Hay from the Haystack: Explicit Examples of Exponential Quantum Circuit Complexity

Abstract

The vast majority of quantum states and unitaries have circuit complexity exponential in the number of qubits. In a similar vein, most of them also have exponential minimum description length, which makes it difficult to pinpoint examples of exponential complexity. In this work, we construct examples of constant description length but exponential circuit complexity. We provide infinite families such that each element requires an exponential number of two-qubit gates to be generated exactly from a product and where the same is true for the approximate generation of the vast majority of elements in the family. The results are based on sets of large transcendence degree and discussed for tensor networks, diagonal unitaries and maximally coherent states.

A. Results in Transcendental Number Theory

In this section, we collect results in transcendental number theory that are used in the main text. For a general overview of the topic we refer to the textbooks of Baker [Bak75] or Murty and Rath [MR14].

Theorem A.1

(Lindemann–Weierstrass, Thm. 1.4 in [Bak75]). If $\alpha _1,\dots ,\alpha _n$ are algebraic numbers that are linearly independent over $\mathbb {Q}$, then $e^{\alpha _1}, \dots , e^{\alpha _n}$ are algebraically independent.

Corollary A.2

If $S\subset {\overline{\mathbb {Q}}}$ is an n-dimensional vector space over $\mathbb {Q}$ and $E:=\{e^\lambda |\lambda \in S\}$, then $\gamma (E)=n$.

Proof

The Lindemann–Weierstrass Theorem A.1 implies that $\gamma (E)\ge n$. For the converse inequality, suppose that $\{a_1,\ldots ,a_m\}\subset S$ fulfill a non-trivial linear relation $\sum _{k=1}^m c_k a_k=0$ for some $c_k\in \mathbb {Q}$. Then

$$\begin{aligned} \prod _{k=1}^m \left( e^{a_k}\right) ^{c_k}=1, \end{aligned}$$

which implies that the $\{e^{a_k}\}_{k=1}^m$ are algebraically dependent. Therefore, $\gamma (E)$ cannot exceed n. $\square$

Theorem A.3

(Besicovitch, [Bes40]). Let $p_1,p_2,\dots ,p_s$ be distinct primes, $b_1,b_2,\dots ,b_s$ positive integers not divisible by any of these primes and $a_i:=(b_i p_i)^{1/n_i}$ positive roots for $i=1,\ldots ,s$ and $n_i\in \mathbb {N}$. If $P\in \mathbb {Q}[x_1,x_2,\dots ,x_s]$ is a polynomial with rational coefficients of degree less than or equal to $n_1-1$ with respect to $x_1$, less than or equal to $n_2-1$ with respect to $x_2$, and so on, then $P(a_1,a_2,\dots ,a_s)=0$ can hold only if $P=0$.

Corollary A.4

(see [Ric74] for a proof based on Galois theory). Let $n,d\in \mathbb {N}$. For distinct prime numbers $p_1, \dots ,p_n$, the following $d^n$ algebraic numbers are linearly independent over $\mathbb {Q}$:

$$\begin{aligned} \phi (j):=\left( \prod \limits _{k=1}^{n}p_k^{j_k}\right) ^{\frac{1}{d}},\quad j\in \mathbb {Z}_d^n. \end{aligned}$$

(39)

Proof

This follows immediately from Besicovitch’s theorem (A.3) when setting $b_i=1$ and arguing by contradiction: suppose there would be a non-trivial linear relation of the form $\sum _{j\in \mathbb {Z}_d^n} c_j \phi (j)=0$ with $c_j\in \mathbb {Q}$, then a non-zero polynomial P of the form that is excluded by Theorem A.3 would exist. $\square$

For the following theorem, recall that the degree of an algebraic number is the minimal degree of a monic polynomial $p\in \mathbb {Q}[x]$ that has the number as a root.

Theorem A.5

(Diaz [Dia89], Philippon [Phi86]). If $\alpha \ne 0,1$ is algebraic and $\beta \in {\overline{\mathbb {Q}}}$ has degree $d\ge 2$, then $S:=\{\alpha ^\beta , \ldots , \alpha ^{\beta ^{d-1}}\}$ has $\gamma (S)\ge d/2$.

In fact, according to the Gel’fond-Schneider conjecture (see Chap.24 in [MR14]) it might be $\gamma (S)\ge d-1$.