Universal MBQC with generalised parity-phase interactions and Pauli measurements

Aleks Kissinger and John van de Wetering

Radboud University Nijmegen

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We introduce a new family of models for measurement-based quantum computation which are deterministic and approximately universal. The resource states which play the role of graph states are prepared via 2-qubit gates of the form $\exp(-i\frac{\pi}{2^{n}} Z\otimes Z)$. When $n = 2$, these are equivalent, up to local Clifford unitaries, to graph states. However, when $n \gt 2$, their behaviour diverges in two important ways. First, multiple applications of the entangling gate to a single pair of qubits produces non-trivial entanglement, and hence multiple parallel edges between nodes play an important role in these generalised graph states. Second, such a state can be used to realise deterministic, approximately universal computation using only Pauli $Z$ and $X$ measurements and feed-forward. Even though, for $n \gt 2$, the relevant resource states are no longer stabiliser states, they admit a straightforward, graphical representation using the ZX-calculus. Using this representation, we are able to provide a simple, graphical proof of universality. We furthermore show that for every $n \gt 2$ this family is capable of producing all Clifford gates and all diagonal gates in the $n$-th level of the Clifford hierarchy.

Much like in classical computation, a quantum program can be described using different programming paradigms. By far the most common such paradigm in the quantum world is the circuit model, which is roughly the quantum analogue of a low-level procedural language. Here, one or more quantum registers are prepared in a fixed initial state, acted upon by a sequence of primitive quantum gates, and measured at the end to produce the output of a computation.

However, other paradigms exist for defining quantum programs, some of which having no classical analogue. One such paradigm is measurement-based quantum computing (MBQC). In MBQC, one starts with a fixed state on many qubits and simply measures it one qubit at a time. The key point, however, is one is free to choose single-qubit measurements (which can be pictured as an axis in 3D space) on the fly, and adapt those choices based on previous measurement outcomes. Using this adaptation, known as feed-forward, one can obtain models for computation which are deterministic (always perform the desired computation) and universal (just as powerful as the circuit model).

Since its introduction in 2001, the one-way model has been a paradigmatic example of a scheme for universal MBQC. It relies crucially on the initial state being a certain kind of highly-entangled stabilizer state called a graph state. Stabilizer states are a well-understood class of quantum states which can be efficiently characterised and manipulated in terms of an associated group called its stabilizer subgroup. In particular, the generators of this group can be used to compute how measurements should be adapted during a computation to obtain the desired outcome with 100% success.

While stabilizer techniques are powerful and very well-understood, once one considers models based on more general families of (non-stabilizer) states, these techniques no longer apply. Furthermore, if one restricts to "stabilizer" measurements (i.e. qubit measurements along the X, Y, and Z axes), the outcome of the entire computation is efficient to compute on a classical computer. Hence, all of the "quantumness" on the one-way model comes from choosing measurements which do not lie on the X, Y, or Z axes.

In this paper, we introduce a new family of models for MBQC where this extra "quantumness" is baked into the initial state, and hence it is no longer a stabilizer state. As a result, this model is universal even when restricting measurements to just the X and Z axes. Moreover, our model is still deterministic, and can implement the popular Clifford+T gate set natively.

In order to reason with non-stabilizer states, we need a more powerful language then just the stabilizer formalism: the ZX-calculus. The ZX-calculus can be seen as a sort of "supercharged stabilizer theory", which represents quantum states and transformations using particular tensor networks called ZX-diagrams. These diagrams satisfy graphical transformation rules which subsume the equations provable by stabilizer techniques and go quite some distance beyond. For our purposes, the ZX-calculus gives us a much richer and more flexible notion of "feed-forward" than the one provided by stabilizer theory and opens the door to previously unconsidered ways of doing universal, deterministic quantum computations.

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[5] Sahar Atallah, Michael Garn, Sania Jevtic, Yukuan Tao, and Shashank Virmani, "Efficient classical simulation of cluster state quantum circuits with alternative inputs", Quantum 8, 1243 (2024).

[6] Aleks Kissinger and John van de Wetering, "PyZX: Large Scale Automated Diagrammatic Reasoning", Electronic Proceedings in Theoretical Computer Science 318, 229 (2020).

[7] Bob Coecke, Dominic Horsman, Aleks Kissinger, and Quanlong Wang, "Kindergarden quantum mechanics graduates ...or how I learned to stop gluing LEGO together and love the ZX-calculus", Theoretical Computer Science 897, 1 (2022).

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[9] Masahito Hayashi and Yuki Takeuchi, "Verifying commuting quantum computations via fidelity estimation of weighted graph states", New Journal of Physics 21 9, 093060 (2019).

[10] Boldizsár Poór, Quanlong Wang, Razin A. Shaikh, Lia Yeh, Richie Yeung, and Bob Coecke, 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) 1 (2023) ISBN:979-8-3503-3587-3.

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The above citations are from Crossref's cited-by service (last updated successfully 2024-06-22 07:06:58) and SAO/NASA ADS (last updated successfully 2024-06-22 07:06:59). The list may be incomplete as not all publishers provide suitable and complete citation data.