No quantum advantage (yet) in the world of tensor networks

Tensor networks pictographic scheme Each tensor is represented by an array of dimension N, that matches its number of indices. By joining the legs (indices) of each tensor, it is possible to encode the Hamiltonian or the state of a many-particle system. (Courtesy: Lucy Reading-Ikkanda/Simons Foundation) Researchers at the Flatiron Institute in New York City have used a new tensor network (TN) scheme to simulate how Ising spin glasses evolve in time. In many cases, their classical method is more accurate than the latest quantum annealers running the same problem. The results scale in two and three dimensions and raise the bar in the ongoing competition between classical and quantum computers for simulating many-particle quantum systems. Classical versus quantum Quantum many-particle systems provide a good way of comparing computational techniques because their complexity grows exponentially with the number of particles. In theory, quantum computers hold an advantage rooted in entanglement and superposition. In a classical computer, the spin of a particle (up or down) is encoded in bit that is either zero (down) or one (up), but never both. In a quantum computer, the spin can be encoded in a qubit holding a quantum superposition of up and down. This is a much more natural way to represent what the simulated system is actually doing. Recently, a team of quantum computing researchers used the D-Wave’s Advantage2 quantum annealer to simulate Ising spin glass dynamics and claimed that classical computers could not match their results. But now, Joseph Tindall and colleagues at the Flatiron Institute have shown that a classical scheme can do just as well -and sometimes better- than the annealer. In this occasion, the competition involved the simulation of the Ising spin glass model, which describes spins on a lattice pointing in random directions. This disordered state is product of different nearest-neighbor interactions for different pairs of spins. The difficulty of this famous problem grows with the system size, which makes it a good benchmark for comparing classical and quantum performance. Tensor networks The tensors used by Tindall and colleagues can be thought of as LEGO pieces. Just as LEGO bricks have knobs and tubes that snap together to build a bigger array, tensors have legs (bond indices) that join through a process called contraction to form networks. In a many-particle simulation, each tensor represents a site along with its physical attributes: for instance, the type of particle it hosts, which could be a spin or an electron. Connected together, the tensors form a network that encodes how those particles interact and how entangled they are. Widening the legs (increasing the bond dimension in technical terms), allows the network to capture the correlations more accurately. In simple terms, the pattern of connections describes how particles can move or interact, and what Tindall and colleagues built is a tensor network representation of both the Ising spin glass Hamiltonian and wave function. The first specifies the interactions between spins and the second describes the state of spins. Writing quantum states this way has proved enormously useful for accessing large systems, where the sheer number of particles rules out any exact solution. Tindall’s group used these tensor network ideas to simulate the spin glass model’s dynamics. But there is a catch: as a TN wave function evolves forward in time, entanglement builds up between different parts of the system and the bond indices must widen to keep track of the growing correlations. This increases the computational cost of finding the ground state of the system at different times, making it more difficult and sometimes impossible. Belief propagation To compensate for this, the team used a belief propagation (BP) message passing approach to perform the contraction. In this scheme, each tensor receives a compact summary of what the rest of the network “looks like” from its perspective (the effective environment of each tensor). This unlike the conventional approach of accounting for every single contribution exactly. As Tindall points out, their approach can be understood as a mean field approximation on each tensor’s environment since the pieces of information reaching a particular given tensor from each of its neighbours are assumed to be independent. D-Wave Systems claims quantum advantage, but some physicists are not convinced Read more By implementing this TN–BP approach, their classical implementation could evolve the system to much longer times than conventional TN schemes and manage far enough to reach the regime the quantum annealer operates. Conceptually, passing messages about an effective environment lets the computation to keep pace with the entanglement inherent to the system’s time evolution. The researchers then measured how the spin at one site relates to the spin elsewhere on the lattice (the two-point correlator) at various times and system sizes on two and three-dimensional lattices with cylindrical, diamond and cubic geometries. Starting with a cylindrical lattice, the team compared the error in the two-point correlator error from the annealer with that from the TN-BP message passing approach. Once the bond dimension was large enough, the classical error was markedly lower error than the quantum counterpart at different annealing. On the diamond lattice, the classical error obtained again came in below the annealer’s, while on the cubic lattice the error was of the same order at a fixed annealing time. What comes next This work, which is described in Science, shows how fast classical simulation of quantum physics is advancing and how the on-going competition between classical and quantum computation is driving this progress. In the future, Tindall and colleagues want to apply the TB-BP to interacting electronic systems such the Hubbard model. Additionally, the group is also extending their scheme to finite temperature problems as well as maintaining their open-source tensor network quantum simulator library. Want to read more? Registration is free, quick and easy Note: The verification e-mail to complete your account registration should arrive immediately. However, in some cases it takes longer. Don't forget to check your spam folder. If you haven't received the e-mail in 24 hours, please contact customerservices@ioppublishing.org. E-mail Address Register