Quantum Circuit Approximations and Entanglement Renormalization for the Dirac Field in 1+1 Dimensions

Abstract

The multiscale entanglement renormalization ansatz describes quantum many-body states by a hierarchical entanglement structure organized by length scale. Numerically, it has been demonstrated to capture critical lattice models and the data of the corresponding conformal field theories with high accuracy. However, a rigorous understanding of its success and precise relation to the continuum is still lacking. To address this challenge, we provide an explicit construction of entanglement-renormalization quantum circuits that rigorously approximate correlation functions of the massless Dirac conformal field theory. We directly target the continuum theory: discreteness is introduced by our choice of how to probe the system, not by any underlying short-distance lattice regulator. To achieve this, we use multiresolution analysis from wavelet theory to obtain an approximation scheme and to implement entanglement renormalization in a natural way. This could be a starting point for constructing quantum circuit approximations for more general conformal field theories.

Proofs of Wavelet Lemmas

In this section we will prove some technical lemmas involving wavelets, amongst which Lemma 3.1, Lemma 3.2, Lemma 3.3. We first state a simple Lipschitz bound for the Fourier transforms of wavelet and scaling filters.

Lemma A.1

Let $g_s$ be scaling filter supported in $\{0,\dots ,M-1\}$. Then the corresponding wavelet filter $g_w$, defined in Eq. (3.3), is supported in $\{2-M,\dots ,1\}$ and we have that

$$\begin{aligned} |{\hat{g}}_s(\theta ) - \sqrt{2} |&\leqslant \frac{M^2}{\sqrt{2}} \, |\theta |, \\ |{\hat{g}}_w(\theta ) |&\leqslant \frac{M(M+1)}{\sqrt{2}} \, |\theta |. \end{aligned}$$

for all $\theta \in [-\pi ,\pi ]$.

Proof

By Eq. (3.2), $\Vert {\hat{g}}_s \Vert _\infty =\sqrt{2}$ and ${\hat{g}}_s(0) = \sqrt{2}$. Hence,

$$\begin{aligned} \Vert {\hat{g}}_s' \Vert _\infty \leqslant \left( \sum _{n=0}^{M-1} n \right) \Vert g_s\Vert _\infty = \frac{M(M-1)}{2} \Vert g_s\Vert _\infty \leqslant \frac{M(M-1)}{2} \Vert {\hat{g}}_s\Vert _\infty \leqslant \frac{M^2}{\sqrt{2}}, \end{aligned}$$

where we used that $\Vert f\Vert _\infty \leqslant \frac{1}{2\pi } \Vert {\hat{f}}\Vert _1 \leqslant \Vert {\hat{f}}\Vert _\infty$ for any trigonometric polynomial. Therefore,

$$\begin{aligned} |{\hat{g}}_s(\theta ) - \sqrt{2} |\leqslant |{\hat{g}}_s(\theta ) - {\hat{g}}_s(0) |\leqslant \Vert {\hat{g}}_s' \Vert _\infty \, |\theta |\leqslant \frac{M^2}{\sqrt{2}} \, |\theta |. \end{aligned}$$

Now consider the corresponding wavelet filter $g_w$ which by Eqs. (3.3) and (3.2) satisfies $\Vert {\hat{g}}_w \Vert _\infty =\sqrt{2}$ and ${\hat{g}}_w(0) = 0$ and is supported in $\{2-M,\dots ,1\}$. Then, similarly as above,

$$\begin{aligned} \Vert {\hat{g}}_w' \Vert _\infty \leqslant \left( \sum _{n=2-M}^1 |n|\right) \Vert g_w\Vert _\infty \leqslant \frac{M(M+1)}{2} \Vert {\hat{g}}_w\Vert _\infty \leqslant \frac{M(M+1)}{\sqrt{2}}, \end{aligned}$$

so we obtain

$$\begin{aligned} |{\hat{g}}_w(\theta ) |= |{\hat{g}}_w(\theta ) - {\hat{g}}_w(0) |\leqslant \Vert {\hat{g}}_w' \Vert _\infty \, |\theta |\leqslant \frac{M(M+1)}{\sqrt{2}} \, |\theta |. \end{aligned}$$

$\square$

In practice, the bounds in Lemma A.1 can be pessimistic. In principle, if the number of vanishing moments of the wavelets increase, one expects better dependence of the bounds on the size of the support, although we are not aware of better bounds than those in Lemma A.1 for approximate Hilbert pair wavelets.

We now proceed to prove the lemmas in Sect. 3.4. Our main tool is the following technical lemma.

Lemma A.2

Let $\chi \in H^{-K}({\mathbb {R}})$ such that ${\hat{\chi }}\in L^\infty ({\mathbb {R}})$ and there exists a constant $C>0$ such that $|{\hat{\chi }}(\omega ) |\leqslant C |\omega |^K$ for all $|\omega |\leqslant \pi$. Define $C_\chi := (C^2 + \Vert {\hat{\chi }}\Vert ^2_\infty / 3)^{1/2}$. Then, for all $f\in H^K({\mathbb {R}})$ and $j\in {\mathbb {Z}}$ we have that

$$\begin{aligned} \sum _{k\in {\mathbb {Z}}} |\left\langle \chi _{j,k}, f\right\rangle |^2 \leqslant 2^{-2Kj} C_\chi ^2 \Vert f^{(K)} \Vert ^2, \end{aligned}$$

where $\chi _{j,k}(x) := 2^{\frac{j}{2}} \chi (2^j x - k)$. Similarly, for all $f\in H^K({\mathbb {S}}^1)$ and $j\geqslant 0$ we have that

$$\begin{aligned} \sum _{k=1}^{2^j} |\left\langle \chi _{j,k}^{{{\,\mathrm{per}\,}}}, f\right\rangle |^2 \leqslant 2^{-2Kj} C_\chi ^2 \Vert f^{(K)} \Vert ^2, \end{aligned}$$

where $\chi ^{{{\,\mathrm{per}\,}}}_{j,k}(x) = \sum _{m \in {\mathbb {Z}}} \chi _{j,k}(x + m)$.

Proof

For $f\in H^K({\mathbb {R}})$, we start with

$$\begin{aligned} \sum _{k \in {\mathbb {Z}}} \left|\left\langle \chi _{j,k}, f\right\rangle \right|^2&= \sum _{k \in {\mathbb {Z}}} \left|\frac{1}{2\pi } \left\langle \widehat{\chi _{j,k}}, {\widehat{f}}\right\rangle \right|^2 \nonumber \\&= \sum _{k \in {\mathbb {Z}}} \left|\frac{1}{2\pi } \int _{-\infty }^{\infty } 2^{-j/2}e^{i\omega 2^{-j}k} \overline{{\hat{\chi }}(2^{-j}\omega )} {\hat{f}}(\omega ) d\omega \right|^2 \nonumber \\&= \sum _{k\in {\mathbb {Z}}} \left|\frac{1}{2\pi } \int _{-\infty }^\infty 2^{j/2} \overline{{\hat{\chi }}(\omega )} {\hat{f}}(2^j\omega ) e^{i\omega k} d\omega \right|^2. \end{aligned}$$

(A.1)

We can interpret this as the squared norm of the Fourier coefficients of the $2\pi$-periodic function defined by

$$\begin{aligned} F(\theta ) := \sum _{m\in {\mathbb {Z}}} 2^{j/2} \overline{{\hat{\chi }}(\theta + 2\pi m)} {\hat{f}}(2^j(\theta + 2\pi m)), \end{aligned}$$

provided the latter is square integrable. To see this and obtain a quantitative upper bound, we note that, for every $\theta \in [-\pi ,\pi ]$,

$$\begin{aligned} |F(\theta )|^2&\leqslant 2^j \sum _{m\in {\mathbb {Z}}} \left|\frac{{\hat{\chi }}(\theta + 2\pi m)}{(\theta + 2\pi m)^K} \right|^2 \sum _{m \in {\mathbb {Z}}} \left|(\theta + 2\pi m)^K {\hat{f}}(2^j(\theta + 2\pi m)) \right|^2 \nonumber \\&= 2^{-(2K-1)j} \sum _{m\in {\mathbb {Z}}} \left|\frac{{\hat{\chi }}(\theta + 2\pi m) }{(\theta + 2\pi m)^K} \right|^2 \sum _{m\in {\mathbb {Z}}} \left|(2^j(\theta + 2\pi m))^K {\hat{f}}(2^j(\theta + 2\pi m)) \right|^2 \end{aligned}$$

(A.2)

by the Cauchy-Schwarz inequality. To bound the left-hand side series, we split off the term for $m=0$ and use the assumptions on ${\hat{\chi }}$ to bound, for $|\theta |\leqslant \pi$,

$$\begin{aligned} \sum _{m\in {\mathbb {Z}}} \left|\frac{{\hat{\chi }}(\theta + 2\pi m)}{(\theta + 2\pi m)^K} \right|^2&= \left|\frac{{\hat{\chi }}(\theta )}{\theta ^K} \right|^2 + \sum _{m\ne 0} \left|\frac{{\hat{\chi }}(\theta + 2\pi m)}{(\theta + 2\pi m)^K} \right|^2 \leqslant C^2 + \sum _{m\ne 0} \frac{|{\hat{\chi }}(\theta + 2\pi m)|^2}{|\theta + 2\pi m|^{2K}} \nonumber \\&\leqslant C^2 + \Vert {\hat{\chi }}\Vert _\infty ^2 \sum _{m=1}^\infty \frac{2}{(\pi m)^{2K}} \leqslant C^2 + \frac{\Vert {\hat{\chi }}\Vert ^2_\infty }{3} = C_\chi ^2 \end{aligned}$$

(A.3)

If we plug this into Eq. (A.2) then we obtain

$$\begin{aligned} |F(\theta )|^2 \leqslant 2^{-(2K-1)j} C_\chi ^2 \sum _{m\in {\mathbb {Z}}} \left|(2^j(\theta + 2\pi m))^K {\hat{f}}(2^j(\theta + 2\pi m)) \right|^2 \end{aligned}$$

and hence

$$\begin{aligned} \frac{1}{2\pi } \int _{-\pi }^\pi |F(\theta )|^2 d\theta&\leqslant 2^{-(2K - 1)j} \frac{C_\chi ^2}{2\pi } \int _{-\infty }^\infty \left|(2^j \omega )^K {\hat{f}}(2^j\omega ) \right|^2 d\omega \\&= 2^{-2Kj} \frac{C_\chi ^2}{2\pi } \int _{-\infty }^\infty \left|\omega ^K {\hat{f}}(\omega ) \right|^2 d\omega = 2^{-2Kj} C_\chi ^2 \Vert f^{(K)} \Vert ^2, \end{aligned}$$

which is finite since $f\in H^K({\mathbb {R}})$. This shows that $F \in L^2({\mathbb {R}}/2\pi {\mathbb {Z}})$. By Parseval’s theorem we can thus bound Eq. (A.1) by

$$\begin{aligned} \sum _{k \in {\mathbb {Z}}} \left|\left\langle \chi _{j,k}, f\right\rangle \right|^2 \leqslant 2^{-2Kj} C_\chi ^2 \Vert f^{(K)} \Vert ^2 \leqslant 2^{-2Kj} C_\chi ^2 \Vert f^{(K)} \Vert ^2 \end{aligned}$$

as desired.

The proof for $f\in H^K({\mathbb {S}}^1)$ proceeds similarly. First note that $\widehat{g^{{{\,\mathrm{per}\,}}}}(m) = {\hat{g}}(2\pi m)$ if we periodize a function $g\in L^2({\mathbb {R}})$ by $g^{{{\,\mathrm{per}\,}}}(x) := \sum _{n\in {\mathbb {Z}}} g(x+n)$, so

$$\begin{aligned} \sum _{k=1}^{2^j} \left|\left\langle \chi ^{{{\,\mathrm{per}\,}}}_{j,k}, f\right\rangle \right|^2 = \sum _{k=1}^{2^j} \left|\left\langle \widehat{\chi ^{{{\,\mathrm{per}\,}}}_{j,k}}, {\widehat{f}}\right\rangle \right|^2 = \sum _{k=1}^{2^j} \left|\sum _{m\in {\mathbb {Z}}} 2^{-j/2}e^{i2\pi m2^{-j}k} \overline{{\hat{\chi }}(2^{-j}2\pi m)} {\hat{f}}(m) \right|^2 \end{aligned}$$

(A.4)

which we recognize as squared norm of the inverse discrete Fourier transform of a vector v with $2^j$ components

$$\begin{aligned} v_l := 2^{j/2} \sum _{m\in {\mathbb {Z}}} \overline{{\hat{\chi }}(2\pi m + 2\pi 2^{-j} l)} {\hat{f}}(2^j m + l), \end{aligned}$$

where it is useful to take $l\in \{-2^{j-1}+1,\dots ,2^{j-1}\}$. To see that the components of this vector are well-defined and obtain a quantitative bound, we estimate

$$\begin{aligned} |v_l|^2&= 2^j \left|\sum _{m\in {\mathbb {Z}}} \overline{{\hat{\chi }}(2\pi m + 2\pi 2^{-j} l)} {\hat{f}}(2^j m + l) \right|^2 \\&\leqslant 2^j \sum _{m\in {\mathbb {Z}}} \left|\frac{{\hat{\chi }}(2\pi m + 2\pi 2^{-j} l)}{(2\pi m + 2\pi 2^{-j} l)^K} \right|^2 \sum _{m\in {\mathbb {Z}}} \left|(2\pi m + 2\pi 2^{-j} l)^K {\hat{f}}(2^j m + l) \right|^2 \\&= 2^{-(2K-1)j} \sum _{m\in {\mathbb {Z}}} \left|\frac{{\hat{\chi }}(2\pi m + 2\pi 2^{-j} l)}{(2\pi m + 2\pi 2^{-j} l)^K} \right|^2 \sum _{m\in {\mathbb {Z}}} \left|(2\pi (2^j m + l))^K {\hat{f}}(2^j m + l) \right|^2. \end{aligned}$$

Since $|2\pi 2^{-j} l|\leqslant \pi$, we can upper-bound the left-hand side series precisely as in Eq. (A.3),

$$\begin{aligned} |v_l|^2 \leqslant 2^{-(2K-1)j} C_\chi ^2 \sum _{m\in {\mathbb {Z}}} \left|(2\pi (2^j m + l))^K {\hat{f}}(2^j m + l) \right|^2, \end{aligned}$$

and obtain

$$\begin{aligned} \Vert v\Vert _2^2 \leqslant 2^{-(2K-1)j} C_\chi ^2 \sum _{n\in {\mathbb {Z}}} \left|(2\pi n)^K {\hat{f}}(n) \right|^2 = 2^{-(2K-1)j} C_\chi ^2 \Vert f^{(K)} \Vert ^2, \end{aligned}$$

which is finite since $f\in H^K({\mathbb {S}}^1)$. As before we conclude by using the Plancherel formula in Eq. (A.4) and plugging in the upper bound.

$$\begin{aligned} \sum _{k=1}^{2^j} \left|\left\langle \chi ^{{{\,\mathrm{per}\,}}}_{j,k}, f\right\rangle \right|^2 = 2^{-j} \sum _{k=1}^{2^j} |v_k|^2 \leqslant 2^{-2Kj} C_\chi ^2 \Vert f^{(K)} \Vert ^2, \end{aligned}$$

which concludes the proof. $\square$

We next use Lemma A.2 to prove Lemma 3.1 and Lemma 3.2, which are wavelet approximation results for sufficiently smooth functions.

Proof of Lemma 3.1

For $f\in H^K({\mathbb {R}})$ and $j\in {\mathbb {Z}}$, we have

$$\begin{aligned} \Vert P_j f - f \Vert ^2 = \sum _{l > j} \sum _{k \in {\mathbb {Z}}} |\left\langle \psi _{l,k},f\right\rangle |^2. \end{aligned}$$

because the wavelets form an orthonormal basis. We would like to bound the inner series by using Lemma A.2. For this, note that since $\hat{g}_s$ is a trigonometric polynomial with a zero of order K at $\theta = \pi$, there exists a constant C such that

$$\begin{aligned} \frac{1}{\sqrt{2}}|\hat{g}_w(\theta ) |= \frac{1}{\sqrt{2}}|\hat{g}_s(\theta + \pi ) |\leqslant C |\theta |^K. \end{aligned}$$

(A.5)

Using Eq. (3.6) and $\Vert {\hat{\phi }}\Vert _\infty =1$, it follows that

$$\begin{aligned} |{\hat{\psi }}(\omega ) |= |\frac{1}{\sqrt{2}}\hat{g}_w(\frac{\omega }{2}) \hat{\phi }(\frac{\omega }{2}) |\leqslant \frac{C}{2^K} |\omega |^K. \end{aligned}$$

Since moreover $\Vert {\hat{\psi }}\Vert _\infty =1$, we can invoke Lemma A.2 with $\chi =\psi$ and obtain that

$$\begin{aligned} \Vert P_j f - f \Vert ^2 \leqslant \sum _{l > j} 2^{-2Kl} C_{{{\,\mathrm{UV}\,}}}^2 \Vert f^{(K)} \Vert ^2 \leqslant 2^{-2Kj} C_{{{\,\mathrm{UV}\,}}}^2 \Vert f^{(K)} \Vert ^2, \end{aligned}$$

where $C_{{{\,\mathrm{UV}\,}}}^2 = C^2/4^K + 1/3 \leqslant C^2 + 1/3$.

In the same way we find that, for any $f\in H^K({\mathbb {S}}^1)$ and $j\geqslant 0$,

$$\begin{aligned} \Vert P^{{{\,\mathrm{per}\,}}}_j f - f \Vert ^2 = \sum _{l > j} \sum _{k=1}^{2^l} |\left\langle \psi ^{{{\,\mathrm{per}\,}}}_{l,k},f\right\rangle |^2 \leqslant 2^{-2Kj} C_{{{\,\mathrm{UV}\,}}}^2 \Vert f^{(K)} \Vert ^2, \end{aligned}$$

again by Lemma A.2.

For the last assertion, we use Lemma A.1 to see that, for $K=1$, Eq. (A.5) always holds with $C=M(M+1)/2$, hence we have $C_{{{\,\mathrm{UV}\,}}} \leqslant 2M^2$. $\square$

Proof of Lemma 3.2

The trigonometric polynomial ${\hat{g}}_s$ satisfies ${\hat{g}}_s(0)=\sqrt{2}$, so there is a constant $C>0$ such that

$$\begin{aligned} |\frac{1}{\sqrt{2}} {\hat{g}}_s(\theta ) - 1|\leqslant C |\theta |\end{aligned}$$

(A.6)

for $\theta \in [-\pi ,\pi ]$. Using the infinite product formula (3.8), it follows that, for all $|\omega |\leqslant \pi$,

$$\begin{aligned} |{\hat{\phi }}(\omega ) - 1 |&= |\prod _{k=1}^\infty \frac{1}{\sqrt{2}} {\hat{g}}_s(2^{-k}\omega ) - 1 |\leqslant \sum _{k=1}^\infty |\frac{1}{\sqrt{2}} {\hat{g}}_s(2^{-k}\omega ) - 1 |\leqslant \sum _{k=1}^\infty \frac{C}{\sqrt{2}} 2^{-k} |\omega |= \frac{C}{\sqrt{2}} |\omega |\end{aligned}$$

(A.7)

using a telescoping sum and the fact that $|{\hat{g}}_s|\leqslant \sqrt{2}$ (in fact, this holds for all $\omega \in {\mathbb {R}}$, but we will not need this). Now recall from Sobolev embedding theory that ${\hat{f}} \in L^1({\mathbb {R}})$ for any $f\in H^1({\mathbb {R}})$. Thus, the continuous representative of f can be computed by the inverse Fourier transform, i.e.,

$$\begin{aligned} f(x) = \frac{1}{2\pi } \int _{-\infty }^\infty {\hat{f}}(\omega ) e^{i\omega x} d\omega \end{aligned}$$

for all $x\in {\mathbb {R}}$. As a consequence,

$$\begin{aligned} \Vert \alpha _j f - f_j \Vert ^2 = \sum _{k\in {\mathbb {Z}}} \left|\left\langle \phi _{j,k}, f\right\rangle - 2^{-j/2} f(2^{-j} k)\right|^2 = \sum _{k\in {\mathbb {Z}}} \left|\left\langle \chi _{j,k}, f\right\rangle \right|^2 \end{aligned}$$

where $\chi := \phi - \delta _0$. Now, ${\hat{\chi }} = {\hat{\phi }} - {\mathbf {1}}$, hence $\Vert {\hat{\chi }}\Vert _\infty \leqslant 2$. Together with the bound in Eq. (A.7) we obtain from Lemma A.2 that

$$\begin{aligned} \Vert \alpha _j f - f_j \Vert ^2 \leqslant 2^{-2j} C_\phi \Vert f' \Vert ^2, \end{aligned}$$

where $C_\phi := C^2 + \frac{4}{3}$. The proof for $H^1({\mathbb {S}}^1)$ proceeds completely analogously. Finally, Lemma A.1 shows that if the scaling filter is supported in $\{0,\dots ,M-1\}$ then Eqs. (A.6) and (A.7) always hold with $C=M^2/2$. Thus, $C_\phi \leqslant 2 M^2$. $\square$

Finally, we prove Lemma 3.3, which is an approximation result for compactly supported functions.

Proof of Lemma 3.3

Let us denote by S the support of f. Since the scaling functions for fixed j form an orthonormal basis of $V_j$, and using Cauchy-Schwarz, we find that

$$\begin{aligned} \Vert P_j f\Vert ^2 = \sum _{k\in {\mathbb {Z}}} |\left\langle \phi _{j,k}, f\right\rangle |^2 \leqslant \Vert f\Vert ^2 \sum _{k\in {\mathbb {Z}}} \int _S |\phi _{j,k}(x) |^2 \, dx = \Vert f\Vert ^2 \int _{2^j S} \sum _{k\in {\mathbb {Z}}} |\phi (y - k)|^2 \, dy. \end{aligned}$$

This allows us to conclude that

$$\begin{aligned} \Vert P_j f\Vert ^2 \leqslant \Vert f\Vert ^2 2^j C_{{{\,\mathrm{IR}\,}}}^2 D(f), \end{aligned}$$

which confirms the claim. If $\phi$ is bounded and supported on an interval of width M, we can bound $\sum _{k\in {\mathbb {Z}}} |\phi (y - k)|^2 \leqslant M \Vert \phi \Vert _\infty ^2$. $\square$