Overlapping qubits

TL;DR

This paper investigates how overlapping qubits in physical systems can drastically reduce the effective dimensionality from exponential to polynomial, impacting quantum power and proposing a test to certify high dimensionality.

Contribution

It introduces the concept of overlapping qubits, showing they can limit system dimensionality, and provides an efficient, though noise-sensitive, test to verify exponential dimensionality.

Findings

Overlapping qubits can fit in polynomially many dimensions.

A method to certify exponential dimensionality is proposed.

The certification test is sensitive to noise.

Abstract

An ideal system of $n$ qubits has $2^n$ dimensions. This exponential grants power, but also hinders characterizing the system's state and dynamics. We study a new problem: the qubits in a physical system might not be independent. They can "overlap," in the sense that an operation on one qubit slightly affects the others. We show that allowing for slight overlaps, $n$ qubits can fit in just polynomially many dimensions. (Defined in a natural way, all pairwise overlaps can be $\leq \epsilon$ in $n^{O(1/\epsilon^2)}$ dimensions.) Thus, even before considering issues like noise, a real system of $n$ qubits might inherently lack any potential for exponential power. On the other hand, we also provide an efficient test to certify exponential dimensionality. Unfortunately, the test is sensitive to noise. It is important to devise more robust tests on the arrangements of qubits in quantum devices.