How are the measurements transformed into the Z basis for different Pauli terms, and why is this transformation necessary in the context of VQE?

In the context of the Variational Quantum Eigensolver (VQE) implemented using TensorFlow Quantum for 2-qubit Hamiltonians, transforming the measurements into the Z basis for different Pauli terms is a important step in the process. This transformation is necessary to accurately estimate the expectation values of the Hamiltonian's components, which are essential for evaluating the cost function in the VQE algorithm.

Understanding Pauli Terms and the Hamiltonian

A 2-qubit Hamiltonian in quantum mechanics can be expressed as a linear combination of tensor products of Pauli operators. The Pauli operators are $\sigma_x$ , $\sigma_y$ , and $\sigma_z$ , along with the identity operator $I$ . For a 2-qubit system, the Hamiltonian $H$ can be written as:

$H = \sum_{i,j} c_{ij} (\sigma_i \otimes \sigma_j)$

where $\sigma_i$ and $\sigma_j$ are Pauli operators or the identity operator acting on the first and second qubits, respectively, and $c_{ij}$ are real coefficients.

Measurement in Quantum Computing

In quantum computing, measurements are typically performed in the computational basis, also known as the Z basis. This means that the measurement outcomes correspond to the eigenvalues of the $\sigma_z$ operator, which are +1 and -1. However, to estimate the expectation values of Pauli terms that are not in the Z basis (such as $\sigma_x$ or $\sigma_y$ ), we need to transform the state of the qubits such that these measurements can be effectively performed in the Z basis.

Basis Transformation

For each Pauli term in the Hamiltonian, we apply a specific unitary transformation to convert the measurement into the Z basis. Here are the transformations for the Pauli operators:

1. Pauli-X ( $\sigma_x$ ): To measure in the $\sigma_x$ basis, we apply a Hadamard gate $H$ before the measurement. The Hadamard gate transforms the basis as follows:

$H \sigma_z H = \sigma_x$

Therefore, applying $H$ to a qubit before measuring it in the Z basis effectively measures it in the $\sigma_x$ basis.

2. Pauli-Y ( $\sigma_y$ ): To measure in the $\sigma_y$ basis, we apply a sequence of gates: a $S^\dagger$ gate followed by a Hadamard gate. The $S^\dagger$ gate is the inverse of the phase gate $S$ , and it transforms the basis as follows:

$H S^\dagger \sigma_z S H = \sigma_y$

Therefore, applying $S^\dagger$ followed by $H$ to a qubit before measuring it in the Z basis effectively measures it in the $\sigma_y$ basis.

3. Pauli-Z ( $\sigma_z$ ): Measurement in the $\sigma_z$ basis does not require any transformation, as it is already in the Z basis.

Example of Basis Transformation

Consider a Hamiltonian for a 2-qubit system given by:

$H = c_1 (\sigma_x \otimes \sigma_z) + c_2 (\sigma_y \otimes \sigma_y) + c_3 (\sigma_z \otimes I)$

To measure the expectation value of this Hamiltonian, we need to transform each term to the Z basis:

1. For the term $\sigma_x \otimes \sigma_z$ :
– Apply a Hadamard gate $H$ to the first qubit.
– Measure both qubits in the Z basis.

2. For the term $\sigma_y \otimes \sigma_y$ :
– Apply $S^\dagger H$ to both qubits.
– Measure both qubits in the Z basis.

3. For the term $\sigma_z \otimes I$ :
– No transformation is needed.
– Measure the first qubit in the Z basis.

Implementation in TensorFlow Quantum

In TensorFlow Quantum, these transformations can be implemented using quantum circuits. Here is a pseudocode example for measuring the Hamiltonian mentioned above:

Necessity of Basis Transformation in VQE

The purpose of the VQE algorithm is to find the ground state energy of a given Hamiltonian. This is achieved by parameterizing a quantum circuit (the ansatz) and optimizing the parameters to minimize the expectation value of the Hamiltonian. The expectation value of the Hamiltonian is computed as a weighted sum of the expectation values of its Pauli terms.

To accurately compute these expectation values, measurements must be performed in the appropriate bases. Since quantum computers typically measure in the Z basis, we transform the measurements for Pauli terms like $\sigma_x$ and $\sigma_y$ into the Z basis. This ensures that we can leverage the quantum hardware's native measurement capabilities while still obtaining the necessary information to evaluate the Hamiltonian's expectation value.

Without these basis transformations, we would not be able to correctly measure the expectation values of Pauli terms that are not in the Z basis, leading to incorrect evaluations of the cost function and, consequently, incorrect optimization of the variational parameters.

EITCA Academy