Determining quantum Monte Carlo simulability with geometric phases

Although stoquastic Hamiltonians are known to be simulable via sign-problem-free quantum Monte Carlo (QMC) techniques, the nonstoquasticity of a Hamiltonian does not necessarily imply the existence of a QMC sign problem. We give a sufficient and necessary condition for the QMC-simulability of Hamiltonians in a given basis: We prove that a QMC simulation will be sign-problem-free if and only if all the overall total phases along the chordless cycles of the weighted graph whose adjacency matrix is the Hamiltonian are zero (modulo $2\pi$). We use our findings to provide a construction for nonstoquastic, yet sign-problem-free and hence QMC-simulable, quantum many-body models. We also demonstrate why the simulation of truly sign-problematic models using the QMC weights of the stoquasticized Hamiltonian is generally suboptimal. We offer a superior alternative.

Figure 1

Every summand $W_{(z,{\mathbf{i}}_q)}$ in the partition function decomposition is associated with a closed walk on the hypercube of basis states (a subgraph of which is shown here for illustration). The graph nodes (in blue) represent computational basis states and edges denote nonzero off-diagonal matrix elements. A walk on the graph (arrows in brown) is determined by the action of the permutation operators of the configuration, represented by the sequence $S_{{\mathbf{i}}_q}=P_{i_q}\cdots P_{i_2}{P}_{i_1}$, on the initial basis state $|z_0\rangle$. Every node $|z_j\rangle$ has an associated classical energy $E_{z_j}$.

Figure 2

Every closed walk (arrows in brown along edges of the graph) has a conjugate closed walk traversing the same path but in the reverse direction (arrows in green). The weight of a walk is the complex conjugate of the weight of its conjugate walk. This observation allows us to disregard the imaginary parts of these partition function summands which cancel out. The permutation operators that generate a closed walk are the inverses of the permutation operators that generate its conjugate and appear in reverse order.

Figure 3

Composite walks can be viewed as a concatenation of cycles (cyclic paths on a graph). Above, the closed walk $|z_0\rangle \to |z_1\rangle \to \cdots \to |z_5\rangle \to |z_1\rangle$ can be decomposed to two cycles: $|z_1\rangle \to |z_2\rangle \to |z_3\rangle \to |z_1\rangle$ and $|z_0\rangle \to |z_1\rangle \to |z_4\rangle \to |z_5\rangle \to |z_0\rangle$. The overall phase associated with a closed walk is the sum of the phases associated with its constituent cycles.

Figure 4

The geometric phase associated with a composite cycle (black) is the total sum of the phases of its subcycles (brown and green).

Figure 5

The frustrated Heisenberg two-leg spin $1/2$ ladder. The Heisenberg-type interactions between the spins (vertices in red) are denoted by the edges connecting the vertices. The coupling strengths associated with the solid, dashed, and dotted edges are $J_{\parallel }$ (horizontal), $J_{\perp }$ (vertical), and $J_{\times }$ (diagonal), respectively. The edges that make up the contour of the shaded area (in green), connecting the spins $(i,+), (i+1,+)$, and $(i,-)$ are an example of a triplet of edges that may yield negative-weight closed walks.

Figure 6

Hamiltonian partitioning by reachability. An example of a $6\times 6$ Hamiltonian containing zero elements. All the matrix elements circled in red are reachable from one another and the same holds for all elements circled in green. Elements from one group cannot be reached from those of the other. The reachability criterion partitions the Hamiltonian into two noninteracting sub-Hamiltonians.