Warm Starting Variational Quantum Algorithms with Near Clifford Circuits

Abstract

As a mainstream approach in the quantum machine learning field, variational quantum algorithms (VQAs) are frequently mentioned among the most promising applications for quantum computing. However, VQAs suffer from inefficient training methods. Here, we propose a pretraining strategy named near Clifford circuits warm start (NCC-WS) to find the initialization for parameterized quantum circuits (PQCs) in VQAs. We explored the expressibility of NCCs and the correlation between the expressibility and acceleration. The achieved results suggest that NCC-WS can find the correct initialization for the training of VQAs to achieve acceleration.

1. Introduction

With the rapid development of quantum computing hardware [1,2,3,4], we are entering the noisy intermediate-scale quantum (NISQ) era [5]. Variational quantum algorithms (VQAs) [6,7,8] are one of the most important applications on NISQ devices, which could be used to demonstrate the quantum advantage over classical computers. VQAs are optimization-based (or learning-based) approaches that use parameterized quantum circuits (PQCs) running on a quantum computer, and then outsource the parameter update to a classical optimizer. As the quantum analog of highly successful machine-learning methods, they have been implemented to solve a wide range of problems, including quantum chemistry [9,10,11], classification [12,13,14,15], image generation and recognition [16,17,18,19], and optimization problems [20,21].

The main obstacles to applying VQAs to achieve quantum advantages are the training efficiency of a PQC. Because the estimation of the expectation values requires repeatedly running and taking measurements of a PQC with the same initial state, the number of the measurements and the number enormously increase with the number of the qubits and trainable parameters. To address the issue of accelerating the training process of a VQA, a variety of improved approaches have been proposed, such as distributed VQA schemes by simultaneously training with multiple quantum processors [22,23], higher-efficiency estimation methods with fewer measurement shots [24,25,26], and parameter initialization by a pretraining process [27,28,29].

In this paper, we propose a new parameter initialization method named near Clifford circuit warm start (NCC-WS) to accelerate the VQAs. We used a near Clifford circuit (NCC) [30,31], which includes Clifford gates and a limited number of T gates, to classically pretrain PQCs with the same structure. Compared with randomly initialized parameters in conventional VQA, the NCC-WS provides a good guess of the initial parameters, which can significantly increase the convergence rate of VQAs. Meanwhile, for an NCC that can be efficiently simulated by a classical computer, the computational overhead of the NCC-WS is negligible. We investigated the relationship between T gate ratio in NCCs and its expressibility, and we explained the role of T gate in enhancing the expressibilities of the NCCs from the perspective of numerical experiments. We further investigated the correlation between the pretraining effect and the expressivity of NCCs, which demonstrates the advantage of NCC-WS in accelerating VQAs.

2. Method

We consider a “composite" training process consisting of two pieces, as schematically depicted in Figure 1. We first perform a discrete optimization training with an NCC, and then transform it into a PQC with continuous parameters for a secondary optimization that has the same structure as the NCC.

In the NCC-WS stage, we leverage a classical optimization algorithm to pretrain an NCC consisting of discrete quantum gates. The Clifford circuit (CC) is composed of a Hardmard gate (H gate), phase gate (S gate), and two qubit control-NOT (CNOT) gates. To further increase the expressibility, we usually considers a near Clifford circuit (NCC) constructed by Clifford gates and a few T gates. Gottesman–Knill theorem [32], as well as more recent studies [33,34], highlights that an NCC with n qubits and $O\left(1\right)$ T gates can be classically simulated in time $\mathrm{poly}\left(n\right)$. Therefore, the NCC-WS can be performed on a classical computer with a low computational cost. The classical training strategy of an NCC is not unique. In this study, we employed the simulated annealing algorithm to train the NCC. Specifically, we considered a random NCC consisting of trainable single qubit layers interlaced with CNOT layers. The single qubit gates are chosen from the gate set $\{H,S,I\left(\mathrm{identity}\right),T\}$ with a certain probability distribution. In each iteration, we randomly change part of single qubit gates with the same probability, and compare the new cost value $C_{\mathrm{new}}$ to the previous one $C_{\mathrm{previous}}$. Based on the idea of annealing, we accept the updated NCC with certainty when $C_{\mathrm{new}}$ is better than $C_{\mathrm{previous}}$; otherwise, we accept with probability $e^{-\beta \left(C_{\mathrm{new}}-C_{\mathrm{previous}}\right)}$ in the opposite case, where the inverse temperature $\beta$ increases with iterations.

After the NCC-WS, we obtain a relatively optimal quantum circuit. However, because the CCs and NCCs are not sufficient for universal quantum computation, the cost function may not converge to the optimal value. Therefore, we needed to continue to optimize the training results with PQC on this basis. We can use $Z-Y$ decomposition [35] to convert the single qubit gate U in the NCC into the rotation gates about the z and y axes, i.e., $U=e^{i\alpha }{R}_z\left(\beta \right)R_y\left(\gamma \right)R_z\left(\delta \right),$ where $R_n\left(\phi \right)=e^{-\mathrm{i}\phi {\sigma }_n/2}$ (for $n=y,z$). In this way, an NCC can be converted into an equivalent hardware-efficient PQC, where circuits of this form are a natural choice due to a straightforward evaluation of the gradient with respect to most objective functions [36,37], as shown in Figure 1b.

Compared with random initializations, the training of VQAs after NCC-WS starts from a better initial point, which directly accelerates the convergence of VQAs. A well set of initial parameters will allow the cost function to achieve optimal value in fewer iterations. Another potential advantage of NCC-WS is the ability to escape from local traps in the landscape of cost function. The landscape of cost function usually becomes more rugged when parameter space increases, such that the cost function more easily fall into the trap of local minima [38]. The simulated annealing algorithm [39] is a probability check method that may accept a new solution that is even worse than previous solution, so it is possible to move away from the local optimal solution and thus search for the global minimum cost function.

3. Expressibility

An empirical intuition is that the better the expressibility of an NCC, the more likely we are able to obtain better initial parameters for the training of VQAs. Therefore, to evaluate the quality of initial parameters obtained by NCC-WS, we first need to investigate the expressibility of an NCC. We introduce a descriptor $\mathrm{Expr}$ [40] to quantify the expressibility of a quantum circuit. $\mathrm{Expr}$ is defined as a circuit’s ability to generate states that are well representative of the Hilbert space, which can be quantified by the Kullback–Leibler (KL) divergence $D_{\mathrm{KL}}$ between the probability distributions,

$$\begin{array}{c}\hfill \mathrm{Expr}=D_{\mathrm{KL}}({\widehat{P}}_{\mathrm{circuit}}\left(F\right)\Vert P_{\mathrm{Haar}}\left(F\right)),\end{array}$$

where ${\widehat{P}}_{\mathrm{circuit}}\left(F\right)$ is the estimated probability distribution of fidelities resulting from sampling states from a quantum circuit with initial state $|0\rangle$ for each qubit, and $P_{\mathrm{Haar}}\left(F\right)=(N-1){(1-F)}^{N-2}$ is the probability density function of fidelities for the uniform distribution of states, i.e., the ensemble of Haar random states, where F is fidelity, and N corresponds to the dimension of Hilbert space.

We were particularly interested in the additional T gates in an NCC, which improves the expressibility of pure Clifford circuits [31]. However, the number of T gates in an NCC that can drive the best expressibility is not logically and tightly answered. We numerically simulated the variation in the expressibility with the probability of initializing a single qubit gate to T gate on a six-qubits NCC, as shown in Figure 2a. To investigate the relationship between non-Cliffordness and expressivity, we considered a more general case, that the probability of T gate ranges from 0 to $40\%$ in steps of $10\%$.

Figure 3 displays the simulated data of the estimated fidelities, which were normalized to a discrete probability density function. As a reference, we show the curve of $P_{\mathrm{Haar}}\left(F\right)$ when $N=2^6$. A bin number of 200 was used to generate the histogram Figure 3a–f. Above subgraphs, the KL divergence (Here, the area of each bar in histogram represents the probability of occurrence of the estimated fidelity in corresponding bin. Similarly, the integration of the curve representing $P_{\mathrm{Haar}}\left(F\right)$ over each interval also has the same meaning. The vectors for acquiring KL divergence were obtained by arranging the probabilities calculated from above two ways according to the fidelity interval from smallest to largest.) between ${\widehat{P}}_{\mathrm{circuit}}\left(F\right)$ and $P_{\mathrm{Haar}}\left(F\right)$ are reported to quantify the expressibility, in which a smaller value corresponds to more favorable expressibility. Although the KL divergence varies with bin number, the observations coming from the relative quantitative comparisons among circuits remain the same. Additionally, the degree of fitting between the histogram and curve in each subgraph visually demonstrates the appropriateness of the expressibility of the corresponding circuit.

As shown in Figure 3a–f, under the condition that the total number of gates is constant, we see that the expressibility of NCCs with a probability of a $10\%$ T gate is closest to that of Haar states in addition to PQCs. It means that there is no strict positive correlation between NCC expressivity and the number of T gates, and only the correct proportion of T gate can maximize the expressibility of an NCC.

4. Numerical Simulation

In this section, we demonstrate the application of NCC-WS VQA for classifying the handwritten digits 0 and 1 on MNIST, which contains a total of 3112 examples, to further explore whether there is a link between the acceleration effect and the expressibility. The settings on NCCs and PQC remained the same as those in the previous section.

We first encoded a classical example (a $8\times 8$ matrix) in the MNIST dataset into the quantum state (6 qubits) by amplitude encoding, i.e.,

$$\begin{array}{c}\hfill {\mid \phi \rangle }_k=\frac{a_{k,1}\mid 000000\rangle +\cdots +a_{k,64}\mid 111111\rangle }{\sqrt{|a_{k,1}{|}^2+\cdots +{\left|a_{k,64}\right|}^2}},\end{array}$$

where $(a_{k,1},\cdots ,a_{k,64})$ represents the kth sample in training set, which arranges the elements in the matrix into a vector.

To calibrate the training effect, we employed the mean square error (MSE) [41] cost function C, defined as

$$\begin{array}{c}\hfill C=\frac{1}{2}\sum _k{(\widehat{y_k}-y_k)}^2,\end{array}$$

where $\widehat{y_k}=(\langle Z^{\otimes 6}\rangle +1)/2$ is the predicted label, and $y_k$ is the real label.

According to parameter-shift rule, ${\nabla }_{j}C\left({\theta }^{\left(\mathit{t}\right)}\right)$ (jth component of parameter vector ${\theta }^{\left(\mathit{t}\right)}$) satisfies

$$\begin{array}{c}\hfill ({\widehat{y_k}}^{\left(t\right)}-y_k)\frac{{\widehat{y_k}}^{(t,+j)}-{\widehat{y_k}}^{(t,-j)}}{2},\end{array}$$

where ${\widehat{y_k}}^{(t,\pm j)}$ donates the output of VQA with shifted parameter ${\theta }^{\left(\mathit{t}\right)}\pm \frac{\pi }{2}{\mathit{e}}_{\mathit{j}}$ at the tth iteration, and ${\mathit{e}}_{\mathit{j}}$ donates the unit vector.

In NCC-WS phase, we utilize an annealing algorithm (When $C_{\mathrm{new}}>C_{\mathrm{previous}}$, the probability of accepting $C_{\mathrm{new}}$ satisfies $p=e^{-\beta (C_{\mathrm{new}}-C_{\mathrm{previous}})}$, where $\beta$ is inverse temperature.) for discrete gate sequence updates, where $\beta$ increases by 1.25 times per iteration, varying from 10 to ${10}^{15}$. In the VQA phase after NCC-WS, we leveraged the vanilla SGD (Vanilla SGD as a common classical optimizer to update the parameters in PQC from ${\theta }^{\left(\mathit{t}\right)}$ to ${\theta }^{\left(\mathit{t}\right)}$, i.e., ${\theta }^{\left(\mathit{t}\right)}={\theta }^{\left(\mathit{t}\right)}-\eta \nabla C\left({\theta }^{\left(\mathit{t}\right)}\right)$, where $\eta$ is the fixed learning rate.) with $\eta =0.1$ to update the parameters of rotation gates in PQC, and the batch size was set as 10. To ensure the accuracy of the experimental results, a total of 30 independent experiments were run by setting random initialization (random NCC gate sequence for NCC-WS VQA and random initial parameter vector for conventional VQA) for each setting.

Figure 4 displays the simulation results. In Figure 4a, the yellow C curve, representing NCC with $10\%$ T gates, has the minimum convergence value in all curves; the purple curve, representing NCC with the worst expressibility, is just the opposite, which reveals that there is a positive correlation between the expressibility of an NCC in NCC-WS and the convergence value of cost function C, i.e., the better the expressibility, the smaller the convergence value of C. This is easy to understand because the NCCs with good expressibility can characterize more quantum states to be retrieved, which makes the output of NCCs closer to the target quantum states.

As shown in Figure 4b, comparing the second stage training in the NCC-WS VQA and the conventional VQA, the cost function C of the former is closer to convergence at the beginning, which demonstrates the powerful acceleration effect of NCC-WS. Noticeably, although NCCs with different expressibilities are capable of accelerating VQAs, the quality of initializations searched by NCC-WS are closely related to the expressibility of an NCC. NCCs with better expressibility have more potential to find better initializations.

Additionally, NCC-WS VQA has good scalability. The reasons are as follows: (1) The complexity of simulating an NCC is $poly\left(n\right)$ when the number of T gates is limited [42,43], which means that it is efficient when the scale of the system is large. (2) From the perspective of the principle of NCC-WS, this approach is applicable to larger systems. Essentially, NCC-WS will firstly determine an approximate range of the target quantum state by roughly searching in Hilbert space, then VQAs continues to perform a precise search, where these two phases are implemented on a clssical computer and quantum computer, respectively. Therefore, the feasibility of NCC-WS VQA is independent of system size.

5. Conclusions and Outlook

In summary, our results showed that warm start using NCCs with good expressibility can dramatically accelerate the training of VQAs by achieving a “good" initialization for PQCs used in VQAs. Another elegant feature of our method is its generality, as our approach is independent of the VQAs, which is promising for application to a wide range of problems.

Furthermore, we would like to emphasize the issue of T gates so that the readers can make the appropriate trade-offs when employing this method. Our results showed that the addition of the T gates is just an additional feature accelerating the conventional VQAs, because it also exponentially increases the difficulty of the simulation [42,43], which means that we need to balance the pros and cons, rather than adding $poly\left(n\right)$ T gates to an $n-qubit$ system.