Table 3 Overview of our solvers for Equation (6)

From: Hardware-tailored diagonalization circuits

name

runtime

advantage

BF

exp.

yields conclusive answer

restr. BF

poly.

–

exh. alg.

exp.

as BF but often faster

restr. alg.

poly.

performs well in practice

naive num.

limited by IQP

CZ count minimization

informed num.

limited by IQP

faster than naive num.

  1. The brute-force (BF) solver loops over all 6n2e choices of (A, Γ) and has an exponential (exp.) runtime in n. When restricted to a polynomial subset of choices for (A, Γ), the restricted (restr.) BF solver has a polynomial (poly.) runtime but a vanishingly low success probability. The exhaustive (exh.) algebraic (alg.) solver loops over all 2e subgraphs of Γcon and, in the worst case, through all 4n intersections in Fig. 2; it can be fast as it terminates prematurely when either a solution \({\boldsymbol{\lambda }}\in {\mathcal{L}}\) is found or \({\mathcal{L}}=\varnothing\) is concluded. The restricted algebraic solver loops over a random selection of s(n) = poly(n) subgraphs of Γcon and employs a cutoff c(n) as described in the methods section; it performs well in practice (see Fig. 5) and has poly. runtime by design. The naive numerical (num.) solver leverages an IQP solver with binary variables a, Γ and integer slack variables νi,j as described in the main text; it is not implemented here due to the large number of variables. The informed num. solver reduces the number of variables by leveraging our knowledge about the geometry of Eq. (6); this solver loops over a random selection of s(n) = poly(n) subgraphs, for each of which it solves an IQP problem with binary variables λj and integer slack variables μi as described in the main text. For both numerical solvers, the runtime is limited by the leveraged IQP solver which can be fast in practice.
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