Monte Carlo sampling from a projected entangled-pair state in simulations of quantum annealing in the three dimensional random Ising model

Abstract

Quantum annealing with the D-Wave Advantage system in the random Ising model on a cubic lattice is simulated using a three-dimensional (3D) tensor network. The Hamiltonian is driven across a quantum phase transition from a paramagnetic phase to a spin-glass phase. The network is represented as a tensor product state, also known-particularly in two dimensions-as a projected entangled-pair state (PEPS). The annealing procedure is repeated for a range of annealing times in order to test the Kibble-Zurek (KZ) power law governing the residual energy at the end of the annealing ramp. For an infinite lattice with periodic nearest-neighbor random Ising couplings, the final energy is evaluated using a deterministic method. For a finite lattice with open boundaries, we introduce a more efficient Monte Carlo sampling approach. In both cases, the residual energy as a function of annealing time approaches the KZ power law as the annealing time increases.

Figures (8)

• Figure 1: Quantum annealing ramp. The energy scales $\Gamma$ and ${\cal J}$ (in GHz) in the Hamiltonian \ref{['eq:H']} are shown as functions of the dimensionless ramp parameter $s=t/t_a$, where $t_a$ is the annealing time. This corresponds to the annealing schedule of the D-Wave Advantage system used in Ref. Science_Dwave.
• Figure 2: Deterministic expectation values in iPEPS. (a) A 3D infinite PEPS (iPEPS) tensor network with bond dimension $D$. The red lines denote physical indices. (b) The norm squared of the state in (a), obtained by contracting the TN with its complex conjugate (indicated by stars) over the physical indices. Each horizontal layer at position $z$ can be interpreted as a transfer matrix. (c) The upper boundary at layer $z$. The black dots denote diagonal matrices of singular values. The $z$-th boundary is obtained from the $(z+1)$-th boundary by applying the $z$-th transfer matrix, as illustrated in (d-g). (d) The $z$-th iPEPS layer is applied to the $(z+1)$-th boundary, resulting in the $z$-th mixed boundary shown in (f), which carries additional physical indices. (e) The bond dimension of the mixed boundary is truncated back to $d$ using the simple update, by truncating the singular values on each bond. (g) The conjugate $z$-th iPEPS layer is applied to the mixed boundary (f), yielding the upper boundary at layer $z$. The bond dimension is again truncated to $d$. (i) The lower mixed boundary at layer $z$ is obtained from the $(z-1)$-th lower boundary by applying the conjugate $z$-th iPEPS layer.
• Figure 3: Trotter gate. (a, left) A fragment of the iPEPS shown in Fig. \ref{['fig:ipeps']}(a). The bluish tensors in the center have been acted upon by a two-site Trotter gate, which doubles the dimension of their shared bond, as indicated by the thicker lines. In this diagram, the doubled bond indices are left uncontracted. (a, right) The Gram-Schmidt metric corresponding to the doubled indices, obtained as the overlap between the diagram on the left and its complex conjugate. The large bullets represent double PEPS tensors, each formed by contracting a PEPS tensor with its complex conjugate over the physical index and a subset of the corresponding bond indices. This metric is used to compress the doubled bond dimension back to $D$. (b) A larger fragment of the network and the corresponding metric tensor. To contract the metric efficiently, some double PEPS tensors and bonds are truncated via singular value decomposition, retaining only the leading singular value ($SVD_1$). This method is used in the present paper.
• Figure 4: Excitation energy - deterministic. Average residual excitation energy per bond, $Q$ (in GHz), versus annealing time $t_a$ (in ns). The dashed line shows the slope predicted by the Kibble-Zurek power law \ref{['eq:Q_KZ']}. The energies were computed using the deterministic method for an infinite lattice with a periodic unit cell of size $L^3=8^3$.
• Figure 5: Monte Carlo sampling in a 3D PEPS. (a) A projected PEPS in which all physical indices have been assigned definite values. Unlike the full PEPS shown in Fig. \ref{['fig:ipeps']}(a), the fixed physical indices are not displayed. Here, a $3\times3\times4$ PEPS is shown for illustration. (b) The upper and lower projected PEPS boundaries (blue) are used to compute the probability amplitudes for two selected sites with the full PEPS tensors. These amplitudes depend explicitly on the physical indices of the sites (red). (c) At each site, a triple tensor is defined, which may include or exclude the physical index depending on whether the full or projected PEPS tensor is considered.