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arXiv:2304.03559 · Physical review. D/Physical review. D.

Quantum circuits for the Ising spin networks

Grzegorz Czelusta, Jakub Mielczarek A B

Published Oct 24, 2023• 2 citations
arXivHTMLPDF

Abstract

Spin network states are a powerful tool for constructing the $SU(2)$ gauge theories on a graph. In loop quantum gravity (LQG), they have yielded many promising predictions, although progress has been limited by the computational challenge of dealing with high-dimensional Hilbert spaces. To explore more general configurations, quantum computing methods can be applied by representing spin network states as quantum circuits. In this article, we introduce an improved method for constructing quantum circuits for 4-valent Ising spin networks, which utilizes a smaller number of qubits than previous approaches. This has practical implications for the implementation of quantum circuits. We also demonstrate the procedure with various examples, including the construction of a 10-node Ising spin network state. The key ingredient of the method is the variational transfer of partial states, which we illustrate through numerous examples. Our improved construction provides a promising avenue for further exploring the potential of quantum computing methods in quantum gravity research.

DOI
10.1103/physrevd.108.086027

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Spin network states are a powerful tool for constructing the SU(2)SU(2)SU(2) gauge theories on a graph. In loop quantum gravity (LQG), they have yielded many promising predictions, although progress has been limited by the computational challenge of dealing with high-dimensional Hilbert spaces. To explore more general configurations, quantum computing methods can be applied by representing spin network states as quantum circuits. In this article, we introduce an improved method for constructing quantum circuits for 4-valent Ising spin networks, which utilizes a smaller number of qubits than previous approaches. This has practical implications for the implementation of quantum circuits. We also demonstrate the procedure with various examples, including the construction of a 10-node Ising spin network state. The key ingredient of the method is the variational transfer of partial states, which we illustrate through numerous examples. Our improved construction provides a promising avenue for further exploring the potential of quantum computing methods in quantum gravity research.
In loop quantum gravity approach to Planck scale physics, quantum geometry is represented by superposition of the so-called spin network states. In the recent literature, a class of spin networks promising from the perspective of quantum simulations of quantum gravitational systems has been studied. In this case, the spin network states are represented by graphs with four-valent nodes, and two dimensional intertwiner Hilbert spaces (qubits of space) attached to them. In this article, construction of quantum circuits for a general intertwiner qubit is presented. The obtained circuits are simulated on 5-qubit (Yorktown) and 15-qubit (Melbourne) IBM superconducting quantum computers, giving satisfactory fidelities. The circuits provide building blocks for quantum simulations of complex spin networks in the future. Furthermore, a class of maximally entangled states of spin networks is introduced. As an example of application, attempts to determine transition amplitudes for a monopole and a dipole spin network with the use of superconducting quantum processor are made.
Loop Quantum Gravity (LQG) is one of the leading approaches to unify quantum physics and General Relativity (GR). The Hilbert space of LQG is spanned by spin-networks which describe the local geometry of quantum space-time. Simulation of LQG spin-network states and their dynamics is classically intractable and is widely believed to fall in the Bounded Quantum Polynomial (BQP) time complexity class. There have been many recent attempts to simulate these states using novel and off the shelf quantum computing technologies. In this article, we review three such efforts which utilize superconducting qubits, linear optical qubits and Nuclear Magnetic Resonance (NMR) qubits respectively. The articles chosen for this review represent state of the art in quantum simulations of LQG.
Abstract Variational algorithms require architectures that naturally constrain the optimization space to run efficiently. Geometric quantum machine learning achieves this goal by encoding group structure into parameterized quantum circuits to include the symmetries of a problem as an inductive bias. However, constructing such circuits is challenging as a concrete guiding principle has yet to emerge. In this paper, we propose the use of spin networks , a form of directed tensor network invariant under a group transformation, to devise SU(2) equivariant quantum circuit ansätze—circuits possessing spin-rotation symmetry. By changing to the basis that block diagonalizes the SU(2) group action, these networks provide a natural building block for constructing parameterized equivariant quantum circuits. We prove that our construction is mathematically equivalent to other known constructions, such as those based on twirling and generalized permutations, but more direct to implement on quantum hardware. The efficacy of our constructed circuits is tested by solving the ground state problem of SU(2) symmetric Heisenberg models on the one-dimensional triangular lattice and the Kagome lattice. Our results highlight that our equivariant circuits boost the performance of quantum variational algorithms, indicating broader applicability to other real-world problems.
The accurate simulation of quantum circuits remains a significant challenge due to the exponential scaling of computational resources with circuit size. This work explores the application of tensor network methods, specifically Matrix Product States (MPS), to provide a scalable approach for simulating quantum circuits. We demonstrate how tensor networks can efficiently represent and manipulate the states of quantum circuits, reducing the computational burden associated with traditional methods. The core claim of this research is that tensor networks offer a viable path to accurately simulating larger and more complex quantum systems. This research introduces a novel framework for symbolic simulation, leveraging the power of tensor network representations to overcome the limitations of conventional simulation techniques. The effectiveness of this approach is illustrated through theoretical analysis and conceptual examples.
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Tensor network theory and quantum simulation are, respectively, the key classical and quantum computing methods in understanding quantum many-body physics. Here, we introduce the framework of hybrid tensor networks with building blocks consisting of measurable quantum states and classically contractable tensors, inheriting both their distinct features in efficient representation of many-body wave functions. With the example of hybrid tree tensor networks, we demonstrate efficient quantum simulation using a quantum computer whose size is significantly smaller than the one of the target system. We numerically benchmark our method for finding the ground state of 1D and 2D spin systems of up to 8×8 and 9×8 qubits with operations only acting on 8+1 and 9+1 qubits, respectively. Our approach sheds light on simulation of large practical problems with intermediate-scale quantum computers, with potential applications in chemistry, quantum many-body physics, quantum field theory, and quantum gravity thought experiments.