The one-body tensor \(\kappa\).
+np.ndarray
+The diagonal Coulomb matrices.
+np.ndarray
+The orbital rotations.
+np.ndarray
+The constant.
+float
+Whether the Hamiltonian is in the “Z” representation rather -than the “number” representation.
+Whether the Hamiltonian is in the “Z” representation +rather than the “number” representation.
+bool
+The number of spatial orbitals.
The one-body tensor.
+np.ndarray
+The two-body tensor.
+np.ndarray
+The constant.
+float
+The number of spatial orbitals.
The one-body tensor \(\kappa\).
+np.ndarray
+The diagonal Coulomb matrices.
+np.ndarray
+The orbital rotations.
+np.ndarray
+The constant.
+float
+Whether the Hamiltonian is in the “Z” representation rather -than the “number” representation.
+Whether the Hamiltonian is in the “Z” representation +rather than the “number” representation.
+bool
+The one-body tensor.
+np.ndarray
+The two-body tensor.
+np.ndarray
+The constant.
+float
+
-Fidelity of Trotter-evolved state with exact state: 0.9999906233109686
+Fidelity of Trotter-evolved state with exact state: 0.9999906233109683
As mentioned above, ffsim already includes functionality for Trotter simulation of double-factorized Hamiltonians. The implementation in ffsim includes higher-order Trotter-Suzuki formulas. The first-order asymmetric formula that we just implemented corresponds to order=0 in ffsim’s implementation. order=1 corresponds to the first-order symmetric (commonly known as the second-order) formula, order=2 corresponds to the second-order symmetric (fourth-order) formula, and so on.
-Fidelity of Trotter-evolved state with exact state: 0.9999906233109686
+Fidelity of Trotter-evolved state with exact state: 0.9999906233109683
A higher order formula achieves a higher fidelity with fewer Trotter steps:
@@ -435,7 +435,7 @@
-Fidelity of Trotter-evolved state with exact state: 0.9999999336740067
+Fidelity of Trotter-evolved state with exact state: 0.9999999336740071
-converged SCF energy = -77.4456267643962
+converged SCF energy = -77.4456267643961
CASCI E = -77.6290254326717 E(CI) = -3.57322412553862 S^2 = 0.0000000
-E(CCSD) = -77.49387212754471 E_corr = -0.04824536314851481
-Energy at initialialization: -77.46975600021685
+E(CCSD) = -77.49387212754462 E_corr = -0.04824536314851294
+Energy at initialialization: -77.46975600021707
To facilitate variational optimization of the ansatz, UCJOperator implements methods for conversion to and from a vector of real-valued parameters. The precise relation between a parameter vector and the matrices of the UCJ operator is somewhat complicated. In short, the parameter vector stores the entries of the UCJ matrices in a non-redundant way (for the orbital rotations, the parameter vector actually stores the entries of their generators.)
-Energy at initialialization: -75.68366174447618
+Energy at initialialization: -75.68366174447621
FermionOperator({
- (cre_b(1), des_b(5), cre_a(4)): 1+1j,
(cre_a(0), des_a(3)): 0.5,
- (cre_a(3), des_a(0)): -0.25
+ (cre_a(3), des_a(0)): -0.25,
+ (cre_b(1), des_b(5), cre_a(4)): 1+1j
})
-'FermionOperator({((True, True, 1), (False, True, 5), (True, False, 4)): 1+1j, ((True, False, 0), (False, False, 3)): 0.5+0j, ((True, False, 3), (False, False, 0)): -0.25+0j})'
+'FermionOperator({((True, False, 0), (False, False, 3)): 0.5+0j, ((True, False, 3), (False, False, 0)): -0.25+0j, ((True, True, 1), (False, True, 5), (True, False, 4)): 1+1j})'
FermionOperators support arithmetic operations. Note that when multiplying a FermionOperator by a scalar, the scalar must go on the left, i.e. 2 * op and not op * 2.
FermionOperator({
- (cre_b(1), des_b(5), cre_a(4), des_a(3), des_b(3)): -0.25-0.25j,
- (cre_b(2)): 0-0.25j,
+ (cre_a(0), des_a(3), cre_b(2)): 0+0.5j,
(cre_a(0), des_a(3)): 1,
+ (cre_a(3), des_a(0)): -0.5,
(cre_b(1), des_b(5), cre_a(4), cre_b(2)): -1+1j,
- (cre_a(3), des_a(0), des_a(3), des_b(3)): 0.0625,
+ (cre_a(3), des_a(0), cre_b(2)): 0-0.25j,
+ (cre_b(2)): 0-0.25j,
+ (cre_b(1), des_b(5), cre_a(4)): 2+2j,
(des_a(3), des_b(3)): 0.0625,
+ (cre_a(3), des_a(0), des_a(3), des_b(3)): 0.0625,
(cre_a(0), des_a(3), des_a(3), des_b(3)): -0.125,
- (cre_b(1), des_b(5), cre_a(4)): 2+2j,
- (cre_a(0), des_a(3), cre_b(2)): 0+0.5j,
- (cre_a(3), des_a(0)): -0.5,
- (cre_a(3), des_a(0), cre_b(2)): 0-0.25j
+ (cre_b(1), des_b(5), cre_a(4), des_a(3), des_b(3)): -0.25-0.25j
})
FermionOperator({
- (cre_b(1), des_b(5), cre_a(4), des_a(3), des_b(3)): -1+1j,
- (cre_b(2)): -5,
+ (cre_a(0), des_a(3), cre_b(2)): 2,
(cre_a(0), des_a(3)): 0-6j,
+ (cre_a(3), des_a(0)): 0+3j,
(cre_b(1), des_b(5), cre_a(4), cre_b(2)): 4+4j,
- (cre_a(3), des_a(0), des_a(3), des_b(3)): 0-0.25j,
+ (cre_a(3), des_a(0), cre_b(2)): -1,
+ (cre_b(2)): -5,
+ (cre_b(1), des_b(5), cre_a(4)): 12-12j,
(des_a(3), des_b(3)): 0-1.25j,
+ (cre_a(3), des_a(0), des_a(3), des_b(3)): 0-0.25j,
(cre_a(0), des_a(3), des_a(3), des_b(3)): 0+0.5j,
- (cre_b(1), des_b(5), cre_a(4)): 12-12j,
- (cre_a(0), des_a(3), cre_b(2)): 2,
- (cre_a(3), des_a(0)): 0+3j,
- (cre_a(3), des_a(0), cre_b(2)): -1
+ (cre_b(1), des_b(5), cre_a(4), des_a(3), des_b(3)): -1+1j
})
FermionOperator({
(cre_b(1), cre_a(4), des_b(5), des_b(3), des_a(3)): -1+1j,
- (cre_b(1), cre_a(4), des_b(5)): -12+12j,
- (cre_a(3), des_b(3), des_a(3), des_a(0)): 0+0.25j,
- (cre_a(0), des_a(3)): 0-6j,
(cre_b(2), cre_a(0), des_a(3)): 2,
- (des_b(3), des_a(3)): 0+1.25j,
+ (cre_b(2)): -5,
+ (cre_b(2), cre_b(1), cre_a(4), des_b(5)): 4+4j,
(cre_b(2), cre_a(3), des_a(0)): -1,
+ (des_b(3), des_a(3)): 0+1.25j,
+ (cre_b(1), cre_a(4), des_b(5)): -12+12j,
(cre_a(3), des_a(0)): 0+3j,
- (cre_b(2), cre_b(1), cre_a(4), des_b(5)): 4+4j,
- (cre_b(2)): -5
+ (cre_a(0), des_a(3)): 0-6j,
+ (cre_a(3), des_b(3), des_a(3), des_a(0)): 0+0.25j
})
@@ -271,11 +271,11 @@
-array([0. +0.j , 0. +0.j ,
- 0. +0.j , 0. +0.j ,
- 0.14019459+0.05152728j, 0. +0.j ,
- 0. +0.j , 0. +0.j ,
- 0. +0.j ])
+array([ 0. +0.j , 0. +0.j ,
+ 0. +0.j , 0. +0.j ,
+ -0.11830879-0.02112505j, 0. +0.j ,
+ 0. +0.j , 0. +0.j ,
+ 0. +0.j ])
It can also be passed into most linear algebra routines in scipy.sparse.linalg.