What is Quantum Parallelism, Anyhow?

TL;DR

This paper investigates the nature of quantum parallelism, clarifying how it arises from $2^N$-state superpositions and is shaped by interference rather than simple parallel execution. It introduces quantum dataflow diagrams to quantify parallelism through metrics like $T_W$, $T_\infty$, $P$, $\eta_P$, and $\eta_{DI}$, and applies the framework to the Quantum Fourier Transform and Amplitude Amplification to reveal how input encoding and circuit structure govern realized parallelism. It further reevaluates classical parallelism laws, arguing that Amdahl's and Gustafson's laws require quantum-context reinterpretation due to the classical-quantum I/O bottleneck and the central role of destructive interference. The work provides a structured method to analyze quantum parallelism and offers guidance for quantum algorithm design and HPC performance modeling in the presence of quantum I/O and interference effects.

Abstract

Central to the power of quantum computing is the concept of quantum parallelism: quantum systems can explore and process multiple computational paths simultaneously. In this paper, we discuss the elusive nature of quantum parallelism, drawing parallels with classical parallel computing models to elucidate its fundamental characteristics and implications for algorithmic performance. We begin by defining quantum parallelism as arising from the superposition of quantum states, allowing for the exploration of multiple computational paths in parallel. To quantify and visualize quantum parallelism, we introduce the concept of quantum dataflow diagrams, which provide a graphical representation of quantum algorithms and their parallel execution paths. We demonstrate how quantum parallelism can be measured and assessed by analyzing quantum algorithms such as the Quantum Fourier Transform (QFT) and Amplitude Amplification (AA) iterations using quantum dataflow diagrams. Furthermore, we examine the interplay between quantum parallelism and classical parallelism laws, including Amdahl's and Gustafson's laws. While these laws were originally formulated for classical parallel computing systems, we reconsider their applicability in the quantum computing domain. We argue that while classical parallelism laws offer valuable insights, their direct application to quantum computing is limited due to the unique characteristics of quantum parallelism, including the role of destructive interference and the inherent limitations of classical-quantum I/O. Our analysis highlights the need for an increased understanding of quantum parallelism and its implications for algorithm design and performance.

Figures (9)

• Figure 1: A classical deterministic NOT operation can be decomposed in two quantum ROOT-of-NOT operations, exploiting quantum parallelism. $p$ denotes the probability to transition from state 0 to 1 and vice versa. Note that quantum probability addition (instead of classical probability addition) results in a deterministic output. The computation is quantum parallel as it allows for a superposition of states 0 and 1 as execution paths.
• Figure 2: Some of the original quantum algorithms were inspired by antenna arrays, where several antennae emit electromagnetic waves with different phases and amplitudes to create directed single or multiple beams. Parallelism arises from utilizing the signal from different antennas. Similarly to quantum computing, beamforming employs parallelism and interference phenomena to cancel wave propagation in undesired directions.
• Figure 3: A local gate operation $U$ on a qubit is effectively carried out across all $2^N$ quantum states. While $U$ corresponds to a $2 \times 2$ matrix operation in a single-qubit system, it extends to a $2^N \times 2^N$ matrix operation in a multi-qubit system. This principle is often referred to as the principle of local operations.
• Figure 4: Diagram illustrating a prototypical quantum application workflow. Traditional quantum algorithms typically begin by initializing a classical state, followed by the parallel spawning of quantum parallelism by applying Hadamard gates ($H^{\otimes N}$). Subsequently, the input data is encoded, typically in the quantum state's amplitude and phases, or an oracle is applied. Computational processes then occur in superposition before concluding with a READ operation (measurement). Notably, while the initial phase of the algorithms maximizes quantum parallelism, extracting meaningful results often relies on pruning erroneous outcomes via destructive interference.
• Figure 5: Examples of single-qubit gate operations capable of generating quantum parallelism from a classical state, where no superposition exists. In classical terms, these operations correspond to a fork operation, akin to spawning tasks across multiple processing units.