Variational Power of Quantum Circuit Tensor Networks

Abstract

We characterize the variational power of quantum circuit tensor networks in the representation of physical many-body ground states. Such tensor networks are formed by replacing the dense block unitaries and isometries in standard tensor networks by local quantum circuits. We explore both quantum circuit matrix product states and the quantum circuit multiscale entanglement renormalization Ansatz, and introduce an adaptive method to optimize the resulting circuits to high fidelity with more than 10 4 parameters. We benchmark their expressiveness against standard tensor networks, as well as other common circuit architectures, for the 1D and 2D Heisenberg and 1D Fermi-Hubbard models. We find quantum circuit tensor networks to be substantially more expressive than other quantum circuits for these problems, and that they can even be more compact than standard tensor networks. Extrapolating to circuit depths which can no longer be emulated classically, this suggests a region of advantage in quantum expressiveness in the representation of physical ground states.

Algebraic Bethe Circuits

“…Hence the ABCs could prove relevant, not only for the preparation of eigenstates of integrable spin chains, but also for general MPS of low bond dimension. We note that the implementation of tensor-network states on a quantum computer has been addressed in [22,34,35,[35][36][37][38].…”

“…Hence our method to obtain ABCs should also prove relevant for the circuit implementation of general MPS with low bond dimension. Let us mention that several works have considered the implementation of tensor-network states on a quantum computer [22,34,35,[35][36][37][38].…”

Quantum machine learning for chemistry and physics

“…The goal is to prepare variational states using quantum circuits which are more expressive than tensor networks or any other classical ansatz and also are difficult to simulate on classical computers. In a recent report, the authors 549 use this idea to represent quantum states with variational parameters of quantum circuit defined on a few qubits instead of standard parameterized tensor used in DMRG (see Section 3.4.5). They show that sparsely parameterized quantum circuit tensor networks are capable of representing physical states more efficiently than the dense tensor networks.…”

Quantum Annealing for Neural Network optimization problems: a new approach via Tensor Network simulations

“…The mapping onto a quantum circuit can be achieved by exploiting the MPS nature of the final state. It is indeed well-known that, in a N qubits system, any MPS having maximum bond dimension χ = 2 n can be obtained from the trivial state 2 |0 0 0〉 = |0...0〉 by applying sequentially N unitary gates, each acting (at most) on log 2 χ + 1 = n + 1 qubits [21,[41][42][43] (see Appendix B for additional information). These unitaries can be further decomposed into two-qubits gates.…”

“…In the simplest case of an MPS with bond dimension χ = 2, equal to the physical dimension of the local Hilbert space, the MPS can be exactly mapped to a "staircase" of two-qubits unitary gates, acting sequentially on |0 0 0〉. This mapping is well-established [21,[41][42][43] and relies on the following fact: MPS local tensors can always be chosen as left or right isometries, which can be completed to unitary matrices, in turn decomposed into quantum gates. These tricks can be easily extended to the general case of an MPS with maximum bond dimension χ = 2 n , consequently obtaining a quantum circuit defined by a sequence of unitaries acting (at most) on log 2 χ+1 = n+1 qubits [41-Figure 13: Exact mapping of a right-normalized MPS -with maximum bond dimension χ = 2 n (n = 3 in the figure) -to a staircase quantum circuit involving N unitaries, each acting non-trivially on up to log 2 χ + 1 = n + 1 qubits.…”