Algebraic Bethe Circuits

Abstract

The Algebraic Bethe Ansatz (ABA) is a highly successful analytical method used to exactly solve several physical models in both statistical mechanics and condensed-matter physics. Here we bring the ABA into unitary form, for its direct implementation on a quantum computer. This is achieved by distilling the non-unitary R matrices that make up the ABA into unitaries using the QR decomposition. Our algorithm is deterministic and works for both real and complex roots of the Bethe equations. We illustrate our method on the spin-12 XX and XXZ models. We show that using this approach one can efficiently prepare eigenstates of the XX model on a quantum computer with quantum resources that match previous state-of-the-art approaches. We run small-scale error-mitigated implementations on the IBM quantum computers, including the preparation of the ground state for the XX and XXZ models on 4 sites. Finally, we derive a new form of the Yang-Baxter equation using unitary matrices, and also verify it on a quantum computer.

Large-scale simulations of Floquet physics on near-term quantum computers

“…In that case, Ĥ0 can be simulated using a constant depth circuit. Similar cases of known implementations of Ĥ0 are 1D XY-models [28], Bethe diagonalizable models [71], and translational invariant models [34].…”

“…The transversely driven Ising model (independent of topology and dimension) falls into this category. More generally, one could use known methods for simulating the time-independent 1D XY -model [28], Bethe diagonalizable models [71] and translationally symmetric models [34], to simulate the effect of local driving on such systems. We note that in classical simulations of Floquet systems, high frequency expansions are known and commonly used [63,[65][66][67][68] both for simulation and for Floquet engineering.…”

The Bethe Ansatz as a Quantum Circuit

“…The quantum gates of Figure 3 ultimately follow from a recurrence relation. The recurrence is involved, and [15] provided analytical expressions for M = 1, 2, whereas it computed the quantum gates numerically for M > 2. This problem served as a starting point for [16], which obtained closed formulae for the quantum gates of the ABC.…”

“…Reference [15] alternatively developed a deterministic quantum algorithm, called algebraic Bethe circuit (ABC), that prepares Bethe states based on the ABA. The number of qubits and quantum gates scale polynomially with N .…”

“…The number of qubits and quantum gates scale polynomially with N . Reference [15] showed that quantum gates decompose into a layer of matchgates at the non-interacting point, where the XXZ model reduces to the XX model. Nonetheless, no conclusive answer was obtained for the XXZ in general.…”