[Paper Review] Measurement-based quantum computation--a quantum-mechanical toy model for spacetime?

Robert Raussendorf, Pradeep Kiran Sarvepalli+2 more|arXiv (Cornell University)|Aug 29, 2011

Quantum Mechanics and Applications5 references3 citations

TL;DR

This paper proposes measurement-based quantum computation (MBQC) as a quantum-mechanical toy model for spacetime, demonstrating that the inherent randomness of quantum measurements enforces a causal temporal order. It identifies event horizons, light cones, and closed timelike curves as relativistic phenomena with analogues in MBQC, and classifies valid temporal orders using a matroid structure derived from stabilizer states and measurement bases.

ABSTRACT

We propose measurement-based quantum computation (MBQC) as a quantum mechanical toy model for spacetime. Within this framework, we discuss the constraints on possible temporal orders enforced by certain symmetries present in every MBQC. We provide a classification for all MBQC temporal relations compatible with a given initial quantum state and measurement setting, in terms of a matroid. Further, we find a symmetry transformation related to local complementation that leaves the temporal relations invariant. After light cones and closed time-like curves have previously been found to have MBQC counterparts, we identify event horizons as a third piece of the phenomenology of General Relativity that has an analogue in MBQC.

Motivation & Objective

• To explore whether quantum mechanics can generate spacetime structure, particularly temporal order, through measurement-induced constraints.
• To identify and classify relativistic spacetime phenomena—such as light cones, closed timelike curves, and event horizons—within the framework of MBQC.
• To formalize the constraints on measurement order in MBQC using a matroid structure that captures the resource state and measurement basis information.
• To uncover symmetry transformations, including local complementation, that preserve temporal order and reveal deeper structural invariances in MBQC.
• To investigate whether the classical processing rules in MBQC, which prevent measurement randomness from affecting computation, can be interpreted as fundamental laws in a quantum spacetime model.

Proposed method

• Utilize the stabilizer formalism to describe resource states and measurement bases in MBQC, focusing on their role in determining temporal order.
• Introduce a matroid ${\cal{G}}(|\Psi\rangle)$ that encodes all classical processing relations and temporal order information, with bases of the matroid in one-to-one correspondence with valid temporal orders.
• Identify a gauge transformation group that preserves the classical output of MBQC while modifying processing relations, and show that these relations fully specify the resource state and measurement setting up to equivalence.
• Define a symmetry group generated by flipping measurement planes (related to local complementation), which preserves temporal order and explains how such orders arise as group representations.
• Analyze the influence matrix $T$ and additional matrices $H$, $R$, $Z$ associated with boundary sets $I_{\text{gauge}}$ and $O_{\text{comp}}$ to encode classical processing and causal dependencies.
• Draw analogies to Malament’s theorem by suggesting that temporal structure in MBQC, encoded in processing relations, may determine a topological notion of 'space' via cellular complexes such as the planar code or 3D/4D cluster states.

Experimental results

Research questions

• RQ1 Can the two symmetry groups—gauge transformations and measurement-plane flipping—be unified into a single algebraic structure preserving temporal order?
• RQ2 For a given stabilizer state and measurement planes, can a structure be found that restricts temporal orders to only partial orders, excluding closed timelike curves, and can the minimal $O_{\text{comp}}$ partial order be efficiently computed?
• RQ3 How does the reducibility of temporal order representations under the local complementation group affect the decomposition into irreducible representations, and where is the temporal order encoded in such a decomposition?
• RQ4 What is an appropriate notion of 'space' in MBQC that admits topological characterization, and how can it be linked to cellular complexes associated with resource states?
• RQ5 In a broader physical context beyond quantum computation, what fundamental principle could replace the classical processing rules of MBQC, which currently serve only to shield computation from measurement randomness?

Key findings

• The matroid ${\cal{G}}(|\Psi\rangle)$ fully captures all classical processing relations and temporal orders compatible with a given stabilizer state and measurement basis setting, with bases of the matroid in one-to-one correspondence with valid temporal orders.
• A gauge transformation group exists that preserves the classical output of MBQC while modifying processing rules, and this group allows full reconstruction of the resource state and measurement setting up to equivalence.
• A symmetry group generated by flipping measurement planes—related to local complementation—preserves temporal order, showing that temporal orders in MBQC arise as representations of this group.
• Event horizons are identified as a third relativistic phenomenon with a counterpart in MBQC, following light cones and closed timelike curves, completing a phenomenological analogy to general relativity.
• The classical processing rules in MBQC, while currently seen as operational constraints, can be interpreted as fundamental laws in a dedicated MBQC device, analogous to Newton’s or Maxwell’s laws, and are not arbitrarily violable by conscious agents.
• The influence matrix $T$ and associated matrices $H$, $R$, $Z$ encode the full causal structure of MBQC, and together with the matroid, determine the computation up to measurement angles, suggesting a deep link to spacetime geometry akin to Malament’s theorem.