Fermionic Insights into Measurement-Based Quantum Computation: Circle Graph States Are Not Universal Resources

TL;DR

The paper addresses whether circle graph states can serve as a universal resource for measurement-based quantum computation. It develops a novel connection between circle graph states and fermionic Gaussian states via Kitaev’s four-Majorana-per-qubit mapping on 4-regular multigraphs, enabling an explicit, efficient classical description of circle-graph resource states and their LU-equivalent relatives. The main result shows circle graph states are not efficiently universal for MBQC (assuming $\mathsf{BQP} \neq \mathsf{BPP}$), by deriving polynomial-time Pfaffian-based formulas to compute all marginal measurement probabilities, thus enabling complete classical simulation. This finding links MBQC simulability to free-fermion solvability and suggests broader implications for identifying other classically simulable resource families and for understanding the role of entanglement width in quantum computation. The work also opens up avenues to explore connections with surface codes and vertex-minor theory, and to extend the fermionic–graph correspondence to other graph-state families."

Abstract

Measurement-based quantum computation (MBQC) is a strong contender for realizing quantum computers. A critical question for MBQC is the identification of resource graph states that can enable universal quantum computation. Any such universal family must have unbounded entanglement width, which is known to be equivalent to the ability to produce any circle graph state from the states in the family using only local Clifford operations, local Pauli measurements, and classical communication. Yet, it was not previously known whether or not circle graph states themselves are a universal resource. We show that, in spite of their expressivity, circle graph states are not efficiently universal for MBQC (i.e., assuming $\mathsf{BQP} \neq \mathsf{BPP}$). We prove this by articulating a precise graph-theoretic correspondence between circle graph states and a certain subset of fermionic Gaussian states. This is accomplished by synthesizing a variety of techniques that allow us to handle both stabilizer states and fermionic Gaussian states at the same time. As such, we anticipate that our developments may have broader applications beyond the domain of MBQC as well.

Figures (8)

• Figure 1: Graphical representation of a GHZ-like state under our mapping. The black squares represent nodes (qubits), and the circular corners correspond to half-edges (Majorana operators). The blue arrows represent edges of a $4$-regular multigraph, with directions given by the orientation of the corresponding Eulerian tour. The top and bottom loops yield the stabilizer $X^{\otimes n}$, while the parallel edges between any two adjacent nodes yield the stabilizer $Z \otimes Z$ (up to sign conventions).
• Figure 2: A chord diagram (left) and its corresponding circle graph (right). Chords and their intersections in the chord diagram correspond to vertices and edges, respectively, in the circle graph. Note that we label chords by color and intersections by letters.
• Figure 3: Relevant graphs for construction of fermionic matching state corresponding to $C_4$.
• Figure 4: An example 5-node graph
• Figure 5: An example of a subcubic tree graph.

Theorems & Definitions (58)

• Theorem : Informal
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