability distributions over stabilizer operations, which are then sampled, simulated, and weighted to yield unbiased statistical estimates of circuit outputs and other observables. As a demonstration of these techniques, we simulate a Steane [[7,1,3]]-encoded logical operation with non-Clifford errors and compute its fault tolerance error threshold. We expect that the method presented here will enable studies of much larger and more realistic quantum circuits than was previously possible.
Calderbank--Shor--Steane (CSS) codes support decoding and fault-tolerant logical operations through separate pure X and pure Z stabilizer checks. Generator choices and single-qubit Clifford transformations can obscure this structure. We give a deterministic polynomial-time algorithm that decides whether an arbitrary stabilizer code is locally Clifford-equivalent to a CSS code and constructs the required transformation and CSS checks whenever such a form exists. For m possibly redundant generator rows on n qubits, the algorithm uses O(mn2+n4) binary field operations. We reduce the problem to affine linear systems by characterizing local projectors that preserve the span of the binary stabilizer labels. The conversion preserves code parameters and check sparsity for quantum low-density parity-check (qLDPC) codes. We demonstrate CSS recovery and decoder reuse and resolve previously open LC-to-CSS classification questions for published code constructions. For graph states, LC-to-CSS equivalence corresponds to the existence of a bipartite graph in the local-complementation orbit. We derive a necessary-and-sufficient adjacency-matrix criterion for this property and construct such a graph whenever one exists. Our algorithm makes CSS decoding and fault-tolerant constructions accessible throughout the local Clifford equivalence classes of CSS codes.
Modeling and simulation is essential for predicting and verifying the behavior of fabricated quantum circuits, but existing simulation methods are either impractically costly or require an unrealistic simplification of error processes. We present a method of simulating noisy Clifford circuits that is both accurate and practical in experimentally relevant regimes. In particular, the cost is weakly exponential in the size and the degree of non-Cliffordness of the circuit. Our approach is based on the construction of exact representations of quantum channels as quasiprobability distributions over stabilizer operations, which are then sampled, simulated, and weighted to yield statistical estimates of circuit outputs and other observables. As a demonstration of these techniques we simulate a Steane [[7,1,3]]-encoded logical operation with non-Clifford errors and compute its fault tolerance error threshold. We expect that the method presented here will enable studies of much larger and more realistic quantum circuits than was previously possible.