Learning bounds for quantum circuits in the agnostic setting

Abstract

In this paper, we investigate the learnability of some hypothesis sets for regression and binary classification defined by quantum circuits. The analysis is based on concepts and results from quantum computing (Solovay–Kitaev theorem) and statistical learning theory (covering numbers and Rademacher complexity). The obtained learning bounds depend polynomially on the parameters defining the circuits set, namely the number of qubits and the number of 1 and 2 qubits gates used for their implementation. Our setting is quite general: no realisability assumptions are made, and any 1 and 2 qubits gates are allowed. Finally, we compare the current bounds with others found in the literature and discuss their implications for classification and regression on quantum data.

Appendices

Additional proofs

In this appendix, we present the proofs of the theorems from Sect. 4 which were not proved there. For convenience, the theorems are restated.

Theorem 5

Let $\mathcal {H}_{d,\gamma }$ be the hypothesis set defined by Eq. (3) and S a sample of size n. Then, we have

$$\begin{aligned} log N(\epsilon , \mathcal {H}_{d,\gamma }, S) = O(\gamma log^{c}(\frac{\gamma \sqrt{n}}{\epsilon })log(q)), \end{aligned}$$

(28)

where $3 < c \le 4$ is the constant from Solovay–Kitaev theorem.

Proof

For any family of circuits $\mathcal {C}_{d,\gamma }$, we will define a finite family $\mathcal {C}_{d,\gamma , \epsilon '}$, for some $\epsilon >0$, in the following way: $\mathcal {C}_{d,\gamma , \epsilon '}$ is the finite set of circuits of minimum cardinality such that for any circuit C in $\mathcal {C}_{d,\gamma }$, there is a circuit $C_{\epsilon '}$ in $\mathcal {C}_{d,\gamma , \epsilon '}$ that has the same positions for all the gates, and for each gate of C the analog gate in $C_\epsilon '$ is an $\epsilon '$ approximation of it (in operator norm). By universality property [48], such a construction is possible. The circuits in $\mathcal {C}_{d,\gamma , \epsilon '}$ will be called approximation circuits.

Let as call the architecture of a circuit a specific placement of each gate. For the set of all possible architectures for circuits of size at most $\gamma$ on $q = log_{2}d$ qubits we use the notation $\mathcal {M}_{d,\gamma }$. We will now focus our attention on circuits with a fixed architecture $\mathcal {N} \in \mathcal {M}_{d,\gamma }$, denoted by $\mathcal {C}_{d,\gamma }^{\mathcal {N}}$.

By a method analog to the one presented in the first paragraph, we can define for a fixed architecture $\mathcal {N} \in \mathcal {M}_{d,\gamma }$, the corresponding family of approximation circuits $\mathcal {C}_{d,\gamma , \epsilon '}^{\mathcal {N}}$. For each set of circuits, we have also an associated family of unitary operators, $\mathcal {U}_{d,\gamma }^{\mathcal {N}}$ and $\mathcal {U}_{d,\gamma ,\epsilon '}^{\mathcal {N}}$, respectively (see Eq. (2)).

For a sample S, the hypothesis sets defined based on the two families of circuits, $\mathcal {C}_{d,\gamma }^{\mathcal {N}}$ and $\mathcal {C}_{d,\gamma , \epsilon '}^{\mathcal {N}}$, will generate the following subsets of $[0,1]^{n}$:

$$\begin{aligned}&\mathcal {A}_{d,\gamma ,S}^{\mathcal {N}} =\left\{ u \in [0,1]^{n}| u_i = |{\langle {0}|}^{\otimes q}U{|{\phi _i}\rangle }|^2, U \in \mathcal {U}_{d,\gamma }^{\mathcal {N}}, i \in [n] \right\} . \end{aligned}$$

(29)

$$\begin{aligned}&\mathcal {A}_{d,\gamma ,\epsilon ',S}^{\mathcal {N}} = \left\{ u \in [0,1]^{n}| u_i = |{\langle {0}|}^{\otimes q}U'{|{\phi _i}\rangle }|^2, U' \in \mathcal {U}_{d,\gamma ,\epsilon '}^{\mathcal {N}}, i \in [n] \right\} . \end{aligned}$$

(30)

A similar set can be defined for $\mathcal {C}_{d,\gamma }$:

$$\begin{aligned} \mathcal {A}_{d,\gamma ,S} =\left\{ u \in [0,1]^{n}| u_i = |{\langle {0}|}^{\otimes q}U{|{\phi _i}\rangle }|^2, U \in \mathcal {U}_{d,\gamma }, i \in [n] \right\} . \end{aligned}$$

(31)

For a fixed $\epsilon$, let us take $\epsilon ' = \frac{\epsilon }{2\sqrt{n}\gamma }$. For some $u \in \mathcal {A}_{d,\gamma ,S}^{\mathcal {N}}$, we choose $u' \in \mathcal {A}_{d,\gamma ,\epsilon ',S}^{\mathcal {N}}$ to be the closest vector to it (in Euclidean metric). By the definition of Euclidean metric, we have:

$$\begin{aligned} {\Vert u-u'\Vert }^2 = \sum _{i=1}^{n}\left( |{\langle {0}|}^{\otimes q}U{|{\phi _i}\rangle }|^2-|{\langle {0}|}^{\otimes q}U'{|{\phi _i}\rangle }|^2\right) ^2. \end{aligned}$$

Using elementary algebra, we get

$$\begin{aligned} {\Vert u-u'\Vert }^2 = \sum _{i=1}^{n} (|{\langle {0}|}^{\otimes q}U{|{\phi _i}\rangle }|- |{\langle {0}|}^{\otimes q}U'{|{\phi _i}\rangle }|)^{2}(|{\langle {0}|}^{\otimes q}U{|{\phi _i}\rangle }|+|{\langle {0}|}^{\otimes q}U'{|{\phi _i}\rangle }|)^2 \nonumber \\ \end{aligned}$$

(32)

The first factor from each term of the right hand side of Eq. (32) can be bounded as follows:

$$\begin{aligned}&|{\langle {0}|}^{\otimes q}U{|{\phi _i}\rangle }|-|{\langle {0}|}^{\otimes q}U'{|{\phi _i}\rangle }|\\&\quad \le |{\langle {0}|}^{\otimes q}U{|{\phi _i}\rangle }-{\langle {0}|}^{\otimes q}U'{|{\phi _i}\rangle }| = \\&\quad =|{\langle {0}|}^{\otimes q}U-U'{|{\phi _i}\rangle }| \le {\Vert U-U'\Vert }, \end{aligned}$$

where the first inequality is based on triangle inequality, the equality is by distributivity of inner product, and the last inequality follows from the definition of operator norm, taking into account that the vectors ${\langle {0}|}^{\otimes q}$ and ${|{\phi _i}\rangle }$ have unit length.

The matrices U and $U'$ can be expressed as $U = U'_\gamma U_{\gamma - 1}\ldots U_{1}$ and $U' = U'_\gamma U'_{\gamma - 1}\ldots U'_{1}$, respectively, where $U_i$ and $U'_i$ are the transformations corresponding to the layer i in the two circuits. By our assumption that on each layer we only have one 2-qubits local gate, any $U_i$ can be written as

$$\begin{aligned} U_i = \underbrace{I_2 \otimes I_2 \otimes \ldots \otimes I_2}_{\text {j-2 factors}} \otimes U_{gi} \underbrace{\otimes I_2 \otimes \ldots \otimes I_2 \otimes I_2}_{\text {q-j factors}}, \text { for some j}\in [2,q], \end{aligned}$$

(33)

where $U_{gi}$ is the 4-dimensional unitary matrix associated with the gate of the layer i.

By the same considerations, we have

$$\begin{aligned} U'_i = \underbrace{I_2 \otimes I_2 \otimes \ldots \otimes I_2}_{\text {j-2 factors}} \otimes U'_{gi} \underbrace{\otimes I_2 \otimes \ldots \otimes I_2 \otimes I_2}_{\text {q-j factors}}, \text { for some j}\in [2,q], \end{aligned}$$

(34)

$U'_{gi}$ being the unitary matrix associated to the gate of the layer i in the approximation circuit.

Introducing the notation $E_i = U_i - U'_i$, and using the distributivity of tensor product over sums, the property ${\Vert A \otimes B\Vert } = {\Vert A\Vert }{\Vert B\Vert }, \forall A,B$ arbitrary matrices, and the fact that the operator norm of the identity matrix is 1, we can write

$$\begin{aligned} {\Vert E_i\Vert }&= {\Vert I_2 \otimes \ldots \otimes I_2 \otimes U_{gi} \otimes I_2 \otimes \ldots \otimes I_2 - I_2 \otimes \ldots \otimes I_2 \otimes U'_{gi} \otimes I_2 \otimes \ldots \otimes I_2\Vert } = \\&= {\Vert I_2 \otimes \ldots \otimes I_2 \otimes (U_{gi}-U'_{gi}) \otimes I_2 \otimes \ldots \otimes I_2 \Vert } = {\Vert I_{2^{j-2}}(U_{gi}-U'_{gi})I_{2^{q-j}}\Vert } \le \\&\le {\Vert I_{2^{j-2}}\Vert }{\Vert U_{gi}-U'_{gi}\Vert }{\Vert I_{2^{q-j}}\Vert } = {\Vert U_{gi}-U'_{gi}\Vert } \le \epsilon '. \end{aligned}$$

The last inequality is by construction.

Using the notation $U_{\gamma 2} = U_\gamma U_{\gamma - 1}\ldots U_{2}$ and $U'_{\gamma 2} = U'_\gamma U'_{\gamma - 1}\ldots U'_{2}$, we have the following chain of equalities (the strategy is inspired by [48], Section 4.5.3)

$$\begin{aligned} {\Vert U-U'\Vert }&= {\Vert U_{\gamma 2}U_{1} - U'_{\gamma 2}U'_{1}\Vert } ={\Vert U_{\gamma 2}U_{1} - U_{\gamma 2}U'_{1} + U_{\gamma 2}U'_{1} - U'_{\gamma 2}U'_{1}\Vert } = \\&= {\Vert U_{\gamma 2}(U_{1}-U'_{1}) + (U_{\gamma 2} - U'_{\gamma 2})U'_{1}\Vert } \le \\&\le {\Vert U_{1}-U'_{1}\Vert } + {\Vert U_{\gamma 2} - U'_{\gamma 2}\Vert } \le \epsilon ' + {\Vert U_{\gamma 2} - U'_{\gamma 2}\Vert } \end{aligned}$$

By induction, we have

$$\begin{aligned} {\Vert U-U'\Vert } \le \gamma \epsilon '. \end{aligned}$$

(35)

Going back to Eq. (32), we observe that the second factor from each term of the right hand side can be bonded as

$$\begin{aligned} |{\langle {0}|}^{\otimes q}U{|{\phi _i}\rangle }|+|{\langle {0}|}^{\otimes q}U'{|{\phi _i}\rangle }| \le 2, \end{aligned}$$

(36)

and using Eq. (35), the Eq. (32) reduces to

$$\begin{aligned} {\Vert u-u'\Vert }^2 = \sum _{i=1}^{n}4\gamma ^{2}\epsilon '^{2} \le 4n\gamma ^{2} \epsilon '^{2}. \end{aligned}$$

(37)

Therefore, remembering that $\epsilon ' = \frac{\epsilon }{2\sqrt{n}\gamma }$, we have

$$\begin{aligned} {\Vert u-u'\Vert } \le 2\sqrt{n}\gamma \epsilon ' = \epsilon . \end{aligned}$$

(38)

We have shown that the set $\mathcal {A}_{d,\gamma ,\epsilon ',S}^{\mathcal {N}}$ is a covering for $\mathcal {A}_{d,\gamma ,S}^{\mathcal {N}}$ at scale $\epsilon$. From this point all that remains is to find the size of the minimum set of approximation circuits.

By Solovay–Kitaev theorem ([39], see also Theorem 1), it is known that to approximate a 2-qubits gate to an error of at most $\epsilon '$, $C_{1}log^{c}\frac{1}{\epsilon '}$ universal gates are needed (where $C_1$ is some constant). Since for a fixed architecture we have $\gamma$ 2-qubits gates in the circuit, the total number of elements from the universal set needed will be:

$$\begin{aligned} l = C_{1}\gamma log^{c}\frac{1}{\epsilon '}. \end{aligned}$$

(39)

Because a universal set of gates with 8 elements exists (see Sect. 3.2.1), the set $\mathcal {A}_{d,\gamma ,S}^{\mathcal {N}}$ can be covered by a set $\mathcal {A}_{d,\gamma ,\epsilon ',S}^{\mathcal {N}}$, having the cardinality

$$\begin{aligned} N \left( \epsilon , \mathcal {A}_{d,\gamma ,S}^{\mathcal {N}}\right) = 8^l. \end{aligned}$$

(40)

The number of architectures is upper bounded by $q^{\gamma }$, because on each layer we have one 2-qubits gate that can be placed in $q-1$ ways (the gates are restricted to act on consecutive qubits). Therefore, a minimum covering of set $\mathcal {A}_{d,\gamma ,S}$ will have its cardinality bounded by the product between the number of architectures and the covering number of the set associated with a fixed architecture:

$$\begin{aligned} N\left( \epsilon , \mathcal {A}_{d,\gamma ,S}\right) = q^{\gamma }8^l \le q^{\gamma + l}, \end{aligned}$$

(41)

where we have used the fact that $q \ge 8$. By taking the logarithm in Eq. (41), we have

$$\begin{aligned} log N(\epsilon , \mathcal {A}_{d,\gamma ,S})&\le log (q^{\gamma + l}) = (\gamma + C_{1}\gamma log^{c}\frac{1}{\epsilon '})log(q) = \\&= (\gamma + C_{1}\gamma log^{c}\frac{2\sqrt{n}\gamma }{\epsilon })log(q), \end{aligned}$$

and this inequality implies the conclusion. $\square$

In order to prove Theorem 6, we make use of the following lemma:

Lemma 1

With the notation and assumptions from Sect. 3, it holds that

$$\begin{aligned} \sqrt{log N(C_2\sqrt{n}2^{-k}, \mathcal {H}_{d,\gamma }, S)} = O(k^2\sqrt{\gamma log^{c}(\gamma )log(q)}), \end{aligned}$$

(42)

for an arbitrary $0 < C_2 \le 1$, and $k \in {\mathbb {N}}^{*}$.

Proof

By Theorem 5, there is a constant $C_3>0$ such that

$$\begin{aligned} \sqrt{log N(C_2\sqrt{n}2^{-k}, \mathcal {H}_{d,\gamma }, S)} \le \sqrt{C_3 \gamma log^{c}(\frac{\gamma \sqrt{n}}{C_{2}\sqrt{n}2^{-k}})log(q)} \end{aligned}$$

(43)

Taking into consideration that $c \le 4$, and the elementary inequality $log^{c}(ab) = (log(a) + log(b))^c \le log^{c}(a)log^{c}(b)$, true for any $a,b>0$ and large enough, we have the following chain of inequalities that implies the conclusion ($C_5$ is just another constant):

$$\begin{aligned} \sqrt{log N(C_2\sqrt{n}2^{-k}, \mathcal {H}_{d,\gamma },S)}&\le \sqrt{C_3 \gamma log^{c}(\frac{\gamma 2^{k}}{C_{2}})log(q)} \le \sqrt{C_5 \gamma k^{c} log^{c}(\frac{\gamma }{C_{2}})log(q)} \le \\&\le k^{2} \sqrt{C_5 \gamma log^{c}(\frac{\gamma }{C_{2}})log(q)}. \end{aligned}$$

$\square$

Theorem 6

The empirical Rademacher complexity of the class of functions $\mathcal {H}_{d,\gamma }$, defined by Eq. (3) is asymptotically bounded as follows:

$$\begin{aligned} \mathcal {R}_{S}(\mathcal {H}_{d,\gamma }) = O \big ( \sqrt{\frac{\gamma log^{c}(\gamma )log(q)}{n}}\big ), \end{aligned}$$

(44)

where $3 < c \le 4$ is the constant from Solovay–Kitaev theorem.

Proof

By applying Theorem 2, we have

$$\begin{aligned} \mathcal {R}_{S}(\mathcal {H}_{d,\gamma }) \le \frac{s2^{-M}}{\sqrt{n}} + \frac{6s}{n}\sum _{k=1}^{M}2^{-k}\sqrt{log(N(s2^{-k}, \mathcal {H}_{d,\gamma }, S))}, \end{aligned}$$

(45)

for any $M \in {\mathbb {N}}^*$, and $s = sup_{a \in \mathcal {A}_{d,\gamma ,S}}{\Vert a\Vert }$.

Taking $M \rightarrow \infty$, the inequality becomes

$$\begin{aligned} \mathcal {R}_{S}(\mathcal {H}_{d,\gamma }) \le \frac{6s}{n}\sum _{k=1}^{\infty }2^{-k}\sqrt{log(N(s2^{-k}, \mathcal {H}_{d,\gamma }, S))}. \end{aligned}$$

(46)

Because for any $h \in \mathcal {H}_{d,\gamma }$ and ${|{\phi }\rangle } \in \mathbb {C}^d$, $0 \le h({|{\phi }\rangle }) \le 1$, and the hypothesis set is rich enough, we have $s = C_2\sqrt{n}$, for some constant $0 < C_2 \le 1$. The inequality becomes

$$\begin{aligned} \mathcal {R}_{S}(\mathcal {H}_{d,\gamma }) \le \frac{6C_2}{\sqrt{n}}\sum _{k=1}^{\infty }2^{-k}\sqrt{log(N(C_{2}\sqrt{n}2^{-k}, \mathcal {H}_{d,\gamma }, S))}. \end{aligned}$$

(47)

Using Lemma 1, the inequality reduces to

$$\begin{aligned} \mathcal {R}_{S}(\mathcal {H}_{d,\gamma }) \le \frac{C_4}{\sqrt{n}}\sqrt{\gamma log^{c}(\gamma )log(q)}\sum _{k=1}^{\infty }k^{2}2^{-k} \end{aligned}$$

(48)

($C_4>0$ is another constant).

Using the fact that $\sum _{k=1}^{\infty }k^{2}2^{-k} = 6$, we arrive at the conclusion:

$$\begin{aligned} \mathcal {R}_{S}(\mathcal {H}_{d,\gamma }) \le 6C_4 \sqrt{\frac{\gamma log^{c}(\gamma )log(q)}{n}}. \end{aligned}$$

(49)

$\square$

Notations

Table 2 summarises the notation used in this paper.

Table 2 Notation summary