Parameter-Shift Rule
Let us describe below how to act the parameter-shift rules on the UCC circuit ansatz. In the present work, we use a UCC ansatz truncated at the single and double excitation levels with a single Trotter step. Off course, one can go further to triple, quadruple and further excitation, however our goal in this section is to show how to link the parameter shift rule on a unitary coupled cluster gates in a circuit, which is the most important key point. Once these approximations are performed, each exponential of a fermionic excitation operator can be directly implemented as a sequence of gates. This in practice, can be done by using the standard CNOT staircase method (described in the previous chapter). The unitary transformation carried out by the circuit can thus be broken down into a product of local unitaries
$$ U\left(x;\bm{\theta}\right)=U_N\left(\bm{\theta}_N\right)U_{N-1}\left(\bm{\theta}_{N-1}\right)\cdots U_i\left(\bm{\theta}_i\right) \cdots U_1\left(\bm{\theta}_1\right)U_0\left(x\right) $$Each of these gates is unitary, and therefore must have the form
$$ U_j\left(\gamma_j\right)=\exp{\left(i\gamma_jH_j\right)} $$where $H_j$ is a Hermitian operator which generates the gate and $\gamma_j$ is the gate parameter. When using CNOT staircase method, $U_j\left(\gamma_j\right)$ are typically equivalent to $R_z(\bm{\theta})$ gates. Since for any type of coupled cluster excitations, the circuit, in practice should be composed of multiple parameterized $R_z(\bm{\theta})$ gates, and some other non-parameterized gates such as (CNOT, Hadamard etc.). There should be then a tool how to make the quantum circuit function to compute gradient ($\nabla{\bm{\theta}_i}{f}(x;\bm{\theta}_i$)) for a certain $\bm{\theta}_i$:\ in fact any gates applied before gate $i$ into the initial state, can be expressed as:
$$ \left|\psi_{i-1}\right\rangle=U_{i-1}\left(\bm{\theta}_{i-1}\right)\cdots U_1\left(\bm{\theta}_1\right)U_0\left(x\right)\left|0\right\rangle $$Similarly, any gates applied after gate $i$ are combined with the observable, which is the Hamiltonian in our case
$$ \hat{H}_{i+1}=U_N^\dag(\bm{\theta}_N)\cdot U^\dag_{i+1}(\bm{\theta}_{i+1})\hat{H}U_{i+1}(\bm{\theta}_{i+1})\cdots U_N(\bm{\theta}_N) $$With this simplification, the quantum circuit function becomes
$$ f(x;\bm{\theta})=\left\langle\psi_{i-1}\left|U_i^\dag(\bm{\theta}_i)\hat{H}_{i+1}U_i(\bm{\theta}_i)\right|\psi_{i-1}\right\rangle $$now suppose
$$\mathcal{M}_{\bm{\theta}_i}(\hat{H}_{i+1}) = U_i^\dag(\bm{\theta}_i)\hat{H}_{i+1}U_i(\bm{\theta}_i) $$then its gradient takes the form
$$ \nabla_{\bm{\theta}_i}f\left(x;\bm{\theta}\right)=\left\langle\psi_{i-1}\left|\nabla_{\bm{\theta}_i}\mathcal{M}_{\bm{\theta}_i}\left(\hat{H}_{i+1}\right)\right|\psi_{i-1}\right\rangle $$as is seen in the expression above, in terms of the circuit, one can leave all other gates as they are, and only gate
$U_i\left(\bm{\theta}_i\right)$ should be changed
when question comes to differentiate it with respect to the parameter
$\bm{\theta}_i$. Let us now apply the equations above into $R_z$ gate.
Consider a quantum computer with parameterized gates of the form
where $\widehat{Z_i}={\widehat{Z}_i}^ \dag$ is a Pauli operator. The gradient of $R_z$ is
$$ \nabla_{\bm{\theta}_i}R_i\left(\bm{\theta}_i\right)=-\frac{i}{2}\widehat{Z_i}R_i\left(\bm{\theta}_i\right)=-\frac{i}{2}R_i\left(\bm{\theta}_i\right)\widehat{Z_i} $$By substituting this expression into the quantum circuit function $f(x;\bm{\theta})$, we get
$$ \nabla_{\bm{\theta}_i} f\left(x;\bm{\theta}\right) = \frac{i}{2} \left\langle \psi_{i-1} \left| R_i\left(\bm{\theta}_i\right)^\dag \left( Z_i \widehat{H}_{i+1} - \widehat{H}_{i+1} Z_i \right) R_i\left(\bm{\theta}_i\right) \right| \psi_{i-1} \right\rangle $$ $$= \frac{i}{2} \left\langle \psi_{i-1} \left| R_i\left(\bm{\theta}_i\right)^\dag \left[ Z_i, \widehat{H}_{i+1} \right] R_i\left(\bm{\theta}_i\right) \right| \psi_{i-1} \right\rangle$$where $\left[X,Y\right]=XY-YX$ is the commutator. \ \ We now make use of the following mathematical identity for commutators involving Pauli operators
$$ \left[\widehat{Z_i},\hat{H}\right] = -i\left(R_i^\dag\left(\frac{\pi}{2}\right)\hat{H}R_i\left(\frac{\pi}{2}\right) - R_i^\dag\left(-\frac{\pi}{2}\right)\hat{H}R_i\left(-\frac{\pi}{2}\right)\right). $$Substituting this into the previous equation, we obtain the gradient expression
$$ \nabla_{\bm{\theta}_i} f\left(x;\bm{\theta}\right)= \frac{1}{2}\left\langle\psi_{i-1}\left|R_i^\dag\left(\bm{\theta}_i+\frac{\pi}{2}\right){\widehat{H}}_{i+1}R_i\left(\bm{\theta}_i+\frac{\pi}{2}\right)\right|\psi_{i-1}\right\rangle $$ $$ -\frac{1}{2}\left\langle\psi_{i-1}\left|R_i^\dag\left(\bm{\theta}_i-\frac{\pi}{2}\right){\widehat{H}}_{i+1}R_i\left(\bm{\theta}_i-\frac{\pi}{2}\right)\right|\psi_{i-1}\right\rangle $$Finally, this gradient can be rewritten in terms of quantum functions:
$$ \nabla_{\bm{\theta}_i} f\left(x;\bm{\theta}\right)=\frac{1}{2}\left[f\left(x;\bm{\theta}+\frac{\pi}{2}\right)-f\left(x;\bm{\theta}-\frac{\pi}{2}\right)\right] $$We recognize from the comparison of equations above, that these unitaries represent instances of the initial gate, and thus shift the gate’s parameter. This leads to the parameter shift rule for circuit gradients. Below is the code for PMRS for LiH in with CAS method reduced to 6 qubits
1import matplotlib.pyplot as plt
2import numpy as np
3import scipy.optimize
4
5# Assuming the necessary functions are defined:
6# get_optimization_func, get_grad_func, and the variables circ, qpu, H_sp, nqbits, psi0, theta_0, E0
7
8# Initialize dictionaries to store results
9res = {}
10energy_list, fid_list = {}, {}
11
12# Define the optimization methods to use
13methods = ["CG", "BFGS", "COBYLA"]
14
15# Perform optimization for each method
16for method in methods:
17 energy_list[method], fid_list[method] = [], []
18 my_func = get_optimization_func(circ, qpu, H_sp, method, nqbits, psi0, energy_list, fid_list)
19 my_grad = get_grad_func(circ, qpu, H_sp)
20 res[method] = scipy.optimize.minimize(my_func, jac=my_grad, x0=theta_0, method=method, options={"maxiter": 50000, "disp": True})
21
22fig, ax = plt.subplots(figsize=(11, 7))
23steps = np.arange(0, 700)
24
25col = {"CG": "orange", "COBYLA": "firebrick", "BFGS": "darkcyan"}
26
27for method in ["CG", "BFGS", "COBYLA"]:
28 energy_list[method] = np.array(energy_list[method])
29 error = np.maximum(energy_list[method] - E0, 1e-6) # Ensure no values less than 1e-16
30 print(f"The error for the {method} method:", error)
31 ax.plot(error, "-o", color=col[method], label=f" Subtracted energy [{method}]")
32
33 if method == "COBYLA" and error.size > 0:
34 ax.fill_between(steps[:len(error)], 1e-6, 1e-3, color="cadetblue", alpha=0.2, interpolate=True, label="Chemical Accuracy")
35
36
37
38ax.set_xlabel("optimization step")
39ax.set_xlim(0,128)
40
41ax.set_yscale('log')
42ax.legend()
43ax.grid(True, which="both", ls="--")
44plt.savefig("error_H4.pdf")
45 # Grid lines for both major and minor ticks
46
47plt.show()
1fig, ax = plt.subplots(figsize=(11, 7))
2steps = np.arange(0, 700)
3
4col = {"CG": "orange", "COBYLA": "firebrick", "BFGS": "darkcyan"}
5
6for method in ["CG", "BFGS", "COBYLA"]:
7 energy_list[method] = np.array(energy_list[method])
8 error = np.maximum(energy_list[method] - E0, 1e-6) # Ensure no values less than 1e-16
9 print(f"The error for the {method} method:", error)
10 ax.plot(error, "-o", color=col[method], label=f" Subtracted energy [{method}]")
11
12 if method == "COBYLA" and error.size > 0:
13 ax.fill_between(steps[:len(error)], 1e-6, 1e-3, color="cadetblue", alpha=0.2, interpolate=True, label="Chemical Accuracy")
14
15
16
17ax.set_xlabel("optimization step")
18ax.set_xlim(0,128)
19
20ax.set_yscale('log')
21ax.legend()
22ax.grid(True, which="both", ls="--")
23plt.savefig("error_LiH.pdf")
24 # Grid lines for both major and minor ticks
25
26plt.show()
1fig, ax = plt.subplots(figsize=(11, 7))
2steps = np.arange(0, 700)
3
4col = {"CG": "orange", "COBYLA": "firebrick", "BFGS": "darkcyan"}
5
6for method in ["CG", "BFGS", "COBYLA"]:
7 fid_list[method] = np.array(fid_list[method])
8
9 ax.plot(fid_list[method], "-o", color=col[method], label= f"fidelity w.r.t true ground state [{method}]")
10
11
12ax.set_xlabel("optimization step")
13
14ax.legend()
15ax.grid(True, which="both", ls="--")
16plt.savefig("fidelity_H4_.pdf")
17 # Grid lines for both major and minor ticks
18
19plt.show()
The energy according to the chosen optimization algorithm.
