Efficient Postprocessing Procedure for Evaluating Hamiltonian Expectation Values in Variational Quantum Eigensolver.

Variational quantum eigensolvers (VQEs)[1, 2, 3] have been considered one of the main applications of quantum computers in the noisy intermediate-scale quantum (NISQ) era[4]. VQEs can be applied to a broad type of tasks, such as chemistry, optimization, and machine learning. A common goal of most of these applications is to minimize an objective function, which usually is represented by a Hamiltonian that can be decomposed into a sum of tensor products of Pauli operators, i.e.

where $O_{i}\in\{I,\sigma_{X},\sigma_{Y},\sigma_{Z}\}^{\otimes N}$, and $N$ is the number of qubits. In VQE, the goal is to minimize the Hamiltonian expectation value, obtained from a statistical average over a large number of measurements. The general process of VQEs can be summarized as follows:

1. 1.

   Design the parameterized ansatz $U(\mathbf{\theta})$.

2. 2.

   Choose the initial parameter vector $\mathbf{\theta}$.

3. 3.

   Calculate the gradient of $f(\mathbf{\theta})=\expectationvalue{U^{\dagger}(\mathbf{\theta})HU(\mathbf{\theta})}{i}$, where $\ket{i}$ is the initial state.

4. 4.

   Update $\mathbf{\theta}$ according to the gradient, and check if $f(\theta)$ has decreased.

5. 5.

   Go back to step 3 until $f(\theta)$ cannot be minimized anymore or has reached the maximum iterations.

Assuming choosing a gradient-based classical optimizer, the gradient can be calculated analytically with the parameter shift rule [5] for a quantum circuit. However, this process also requires evaluating the expectation value $2$ times at $f(\theta+\frac{\pi}{2}\hat{\theta_{x}})$ and $f(\theta-\frac{\pi}{2}\hat{\theta_{x}})$ for each dimension $\theta_{x}$ in $\mathbf{\theta}$. This results in $2d$ times of evaluations of Hamiltonian expectation value for step 3, where $d$ is the dimension of $\mathbf{\theta}$. Then we evaluate $f(\mathbf{\theta})$ once again with the updated $\mathbf{\theta}$ in step 4, making it a total of $2d+1$ times of evaluation in every optimization step, and thus $(2d+1)T$ evaluations are required for a whole VQE process with $T$ iterations. This number increases as the problem size grows since the number of parameters should grow, and the number of iterations may also grow. It is well known that evaluating a Hamiltonian expectation value once already requires a large number of circuit repetitions [6]. However, as stated above, we even need to repeat this task over and over again to complete the whole VQE process.

There have been many elaborated methods proposed to reduce the circuit repetition for evaluating the Hamiltonian expectation value, such as efficient state tomography [7, 8, 9], measurement distribution [6, 10, 11], and Hamiltonian partitioning [1, 12, 13, 14, 15, 11]…etc. However, the measurement needed to reach a specific error is still very large. For example, it still requires hundreds of millions of circuit repetitions to evaluate even small molecules [11] (e.g., H₂O, NH₃) to chemical accuracy. Moreover, with such a large number of measurements, the classical overhead of postprocessing becomes nontrivial.

First, we started from a Hamiltonian that is partitioned into commuting groups, i.e.,

Since commuting operators share common eigenstates, the Pauli operators in the same group can be evaluated with the same set of measurement outcomes $\{b\}$ in the corresponding measurement basis. The classical cost of evaluating a bit string depends on the number of terms in the group. For example, a fully connected Ising Hamiltonian has all of its operators in the same group, since they are all composed of Pauli Z, and thus requires $O(N^{2})$ scaling of cost. A molecular Hamiltonian is known to have $O(N^{4})$ operators, and a grouping strategy with $O(N^{3})$ groups will thus result in linear scaling of operators in each group, which requires $O(N)$ cost to evaluate. Observing the fact that $\expectationvalue{G}{b}$ is fixed and needs to be calculated every time $b$ occurs, a straightforward strategy is to memorize it. Since looking up a bit string with length $N$ in a dictionary requires only $O(1\times N)$ on average. This is the main idea of measurement memory (MM).

We implemented MM on fully connected Ising Hamiltonians up to 20 qubits, finding an $O(N^{2})$ speedup in terms of the percentage of time saved. MM is also tested on Molecular Hamiltonian of H₂, H₄, LiH, and H₂O, each with 3 different grouping strategies: qubit-wise commuting (QWC) grouping [1, 13], general commuting (GC) grouping [14], and Fermion grouping (FG) [16]. Each grouping method has different scaling of groups and operators in each group, thus leading to a different degree of improvement.

For a Hamiltonian $H$ that has been divided into $K$ commuting operator groups, i.e.,

where $O_{j}^{k}\in\{I,\sigma_{X},\sigma_{Y},\sigma_{Z}\}^{\otimes N}$, and

The operators in the same commuting groups are simultaneously diagonalizable and can be evaluated with the same set of measurements in the corresponding basis. Measurement Memory (MM) creates an independent dictionary $\mathcal{M}_{k}$ for each group $G_{k}$ at the beginning. During the future process, every time we evaluate the Hamiltonian expectation value via quantum computer measurement results (including calculating the gradient with the parameter shift rule), we try to find the value $\expectationvalue{G_{k}}{b}$ in the corresponding dictionary $\mathcal{M}_{k}$ for each group with key $b$. If it doesn’t exist, we evaluate it as the normal process and store $b$ and $\expectationvalue{G_{k}}{b}$ in the dictionary as key and value.

The pseudocode shown in Algorithm 1 represents the process of evaluating $\expectationvalue{H}$ once. For each measurement we evaluate, $\mathcal{M}$ accumulates and starts to develop potential to reduce more and more computational cost.

Note that one does not have to sort the measured bit string $\{b_{1},b_{2},\ldots,b_{m}\}$ into probability or count dictionaries in advance, as we usually do in regular procedures when applying MM. We will discuss the complexity of sorting and MM further in the next chapter. We also analyzed the time complexity in the appendix, showing that the potential performance improvement of MM is mainly determined by the scaling of the number of operators in each group. Thus, the improvement will be very limited for an MM created for a sublinear scaling operator group. On the other hand, we should expect fruitful performance improvement for superlinear scaling operator groups.

For every commuting group $G_{k}$ in a Hamiltonian, we set up the circuit in the corresponding measurement basis and repeat the circuit, say $m$ times, to obtain a collection of measured bit strings $\{b_{1},b_{2},\ldots,b_{m}\}_{k}$. Typically, one would sort the set into distinct terms with corresponding counts or probabilities, such as $\{b_{1}:Pr(b_{1}),b_{2}:Pr(b_{2}),\ldots,b_{L}:Pr(b_{L})\}_{k}$, where $L\leq m$. For simplicity, we refer to the process of evaluating $\expectationvalue{G}$ from $\{b_{1}:Pr(b_{1}),b_{2}:Pr(b_{2}),\ldots,b_{L}:Pr(b_{L})\}_{k}$ as the ”sort-and-evaluate” procedure and the process of directly evaluating every bit string in $\{b_{1},b_{2},\ldots,b_{m}\}_{k}$ as the ”naive evaluation”. The sort-and-evaluate procedure reduces the evaluation of $\expectationvalue{G}{b}$ by $m-L$ times. However, the sorting process is not free and might waste a lot of time in the worst case (i.e., $L=m$). Nevertheless, we showed in the Appendix that in general and practical cases, one will still prefer the sort-and-evaluate procedure over naive evaluation.

One thing to mention here is that the MM procedure has already incorporated the process of sorting raw measurement results into probability dictionaries. Thus, we do not have to sort the measured result in advance when applying MM. The main difference is that we are now storing their eigenvalues ($\expectationvalue{G}{b}$) instead of probabilities. While a probability dictionary is useless except for one specific evaluation of the expectation value, the information of MM is able to sustain through the whole VQE and develop the potential to save more and more cost as the iteration goes on. We further proved that the computational cost of the worst-case scenario of MM, which is the case that no eigenvalue of measured bit strings is stored in memory, is identical to the sort-and-evaluate procedure. This indicates that there are no downside trade-offs for adopting MM in terms of time complexity.

In this article, we introduced a Measurement Memory (MM) dictionary designed to store a measured bit string $b$ and its eigenvalue $\expectationvalue{G}{b}$ of a commuting group in a Hamiltonian. Throughout the VQE process, MM accumulates records, reducing the computational cost of evaluating $b$ over all terms in $G$ each time $b$ occurs after the first instance. We achieved $O(N^{2})$ percentage time savings for a fully connected Ising Hamiltonian, providing an additional benefit of increasing the probability of finding a low-energy solution. MM also provides over $O(N)$ time savings for molecular Hamiltonian depending on the grouping method. With careful selection of grouping elements, one may be able to reduce both circuit repetition and postprocessing classical overhead. We further compare the time complexity of regular sort-and-evaluate procedure with MM, finding that that MM is a more efficient way of storing useful information for evaluating Hamiltonian expectation values in VQE, which sustains through the whole process. Moreover, the worst case scenario of MM is identical to the regular procedure. Thus, MM is a more efficient procedure of doing post-processing for evaluating Hamiltonian expectation value in VQE, and employing MM has no down side trade offs. For future applications, MM can also be applied to other deterministic measurement schemes, such as other grouping methods, and derandomized shadow[9] as long as the measurement basis are fixed.