How is the expectation value of an operator ( A ) in a quantum state described by ( ρ ) calculated, and why is this formulation important for VQE?

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The expectation value of an operator A in a quantum state described by the density matrix ρ is a fundamental concept in quantum mechanics, particularly relevant in the context of the Variational Quantum Eigensolver (VQE). To calculate this expectation value, the following procedure is employed:

Given a quantum state ρ and an observable A, the expectation value \langle A \rangle is defined as:

    \[ \langle A \rangle = \text{Tr}(ρA) \]

Here, \text{Tr} denotes the trace operation, which is the sum of the diagonal elements of a matrix. The density matrix ρ represents the state of the quantum system, and A is a Hermitian operator corresponding to the observable whose expectation value we wish to compute.

Detailed Explanation:

Quantum State Representation:

In quantum mechanics, the state of a system can be described by a wavefunction |\psi\rangle in the case of a pure state or by a density matrix ρ in the case of a mixed state. The density matrix ρ is a positive semi-definite matrix with unit trace, providing a comprehensive description of the statistical properties of the quantum system.

For a pure state |\psi\rangle, the density matrix is given by:

    \[ ρ = |\psi\rangle \langle \psi| \]

For a mixed state, which is a statistical ensemble of pure states |\psi_i\rangle with probabilities p_i, the density matrix is:

    \[ ρ = \sum_i p_i |\psi_i\rangle \langle \psi_i| \]

Observable and Expectation Value:

An observable A in quantum mechanics is represented by a Hermitian operator, meaning A = A^\dagger, where A^\dagger is the conjugate transpose of A. The expectation value \langle A \rangle is a measure of the average outcome of measurements of A on the quantum state ρ.

The trace operation \text{Tr}(ρA) is computed as follows:

1. Matrix Multiplication: Compute the product of the density matrix ρ and the observable A.
2. Trace Calculation: Sum the diagonal elements of the resulting matrix.

Mathematically, if ρ and A are represented in a basis \{|i\rangle\}, the expectation value can be expressed as:

    \[ \langle A \rangle = \sum_{i} \langle i | ρ A | i \rangle \]

For a pure state |\psi\rangle, this simplifies to:

    \[ \langle A \rangle = \langle \psi | A | \psi \rangle \]

Importance for VQE:

The Variational Quantum Eigensolver (VQE) is an algorithm used to find the ground state energy of a Hamiltonian H. It leverages both quantum and classical computations to optimize a parameterized quantum circuit to minimize the expectation value of H. The Hamiltonian H is typically expressed as a sum of Pauli operators:

    \[ H = \sum_j h_j P_j \]

where h_j are real coefficients and P_j are tensor products of Pauli matrices.

In VQE, the quantum state ρ(\boldsymbol{\theta}) is generated by a parameterized quantum circuit U(\boldsymbol{\theta}) acting on an initial state |\psi_0\rangle:

    \[ |\psi(\boldsymbol{\theta})\rangle = U(\boldsymbol{\theta})|\psi_0\rangle \]

The goal is to find the optimal parameters \boldsymbol{\theta} that minimize the expectation value of the Hamiltonian:

    \[ E(\boldsymbol{\theta}) = \langle \psi(\boldsymbol{\theta}) | H | \psi(\boldsymbol{\theta}) \rangle = \sum_j h_j \langle \psi(\boldsymbol{\theta}) | P_j | \psi(\boldsymbol{\theta}) \rangle \]

This expectation value is computed on a quantum computer, while the optimization of \boldsymbol{\theta} is performed using classical optimization algorithms.

Example:

Consider a simple example where the Hamiltonian H is given by:

    \[ H = Z_1 Z_2 + X_1 X_2 \]

where Z and X are Pauli matrices. If the quantum state is described by the density matrix ρ, the expectation value of H is:

    \[ \langle H \rangle = \langle Z_1 Z_2 \rangle + \langle X_1 X_2 \rangle \]

Each term \langle Z_1 Z_2 \rangle and \langle X_1 X_2 \rangle is computed as:

    \[ \langle Z_1 Z_2 \rangle = \text{Tr}(ρ Z_1 Z_2) \]

    \[ \langle X_1 X_2 \rangle = \text{Tr}(ρ X_1 X_2) \]

The overall expectation value \langle H \rangle is then the sum of these individual terms.

Importance of Expectation Value Calculation in VQE:

1. Energy Estimation: The primary objective of VQE is to estimate the ground state energy of a Hamiltonian. The expectation value \langle H \rangle provides an estimate of the energy for a given set of parameters \boldsymbol{\theta}.

2. Optimization: The expectation value serves as the objective function for the classical optimization algorithm. By minimizing \langle H \rangle, the algorithm iteratively updates the parameters \boldsymbol{\theta} to approach the ground state energy.

3. Quantum-Classical Hybrid Approach: VQE exemplifies the synergy between quantum and classical computations. The quantum computer evaluates the expectation values, while the classical computer performs the optimization, leveraging the strengths of both computational paradigms.

4. Scalability: The expectation value calculation is efficient on a quantum computer, even for large systems, due to the inherent parallelism of quantum operations. This scalability is important for tackling complex quantum systems that are intractable for classical methods.

Rotosolve Optimization:

Rotosolve is a specific optimization technique used in the context of VQE. It optimizes the parameters of the quantum circuit by iteratively solving for the optimal rotation angles. The key idea is to decompose the parameter space into individual rotations and solve for the optimal angle for each rotation while keeping the other parameters fixed.

The expectation value calculation plays a important role in Rotosolve, as it provides the necessary feedback to update the rotation angles. By efficiently computing the expectation values, Rotosolve can converge to the optimal parameters more rapidly.

Conclusion:

The expectation value of an operator A in a quantum state described by ρ is computed using the trace operation \text{Tr}(ρA). This formulation is essential for the Variational Quantum Eigensolver (VQE), as it enables the estimation of the ground state energy of a Hamiltonian. The expectation value serves as the objective function for classical optimization algorithms, facilitating the hybrid quantum-classical approach of VQE. The efficient calculation of expectation values on a quantum computer is a key factor in the scalability and effectiveness of VQE, particularly when combined with optimization techniques like Rotosolve.

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