Preparing Bethe Ansatz Eigenstates on a Quantum Computer

Several quantum many-body models in one dimension possess exact solutions via the Bethe ansatz method, which has been highly successful for understanding their behavior. Nevertheless, there remain physical properties of such models for which analytic results are unavailable and which are also not well described by approximate numerical methods. Preparing Bethe ansatz eigenstates directly on a quantum computer would allow straightforward extraction of these quantities via measurement. We present a quantum algorithm for preparing Bethe ansatz eigenstates of the spin-1/2 $XXZ$ spin chain that correspond to real-valued solutions of the Bethe equations. The algorithm is polynomial in the number of $T$ gates and the circuit depth, with modest constant prefactors. Although the algorithm is probabilistic, with a success rate that decreases with increasing eigenstate energy, we employ amplitude amplification to boost the success probability. The resource requirements for our approach are lower than for other state-of-the-art quantum simulation algorithms for small error-corrected devices and thus may offer an alternative and computationally less demanding demonstration of quantum advantage for physically relevant problems.

Popular Summary

Problems with known exact solutions are rare in physics–instead, the norm is to use approximations that capture the essence of the phenomena under investigation. When such solutions do arise, exactly-solvable problems are helpful because they provide rigorous conclusions that can establish the validity (or not) of approximate methods. The Bethe ansatz is a powerful mathematical technique for exactly solving a number of different models of quantum many-body physics in one dimension. But even with the exact solution in hand, there are aspects of these systems which are not fully understood due to the complexity of the quantum wave function. This work presents an algorithm for preparing the wave function of a particular Bethe ansatz-solvable model on a quantum computer. The approach has several benefits: first, it allows one to compute quantities that are not easily obtained by classical methods, and second, it can be used to verify that the quantum device is behaving as intended, through the calculation of quantities that are already known from the Bethe ansatz itself. More specifically, the proposed algorithm allows one to prepare eigenstates of the spin-1/2 XXZ chain, a paradigmatic model of quantum magnetism. The algorithm works for states throughout the energy spectrum, including highly entangled ones that are less accessible using classical computational methods. Having prepared the state, one can then measure the system directly to obtain important quantities such as correlation functions, which can characterize the behavior of the model throughout the phase diagram, from weak to strong coupling. This work paves the way for future algorithms that prepare eigenstates of other interacting exactly-solvable models relevant to high-energy or condensed matter physics, such as the one-dimensional Hubbard model used to describe strongly-correlated electrons.

Figure 1

The circuit to prepare the permutation-label ancilla state for $M=3$. When the gates inside the red dashed rectangles are excluded, the circuit realizes the equally weighted superposition encoding the permutations of three objects. With these gates included, the circuit applies the additional permutation-dependent phases $A_P$. The three subregisters are indicated by shading and the initial state of each qubit is $\left|0\right.⟩$. The alternating cnot gates interleaved with $R_y$ and $R_z$ rotations realize the aswap gates.

Figure 2

A schematic diagram illustrating the idea of the faucet method. Phases $e^{i{k}_{Pj}}$ are applied for each system qubit while traversing the bit string. When a “1” is encountered, the next faucet in the list is turned off, so that no more phases with the given $k_{Pj}$ value are applied.

Figure 3

(a) The success probabilities for the preparation of selected eigenstates as a function of their energies, when $L=2M$ and $M=2,3,4$. (b) The success probabilities of selected eigenstates for $M=3$ and varying $L$. The Hamiltonian parameter values are $J_{xy}=1$ and $J_z=-1/2$.

Figure 4

The Bethe-state-preparation (a) circuit depth, (b) number of Toffoli gates, (c) number of controlled-phase gates, and (d) number of qubits versus the system size $L$, for different numbers of down spins $M$.

Figure 5

The Bethe-state-preparation gate and measurement counts for $M=5$ as a function of $L$.

Figure 6

(a) The success probability for Bethe-state preparation with (solid lines) and without (dashed lines) amplitude amplification (“AA”). In the former case, a single round of amplification is applied. The system parameters are $J_{xy}=1$, $J_z=-1/2$, $L=2M$. (b) The gate counts as a function of the system size $L$ for Toffoli (circles) and controlled-phase (squares) gates (“Cphase”) without (solid lines) and with one round of amplitude amplification (dashed lines). The inset shows the total number of qubits without (solid) and with (dashed) amplitude amplification.

Figure 7

A comparison between Algorithm 10 and the alternative method of the number of (a) Toffoli and (b) controlled-phase gates used, for $M=5$.

Figure 8

The quantum circuit to prepare the Bethe ansatz eigenstate with $L=4$, $M=2$, $k_1=1.14676529$, and $k_2=3.56562369$.

Figure 9

The success probabilities for the preparation of selected eigenstates as a function of their energies, for $L=12$, $M=3$, $J_{xy}=1$, with varying interaction strengths $J_z$.

Figure 10

Bethe-state preparation.