Measuring Expectation Values of Pauli String Hamiltonians

Hello! Welcome back to our deep dive into quantum computing.

In our previous lesson, we tackled the crucial step of translating physical Hamiltonians into the language of quantum computers. We saw how mappings like Jordan-Wigner transform a fermionic or spin Hamiltonian into a weighted sum of Pauli strings, $H={\sum }_k{c}_k{P}_k$.

Today, we will build directly on that foundation to address the next step in the Variational Quantum Eigensolver (VQE) algorithm: actually measuring the energy. This lesson focuses on the learning outcome: For a given Hamiltonian expressed as a sum of Pauli strings, devise a measurement strategy to estimate its expectation value by measuring each term individually.

We will break down how to measure the expectation value of any given Pauli string on a quantum computer and then combine these measurements to estimate the total energy of the state prepared by our ansatz.

1. The Core Task: Measuring Pauli Operator Expectation Values

The expectation value of our Hamiltonian $H$ for a given trial state $|\psi ⟩$ is:

$$⟨H⟩=⟨\psi |H|\psi ⟩=⟨\psi \left|\sum _k{c}_k{P}_k\right|\psi ⟩$$

Due to the linearity of quantum mechanics, this simplifies to:

$$⟨H⟩=\sum _k{c}_k⟨\psi |P_k|\psi ⟩=\sum _k{c}_k⟨P_k⟩$$

The problem of finding the total energy thus reduces to finding the expectation value $⟨P_k⟩$ for each Pauli string in the sum.

The challenge is that most quantum hardware is designed to perform measurements in only one basis, the computational or Z-basis, which distinguishes between the $|0⟩$ and $|1⟩$ states. Our task is to find a way to measure operators like $X$ and $Y$ using only Z-basis measurements.

The following resource provides a clear, mathematical derivation of how this is done.

Let's summarize the key techniques derived in the notes.

Measuring $⟨Z⟩$

This is the most direct case. The Pauli-Z operator has eigenvalues +1 for state $|0⟩$ and -1 for state $|1⟩$. Therefore, its expectation value is the difference between the probabilities of measuring these two outcomes:

$$⟨Z⟩=(+1)\cdot P(|0⟩)+(-1)\cdot P(|1⟩)=P(0)-P(1)$$

To estimate this on a quantum computer, you prepare the state $|\psi ⟩$, measure in the Z-basis many times (taking many "shots"), count the occurrences of '0' ($C_0$) and '1' ($C_1$), and compute $⟨Z⟩\approx (C_0-C_1)/(C_0+C_1)$.

Measuring $⟨X⟩$ and $⟨Y⟩$

To measure operators that are not diagonal in the Z-basis, we perform a change of basis. The key is to find a unitary transformation $U$ that turns the operator we want to measure into $Z$. For the Pauli-X operator, we know that $X=HZH$, where $H$ is the Hadamard gate.

So, the expectation value is:

$$⟨X⟩=⟨\psi |X|\psi ⟩=⟨\psi |HZH|\psi ⟩=⟨\psi |H^{†}ZH|\psi ⟩=⟨{\psi }^{\prime }|Z|{\psi }^{\prime }⟩$$

where $|{\psi }^{\prime }⟩=H|\psi ⟩$. This shows that measuring $⟨X⟩$ for the state $|\psi ⟩$ is equivalent to measuring $⟨Z⟩$ for the state $H|\psi ⟩$. Operationally, this means we simply apply a Hadamard gate to our qubit just before performing the standard Z-basis measurement.

Similarly, for the Pauli-Y operator, we can use the identity $Y=(H{S}^{†}{)}^{†}Z(H{S}^{†})$. This means to measure $⟨Y⟩$, we apply the sequence $S^{†}$ followed by $H$ just before the Z-measurement.

The general strategy is summarized in this table:

To Measure	Basis Transformation $U$	Circuit Operation (before measurement)
$⟨Z⟩$	$I$	None
$⟨X⟩$	$H$	Apply $H$
$⟨Y⟩$	$H{S}^{†}$	Apply $S^{†}$, then $H$

2. Measuring Multi-Qubit Pauli Strings

The strategy for single qubits extends naturally to multi-qubit Pauli strings like $P_k=P_{k,0}\otimes P_{k,1}\otimes \cdots \otimes P_{k,N-1}$. The required basis transformation is simply the tensor product of the individual single-qubit transformations.

For example, to measure $⟨X_0\otimes Y_1\otimes Z_2⟩$:

1. On qubit 0, apply $H$.
2. On qubit 1, apply $S^{†}$ then $H$.
3. On qubit 2, do nothing.
4. Perform a Z-basis measurement on all three qubits.

After collecting measurement statistics for the bit string outcomes $s=s_0{s}_1{s}_2$, we calculate the expectation value. The eigenvalue ${\lambda }_s$ of the Pauli string for a basis state $|s⟩$ is the product of the individual eigenvalues. For $X_0\otimes Y_1\otimes Z_2$, the eigenvalues are not as simple as for $Z\otimes Z$, but after the basis rotation, we are effectively measuring $Z_0\otimes Z_1\otimes Z_2$. The eigenvalue for an outcome $s_0{s}_1{s}_2$ is $(-1{)}^{s_0}\cdot (-1{)}^{s_1}\cdot (-1{)}^{s_2}=(-1{)}^{s_0\oplus s_1\oplus s_2}$.

The expectation value is then the weighted average of these eigenvalues:

$$⟨P_k⟩=\sum _{s\in \{0,1{\}}^N}{\lambda }_{s}P(s)$$

where $P(s)$ is the probability of measuring the bit string $s$.

Let's return to the notes to see a concrete example.

3. Devising a Measurement Strategy: From Naive to Efficient

We now have all the tools to devise a measurement strategy.

Strategy 1: The Naive Approach

The most direct strategy, which directly follows from the "measure each term individually" part of our goal, is as follows:
For each term $c_k{P}_k$ in the Hamiltonian $H={\sum }_k{c}_k{P}_k$:

1. Prepare the ansatz state $|\psi (\vec{\theta })⟩$.
2. Apply the tensor product of single-qubit gates corresponding to the Pauli string $P_k$ to perform the basis change.
3. Measure all qubits in the computational basis.
4. Repeat steps 1-3 for a sufficient number of shots to get good statistics for $⟨P_k⟩$.
5. After iterating through all $k$, compute the total energy $E={\sum }_k{c}_k⟨P_k⟩$.

This approach is correct, but it can be highly inefficient. As you know from your work with numerical simulations, efficiency matters. For molecular Hamiltonians, the number of Pauli terms can be in the thousands or more, and running a separate quantum circuit for each term is prohibitively expensive.

Strategy 2: An Efficient Approach via Grouping

As the video suggests, we can do better. The key insight comes from a fundamental principle of quantum mechanics: commuting observables can be measured simultaneously. This is because they share a common set of eigenvectors.

Our goal is to partition the set of Pauli strings $\{P_k\}$ into the smallest possible number of groups, where all operators within each group commute with one another. We can then measure all operators in a group with a single experimental setup.

While general commutativity is powerful, finding the joint measurement basis can be complex. A more practical and widely used approach is to group terms based on qubit-wise commutativity.

Two Pauli strings $P={⨂}_i{P}_i$ and $Q={⨂}_i{Q}_i$ are qubit-wise commuting if $[P_i,Q_i]=0$ for all qubits $i$. Since non-identical Pauli operators anti-commute, this condition simplifies: for each qubit $i$, either $P_i=Q_i$ or at least one of them is the identity $I$.

This leads to a much more efficient strategy:

1. Partition: Group the Pauli terms $\{P_k\}$ of the Hamiltonian into sets where all terms in a set are mutually qubit-wise commuting. (As the resource notes, this is equivalent to solving the minimum clique cover problem on the graph of commuting operators).
2. Determine Basis: For each group, determine the single measurement basis. For any qubit $j$, if there is a non-identity operator (e.g., $X_j$) acting on it anywhere in the group, that determines the measurement basis for that qubit ($X$-basis). If all terms have $I_j$, that qubit is measured in the Z-basis by convention.
3. Measure: For each group:
   a. Prepare the state $|\psi (\vec{\theta })⟩$.
   b. Apply the single set of basis-change gates for the entire group.
   c. Measure all qubits and repeat for many shots.
   d. From this single data set, you can compute the expectation value for every operator in the group.
4. Combine: Sum the results: $E={\sum }_{\text{groups}}{\sum }_{P_k\in \text{group}}{c}_k⟨P_k⟩$.

For example, the terms $Z_0$, $Z_1$, and $Z_0{Z}_1$ are all qubit-wise commuting. They can all be measured simultaneously in the $Z\otimes Z$ basis. From one set of shots, you can calculate $⟨Z_0⟩$, $⟨Z_1⟩$, and $⟨Z_0{Z}_1⟩$. This is a significant saving compared to running three separate experiments.

Conclusion

In this lesson, we have developed a complete and practical strategy for one of the most critical parts of the VQE algorithm: estimating the Hamiltonian's expectation value.

Key Takeaways:

• The total energy $⟨H⟩$ is the weighted sum of the expectation values of the individual Pauli strings, ${\sum }_k{c}_k⟨P_k⟩$.
• The expectation value of any Pauli string can be measured by applying a specific unitary transformation (composed of $I,H,S^{†}$) to each qubit and then performing a measurement in the computational (Z) basis.
• A naive strategy of measuring each Pauli term in a separate experiment is correct but inefficient.
• A far more efficient strategy involves grouping qubit-wise commuting Pauli strings. All terms in such a group can be measured simultaneously, drastically reducing the number of required quantum circuit executions.

Preview of the Next Lesson

We have now assembled all the quantum components of the VQE algorithm. We know how to build a parameterized ansatz to prepare a trial state $|\psi (\vec{\theta })⟩$, and we know how to measure its energy $E(\vec{\theta })$. The final step is to use a classical computer to perform the optimization.

In our next lesson, Implement the VQE algorithm to find the ground state of a simple molecular Hamiltonian like H2, we will combine these pieces. We will see how a classical optimization algorithm iteratively updates the parameters $\vec{\theta }$, calling upon the quantum computer at each step to evaluate the energy, in order to find the parameter set ${\vec{\theta }}^{\ast }$ that minimizes $E$ and thus approximates the ground state energy of the system.