Table 2 Examples of HT diagonalization circuits and their performance in the context of quantum chemistry

From: Hardware-tailored diagonalization circuits

n

20

40

60

80

100

120

M

7.2 k

117 k

595 k

1.9 M

4.6 M

9.6 M

m

191

781

1771

3161

4951

7141

\({N}_{{\rm{TPB}}}^{{\rm{circs}}}\)

19

24

25

28

30

32

  1. Performance of an educated guess (A, Γ) as measured by the number m of jointly-HT-diagonalizable n-qubit Pauli operators P1, …, Pm occurring in a hydrogen chain Hamiltonian \(O=\mathop{\sum }\nolimits_{i = 1}^{M}{c}_{i}{P}_{i}\) for which (A, Γ) solves Eq. (6). The guess corresponds to the constant-depth circuit \({H}^{\otimes n}(\mathop{\prod }\nolimits_{k = 1}^{n/4}{{\rm{CZ}}}_{4k+1,4k+2}{{\rm{CZ}}}_{4k+2,4k+3}){H}^{\otimes n}\) which makes use of n/2 two-qubit gates and is tailored to a linear hardware connectivity. To diagonalize the other M − m Pauli operators in O, one would need to find additional HT circuits, e.g., by making educated guesses based on careful inspection of the circuits in Tab. VII of SM Sec. XIII. The performance of such guesses can be easily assessed by checking for how many of the remaining Pauli operators Eq. (6) is fulfilled. The advantage of the here-presented HT circuit over tensor product bases is quantified by \({N}_{{\rm{TPB}}}^{{\rm{circs}}}\), which is the number of circuits needed to diagonalize the same operators P1, …, Pm if two-qubit gates are forbidden. See methods for details about Hamiltonians.
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