We introduce a novel tableau-based classical simulation method for quantum computation, formulated within the phase-space framework of the extended stabilizer theory of closed noncontextual operators. This method enables the efficient classical simulation of a broader class of quantum circuits beyond the stabilizer formalism. We identify additional families of input states that are efficiently simulatable within this framework, going beyond stabilizer states, and implement the simulator and benchmark its performance on basic quantum algorithms, including the hidden shift and Deutsch–Jozsa algorithms.
We introduce a novel tableau-based classical simulation method for quantum computation, formulated within the phase-space framework of the extended stabilizer theory of closed noncontextual operators. This method enables the efficient classical simulation of a broader class of quantum circuits beyond the stabilizer formalism. We identify additional families of input states that are efficiently simulatable within this framework, going beyond stabilizer states, and implement the simulator and benchmark its performance on basic quantum algorithms, including the hidden shift and Deutsch–Jozsa algorithms.
Recent work has explored using the stabilizer formalism to classically simulate quantum circuits containing a few non-Clifford gates. The computational cost of such methods is directly related to the notion of \ns\nt\na\nb\ni\nl\ni\nz\ne\nr\n \nrank\n, which for a pure state \nψ\n is defined to be the smallest integer \nχ\n such that \nψ\n is a superposition of \nχ\n stabilizer states. Here we develop a comprehensive mathematical theory of the stabilizer rank and the related approximate stabilizer rank. We also present a suite of classical simulation algorithms with broader applicability and significantly improved performance over the previous state-of-the-art. A new feature is the capability to simulate circuits composed of Clifford gates and arbitrary diagonal gates, extending the reach of a previous algorithm specialized to the Clifford+T gate set. We implemented the new simulation methods and used them to simulate quantum algorithms with 40-50 qubits and over 60 non-Clifford gates, without resorting to high-performance computers. We report a simulation of the Quantum Approximate Optimization Algorithm in which we process superpositions of \nχ\n∼\n10\n6\n stabilizer states and sample from the full \nn\n-bit output distribution, improving on previous simulations which used \n∼\n10\n3\n stabilizer states and sampled only from single-qubit marginals. We also simulated instances of the Hidden Shift algorithm with circuits including up to 64 \nT\n gates or 16 CCZ gates; these simulations showcase the performance gains available by optimizing the decomposition of a circuit's non-Clifford components.
Recent work has explored using the stabilizer formalism to classically simulate quantum circuits containing a few non-Clifford gates. The computational cost of such methods is directly related to the notion of s t a b i l i z e r rank , which for a pure state ψ is defined to be the smallest integer χ such that ψ is a superposition of χ stabilizer states. Here we develop a comprehensive mathematical theory of the stabilizer rank and the related approximate stabilizer rank. We also present a suite of classical simulation algorithms with broader applicability and significantly improved performance over the previous state-of-the-art. A new feature is the capability to simulate circuits composed of Clifford gates and arbitrary diagonal gates, extending the reach of a previous algorithm specialized to the Clifford+T gate set. We implemented the new simulation methods and used them to simulate quantum algorithms with 40-50 qubits and over 60 non-Clifford gates, without resorting to high-performance computers. We report a simulation of the Quantum Approximate Optimization Algorithm in which we process superpositions of χ ∼ 10 6 stabilizer states and sample from the full n -bit output distribution, improving on previous simulations which used ∼ 10 3 stabilizer states and sampled only from single-qubit marginals. We also simulated instances of the Hidden Shift algorithm with circuits including up to 64 T gates or 16 CCZ gates; these simulations showcase the performance gains available by optimizing the decomposition of a circuit's non-Clifford components.
One of the core research questions in the theory of quantum computing is to find out to what precise extent the classical simulation of a noisy quantum circuits is possible and where potential quantum advantages can set in. In this work, we introduce a unified framework for the classical simulation of quantum circuits based on frame theory, encompassing and generalizing a broad class of existing simulation strategies. Within this framework, the computational cost of a simulation algorithm is determined by the one-norm of an associated quasi-probability distribution, providing a common quantitative measure across different simulation approaches. This enables a comprehensive perspective on common methods for the simulation of noisy circuits based on different quantum resources, such as entanglement or non-stabilizerness. It further provides a clear scheme for generating novel classical simulation algorithms. Indeed, by exploring different choices of frames within this formalism and resorting to tools of convex optimization, we are able not only to obtain new insights and improved bounds for existing methods -- such as stabilizer state simulation or Pauli back-propagation -- but also to discover a new approach with an improved performance based on a generalization of the Pauli frame. We, thereby, show that classical simulation techniques can directly benefit from a perspective -- that of frames -- that goes beyond the traditional classification of quantum resources.
We present the generalization of the CNC formalism, based on closed and noncontextual sets of Pauli observables, to the setting of odd-prime-dimensional qudits. By introducing new CNC-type phase space point operators, we construct a quasiprobability representation for quantum computation which is covariant with respect to the Clifford group and positivity preserving under Pauli measurements, and whose nonnegative sector strictly contains the subtheory of quantum theory described by nonnegative Wigner functions. This allows for a broader class of magic state quantum circuits to be efficiently classically simulated than those covered by the stabilizer formalism and Wigner function methods.
Efficient simulation of quantum computers relies on understanding and exploiting the properties of quantum states. This is the case for methods such as tensor networks, based on entanglement, and the tableau formalism, which represents stabilizer states. In this Letter, we integrate these two approaches to present a generalization of the tableau formalism used for Clifford circuit simulation. We explicitly prove how to update our formalism with Clifford gates, non-Clifford gates, and measurements, enabling universal circuit simulation. We also discuss how the framework allows for efficient simulation of more states, raising some interesting questions on the representation power of tensor networks and the quantum properties of resources such as entanglement and magic, and support our claims with simulations.
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.