--99.5571707255155
+-99.55717072551582
Time evolution by the Hamiltonian can be computed using expm_multiply:
-/tmp/ipykernel_4332/2190712273.py:2: UserWarning: Trace of LinearOperator not available, it will be estimated. Provide `traceA` to ensure performance.
+/tmp/ipykernel_4308/2190712273.py:2: UserWarning: Trace of LinearOperator not available, it will be estimated. Provide `traceA` to ensure performance.
evolved_vec = scipy.sparse.linalg.expm_multiply(-1j * time * linop, vec)
-Energy at initialialization: -74.20656273321637
+Energy at initialialization: -74.20656273321634
FermionOperator({
- (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_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_b(2)): 0-0.25j,
(cre_b(1), des_b(5), cre_a(4), des_a(3), des_b(3)): -0.25-0.25j,
- (cre_a(3), des_a(0), cre_b(2)): 0-0.25j,
+ (des_a(3), des_b(3)): 0.0625,
+ (cre_a(3), des_a(0)): -0.5,
+ (cre_a(0), des_a(3), des_a(3), des_b(3)): -0.125,
(cre_a(0), des_a(3), cre_b(2)): 0+0.5j,
+ (cre_a(3), des_a(0), cre_b(2)): 0-0.25j,
(cre_a(0), des_a(3)): 1,
- (cre_a(3), des_a(0)): -0.5,
- (cre_b(2)): 0-0.25j,
- (des_a(3), des_b(3)): 0.0625
+ (cre_a(3), des_a(0), des_a(3), des_b(3)): 0.0625,
+ (cre_b(1), des_b(5), cre_a(4), cre_b(2)): -1+1j,
+ (cre_b(1), des_b(5), cre_a(4)): 2+2j
})
FermionOperator({
- (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_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_b(2)): -5,
(cre_b(1), des_b(5), cre_a(4), des_a(3), des_b(3)): -1+1j,
- (cre_a(3), des_a(0), cre_b(2)): -1,
+ (des_a(3), des_b(3)): 0-1.25j,
+ (cre_a(3), des_a(0)): 0+3j,
+ (cre_a(0), des_a(3), des_a(3), des_b(3)): 0+0.5j,
(cre_a(0), des_a(3), cre_b(2)): 2,
+ (cre_a(3), des_a(0), cre_b(2)): -1,
(cre_a(0), des_a(3)): 0-6j,
- (cre_a(3), des_a(0)): 0+3j,
- (cre_b(2)): -5,
- (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_b(1), des_b(5), cre_a(4), cre_b(2)): 4+4j,
+ (cre_b(1), des_b(5), cre_a(4)): 12-12j
})
FermionOperator({
- (cre_a(3), des_a(0)): 0+3j,
- (cre_b(2), cre_b(1), cre_a(4), des_b(5)): 4+4j,
- (des_b(3), des_a(3)): 0+1.25j,
- (cre_a(3), des_b(3), des_a(3), des_a(0)): 0+0.25j,
- (cre_b(1), cre_a(4), des_b(5), des_b(3), des_a(3)): -1+1j,
- (cre_b(2), cre_a(0), des_a(3)): 2,
+ (cre_b(2)): -5,
(cre_a(0), des_a(3)): 0-6j,
+ (cre_b(1), cre_a(4), des_b(5), des_b(3), des_a(3)): -1+1j,
(cre_b(2), cre_a(3), des_a(0)): -1,
- (cre_b(2)): -5,
- (cre_b(1), cre_a(4), des_b(5)): -12+12j
+ (cre_b(2), cre_a(0), des_a(3)): 2,
+ (des_b(3), des_a(3)): 0+1.25j,
+ (cre_a(3), des_b(3), des_a(3), des_a(0)): 0+0.25j,
+ (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
})
-array([ 0. +0.j , 0. +0.j ,
- 0. +0.j , 0. +0.j ,
- -0.13117779+0.11588432j, 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 ,
+ 0.07453016+0.10200987j, 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.
-converged SCF energy = -77.4456267643962
+converged SCF energy = -77.4456267643963
-CASCI E = -77.6290254326717 E(CI) = -3.57322412553863 S^2 = 0.0000000
+CASCI E = -77.6290254326717 E(CI) = -3.57322412553862 S^2 = 0.0000000
-E(CCSD) = -77.49387212754468 E_corr = -0.04824536314851544
+E(CCSD) = -77.49387212754476 E_corr = -0.04824536314851349
-Energy at initialization: -77.46975600021707
+Energy at initialization: -77.46975600021699
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.)
-converged SCF energy = -77.4456267643962
+converged SCF energy = -77.4456267643963
Here, mol_hamiltonian is an instance of MolecularHamiltonian, a dataclass that stores the one- and two-body tensors, and df_hamiltonian is an instance of DoubleFactorizedHamiltonian, a dataclass that stores the updated one-body-tensor, diagonal Coulomb matrices, and orbital rotations. In the cell below, we print out the shapes of the tensors describing the original and double-factorized representations.
-Fidelity of evolved state w.r.t. initial state: 0.931506230139706
+Fidelity of evolved state w.r.t. initial state: 0.93150623014006
Now, let’s test our implementation.
@@ -531,7 +531,7 @@
-Fidelity of Trotter-evolved state with exact state: 0.9928527668210921
+Fidelity of Trotter-evolved state with exact state: 0.9928527668214251
The fidelity of the final result can be improved by increasing the number of Trotter steps.
@@ -558,7 +558,7 @@
-Fidelity of Trotter-evolved state with exact state: 0.999932085128466
+Fidelity of Trotter-evolved state with exact state: 0.9999320851287457
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: 0.999932085128466
+Fidelity of Trotter-evolved state with exact state: 0.9999320851287457
A higher order formula achieves a higher fidelity with fewer Trotter steps:
@@ -615,7 +615,7 @@
-Fidelity of Trotter-evolved state with exact state: 0.9999913261305375
+Fidelity of Trotter-evolved state with exact state: 0.999991326130849