Abstract Equivalence checking plays a crucial role in quantum circuit compilation, optimization, and verification. Stabilizer circuits can be simulated classically by tracking the so-called stabilizer operators in linear time. But the simulation of large stabilizer circuits with thousands of qubits and gates, arising e.g. in the study of novel quantum error-correction protocols, still poses a challenge. In this work, we propose a GPU-based deterministic algorithm for equivalence checking of stabilizer circuits using the stabilizer tableau formalism. We explore various design choices and implement the most efficient version. Our algorithm significantly outperforms the state-of-the-art CCEC checker (which relies on the Stim simulator) in terms of time, memory, and energy. Our approach demonstrates up to two orders of magnitude speedup over existing methods. Notably, previous attempts at GPU acceleration in this area were unsuccessful, making this the first effective implementation.
Abstract Equivalence checking plays a crucial role in quantum circuit compilation, optimization, and verification. Stabilizer circuits can be simulated classically by tracking the so-called stabilizer operators in linear time. But the simulation of large stabilizer circuits with thousands of qubits and gates, arising e.g. in the study of novel quantum error-correction protocols, still poses a challenge. In this work, we propose a GPU-based deterministic algorithm for equivalence checking of stabilizer circuits using the stabilizer tableau formalism. We explore various design choices and implement the most efficient version. Our algorithm significantly outperforms the state-of-the-art CCEC checker (which relies on the Stim simulator) in terms of time, memory, and energy. Our approach demonstrates up to two orders of magnitude speedup over existing methods. Notably, previous attempts at GPU acceleration in this area were unsuccessful, making this the first effective implementation.
We introduce new parallel algorithms for efficiently simulating stabilizer (Clifford) circuits on GPUs, with a focus on data-parallel tableau evolution and scalable handling of projective measurements. Our approach reformulates key bottlenecks in stabilizer simulation -- such as Gaussian elimination and measurement updates -- into GPU-tailored primitives that eliminate sequential dependencies and maximize memory coalescing. We implement these techniques in QuaSARQ, a GPU-accelerated stabilizer simulator designed for large qubit counts and many-shot sampling. Across a broad benchmark suite reaching 180,000 qubits and depth 1,000 (roughly 130M gates), QuaSARQ shows substantial runtime improvements, with up to 105× speedup, and over 80% energy reduction on demanding instances. Moreover, QuaSARQ consistently outperforms Stim, a state-of-the-art CPU-optimized stabilizer simulator, as well as Qiskit-Aer (CPU/GPU), Qibo, Cirq, and PennyLane. Finally, QuaSARQ exhibits a significant advantage in many-shot sampling on large workloads. These results demonstrate that our parallel algorithms can significantly advance the scalability of stabilizer-circuit simulation, particularly for workloads involving extensive measurements and sampling.
This paper presents “Stim", a fast simulator for quantum stabilizer circuits. The paper explains how Stim works and compares it to existing tools. With no foreknowledge, Stim can analyze a distance 100 surface code circuit (20 thousand qubits, 8 million gates, 1 million measurements) in 15 seconds and then begin sampling full circuit shots at a rate of 1 kHz. Stim uses a stabilizer tableau representation, similar to Aaronson and Gottesman's CHP simulator, but with three main improvements. First, Stim improves the asymptotic complexity of deterministic measurement from quadratic to linear by tracking the i n v e r s e of the circuit's stabilizer tableau. Second, Stim improves the constant factors of the algorithm by using a cache-friendly data layout and 256 bit wide SIMD instructions. Third, Stim only uses expensive stabilizer tableau simulation to create an initial reference sample. Further samples are collected in bulk by using that sample as a reference for batches of Pauli frames propagating through the circuit.
The quantum stabilizer formalism became foundational for understanding error correction soon after the realization of the first useful quantum error correction codes. Stabilizers provide a way to describe sets of quantum states which are valid codewords within a quantum error correction (QEC) scheme. Existing stabilizer simulators are single threaded applications used to sample larger codes than is possible with other methods. However, there is an outstanding gap in the scaling and accuracy of current simulators for QEC as quantum computing exceeds hundreds of qubits, along with an under-utilization of the capabilities of highly-efficient stabilizer simulation across other quantum domains. In this work, we present the first GPU-accelerated tableau stabilizer simulator to scale better than CPU methods in QEC workloads, by trivializing Clifford gates and exploiting the large parallelism of dedicated GPUs with CUDA warp-level primitives to quickly overcome costly measurement gates. We then implement a new error model that captures non-unitarity in T1/T2 error channels much faster and with exact accuracy for most physical qubits, demonstrate a chemistry use case, and present a new Clifford+T to Pauli-Based Computing (PBC) transpilation optimization through our simulator.
Simulating Clifford and near-Clifford circuits using the extended stabilizer formalism has become increasingly popular, particularly in quantum error correction. Compared to the state-vector approach, the extended stabilizer formalism can solve the same problems with fewer computational resources, as it operates on stabilizers rather than full state vectors. Most existing studies on near-Clifford circuits focus on balancing the trade-off between the number of ancilla qubits and simulation accuracy, often overlooking performance considerations. Furthermore, in the presence of high-rank stabilizers, performance is limited by the sequential property of the stabilizer formalism. In this work, we introduce a parallelized version of the extended stabilizer formalism, enabling efficient execution on multi-core devices such as GPU. Experimental results demonstrate that, in certain scenarios, our Python-based implementation outperforms state-of-the-art simulators such as Qiskit and Pennylane.
Quantum technologies have progressed beyond the laboratory setting and are beginning to make an impact on industrial development. The construction of practical, general purpose quantum computers has been challenging, to say the least. But quantum cryptographic and communication devices have been available in the commercial marketplace for a few years. Quantum networks have been built in various cities around the world, and plans are afoot to launch a dedicated satellite for quantum communication. Such new technologies demand rigorous analysis and verification before they can be trusted in safety and security-critical applications. In this thesis we investigate the theory and practice of equivalence checking of quantum information systems. We present a tool, Quantum Equivalence Checker (QEC), which uses a concurrent language for describing quantum systems, and performs verification by checking equivalence between specification and implementation. For our process algebraic language CCSq, we define an operational semantics and a superoperator semantics. While in general, simulation of quantum systems using current computing technology is infeasible, we restrict ourselves to the stabilizer formalism, in which there are efficient simulation algorithms and representation of quantum states. By using the stabilizer representation of quantum states we introduce various algorithms for testing equality of stabilizer states. In this thesis, we consider concurrent quantum protocols that behave functionally in the sense of computing a deterministic input-output relation for all interleavings of a concurrent system. Crucially, these input-output relations can be abstracted by superoperators, enabling us to take advantage of linearity. This allows us to analyse the behaviour of protocols with arbitrary input, by simulating their operation on a finite basis set consisting of stabilizer states. We present algorithms for the checking of functionality and equivalence of quantum protocols. Despite the limitations of the stabilizer formalism and also the range of protocols that can be analysed using equivalence checking, QEC is applied to specify and verify a variety of interesting and practical quantum protocols from quantum communication and quantum cryptography to quantum error correction and quantum fault tolerant computation, where for each protocol different sequential and concurrent model are defined in CCSq. We also explain the implementation details of the QEC tool and report on the experimental results produced by using it on the verification of a number of case studies.
The novel field of quantum computation and quantum information has gathered significant momentum in the last few years. It has the potential to radically impact the future of information technology and influence the development of modern society. The construction of practical, general purpose quantum computers has been challenging, but quantum cryptographic and communication devices have been available in the commercial marketplace for several years. Quantum networks have been built in various cities around the world and a dedicated satellite has been launched by China to provide secure quantum communication. Such new technologies demand rigorous analysis and verification before they can be trusted in safety- and security-critical applications. Experience with classical hardware and software systems has shown the difficulty of achieving robust and reliable implementations. We present CCS q , a concurrent language for describing quantum systems, and develop verification techniques for checking equivalence between CCS q processes. CCS q has well-defined operational and superoperator semantics for protocols that are functional , in the sense of computing a deterministic input-output relation for all interleavings arising from concurrency in the system. We have implemented QEC (Quantum Equivalence Checker), a tool that takes the specification and implementation of quantum protocols, described in CCS q , and automatically checks their equivalence. QEC is the first fully automatic equivalence checking tool for concurrent quantum systems. For efficiency purposes, we restrict ourselves to Clifford operators in the stabilizer formalism, but we are able to verify protocols over all input states. We have specified and verified a collection of interesting and practical quantum protocols, ranging from quantum communication and quantum cryptography to quantum error correction.