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arXiv:2401.10079 · Physical Review Letters

Parity Quantum Computing as yz -Plane Measurement-Based Quantum Computing

Isaac D. Smith, Hendrik Poulsen Nautrup, Hans J. Briegel

Published May 28, 2024• 8 citations
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Abstract

We show that universal parity quantum computing employing a recently introduced constant depth decoding procedure is equivalent to measurement-based quantum computation (MBQC) on a bipartite graph using only yz-plane measurements. We further show that any unitary MBQC using only yz-plane measurements must occur on a bipartite graph. These results have a number of consequences and open new research avenues for both frameworks.

DOI
10.1103/physrevlett.132.220602

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We show that universal parity quantum computing employing a recently introduced constant depth decoding procedure is equivalent to measurement-based quantum computation (MBQC) on a bipartite graph using only yz-plane measurements. We further show that any unitary MBQC using only yz-plane measurements must occur on a bipartite graph. These results have a number of consequences and open new research avenues for both frameworks.
We define the class of register-logic graphs and prove that any uniformly deterministic measurement-based quantum computation (MBQC) where the inputs coincide with the outputs must be driven on such graphs by measurements in the YZYZYZ plane of the Bloch sphere. This observation is revisited in the context that goes beyond uniform determinism, where we present a universal YZYZYZ-plane-only measurement pattern and establish a connection between YZYZYZ-plane-only and XZXZXZ-plane-only patterns. These results conclude the line of research on universal patterns with measurements restricted to one of the principal planes of the Bloch sphere. We further demonstrate, within the framework of the Parity Architecture, that YZYZYZ-plane patterns with the register-logic graph can be embedded into another graph with purely local interactions, and we extend this case to the scenario of universal quantum computation.
This paper shows that, in measurement-based quantum computation, it is possible to write any quantum oracle implementing a classical function in constant depth. The result is shown through the equivalence between MBQC and the circuit model where arbitrary rotations along ZZZ axis and unbounded fan-outs are elementary operations. A corollary of this result is that disjunction can be implemented exactly in constant-depth, answering an open question of Høyer and Špalek.
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⁡(−iπ2nZ⊗Z) . When n=2 , these are equivalent, up to local Clifford unitaries, to graph states. However, when n>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>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>2 this family is capable of producing all Clifford gates and all diagonal gates in the n -th level of the Clifford hierarchy.
Measurement-based quantum computing (MBQC) is a model of quantum computation where quantum information is coherently processed by means of projective measurements on highly entangled states. Following the introduction of MBQC, cluster states have been studied extensively both from the theoretical and experimental point of view. Indeed, the study of MBQC was catalysed by the realisation that cluster states are universal for MBQC with (X, Y)-plane and Z measurements. Here we examine the question of whether the requirement for Z measurements can be dropped while maintaining universality. We answer this question in the affirmative by showing that universality is possible in this scenario.
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 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 BQP≠BPP\mathsf{BQP} \neq \mathsf{BPP}BQP=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.
A powerful tool emerging from the study of many-body quantum dynamics is that of dual-unitary circuits, which are unitary even when read "sideways," i.e., along the spatial direction. Here, we show that this provides the ideal framework to understand and expand on the notion of measurement-based quantum computation (MBQC). In particular, applying a dual-unitary circuit to a many-body state followed by appropriate measurements effectively implements quantum computation in the spatial direction. We show how the dual-unitary dynamics generated by the dynamics of the paradigmatic one-dimensional kicked Ising chain with certain parameter choices generate resource states for universal deterministic MBQC. Specifically, after k time steps, equivalent to a depth-k quantum circuit, we obtain a resource state for universal MBQC on ∼3k/4 encoded qubits. Our protocol allows generic quantum circuits to be "rotated" in space-time and gives new ways to exchange between resources like qubit number and coherence time in quantum computers. Beyond the practical advantages, we also interpret the dual-unitary evolution as generating an infinite sequence of new symmetry-protected topological phases with spatially modulated symmetries, which gives a vast generalization of the well-studied one-dimensional cluster state and shows that our protocol is robust to symmetry-respecting deformations.