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arXiv:2207.07725 · Physical Review Research

Preparing valence-bond-solid states on noisy intermediate-scale quantum computers

Bruno Murta, Pedro M. Q. Cruz, Joaquín Fernández-Rossier

Published Mar 20, 2023• 22 citations
arXivHTMLPDF

Abstract

Quantum state preparation is a key step in all digital quantum simulation algorithms. Here we propose methods to initialize on a gate-based quantum computer a general class of quantum spin wave functions, the so-called valence-bond-solid (VBS) states, that are important for two reasons. First, VBS states are the exact ground states of a class of interacting quantum spin models introduced by Affleck, Kennedy, Lieb, and Tasaki (AKLT). Second, the two-dimensional VBS states are universal resource states for measurement-based quantum computing. We find that schemes to prepare VBS states based on their tensor-network representations yield quantum circuits that are too deep to be within reach of noisy intermediate-scale quantum (NISQ) computers. We then apply the general nondeterministic method herein proposed to the preparation of the spin-1 and spin-$3/2$ VBS states, the ground states of the AKLT models defined in one dimension and in the honeycomb lattice, respectively. Shallow quantum circuits of depth independent of the lattice size are explicitly derived for both cases, making use of optimization schemes that outperform standard basis gate decomposition methods. The probabilistic nature of the proposed routine translates into an average number of repetitions to successfully prepare the VBS state that scales exponentially with the number of lattice sites $N$. However, two strategies to quadratically reduce this repetition overhead for any bipartite lattice are devised. Our approach should permit to use NISQ processors to explore the AKLT model and variants thereof, outperforming conventional numerical methods in the near future.

DOI
10.1103/physrevresearch.5.013190

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Quantum state preparation is a key step in all digital quantum simulation algorithms. Here we propose methods to initialize on a gate-based quantum computer a general class of quantum spin wave functions, the so-called valence-bond-solid (VBS) states, that are important for two reasons. First, VBS states are the exact ground states of a class of interacting quantum spin models introduced by Affleck, Kennedy, Lieb, and Tasaki (AKLT). Second, the two-dimensional VBS states are universal resource states for measurement-based quantum computing. We find that schemes to prepare VBS states based on their tensor-network representations yield quantum circuits that are too deep to be within reach of noisy intermediate-scale quantum (NISQ) computers. We then apply the general nondeterministic method herein proposed to the preparation of the spin-1 and spin-3/23/23/2 VBS states, the ground states of the AKLT models defined in one dimension and in the honeycomb lattice, respectively. Shallow quantum circuits of depth independent of the lattice size are explicitly derived for both cases, making use of optimization schemes that outperform standard basis gate decomposition methods. The probabilistic nature of the proposed routine translates into an average number of repetitions to successfully prepare the VBS state that scales exponentially with the number of lattice sites NNN. However, two strategies to quadratically reduce this repetition overhead for any bipartite lattice are devised. Our approach should permit to use NISQ processors to explore the AKLT model and variants thereof, outperforming conventional numerical methods in the near future.
The spin-3/2 Affleck-Kennedy-Lieb-Tasaki (AKLT) valence-bond state on a hexagonal lattice was shown to be a universal resource state for measurement-based quantum computation (MBQC). Can AKLT states of higher spin magnitude support universal MBQC? We demonstrate that several hybrid two-dimensional AKLT states involving a mixture of spin-2 and other lower-spin entities, such as spin-3/2 and spin-1, are also universal for MBQC. This significantly expands universal resource states in the AKLT family. Even though frustration may be a hindrance to quantum computational universality, lattices can be modified to yield AKLT states that are universal. The family of AKLT states thus provides a versatile playground for quantum computation.
The Affleck–Kennedy–Lieb–Tasaki (AKLT) spin interacting model can be defined on an arbitrary graph. We explain the construction of the AKLT Hamiltonian. Given certain conditions, the ground state is unique and known as the valence-bond-solid (VBS) state. It can be used in measurement-based quantum computation as a resource state instead of the cluster state. We study the VBS ground state on an arbitrary connected graph. The graph is cut into two disconnected parts: the block and the environment. We study the entanglement between these two parts and prove that many eigenvalues of the density matrix of the block are zero. We describe a subspace of eigenvectors of the density matrix corresponding to non-zero eigenvalues. The subspace is the degenerate ground states of some Hamiltonian which we call the block Hamiltonian.
This article reviews the quantum entanglement in Valence-Bond-Solid (VBS) states defined on a lattice or a graph. The subject is presented in a self-contained and pedagogical way. The VBS state was first introduced in the celebrated paper by I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki (abbreviation AKLT is widely used). It became essential in condensed matter physics and quantum information (measurement-based quantum computation). Many publications have been devoted to the subject. Recently entanglement was studied in the VBS state. In this review, we start with the definition of a general AKLT spin chain and the construction of VBS ground state. In order to study entanglement, a block subsystem is introduced and described by the density matrix. Density matrices of 1-dimensional models are diagonalized and the entanglement entropies (the von Neumann entropy and Rényi entropy) are calculated. In the large block limit, the entropies also approach finite limits. Study of the spectrum of the density matrix led to the discovery that the density matrix is proportional to a projector.
We study the ground-state properties of two-dimensional quantum spin systems having the valence-bond-solid (VBS) type ground states. The `product-of-tensors' form of the ground-state wavefunction of the system is utilized to associate it with an equivalent classical lattice statistical model which can be treated by the transfer-matrix method. For diagonalization of the transfer matrix, we employ the product-wavefunction renormalization group method, which is a variant of the density-matrix renormalization group method. We obtain the correlation length and the sublattice magnetization accurately. For the anisotropically `deformed' S = 3/2 VBS model on the honeycomb lattice, we find that the correlation length as a function of the deformation parameter behaves very much as that in the S = 3/2 VBS chain.
The AKLT state is the ground state of an isotropic quantum Heisenberg spin-1 model. It exhibits an excitation gap and an exponentially decaying correlation function, with fractionalized excitations at its boundaries. So far, the one-dimensional AKLT model has only been experimentally realized with trapped-ions as well as photonic systems. In this work, we successfully prepared the AKLT state on a noisy intermediate-scale quantum (NISQ) era quantum device. In particular, we developed a non-deterministic algorithm on the IBM quantum processor, where the non-unitary operator necessary for the AKLT state preparation is embedded in a unitary operator with an additional ancilla qubit for each pair of auxiliary spin-1/2’s. Such a unitary operator is effectively represented by a parametrized circuit composed of single-qubit and nearest-neighbor CX gates. Compared with the conventional operator decomposition method from Qiskit, our approach results in a much shallower circuit depth with only nearest-neighbor gates, while maintaining a fidelity in excess of 99.99% with the original operator. By simultaneously post-selecting each ancilla qubit such that it belongs to the subspace of spin-up |↑>, an AKLT state can be systematically obtained by evolving from an initial trivial product state of singlets plus ancilla qubits in spin-up on a quantum computer, and it is subsequently recorded by performing measurements on all the other physical qubits. We show how the accuracy of our implementation can be further improved on the IBM quantum processor with readout error mitigation.
Qutrit-based quantum circuits could help reduce the overall circuit depths, and hence the effect of noise, when the system of interest has a local dimension of three. Accessing second excited states in superconducting transmons provides a straightforward hardware realization of qutrits useful for such ternary encoding. In this work, we successfully calibrate microwave pulse gates to a low error rate to operate transmon qutrits. We use these qutrits to simulate one-dimensional spin-1 AKLT states (Affleck, Kennedy, Lieb, and Tasaki), which exhibit a multitude of interesting phenomena, such as topologically protected ground states, string order, and the existence of a robust Berry phase. We demonstrate the efficacy of qutrit-based simulation by preparing high-fidelity ground states of the AKLT Hamiltonian with open boundaries for various chain lengths. We then use ground state preparations of the perturbed AKLT Hamiltonian with periodic boundaries to calculate the Berry phase and illustrate non-trivial ground state topology. To establish the advantage of qutrits over qubits in the presence of noise, we present scalable methods for preparing the AKLT state and computing its Berry phase using tensor network simulations. Our work provides a pathway toward more general spin-1 physics simulations using transmon qutrits, with applications in chemistry, magnetism, and topological phases of matter.