A simulation scheme first encodes fermionic states in qubits, then acts with the qubit operator representing the fermionic operator (obtained by the associated transformation), then inverts the encoding to obtain the resultant fermionic state. The criterion for a successful simulation scheme is that this procedure reproduces the action of the fermionic operator, i.e., that Path 1 is equivalent to Path 2, for all basis states—in other words, that this diagram commutes.
The matrix β n that transforms occupation number basis vectors of length n into the Bravyi-Kitaev basis. β1 is a (1 × 1) matrix with a single entry of 1. Subsequent iterations of the matrix that act on occupation number basis vectors of length 2 x are constructed by taking and then filling in the top row of the first quadrant of this matrix with 1's. β n for 2 x < n < 2 x+1 is just the (n × n) segment of that includes b 0 through b n−1. The recursion pattern for the inverse transformation matrix is also shown. An entry of 1 in row b i , column f j means that b i is a partial sum including f j .
A demonstration of how to exponentiate tensor products of Pauli matrices. First, the parity of the four qubits is computed with CNOT gates, and then a single-qubit phase rotation R z is applied. Then, we uncompute the parity with three further CNOT gates.
A demonstration of how to exponentiate tensor products of Pauli-X and Y matrices. First, the qubits are put in the correct basis by the application of R x or Hadamard gates. Then, the parity of the four qubits is computed with CNOT gates, and then a single-qubit phase rotation R z is applied. Then, we uncompute the parity with more CNOT gates, and finally change back to the computational (Z) basis.
The approximation to the ground state eigenvalue, for both the Bravyi-Kitaev Hamiltonian (squares) and Jordan-Wigner Hamiltonian (circles), as a function of the number of gates required. The solid curves are the first order Suzuki-Trotter approximations, the dotted-dashed second order, the dotted third order, and the dashed fourth. The dotted horizontal line represents the true eigenvalue, while the solid lines above and below represent the bounds for chemical precision.
The approximation to the ground state eigenvalue, for both the Bravyi-Kitaev Hamiltonian (squares) and Jordan-Wigner Hamiltonian (circles), as a function of the number of gates required. The solid curve is the naïve first order Suzuki-Trotter approximation, while the dashed curve is the result from alternating the noncommuting terms. The dotted horizontal line represents the true eigenvalue, while the solid lines above and below represent the bounds for chemical precision. The ground state eigenvalue of the Bravyi-Kitaev Hamiltonian can be approximated to chemical precision with 222 gates, while it takes 328 gates to do the same for the Jordan-Wigner Hamiltonian.
The gate savings of using the Bravyi-Kitaev method instead of the Jordan-Wigner method, as a function of the precision in the estimate of the ground state eigenvalue for the first four orders of Suzuki-Trotter formulae. The vertical line is the threshold error for chemical precision. The triangle data points are first order, the squares second, the circles third, and the diamonds fourth.
The five classes of Hermitian second quantized operators that appear in electronic Hamiltonians. In general, the overlap integrals h ij and h ijkl may be complex.
The algebraic expressions for general products of the form in the Bravyi-Kitaev basis. These expressions vary in form depending on the parity of the indices i and j, as well as on the overlaps between the parity and update sets of the indices. The notation is shorthand to indicate that the operator O does not operate on the qubits in the set S (i.e., ).
The overlap integrals for molecular hydrogen in a minimal basis. The integrals were obtained through a restricted Hartree-Fock calculation in the PyQuante quantum chemistry package at an internuclear separation of 1.401000 a.u. (7.414 × 10−11 m).
The number of single-qubit gates and CNOT gates required to exponentiate subsets of the electronic Hamiltonian for the hydrogen molecule, represented in terms of spin variables through either the Bravyi-Kitaev transformation or the Jordan-Wigner transformation.
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