We introduce a novel hybrid approach combining tensor network methods with the stabilizer formalism to address the challenges of simulating many-body quantum systems. By integrating these techniques, we enhance our ability to accurately model unitary dynamics while mitigating the exponential growth of entanglement encountered in classical simulations. We demonstrate the effectiveness of our method through applications to random Clifford T-doped circuits and random Clifford Floquet dynamics. This approach offers promising prospects for advancing our understanding of complex quantum phenomena and accelerating progress in quantum simulation.
We introduce a novel hybrid approach combining tensor network methods with the stabilizer formalism to address the challenges of simulating many-body quantum systems. By integrating these techniques, we enhance our ability to accurately model unitary dynamics while mitigating the exponential growth of entanglement encountered in classical simulations. We demonstrate the effectiveness of our method through applications to random Clifford T-doped circuits and random Clifford Floquet dynamics. This approach offers promising prospects for advancing our understanding of complex quantum phenomena and accelerating progress in quantum simulation.
Tensor network methods leverage the limited entanglement of quantum states to efficiently simulate many-body systems. Alternatively, Clifford circuits provide a framework for handling highly entangled stabilizer states, which have low magic and are thus also classically tractable. Clifford tensor networks combine the benefits of both approaches, exploiting Clifford circuits to reduce the classical complexity of the tensor network description of states, with promising effects on simulation approaches. We study the disentangling power of Clifford transformations acting on tensor networks, with a particular emphasis on entanglement cooling strategies. We identify regimes where exact or heuristic Clifford disentanglers are effective, explain the link between the two approaches, and characterize their breakdown as non-Clifford resources accumulate. Additionally, we prove that, beyond stabilizer settings, no Clifford operation can universally disentangle even a single qubit from an arbitrary non-Clifford rotation. Our results clarify both the capabilities and fundamental limitations of Clifford-based simulation methods.
This Letter augments the recently introduced stabilizer tensor network (STN) protocol with magic state injection, reporting a new framework with significantly enhanced ability to simulate circuits with an extensive number of non-Clifford operations. Specifically, for random T-doped N-qubit Clifford circuits the computational cost of circuits prepared with magic state injection scales as O[poly(N)] when the circuit has t≲N T gates compared to an exponential scaling for the STN approach, which is demonstrated in systems of up to 200 qubits. In the case of the hidden bit shift circuit, a paradigmatic benchmarking system for extended stabilizer methods with a tunable amount of magic, we report that our magic state injected STN framework can efficiently simulate 4000 qubits and 320T gates. These findings provide a promising outlook for the use of this protocol in the classical modeling of quantum circuits that are conventionally difficult to simulate efficiently.
This book serves as an introductory yet thorough guide to tensor networks and their applications in quantum computation and quantum information, designed for advanced undergraduate and graduate-level readers. In Part I, foundational topics are covered, including tensor structures and network representations like Matrix Product States (MPS) and Tree Tensor Networks (TTN). These preliminaries provide readers with the core mathematical tools and concepts necessary for quantum physics and quantum computing applications, bridging the gap between multi-linear algebra and complex quantum systems. Part II explores practical applications of tensor networks in simulating quantum dynamics, with a particular focus on the efficiency they offer for systems of high computational complexity. Key topics include Hamiltonian dynamics, quantum annealing, open system dynamics, and optimization strategies using TN frameworks. A final chapter addresses the emerging role of “quantum magic” in tensor networks. It delves into non-stabilizer states and their contribution to quantum computational power beyond classical simulability, featuring methods such as stabilizer-enhanced MPS and the Clifford-dressed TDVP. In presenting tensor networks as tools for understanding quantum complexity, this book aims to foster a deeper collaboration between the many-body physics and quantum computing communities, inviting a broader audience to engage with these recent developments.
The density matrix renormalization group (DMRG) is widely acknowledged as a highly effective and accurate method for solving one-dimensional quantum many-body systems. However, the direct application of DMRG to the study of two-dimensional systems encounters challenges due to the limited entanglement encoded in the underlying wave-function Ansatz, known as the matrix product state. Conversely, Clifford circuits offer a promising avenue for simulating states with substantial entanglement, albeit confined to stabilizer states. In this work, we present the seamless integration of Clifford circuits within the DMRG algorithm, leveraging the advantages of both Clifford circuits and DMRG. This integration leads to a significant enhancement in simulation accuracy with small additional computational cost. Moreover, this framework is useful not only for its current application but also for its potential to be easily adapted to various other numerical approaches.
Recent studies have highlighted the combination of tensor network methods and the stabilizer formalism as a very effective framework for simulating quantum many-body systems, encompassing areas from ground state to time evolution simulations. In these approaches, the entanglement associated with stabilizers is transferred to Clifford circuits, which can be efficiently managed due to the Gottesman-Knill theorem. Consequently, only the non-stabilizerness entanglement needs to be handled, thereby reducing the computational resources required for accurate simulations of quantum many-body systems in tensor network related methods. In this work, we adapt this paradigm for finite temperature simulations in the framework of Time-Dependent Variational Principle, in which imaginary time evolution is performed using the purification scheme. Our numerical results on the one-dimensional Heisenberg model and the two-dimensional J1−J2 Heisenberg model demonstrate that Clifford circuits can significantly improve the efficiency and accuracy of finite temperature simulations for quantum many-body systems. This improvement not only provides a useful tool for calculating finite temperature properties of quantum many-body systems, but also paves the way for further advancements in boosting the finite temperature tensor network calculations with Clifford circuits and other quantum circuits.
We introduce the notion of dismagicker: non-Clifford unitary gate designed to reduce the non-stabilizerness (also called magic) of quantum many-body states. Although both entanglement and non-stabilizerness are fundamental quantum resources, they require distinct control strategies. While disentanglers (unitary operations that lower entanglement) are well-established in tensor network methods, analogous concept for non-stabilizerness suppression has been largely missing. In this work, we define dismagicker as non-Clifford unitary operation that actively suppresses non-stabilizerness, steering states toward classically simulatable stabilizer states. We develop optimization method for constructing dismagickers within the Matrix Product States framework. Our numerical results show that the non-stabilizerness reduction procedure, when combined with entanglement reduction steps with Clifford circuits, significantly improves the accuracy for both classical simulation of many-body systems and quantum state preparation on quantum devices. Dismagicker enriches our toolkit for the manipulation of many-body states by unifying non-stabilizerness and entanglement reduction.