Berry phase estimation in gate-based adiabatic quantum simulation

Gate-based quantum computers can in principle simulate the adiabatic dynamics of a large class of Hamiltonians. Here, we consider the cyclic adiabatic evolution of a parameter in the Hamiltonian. We propose a quantum algorithm to estimate the Berry phase and use it to classify the topological order of both single-particle and interacting models, highlighting the differences between the two. This algorithm is extensible to an interacting topological system. Our results evidence the potential of near-term quantum hardware for the topological classification of quantum matter.

Figure 1

Quantum circuits to measure Berry phase. ${\mathcal{U}}_{\mathrm{init}}$ represents the subcircuit that initializes the $n$-qubit register in an eigenstate of $\mathcal{H}$. ${\mathcal{U}}_{\mathrm{loop}}$ implements the quantum simulation of the adiabatic loop. $H$ are Hadamard gates. (a) Hadamard-test scheme [35]. (b) Iterative phase estimation (IPE) scheme [36]. The quantum circuit shows a $k\mathrm{th}$ iteration. The $R_z\left({\omega }_k\right)$ gate, where ${\omega }_k=-2\pi 0.0\cdots {\gamma }_{k+1}\cdots {\gamma }_R$ and $R$ is the total number of iterations, serves to remove the contribution to the phase from the previously measured bits.

Figure 2

(a) SSH tight-binding chain with intra- and intercell hopping parameters $v$ and $w$. The box delimits a unit cell. (b) Quantum circuit implementing a controlled-$\delta \mathcal{U}$ gate defined in Eq. (5). $\alpha , \beta , \gamma$, and $\delta$ are related to the parameter $\rho$ and the Hamiltonian ${\mathcal{H}}_{\text{SSH}}$ as described in the Supplemental Material [38]. (c) Berry phase, as obtained from Eq. (7) analytically (dashed line), in silico unitary simulation (black pluses), and experimental quantum simulation (red crosses) of the Hadamard-test (HT) circuit [Fig. 1], and quantum simulation of the IPEA circuit [Fig. 1] for $R=4$ iterations and $N=4$ time steps (green dots with respective error bars). Quantum simulations were carried out in the ibmq_16_melbourne device from the IBM Q Experience. Further details about the quantum simulation can be found in the Supplemental Material.

Figure 3

(a) Local Berry phase ${\theta }_B$ obtained via a Hatsugai twist [25] as a local topological marker that reveals the dimer structure of the Heisenberg ring. Strong links ($J_{\mathrm{eff}}\equiv J\pm \delta >J$) have ${\theta }_B=\pi$, while weak ones ($J_{\mathrm{eff}}<J$) have ${\theta }_B=0$. (b) Local Berry phase of a link with coupling $J_{\mathrm{eff}}=J-\delta$ for a dimerized Heisenberg ring of four spins with $J=1$ obtained via a noiseless unitary simulation of the Berry phase estimation circuit shown in Fig. 1. Quantum algorithm results (green dots) deviate slightly from those obtained via exact exponentiation of the full $16\times 16$ Hamiltonian (red pluses) due to shot-noise and Trotterization errors. Both simulations reveal a deviation from the expected steplike pattern (blue dashed line) due to finite-size effects (see Supplemental Material [38]).