Sample complexity of learning parametric quantum circuits

We define the concept class $\mathcal{C}$ as the collection of all the n-qubit parametric quantum circuits with at most n^c unitary gates, each gate acting on at most b qubits (b, c are constant numbers independent of n). We note that $\mathcal{C}$ is general enough to include most quantum circuits in practical applications. Here, we study the learnability of the quantum circuits in $\mathcal{C}$ under the PAC learning framework [18]. Let $C\in \mathcal{C}$ be any n-qubit circuit in this concept class. When we input an n-qubit pure state |ψ⟩_in to C, we will get an output n-qubit pure state |ψ⟩_out = C|ψ⟩_in. Therefore, C can be viewed as a function ${f}_{C}:\mathcal{X}\to \mathcal{Y}$, where its domain $\mathcal{X}$ and range $\mathcal{Y}$ are both the set of all n-qubit pure states. In this work, we write $x\in \mathcal{X}$ as an abbreviation of the n-qubit quantum state |ψ(x)⟩, and similarly for $y\in \mathcal{Y}$. With these notations, we sometimes write y = f_C (x) to denote |ψ(y)⟩ = C|ψ(x)⟩ for simplicity.

We consider the supervised learning scenario [18] and denote the training set of size m as S = {(x₁, y₁), (x₂, y₂), ..., (x_m , y_m )}. Under the PAC learning framework, to learn the unknown circuit C, assume we have m independent n-qubit input samples $\left\{{x}_{1},{x}_{2},\dots ,{x}_{m}\right\}\in {\mathcal{X}}^{m}$, we can input them into C, and obtain output states $\left\{{y}_{1},{y}_{2},\dots ,{y}_{m}\right\}\in {\mathcal{Y}}^{m}$. The essential task of supervised learning is then to learn from S a hypothesis function (here a quantum circuit F) that can approximate the target function f_C (x). This might be accomplished by minimizing certain loss functions over a set of variational model parameters. More concretely, we construct a variational quantum circuit F consisting of multiple gates with some of them having tunable parameters. By tuning these parameters, we can use F to represent different functions ${f}_{h}:\mathcal{X}\to \mathcal{Y}$, and we define our hypothesis space $\mathcal{F}$ as the collection of all the functions f_h that F can represent. Given m independent samples and a tunable quantum circuit F, we can use the following process to make our circuit F become a good approximation of C. We can tune the parameters of F according to the training set S, so that when we put the state x_i (i = 1, 2, ..., m) into the input of F, the output of F will be a good approximation of y_i . By the PAC learning theory, assuming that $\mathcal{F}$ has good generalization power, f_h 's decent performance on the training set can imply its good performance over the whole sample space.

The effectiveness of the above process is based on two assumptions. First, the space of $\mathcal{F}$ should be large enough, so that given any set of samples $S={({x}_{i},{y}_{i})}_{i=1,2,\dots ,m}$, we can always find a function ${f}_{h}\in \mathcal{F}$, such that f_h (x_i ) can approximate y_i with a small error for all i = 1, 2, ..., m. Second, the space of $\mathcal{F}$ should not be too large or complex, so that $\mathcal{F}$ has favorable generalization power to generalize its performance from the training set S to the true probability distribution that S is sampled from. This is a reflection of the Occam's razor principle [18]. Therefore, we need to design a variational quantum circuit class $\mathcal{F}$, which meets the following two requirements simultaneously in order to learn f_C :

• R1: For any $C\in \mathcal{C}$, there exists a hypothesis function ${f}_{h}\in \mathcal{F}$, such that f_h (x) = f_C (x) for all $x\in \mathcal{X}$.
• R2: The hypothesis space $\mathcal{F}$ satisfies the PAC learnability.

We note that the first requirement R1 is stronger than the first assumption, because the function f_h in R1 is the same as f_C . Hence, the training error f_h on the set of samples S is necessarily zero, whereas the first assumption only requires that f_h has a small training error for S. In supervised learning, obtaining high-quality training samples is usually resource-demanding in practice. Thus, studying the sample complexity becomes crucial. In the following, we will study the sample complexity of learning parametric quantum circuits and rigorously prove that any physical quantum circuit is PAC learnable.

To meet R1, we should ensure that F has representation power for all the n-qubit parametric quantum circuits C in $\mathcal{C}$. We observe that any quantum circuit C can be decomposed as a sequence of O(n^c ) number of H gates, R_x (θ) gates, and CNOT gates (see the proposition 1 in appendix A). Thus in our construction of F, we also use these three kinds of gates and arrange them into a uniform pattern, so that its scaling to quantum circuits with more qubits is clear. The construction of F is illustrated in figure 1, which is based on block assembling, i.e., assembling some relatively small gadgets to form a more complicated block. By convention, we denote the three Pauli matrices by X, Y, and Z. A well-known result about quantum circuit states that any single-qubit unitary gate can be expressed as e^iα R_z (β)R_x (γ)R_z (δ), where α, β, γ, and δ are four real numbers, and R_z (θ) = e^−iθZ/2 and R_x (θ) = e^−iθX/2 are the rotation operators along the z-axis and x-axis on the Bloch sphere, respectively [40]. Inspired by this, we define a set of basic gadgets (we call them level-1 blocks) to be L = {HR_x (β)HR_x (γ)HR_x (δ)H|β, γ, δ ∈ [0, 2π)}. In this way, we can tune the parameters of a level-1 block so that it can represent all the single-qubit unitary gates up to an irrelevant global phase factor e^iα . In addition, when we set β = γ = δ = 0, the level-1 block will reduce to the identity gate.

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Using level-1 blocks we can construct level-2 blocks B_i  (i = 2, 3, ..., n). First we put two level-1 blocks (denoted as L) at qubit 1 and qubit i, and then insert two CNOT_1i gates, as shown in figure 1(b). Here, we use CNOT_1i to denote the controlled-NOT gate between the first and ith qubits, with the first qubit being the control qubit and the ith one being the target one. With this, the desired hypothesis quantum circuit F can be constructed as:

$$\begin{equation}F={({B}_{n}{B}_{n-1}\dots {B}_{2})}^{M{n}^{c}},\end{equation} \tag{ 1 }$$

where M is a constant independent of the number of qubits n. We mention that the hypothesis circuit F has a uniform structure for arbitrary system sizes. In addition, only the x-rotations contain variational parameters, and it is straightforward to obtain that the total number of parameters used for defining F scales as O(n^c+1).

For an arbitrary quantum circuit U, we say that it can be represented by the variational circuit F if there exists a solution to the parameters (denoted collectively as θ ) in F such that F = U up to an irrelevant global phase. Now, we are ready to give the first theorem stating that for any $C\in \mathcal{C}$, we can use F to represent it:

Theorem 1. For any $C\in \mathcal{C}$, there exists a hypothesis function ${f}_{h}\in \mathcal{F}$, such that f_h (x) = f_C (x) for all $x\in \mathcal{X}$.

Proof. We first give a high-level intuition for the proof. We note that any gate acting on a constant number of qubits can be decomposed into a quantum circuit with a constant number of elementary gates, namely the CNOT gates, H gates, and R_x (θ) gates. Thus, any $C\in \mathcal{C}$ can be decomposed into a quantum circuit with O(n^c ) elementary gates. In addition, as our hypothesis quantum circuit F consists of Mn^c layers and every two layers can represent one arbitrary elementary gate acting on any pair of qubits through tuning the parameters properly, we can prove that C can be represented by F in an exact fashion.

The complete proof is as follows. First, we note that C consists of O(n^c ) elementary gates by the definition of $\mathcal{C}$. By proposition 1 in appendix A, the circuit C can be written as a product form U_l U_l−1...U₁, where l = O(n^c ), and each U_i is either CNOT_1j or a single-qubit unitary gate V_j on qubit j.

Denoting B_n B_n−1...B₂ as one layer of level-2 blocks, then we define a block string of d layers as follows:

$$\begin{equation*}{({B}_{n}{B}_{n-1}\dots {B}_{2})}^{d}.\end{equation*}$$

We define T as the minimal number so that a block string of T layers can represent U_l U_l−1...U₁. As our hypothesis circuit F contains Mn^c layers in total, we will show that to represent U_l U_l−1...U₁, the minimal number of layers needed in the block string is no greater than Mn^c . Then, when the previous T layers in F have represented C exactly, by the first part of proposition 2 in appendix A, we can set the remaining (Mn^c − T) layers to be the identity gate on the n qubits. Therefore, we can show that F can represent C in an exact fashion and complete the proof of theorem 1.

To show that T ⩽ Mn^c , we will prove that for each U_i , i = 1, 2, ..., l, we only need two layers to represent U_i exactly. Then putting them together, we can prove that we need only 2l layers to represent U_l U_l−1...U₁, which yields T ⩽ 2l.

We fix any i ∈ {1, 2, ..., l}. By proposition 2, one layer B_n B_n−1...B₂ can represent any single-qubit unitary gate V_j acting on any qubit j = 1, 2, ..., n up to a global phase e^iα . Moreover, two layers ${({B}_{n}{B}_{n-1}\dots {B}_{2})}^{2}$ can represent CNOT_1j for any j = 2, 3, ..., n. We note that U_i is either a single-qubit gate V_j acting on some qubit j, or a two-qubit gate CNOT_1j . Therefore, we only need two layers at most to represent U_i exactly.

As we have shown that T ⩽ 2l = O(n^c ), by choosing a large enough constant M, we can prove that T ⩽ Mn^c and complete the proof of the theorem.□

Theorem 1 shows that given any quantum circuit $C\in \mathcal{C}$, there always exists a solution to the parameters, such that the circuit F can simulate the quantum circuit C acting on the n-qubits with zero error. Therefore, given any training set $S={({x}_{i},{y}_{i})}_{i=1,2,\dots ,m}$ sampled independently from some distribution P over $\mathcal{X}\times \mathcal{Y}$, if we have y_i = f_C (x_i ) for all i = 1, 2, ..., m, we can find an instance ${f}_{h}\in \mathcal{F}$ with zero training error. In fact, this theorem has a wider range of applications. When a quantum circuit consists of fewer than n^c gates, we can add some identity gates after it, so our theorem covers all the quantum circuits containing no more than n^c gates. In other words, we can use only O(n^c+1) gates, which are arranged in a uniform pattern, to represent all the circuits with n^c gates or fewer. We remark that the number of gates in many famous quantum circuits, such as quantum support vector machine [14], HHL algorithm [12], and quantum Fourier transform [41], scales polynomially with the number of qubits. Therefore, all of these circuits above can be represented exactly by our circuit F.

In the PAC setting, we usually assume that the input samples are randomly generated from certain unknown probability distribution. As a result, when the hypothesis space covers the underlying distribution and the training dataset is large enough, both the training and generalization error should be small. In this paper, our hypothesis space $\mathcal{F}$ has been proved in theorem 1 to be able to cover all the parametric quantum circuits in $\mathcal{C}$. Now, we study the sample complexity for training a circuit $F\in \mathcal{F}$ to represent $C\in \mathcal{C}$. To this end, we define a measure of the distance between two pure states in $\mathcal{Y}$, which is used as the loss function $\mathcal{L}:\mathcal{Y}\times \mathcal{Y}\to [0,1]$. Specifically, we define the loss function $\mathcal{L}({y}_{1},{y}_{2})$ to be the trace distance of two quantum states |ψ(y₁)⟩ and |ψ(y₂)⟩:

$$\begin{equation*}\mathcal{L}({y}_{1},{y}_{2})=\frac{1}{2}{\Vert}\vert \psi ({y}_{1})\rangle \langle \psi ({y}_{1})\vert -\vert \psi ({y}_{2})\rangle \langle \psi ({y}_{2})\vert {{\Vert}}_{1},\end{equation*}$$

where ||⋅||₁ denotes the trace norm of a matrix. Given a hypothesis function ${f}_{h}\in \mathcal{F}$ and the training set $S={({x}_{i},{y}_{i})}_{i=1,2,\dots ,m}$ sampled independently from some distribution P, we can define the empirical risk of f_h , which is also known as in-sample error:

$$\begin{equation*}\hat{R}({f}_{h})=\frac{1}{m}\sum\limits _{i=1}^{m}\mathcal{L}({y}_{i},{f}_{h}({x}_{i})).\end{equation*}$$

The risk of a hypothesis function ${f}_{h}\in \mathcal{F}$ is then defined as the average loss of f_h over the probability distribution P:

$$\begin{equation*}R({f}_{h})={\int }_{\mathcal{X}\times \mathcal{Y}}\mathcal{L}(y,{f}_{h}(x))\mathrm{d}P(x,y).\end{equation*}$$

Our goal is to find a hypothesis ${f}_{h}\in \mathcal{F}$ and minimize its risk R(f_h ). As the parametric quantum circuit C is a black box in our setting, we do not know the probability distribution P. In the learning process, we use the training set S and find an empirical risk minimizer $\hat{h}\in \mathcal{F}$ over S. For convenience, given the training set S and the probability distribution P, we define $\hat{h}=\mathrm{arg}\underset{{h}^{\prime }\in \mathcal{F}}{\mathrm{min}}\enspace \hat{R}({h}^{\prime })$ to be the empirical risk minimizer, and $h=\mathrm{arg}\underset{{h}^{\prime }\in \mathcal{F}}{\mathrm{min}}\enspace R({h}^{\prime })$ to be the risk minimizer. Now we formally introduce the definition of the PAC learnability for completeness [42]:

Definition 1 (PAC learnability). A hypothesis space $\mathcal{F}$ is PAC learnable, if there exists a function $\nu :{(0,1)}^{2}\to \mathbb{N}$, such that for all (, δ) ∈ (0, 1)² and all probability measures P over $\mathcal{X}\times \mathcal{Y}$, when the size of training set |S| ⩾ ν(, δ), we have

$$\begin{equation}{\mathbb{P}}_{S}\left(R(\hat{h})-R(h)\leqslant {\epsilon}\right)\geqslant 1-\delta .\end{equation} \tag{ 2 }$$

Here ${\mathbb{P}}_{S}(A)$ denotes the probability that event A happens over repeated sampling of the training set S.

We note that when we randomly select a state $x\in \mathcal{X}$, input it into the circuit C and get an output state $y={f}_{C}(x)\in \mathcal{Y}$, the resulting probability distribution P of state pairs (x, y) will satisfy R(h) = 0, because we can find an instance ${f}_{h}\in \mathcal{F}$ equal to f_C by theorem 1. Therefore, if our $\mathcal{F}$ is PAC learnable, after we prepare the training samples S and get an empirical minimizer $\hat{h}$, with probability 1 − δ, the average loss of $\hat{h}$ over P will be no larger than , i.e., $R(\hat{h})\leqslant {\epsilon}$. Now we are going to prove that $\mathcal{F}$ is PAC learnable, and the sample complexity ν is polynomial with n, 1/, and $\mathrm{ln}\enspace \frac{1}{\delta }$.

Theorem 2. The hypothesis space $\mathcal{F}$ satisfies the PAC learnability, with sample complexity $\nu ({\epsilon},\delta )=O(\frac{1}{{{\epsilon}}^{2}}({n}^{c+1}\enspace \mathrm{ln}\enspace \frac{n}{{\epsilon}}+\mathrm{ln}\enspace \frac{1}{\delta }))$.

Proof. The essential idea for the proof relies on the discretization of $\mathcal{F}$. First, we construct a finite set of hypothesis functions ${\mathcal{F}}^{\prime }$, such that for each function ${f}_{h}\in \mathcal{F}$, we can find a function ${f}_{h}^{\prime }\in {\mathcal{F}}^{\prime }$ close enough to f_h . Then we use lemma 2 in appendix B to show that ${\mathcal{F}}^{\prime }$ is PAC learnable. Finally, for any ${f}_{h}\in \mathcal{F}$ and its corresponding ${f}_{h}^{\prime }$, as f_h and ${f}_{h}^{\prime }$ are close enough, we can prove that their risk and empirical risk are close as well. Therefore, we obtain that $\mathcal{F}$ is PAC learnable.

For clarity, we denote l as the total number of R_x (θ) gates in the circuit F, and observe that l ⩽ 12Mn^c+1. We recall that $\mathcal{F}$ is defined as the collection of all the functions ${f}_{h}:\mathcal{X}\to \mathcal{Y}$ that the circuit F can represent by tuning the value of the parameters θ = (θ₁, θ₂, ..., θ_l ), where θ_i ∈ [0, 2π) is the variational parameter characterizing the ith x-rotation. Now we define a finite set ${\mathcal{F}}^{\prime }\subseteq \mathcal{F}$ in this way: ${\mathcal{F}}^{\prime }$ is the collection of all the functions ${f}_{h}^{\prime }:\mathcal{X}\to \mathcal{Y}$ that circuit F can represent by tuning the value of all the θ_i in {0, e, 2e, ..., Ne}, where $e=\frac{{\epsilon}}{6K{n}^{c+1}}$, $N=\lfloor \frac{2\pi }{e}\rfloor$, and K is a large enough constant. As there are l = O(n^c+1) rotational gates in circuit F in total, we have

$$\begin{equation*}\vert {\mathcal{F}}^{\prime }\vert ={\lceil \frac{12\pi K{n}^{c+1}}{{\epsilon}}\rceil }^{O({n}^{c+1})},\end{equation*}$$

which is finite. As a result, we can plug ${\mathcal{F}}^{\prime }$ and ' = /6 into lemma 2 to obtain that when $\vert S\vert \geqslant \frac{18}{{{\epsilon}}^{2}}(\mathrm{ln}\vert {\mathcal{F}}^{\prime }\vert +\mathrm{ln}\enspace \frac{2}{\delta })$,

$$\begin{equation}{\mathbb{P}}_{S}\left(\forall \enspace {f}_{h}^{\prime }\in {\mathcal{F}}^{\prime }:\left\vert \hat{R}\left({f}_{h}^{\prime }\right)-R\left({f}_{h}^{\prime }\right)\right\vert \leqslant \frac{{\epsilon}}{6}\right)\geqslant 1-\delta .\end{equation} \tag{ 3 }$$

We fix all the parameters θ in circuit F, and we will get an arbitrary hypothesis function ${f}_{h}\in \mathcal{F}$. Then we can round all the parameter θ of circuit F into their nearest multiples of e in {0, e, 2e, ..., Ne}, and we will get a new hypothesis function $\tilde{{f}_{h}}\in {\mathcal{F}}^{\prime }$. By proposition 5, we obtain that for any ${f}_{h}\in \mathcal{F}$,

$$\begin{equation}\left\vert R(\tilde{{f}_{h}})-R({f}_{h})\right\vert \leqslant \frac{{\epsilon}}{6},\end{equation} \tag{ 4 }$$

$$\begin{equation}\left\vert \hat{R}(\tilde{{f}_{h}})-\hat{R}({f}_{h})\right\vert \leqslant \frac{{\epsilon}}{6}.\end{equation} \tag{ 5 }$$

Combining the three inequalities (3)–(5) together, we arrive at

$$\begin{equation}{\mathbb{P}}_{S}\left(\forall \enspace {f}_{h}\in \mathcal{F}:\left\vert \hat{R}\left({f}_{h}\right)-R\left({f}_{h}\right)\right\vert \leqslant \frac{{\epsilon}}{2}\right)\geqslant 1-\delta ,\end{equation} \tag{ 6 }$$

when $\vert S\vert \geqslant \frac{18}{{{\epsilon}}^{2}}(\mathrm{ln}\vert {\mathcal{F}}^{\prime }\vert +\mathrm{ln}\enspace \frac{2}{\delta })$. To prove that $\mathcal{F}$ is PAC learnable, we recall our notations that $h=\mathrm{arg}\underset{{h}^{\prime }\in \mathcal{F}}{\mathrm{min}}\enspace R({h}^{\prime })$, and $\hat{h}=\mathrm{arg}\underset{{h}^{\prime }\in \mathcal{F}}{\mathrm{min}}\enspace \hat{R}({h}^{\prime })$. Combining the inequality (6) and proposition 6, we obtain that

$$\begin{equation*}{\mathbb{P}}_{S}\left(R(\hat{h})-R(h)\leqslant {\epsilon}\right)\geqslant 1-\delta ,\end{equation*}$$

when $\vert S\vert \geqslant \frac{18}{{{\epsilon}}^{2}}(\mathrm{ln}\vert {\mathcal{F}}^{\prime }\vert +\mathrm{ln}\enspace \frac{2}{\delta })$. Plugging in $\vert {\mathcal{F}}^{\prime }\vert ={\lceil \frac{12\pi K{n}^{c+1}}{{\epsilon}}\rceil }^{O({n}^{c+1})}$, we can prove that the hypothesis space $\mathcal{F}$ is PAC learnable with sample complexity

$$\begin{equation*}\nu ({\epsilon},\delta )=O\left(\frac{1}{{{\epsilon}}^{2}}\left({n}^{c+1}\enspace \mathrm{ln}\enspace \frac{n}{{\epsilon}}+\mathrm{ln}\enspace \frac{1}{\delta }\right)\right).\end{equation*}$$

This completes the proof of theorem 2.□

We denote $\mathcal{E}$ as the collection of all the functions ${f}_{C}:\mathcal{X}\to \mathcal{Y}$, where $C\in \mathcal{C}$. In fact, we can prove that $\mathcal{E}$ is PAC learnable as well. By theorem 1, our hypothesis space $\mathcal{F}$ can cover all the quantum circuits in $\mathcal{C}$. Thus we can obtain that $\mathcal{E}\subseteq \mathcal{F}$. Using the inequality (6), we will arrive at

$$\begin{equation}{\mathbb{P}}_{S}\left(\forall \enspace {f}_{h}\in \mathcal{E}:\left\vert \hat{R}\left({f}_{h}\right)-R\left({f}_{h}\right)\right\vert \leqslant \frac{{\epsilon}}{2}\right)\geqslant 1-\delta ,\end{equation} \tag{ 7 }$$

when $\vert S\vert \geqslant \frac{18}{{{\epsilon}}^{2}}(\mathrm{ln}\vert {\mathcal{F}}^{\prime }\vert +\mathrm{ln}\enspace \frac{2}{\delta })$. Similarly with the method in theorem 2, we combine the inequality (7) with proposition 6, and we can prove that $\mathcal{E}$ is PAC learnable with sample complexity $\nu ({\epsilon},\delta )=O(\frac{1}{{{\epsilon}}^{2}}({n}^{c+1}\enspace \mathrm{ln}\enspace \frac{n}{{\epsilon}}+\mathrm{ln}\enspace \frac{1}{\delta }))$ as well.

We stress the differences between our results and the previous works [28, 33] in the literature. First, in reference [28] Chung and Lin focused on a finite set of discretized quantum channels, and their algorithm is based on random orthogonal measurements. Whereas, in our settings we focus on a set of unitary quantum circuits with continuous variational parameters, thus the size of our concept class is infinite. Moreover, our proof is based on a family of variational quantum neural networks with an explicit uniform structure, which would be useful in practical applications. Second, in reference [33] Bu et al considered a more general class of quantum channels and their bounds of the sample complexity grow exponentially with the number of qubits n. In contrast, our focus here is variational quantum circuits and the sample complexity we obtained scales only polynomially with the system size. In other words, while reference [33]'s setting is more general, the sample complexity bounds obtained in this work is exponentially tighter. Our work and references [28, 33] are complementary to each other.

It is also worthwhile to clarify that, although we have proved that the sample complexity for learning any physical quantum circuit is low (namely, it only scales polynomially with the number of qubits involved), this does not mean that these circuits can be learned efficiently since the time complexity to learn an unknown circuit can still be exponentially high. In fact, it has been proved recently that training a variational quantum circuit, even for logarithmically many qubits and free fermionic systems, is NP-hard [43]. This implies that although we know for sure that our hypothesis $\mathcal{F}$ can cover all physical quantum circuits and only a polynomial number of samples are needed to train a variational circuit $F\in \mathcal{F}$, how to efficiently solve the optimization problem of minimizing the empirical risk remains unclear and might be an exponentially hard problem in practice.