This module provides the Jordan-Wigner transformation and symbolic derivation of circuit templates for second quantized interaction terms. It is essentially a fully automated version of the calculations from

• James D. Whitfield, Jacob Biamonte, and Alán Aspuru-Guzik. "Simulation of electronic structure Hamiltonians using quantum computers." Molecular Physics 109(5):735–750, 2011. See also http://arxiv.org/abs/1001.3855v3.

For a given tuple of orbital indices, (p,q) in case of one-electron interactions, or (p,q,r,s) in case of two-electron interactions, we first calculate the Jordan-Wigner transformation of the second quantized hermitian interaction terms

a_p^†a_p,

a_p^†a_q + a_q^†a_p,

a_p^†a_q^†a_qa_p,

a_p^†a_q^†a_ra_s + a_s^†a_r^†a_qa_p.

Next, we decompose each operator into a linear combination H = λ₁M₁ + ... + λ_nM_n of mutually commuting hermitian tensors. At this point, each summand M_j in the linear combination will be a tensor product of the following operators (not necessarily in this order):

• an even number (possibly zero) of Pauli X operators;
• an even number (possibly zero) of Pauli Y operators;
• zero or more Pauli Z operators, and
• zero or more D operators, where D = σ⁻σ⁺ = (I−Z)/2.

Note that there may be zero terms in the summation; this happens, for example, for a_p^†a_p^†a_ra_s, because two electrons cannot occupy the same spin orbital due to their fermionic nature. In this case, H = 0 and e^-iθH = I.

Next, we calculate e^-iθH. Because the summands M_j commute, we can exponentiate each summand separately, using the formula e^-iθH = e^{-iθλ₁M₁}⋯e^-iθλ_nM_n.

We then generate the circuit for e^-iθλ_jM_j by applying a sequence of basis changes until the problem is reduced to a controlled rotation. The basis changes are, in this order:

1. Change each Pauli X operator in M_j to a Pauli Z operator, and apply a Hadamard basis change to the corresponding qubit. This uses the relation HXH = Z.
2. Change each Pauli Y operator in M_j to a Pauli Z operator, and apply a Y basis change to the corresponding qubit. Note: the Y basis change operator is defined in [Whitfield et al.] as R_x(-π/2) = (I+iX)/√2, or equivalently Y = SHS, and satisfies Y^†YY = Z. It should not be confused with the Pauli Y operator.
3. If the operator M_j contains one or more Pauli Z operators (including those obtained in steps 1 and 2), then do a basis change by a cascade of controlled-not gates to reduce this to a single Z operator. This uses the relation CNot (Z⊗Z) CNot = I⊗Z.

After these basis changes, the operator M_j consists of exactly zero or one Pauli Z operator, together with zero or more D operators. To see how to translate this into a controlled rotation, note that for any operator A, we have

Therefore, each D operator in M_j turns into a control after exponentiation. The final rotation is then computed by distinguishing two cases:

• If M_j contains a Pauli Z operator, then use the relation e^-iθZ = R_z(2θ). In this case, the circuit for e^-iθM_j is a controlled R_z(2θ) gate in the position of the Z operator, with zero or more controls in the positions of any D operators.
• If M_j does not contain a Pauli Z operator, then the operation to be performed is a phase change e^-iθ, controlled by the qubits in the positions of the D operators. Note that there must be at least one D operator in this case. Also note that a controlled e^-iθ gate is identical to a T(θ) gate.

As outlined above, the functions in this module generate each circuit from first principles, based on the Jordan-Wigner representation of operators and on algebraic transformations. They do not rely on pre-fabricated circuit templates.

Based on the automated calculations provided by this module, we have found small typos in the 5 templates provided by [Whitfield et al.] (Table 3, or Table A1 in the arXiv version).

• The template for the number-excitation operator is missing a control on its rotation gate.
• In the template for the Coulomb operator, the angles are wrong. Moreover, this program finds a simpler template.
• In the template for the double excitation operator, the angles are wrong; they should be ±θ/4 instead of θ.

The corrected templates generated by our code are as follows:

• Number operator h_pp a_p^†a_p.

• Excitation operator h_pq a_p^†a_q.

• Coulomb and exchange operators h_pqqp a_p^†a_q^†a_qa_p.

• Number-excitation operator h_pqqr (a_p^†a_q^†a_qa_r + a_r^†a_q^†a_qa_p). The sign of ±θ depends on the relative ordering of the indices p,q,r.

• Double excitation operator h_pqrs (a_p^†a_q^†a_ra_s + a_s^†a_r^†a_qa_p). The sign of ±θ/4 in each of the eight terms depends on the relative ordering of the indices p,q,r,s.

As noted above, our algorithm found the following template for the Coulomb operator a_p^†a_q^†a_qa_p:

This is simpler than the template given in [Whitfield et al.], even after one accounts for the cost of decomposing the additional controlled T(θ) gate into elementary gates. However, an equivalent circuit can also be given that is more similar to the one in [Whitfield et al.] (but with corrected rotation angles):

We call this the "orthodox" template, because it is closer to the one specified by Whitfield et al. The program will use the orthodox template if the command line option --orthodox is given, and it will use the simplified template otherwise.

The next two functions provide the Jordan-Wigner representation of (Fock-space) annihilation and creation operators.

We provide two functions, accessible via command line options, that allow the user to display individual templates.