Simulation of quantum circuits by low-rank stabilizer decompositions (1808.00128v2)

Abstract: 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 stabilizer rank, which for a pure state $\psi$ is defined to be the smallest integer $\chi$ such that $\psi$ is a superposition of $\chi$ 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 $\chi\sim10^6$ stabilizer states and sample from the full n-bit output distribution, improving on previous simulations which used $\sim 10^3$ 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.

• The paper develops new algorithms for classically simulating quantum circuits with few non-Clifford gates by using low-rank stabilizer decompositions and the concept of stabilizer rank.
• Improved methods enable simulation of circuits with 40-50 qubits and over 60 non-Clifford gates, including complex QAOA simulations, outperforming previous capabilities.
• Theoretical contributions include the Sparsification Lemma and the connection of approximate stabilizer rank to stabilizer fidelity, offering insights for verifying quantum computations.

Simulation of Quantum Circuits by Low-Rank Stabilizer Decompositions

The paper presents an in-depth exploration of an approach to classically simulate quantum circuits using the stabilizer formalism, particularly focusing on circuits containing a small number of non-Clifford gates. Central to this approach is the concept of stabilizer rank, which quantifies the classical computational cost required to represent a quantum state as a superposition of stabilizer states. The authors develop a mathematical framework to understand both exact and approximate stabilizer rank, introducing several innovative algorithms that outperform previously known methods.

The paper introduces a significant extension of stabilizer-based simulation methods. One notable advance includes the ability to simulate circuits comprising Clifford operations and arbitrary diagonal gates. This enhances earlier work that was confined to Clifford+T circuits. Noteworthy achievements using these improved simulation methods include the successful simulation of quantum algorithms with 40-50 qubits and over 60 non-Clifford gates, achievable without relying on high-performance computing resources.

Quantitatively, the authors showcase their method's capability by simulating the Quantum Approximate Optimization Algorithm (QAOA) with significant complexity: processing superpositions with approximately $10^6$ stabilizer states and sampling the complete $n$-bit output distribution. This represents a substantial improvement over earlier efforts, which were restricted to about $10^3$ stabilizer states and only single-qubit marginals.

From a theoretical standpoint, the paper makes several important contributions. A cornerstone of the work is the Sparsification Lemma, which enables the conversion of dense stabilizer decompositions into their sparser counterparts, thereby facilitating the efficient simulation of quantum circuits. The authors connect the approximate stabilizer rank to a quantity called stabilizer fidelity, which measures the overlap of a quantum state with stabilizer states. This fidelity serves as a useful tool for proving several key properties about the stabilizer rank.

A crucial question addressed in the paper is the multiplicativity of stabilizer extent, $\xi(\psi)$, under tensor products. The authors show that for quantum states involving up to three qubits, this extent is indeed multiplicative, and they propose conditions under which such multiplicative properties hold. For instance, they demonstrate that stabilizer alignment ensures this property, thereby providing a criteria to identify when stabilizer fidelity and extent retain their multiplicative form.

Propositions related to the stabilizer rank offer insightful revelations. For instance, the paper argues, backed by numerical evidence, that typical single-qubit states for small tensors exhibit linear scaling in stabilizer rank. It raises intriguing questions about the limits of current lower-bound derivation techniques, and calls for further exploration towards exponential scaling evidence—a nontrivial conjecture with implications for computational complexity theory.

Practically, these advancements suggest new avenues for verifying quantum computations, especially as quantum hardware reaches scales beyond brute-force classical simulations. The findings and methodologies elucidate conditions under which quantum circuits, even when structured as nontrivial compositions of Clifford and other operations, can still yield to efficient classical analysis.

While the results emphasize classical simulation, which remains a necessary tool for current quantum algorithm development and verification, the implications are significant in anticipating the evolving interface of quantum algorithms and classical resources. High-impact queries posed by the work include potential applications of simulation techniques to multi-qubit quantum error correction and other quantum information processes where Clifford operations are prevalent.

The conclusions set forth in the paper are rigorous and the methods afford a detailed pathway for both theoretical inquiry and practical application in the simulation of quantum mechanics—a pressing and ongoing challenge as quantum technology approaches new frontiers. Future work could further elucidate the complex relationships within stabilizer rank, and continue bridging mathematical insights with computational practices in the ever-growing field of quantum simulation.