An efficient superpostional quantum Johnson–Lindenstrauss lemma via unitary t-designs

Abstract

The famous Johnson–Lindenstrauss lemma states that for any set of n vectors $\{v_i\}_{i=1}^n \in \mathbb {C}^{d_1}$ and any $\epsilon > 0$, there is a linear transformation $T: \mathbb {C}^{d_1} \rightarrow \mathbb {C}^{d_2}$, $d_2 = O(\epsilon ^{-2} \log n)$ such that $\left\| { T(v_i) }\right\| _2 \in (1 \pm \epsilon ) \left\| { v_i }\right\| _2$ for all $i \in [n]$. In fact, a Haar random $d_1 \times d_1$ unitary transformation followed by projection onto the first $d_2$ coordinates followed by a scaling of $\sqrt{\frac{d_1}{d_2}}$ works as a valid transformation T with high probability. In this work, we show that the Haar random $d_1 \times d_1$ unitary can be replaced by a uniformly random unitary chosen from a finite set called an approximate unitary t-design for $t = O(d_2)$. Choosing a unitary from such a design requires only $O(d_2 \log d_1)$ random bits as opposed to $2^{\varOmega (d_1^2)}$ random bits required to choose a Haar random unitary with reasonable precision. Moreover, since such unitaries can be efficiently implemented in the superpositional setting, our result can be viewed as an efficient quantum Johnson–Lindenstrauss transform akin to efficient quantum Fourier transforms widely used in earlier work on quantum algorithms. We prove our result by leveraging a method of Low for showing concentration for approximate unitary t-designs. We discuss algorithmic advantages and limitations of our result and conclude with a toy application to private information retrieval.