[Paper Review] A quantum Johnson-Lindenstrauss lemma via unitary t-designs

Pranab Kumar Sen|arXiv (Cornell University)|Jul 23, 2018

Random Matrices and Applications15 references3 citations

TL;DR

This paper presents a quantum Johnson-Lindenstrauss lemma that replaces Haar random unitaries with unitary t-designs for efficient quantum dimensionality reduction. By using an approximate t-design with t = O(d₂), the method achieves the same distortion guarantees as the classical Johnson-Lindenstrauss lemma while requiring only O(d₂ log d₁) random bits and enabling efficient quantum circuit implementation, offering a scalable alternative to full Haar randomness in superpositional settings.

ABSTRACT

The famous Johnson-Lindenstrauss lemma states that for any set of n vectors, there is a linear transformation into a space of dimension O(log n) that approximately preserves all their lengths. In fact, a Haar random unitary transformation followed by projection onto the first O(log n) coordinates followed by a scaling works as a valid transformation with high probability. In this work, we show that the Haar random unitary can be replaced by a uniformly random unitary chosen from a finite set called an approximate unitary t-design for t = O(log n). Choosing a unitary from such a design requires only polylogarithmic random bits as opposed to exponential in dimension 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.

Motivation & Objective

• To develop an efficient quantum analog of the classical Johnson-Lindenstrauss lemma for quantum state embeddings.
• To replace Haar random unitaries—requiring exponential randomness—with finite, efficiently sampleable unitary t-designs.
• To ensure that pairwise inner products between quantum states are preserved up to (1±ε) distortion with high probability.
• To enable practical implementation in quantum algorithms by reducing randomness and circuit depth.
• To explore applications in quantum protocols such as private information retrieval.

Proposed method

• Leverages unitary t-designs with t = Θ(d₂) to approximate Haar randomness in quantum dimensionality reduction.
• Uses the TPE (Tensor Product Extension) framework to bound the difference between t-design and Haar expectation over tensor powers of unitaries.
• Applies Markov’s inequality to control the tail probability of norm distortion after projection onto d₂-dimensional blocks.
• Employs concentration bounds via Low’s method to show that t-designs with t = O(d₂) suffice for high-probability distortion control.
• Constructs the transformation as: apply a random unitary from a t-design, project onto a random block of d₂ coordinates, and scale by √(d₁/d₂).
• Utilizes efficient constructions of local random quantum circuits that form approximate t-designs with polylogarithmic circuit depth and randomness.

Experimental results

Research questions

• RQ1 Can unitary t-designs with t = O(d₂) replace Haar random unitaries in quantum Johnson-Lindenstrauss transforms while preserving distortion guarantees?
• RQ2 What is the minimal t for which a t-design yields a valid quantum Johnson-Lindenstrauss transform with high probability?
• RQ3 How much randomness is required to implement such a transform compared to Haar sampling?
• RQ4 Can the resulting transform be efficiently implemented in a quantum circuit for superpositional states?
• RQ5 What are the algorithmic limitations and potential applications of this transform?

Key findings

• A unitary t-design with t = Θ(d₂) ensures that the projected norm of any quantum state is preserved within (1±ε) of its original norm with high probability.
• The number of random bits required to sample from such a t-design is O(d₂ log d₁), a significant improvement over the 2^Ω(d₁²) bits needed for Haar sampling.
• The transformation can be implemented by a quantum circuit of size polylog(d₁), making it efficient for quantum algorithms.
• The method achieves concentration of measure via the TPE framework, with error decaying exponentially in d₂.
• The distribution over block names is nearly uniform, limiting control in classical settings but enabling new quantum applications.
• A toy application to private information retrieval is demonstrated, highlighting potential for future use in quantum protocols.