Variational Quantum Linear Solver (VQLS) with Linear Combination of Unitaries (LCU) Block Encoding
The Variational Quantum Linear Solver (VQLS) is a quantum algorithm that harnesses the power of Variational Quantum Eigensolvers (VQE) to solve systems of linear equations efficiently:
-
Input: A matrix \(\textbf{A}\) and a known vector \(|\textbf{b}\rangle\).
-
Output: An approximation of a normalized solution \(|x\rangle\) proportional to \(|\textbf{x}\rangle\), satisfying the equation \(\textbf{A} |\textbf{x}\rangle = |\textbf{b}\rangle\).
While the output of VQLS mirrors that of the HHL Quantum Linear-Solving Algorithm, VQLS distinguishes itself by its ability to operate on Noisy Intermediate-Scale Quantum (NISQ) computers. In contrast, HHL necessitates more robust quantum hardware and a larger qubit count, despite offering a faster computation speedup.
This tutorial covers an implementation example of a Variational Quantum Linear Solver [1] using block encoding. In particular, we use linear combinations of unitaries (LCUs) for the block encoding.
As with all variational algorithms, the VQLS is a hybrid algorithm in which we apply a classical optimization on the results of a parametrized (ansatz) quantum program.
Building the Algorithm with Classiq
Quantum Part: Variational Circuit
Given a block encoding of the matrix A:
we can prepare the state
We can approximate the solution \(|x\rangle\) with a variational quantum circuit, i.e., a unitary circuit \(V\), depending on a finite number of classical real parameters \(w = (w_0, w_1, \dots)\):
Our objective is to address the task of preparing a quantum state \(|x\rangle\) such that \(A |x\rangle\) is proportional to \(|b\rangle\); or, equivalently, ensuring that
The state \(|b\rangle\) arises from a unitary operation \(U_b\) applied to the ground state of \(n\) qubits; i.e.,
To maximize the overlap between the quantum states \(|\Psi\rangle\) and \(|b\rangle\), we optimize the parameters, defining a cost function:
At a high level, the above could be implemented as follows:
We construct a quantum model as depicted in the figure below. When measuring the circuit in the computational basis, the probability of finding the system qubits in the ground state (given the ancillary qubits measured in their ground state) is \(|\langle 0 | U_b^\dagger |\Psi \rangle|^2 = |\langle b | \Psi \rangle|^2.\)
To block encode a Variational Quantum Linear Solver as explained above, we can define a high-level block_encoding_vqls function as follows:
from typing import List
import numpy as np
from classiq import *
@qfunc
def block_encoding_vqls(
ansatz: QCallable,
block_encoding: QCallable,
prepare_b_state: QCallable,
) -> None:
ansatz()
block_encoding()
invert(lambda: prepare_b_state())
From here, we only need to define ansatz, block_encoding, and prepare_b_state to fit the specific example above. Now we are ready to build our model, synthesize it, execute it, and analyze the results.
Classical Part: Finding Optimal Parameters
To estimate the overlap of the ground state with the post-selected state, we could directly make use of the measurement samples. However, since we want to optimize the cost function, it is useful to express everything in terms of expectation values through Bayes\' theorem:
To evaluate the conditional probability from the above equation, we construct the following utility function to operate on the measurement results:
To variationally solve our linear problem, we define the cost function \(C = 1- |\langle b | \Psi \rangle|^2\) that we are going to minimize. As explained above, we express it in terms of expectation values through Bayes\' theorem.
We define a classical function that gets the quantum program, minimizes the cost function using the COBYLA optimizer, and returns the optimal parameters.
import random
import matplotlib.pyplot as plt
from scipy.optimize import minimize
class VqlsOptimizer:
def __init__(
self, qprog, ansatz_param_count, ansatz_var_name, aux_var_name, exe_prefs=None
):
self.qprog = qprog
self.ansatz_param_count = ansatz_param_count
self.ansatz_var_name = ansatz_var_name
self.aux_var_name = aux_var_name
if exe_prefs is None:
self.es = ExecutionSession(qprog)
else:
self.es = ExecutionSession(qprog, exe_prefs)
self.intermediate = {}
def get_cond_prop(self, res):
aux_prob_0 = 0
all_prob_0 = 0
for s in res:
if s.state[self.aux_var_name] == 0:
aux_prob_0 += s.shots
if s.state[self.ansatz_var_name] == 0:
all_prob_0 = s.shots
return all_prob_0 / aux_prob_0
def my_cost(self, params):
results = self.es.sample(params)
return 1 - self.get_cond_prop(
results.parsed_counts_of_outputs([self.ansatz_var_name, self.aux_var_name])
)
def f(self, x):
cost = self.my_cost(
{"params_" + str(k): x[k] for k in range(self.ansatz_param_count)}
)
self.intermediate[tuple(x)] = cost
return cost
def optimize(self):
random.seed(1000)
self._out = out = minimize(
self.f,
x0=[
float(random.randint(0, 3000)) / 1000
for i in range(0, self.ansatz_param_count)
],
method="COBYLA",
options={"maxiter": 2000},
)
print(out)
self._out_f = out_f = [out["x"][0 : self.ansatz_param_count]]
print(out_f)
plt.plot(
[l for l in range(len(self.intermediate))], list(self.intermediate.values())
)
return {
"params_" + str(k): list(self.intermediate.keys())[-1][k]
for k in range(self.ansatz_param_count)
}
Once the optimal variational weights w are found, we
can generate the quantum state \(|x\rangle\). By measuring \(|x\rangle\) in
the computational basis we can estimate the probability of each basis
state.
Example Using LCU Block Encoding
We treat a specific example based on a system of three qubits:
where \(Z_j, X_j, H_j\) represent the Pauli \(Z\), Pauli \(X\), and Hadamard gates applied to the qubit with index \(j\).
To block encode the matrix A we use the LCU method. This can be done with the lcu_paulis library function. Note that this function can get a unnormalized Pauli operator, thus we calculate the normalization factor for the post-process analysis.
The LCU quantum circuit looks as follows:
pauli_terms_structs = (
0.55 * Pauli.I(0)
+ 0.225 * Pauli.I(0) * Pauli.Z(1) * Pauli.I(2)
+ 0.225 * Pauli.I(0) * Pauli.I(1) * Pauli.Z(2)
)
normalization = sum([p.coefficient for p in pauli_terms_structs.terms])
num_system_qubits = pauli_terms_structs.num_qubits
num_ancila_qubits = (len(pauli_terms_structs.terms) - 1).bit_length()
ansatz_param_count = 9
Fixed Hardware Ansatz
Let's consider our ansatz \(V(w)\), such that
This allows us to "search" the state space by varying a set of parameters, \(w\).
The ansatz that we use for this three-qubit system implementation takes in nine parameters as defined in the apply_fixed_3_qubit_system_ansatz function:
@qfunc
def apply_ry_on_all(params: CArray[CReal], io: QArray[QBit]):
repeat(count=io.len, iteration=lambda index: RY(params[index], io[index]))
@qfunc
def apply_fixed_3_qubit_system_ansatz(
angles: CArray[CReal], system_qubits: QArray[QBit]
):
apply_ry_on_all([angles[0], angles[1], angles[2]], system_qubits)
repeat(
count=(system_qubits.len - 1),
iteration=lambda index: CZ(system_qubits[0], system_qubits[index + 1]),
)
CZ(system_qubits[1], system_qubits[2])
apply_ry_on_all([angles[3], angles[4], angles[5]], system_qubits)
repeat(
count=(system_qubits.len - 1),
iteration=lambda index: CZ(
system_qubits[system_qubits.len - 1], system_qubits[index]
),
)
CZ(system_qubits[1], system_qubits[0])
apply_ry_on_all([angles[6], angles[7], angles[8]], system_qubits)
To view our ansatz implementation we create a model and view the synthesis result:
@qfunc
def main(
params: CArray[CReal, ansatz_param_count], system_qubits: Output[QArray[QBit]]
):
allocate(3, system_qubits)
apply_fixed_3_qubit_system_ansatz(params, system_qubits)
qprog_1 = synthesize(main)
show(qprog_1)
Quantum program link: https://platform.classiq.io/circuit/30uPGNDLGIpPnlMlpcdLZwvmWqq
This is called a fixed hardware ansatz in that the configuration of quantum gates remains the same for each run of the circuit, and all that changes are the parameters. Unlike the QAOA ansatz, it is not composed solely of Trotterized Hamiltonians. The applications of \(Ry\) gates allow us to search the state space, while the \(CZ\) gates create "interference" between the different qubit states.
Running the VQLS
Now, we can define the main function: we call block_encoding_vqls with the arguments of our specific example.
@qfunc
def main(
params: CArray[CReal, ansatz_param_count],
ancillary_qubits: Output[QNum[num_ancila_qubits]],
system_qubits: Output[QNum[num_system_qubits]],
):
allocate(ancillary_qubits)
allocate(system_qubits)
block_encoding_vqls(
ansatz=lambda: apply_fixed_3_qubit_system_ansatz(params, system_qubits),
block_encoding=lambda: lcu_pauli(
operator=pauli_terms_structs, data=system_qubits, block=ancillary_qubits
),
prepare_b_state=lambda: apply_to_all(H, system_qubits),
)
Constructing the model, synthesizing, and executing on the Classiq simulator:
qprog_2 = synthesize(main)
show(qprog_2)
Quantum program link: https://platform.classiq.io/circuit/30uPHO6okDxm9NGwfx2PHftX009
write_qmod(main, name="vqls_with_lcu", decimal_precision=15, symbolic_only=False)
We run the classical optimizer to get the optimal parameters:
backend_preferences = ClassiqBackendPreferences(backend_name="simulator_statevector")
execution_preferences = ExecutionPreferences(
num_shots=204800, backend_preferences=backend_preferences
)
optimizer = VqlsOptimizer(
qprog_2,
ansatz_param_count,
"system_qubits",
"ancillary_qubits",
execution_preferences,
)
optimal_params = optimizer.optimize()
message: Optimization terminated successfully.
success: True
status: 1
fun: 0.09572291736048333
x: [ 2.183e+00 3.097e+00 9.850e-01 2.482e+00 3.129e+00
6.554e-01 1.201e+00 2.307e+00 9.622e-01]
nfev: 100
maxcv: 0.0
[array([2.18263061, 3.09658041, 0.98498817, 2.48189263, 3.12912539,
0.65539631, 1.20116677, 2.30650636, 0.96216622])]
Measuring the Quantum Solution
Finally, we can apply the optimal parameters to measure the quantum results for \(\vec{x}\):
@qfunc
def main(io: Output[QNum[num_system_qubits]]):
allocate(io)
apply_fixed_3_qubit_system_ansatz(list(optimal_params.values()), io)
qprog_3 = synthesize(main)
with ExecutionSession(qprog_3, execution_preferences) as es:
results = es.sample()
df = results.dataframe
amplitudes = np.zeros(2**num_system_qubits).astype(complex)
amplitudes[df.io] = df.amplitude
# Preprocessed quantum solution: we know the solution is real, and that the last point is positive
global_phase = np.angle(amplitudes[-1])
amplitudes = np.real(amplitudes / np.exp(1j * global_phase))
if (
amplitudes[-1] < 0
): # we can extract the solution up to a sign, align with the expected
amplitudes *= -1
print(amplitudes)
[0.10545817 0.09824632 0.23525783 0.25226615 0.199245 0.30080137
0.46216235 0.71865688]
probabilities = amplitudes**2
Comparing to the Classical Solution
Since the specific problem considered in this tutorial has a small size, we can also solve it in a classical way and then compare the results with our quantum solution.
We use the explicit matrix representation in terms of numerical NumPy arrays.
Classical calculation:
A_num = hamiltonian_to_matrix(pauli_terms_structs) / normalization
b = np.ones(8) / np.sqrt(8)
Calculating the classical \(\vec{x}\) that solves the equation:
A_inv = np.linalg.inv(A_num)
x = np.dot(A_inv, b)
classical_probs = np.real((x / np.linalg.norm(x))) ** 2
classical_probs
array([0.00464634, 0.00464634, 0.0153598 , 0.0153598 , 0.0153598 ,
0.0153598 , 0.46463405, 0.46463405])
To compare the classical to the quantum results we compute the post-processing by applying \(A\) to our optimal vector \(|\psi\rangle_o\), normalizing it, then calculating the inner product squared of this vector and the solution vector, \(|b\rangle\)! We can put this all into code as follows:
print(
"overlap =",
(b.dot(A_num.dot(amplitudes) / (np.linalg.norm(A_num.dot(amplitudes))))) ** 2,
)
overlap = (0.9040151946512618+0j)
import matplotlib.pyplot as plt
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(7, 4))
ax1.bar(np.arange(0, 2**num_system_qubits), classical_probs, color="blue")
ax1.set_xlim(-0.5, 2**num_system_qubits - 0.5)
ax1.set_xlabel("Vector space basis")
ax1.set_title("Classical probabilities")
ax2.bar(np.arange(0, 2**num_system_qubits), probabilities, color="gold")
ax2.set_xlim(-0.5, 2**num_system_qubits - 0.5)
ax2.set_xlabel("Hilbert space basis")
ax2.set_title("Quantum probabilities")
plt.show()
The classical cost function basically agrees with the algorithm result.
References
[1]: Bravo-Prieto et al.,Variational Quantum Linear Solver, 2020.
[2]: Robin Kothari. \"Efficient algorithms in quantum query complexity.\" PhD thesis, University of Waterloo, 2014.