Adiabatic ground state preparation in an expanding lattice

We implement and characterize a numerical algorithm inspired by the $s$ source framework [B. Swingle and J. McGreevy, Phys. Rev. B 93, 045127 (2016)] for building a quantum many-body ground state wave function on a lattice of size $2L$ by applying adiabatic evolution to the corresponding ground state at size $L$, along with $L$ interleaved ancillae. The procedure can in principle be iterated to repeatedly double the size of the system. We implement the algorithm for several one-dimensional (1D) spin model Hamiltonians, and find that the construction works particularly well when the gap is large and, interestingly, at scale-invariant critical points. We explain this feature as a natural consequence of the lattice expansion procedure. This behavior holds for both the integrable transverse-field Ising model and nonintegrable variations. We also develop an analytic perturbative understanding of the errors deep in either phase of the transverse-field Ising model, and suggest how the circuit could be modified to parametrically reduce errors. In addition to sharpening our perspective on entanglement renormalization in 1D, the algorithm could also potentially be used to build states experimentally, enabling the realization of certain long-range correlated states with low-depth quantum circuits.

Figure 1

Circuit diagrams for the $s$ source renormalization procedure. (a) A single layer of the circuit which takes an 8-spin ground state (the large black box) and 8 ancillae (small black boxes) and after applying the turn “on” (yellow) and turn “off” (blue) unitaries produces an approximation to the 16-spin ground state. (b) A two-superlayer circuit which starts from the 4-spin ground state and produces an approximation to the 16-spin ground state; each block outlined by a red dashed line represents a superlayer made up of two on and two off layers.

Figure 2

(a) Relative energy error and (b) infidelity of the TFIM $s$ source ground state as a function of $g$ for several system sizes. Both go to 0 deep in either phase as one would expect, but there is also a local minimum at the critical point due to the scale invariance of the system. (c) Relative energy error and (d) infidelity for $L_{\text{f}}=64$ with with either the normal $s$ source input ($s=1$) or product state input ($s=0$). Using a product input state gives better energies deep in both phases, but has an infidelity over 0.5 in the ferromagnetic phase as long-range entanglement cannot be generated. By either metric, $s=1$ input gives substantially smaller errors near the critical point.

Figure 3

(a) Relative energy error and (b) infidelity of the TFIM for both the “standard” $s$ source circuit with both nearest-neighbor (NN) and next-nearest-neighbor (NNN) unitaries and a simplified circuit with only NN unitaries, both for $L_f=64$. The general shape of the error curves, notably including the local minimum at the critical point, is similar for both circuits. The NN circuit does almost as well as the NNN circuit at the critical point, but the errors fall off more slowly than for the NNN circuit deep in either phase.

Figure 4

Relative energy error in the TFIM for multilayer circuits. (a) Error comparison, starting with $L_0=8$ for one- and two-layer circuits. (b) Error comparison ending with $L_f=32$ for one- and two-layer circuits. Error compounds reasonably with successive layers, and in particular is not much worse than single-layer optimization at the critical point. (c) We hypothesize that the multilayer error is subadditive, as illustrated here. Except where we have failed to find the global minimum, the $L_0=8$ to $L_f=32$ error is bounded by the $L_0=8$ to $L_f=16$ error plus the $L_0=16$ to $L_f=32$ error, with the multilayer circuit substantially outperforming this bound at the critical point.

Figure 5

Relative energy error for the mixed coupling Ising model with $J_x=0.1J_z$. The features of the error curve are qualitatively similar to the transverse-field case, and the local minimum remains at the model's critical point.

Figure 6

Relative energy error for the mixed-field Ising model for $L_f=64$. For sufficiently large $h_z$, there is no longer a local minimum in the error as there is no scale-invariant point.