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arXiv:2209.06191 · Physical Review Letters

Universal Measurement-Based Quantum Computation in a One-Dimensional Architecture Enabled by Dual-Unitary Circuits

David T. Stephen, Wen Wei Ho, Tzu-Chieh Wei, Robert Raussendorf, Ruben Verresen

Published Jun 17, 2024• 30 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.132.250601

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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.
The interplay of unitary evolution and projective measurements is a modern interest in the study of many-body entanglement. On the one hand, the competition between these two processes leads to the recently discovered measurement-induced phase transition (MIPT). On the other hand, measurement-based quantum computation (MBQC) is a well-known computational paradigm where measurements simulate unitary evolution by utilizing the entanglement of special resources such as the two-dimensional (2D) cluster state. The entanglement properties enabling MBQC may be attributed to symmetry-protected topological (SPT) orders, particularly subsystem-symmetry-protected topological (SSPT) orders. It was recently found that the one-dimensional cluster state may be associated with an SPT phase in random circuits respecting a global ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry, and furthermore that all phase transitions in this scenario belong to the same universality class. As resources with greater computational power feature greater symmetry, it is fruitful to investigate further any relationship between levels of symmetry in MIPTs and MBQC. In this paper we investigate MIPTs on a torus with three levels of symmetry-respecting unitary evolution interspersed by measurements. Although we find two area-law phases and one volume-law phase with distinct entanglement structures for each ensemble, the phase transition from the volume-law phase to the area-law phase associated with the 2D SSPT cluster state has variable correlation length exponent $\ensuremath{\nu}$. Whereas $\ensuremath{\nu}\ensuremath{\approx}0.90$ for unconstrained Clifford unitaries and $\ensuremath{\nu}\ensuremath{\approx}0.83$ for globally symmetric Cliffords, subsystem-symmetric Cliffords feature a much smaller value $\ensuremath{\nu}\ensuremath{\approx}0.38$. We discuss how these distinct $\ensuremath{\nu}$'s quantify spacetime response scales where quantum information is manipulated by single-qubit measurements as in MBQC.
While quantum computers can achieve dramatic speedups over the classical computers familiar to us, identifying the origin of this quantum advantage in physical systems remains a major goal of quantum information science. A useful tool here is measurement-based quantum computation (MQC), a computational framework utilizing the quantum entanglement found in many-body resource states. Not all resource states are useful for quantum computation however, so an important question is what properties of many-body entanglement characterize universal resource states, which can implement any quantum computation. Many-body states are also studied in condensed matter physics, where the collective behavior of quantum many-body systems sometimes define topological phases of matter. These phases are defined by nonlocal many-body entanglement, making topologically-ordered states natural candidates for MQC. We might wonder if these topological phases could be organized as phases of quantum computation, so that every state within the phase is universal for MQC. While phases of symmetry-protected topological order (SPTO) have arisen as natural candidates, previous attempts to demonstrate an MQC-SPTO correspondence were mostly limited to nonuniversal 1D spin chains, leaving the important 2D setting wide open. In this dissertation, we explore the wide and varied connections between MQC and SPTO, and obtain new results for 1D and 2D systems. After identifying a new MQC-SPTO correspondence within 1D spin chains, we move up and explore the operational use of 2D states with two complementary forms of SPTO. We create a new Union Jack resource state, whose different form of SPTO than previous 2D resource states permits a hierarchical notion of MQC universality. This state leads us to consider an idealized model of 2D SPTO, where we show that an additional symmetry condition makes these model states form universal resources for MQC only when they have nontrivial SPTO. We finally study the intrinsic complexity of SPTO-inspired states for classically intractable sampling, and identify inherent advantages of MQC for this purpose. Our work highlights the rich complexity available in states of entangled quantum matter, providing new evidence which sharpens our understanding of the diverse connections between MQC and SPTO.
We present the first examples of topological phases of matter with uniform power for measurement-based quantum computation (MBQC). This is possible due to a new framework for analyzing the computational properties of phases of matter that is more general than previous constructions, which have been limited to short-range entangled phases in one dimension. We show that ground states of the toric code in an anisotropic magnetic field yield a natural, albeit noncomputationally universal, application of our framework. We then present a new model with topological order the ground states of which are universal resources for MBQC. Both topological models are enriched by subsystem symmetries and these symmetries protect their computational power. Our framework greatly expands the range of physical models that can be analyzed from the computational perspective.
In measurement-based quantum computation (MBQC), elementary quantum operations can be more parallelized than the quantum circuit model by employing a larger Hilbert space of graph states used as the resource. Thus MBQC can be regarded as a method of quantum computation where the temporal resource described by the depth of quantum operations can be reduced compared to the quantum circuit model by using the extra spatial resource described by graph states. To analyze the trade-off relationship of the spatial and temporal resources, we consider a method to obtain quantum circuit decompositions of general unitary transformations represented by MBQC on graph states with a certain underlying geometry called generalized flow. We present a method to translate any MBQC with generalized flow into quantum circuits without extra spatial resource. We also show an explicit way to unravel acausal gates without postselection that appear in the quantum circuit decomposition derived by a translation method [V. Danos and E. Kashefi, Phys. Rev. A 74, 052310 (2006)] and that represents an effect of the reduction of the temporal resource in MBQC. Finally, by considering a way to deterministically simulate these acausal gates, we investigate a general framework to analyze the trade-off between the spatial and temporal resources for quantum computation.
Measurement-based quantum computation (MBQC) is a universal platform to realize unitary gates, only using measurements that act on a preprepared entangled resource state. By deforming the measurement bases, as well as the geometry of the resource state, we show that MBQC circuits always transmit and act on the input state but generally realize nonunitary logical gates. In contrast to the stabilizer formalism that is often used for unitary gates, we find that ZX-calculus is an ideal computation method for these nonunitary gates. As opposed to unitary gates, nonunitary gates cannot be applied with certainty, due to the randomness of quantum measurements. We maximize the success probability of realizing nonunitary gates and discuss applications including imaginary time evolution, which we demonstrate on a noisy intermediate-scale quantum device.
Measurement-based quantum computation (MBQC) performs quantum algorithms by local, adaptive measurements on an initially entangled many-body "resource" state. This thesis investigates MBQC through two tightly coupled lenses: (i) which physical states and phases of matter constitute universal resources, and (ii) how the temporal order induced by measurement adaptivity constrains and, in turn, is constrained by those resources. Resources from valence-bond constructions. We develop hybrid AKLT-type (valence-bond) resources on two-dimensional lattices and show how a generalized POVM reduces higher-spin degrees of freedom to effective qubits while preserving computational viability. We analyze worst-case and frustrated geometries (e.g., kagome and decorated lattices), identify planarity-restoring local operations, and use percolation-style diagnostics on the induced graph states to certify universality beyond idealized cluster-state settings. A two-parameter phase with noisy POVMs. Extending ground states across a phase connected to the AKLT point, we introduce a measurement-noise–aware framework with probabilistic restoration and closed-form weight formulae derived via a graphical (g-)calculus. Large-scale numerics that account for correlations map a two-parameter phase diagram and delineate regimes of universal vs. non-universal post-measurement graphs. These results exhibit a robust universal region that persists away from ideal measurements and clarify how noise and Hamiltonian deformation jointly control computational power. Temporal order and determinism. We develop a stabilizer-and-influence-matrix formalism that places temporal dependencies on the same footing as resource structure. Four theorems establish a two-way determination between (a) the classical processing relations (including the influence matrix that generates the partial order of measurements) and (b) the resource stabilizer in a normal form, up to local equivalence. This links gauge freedoms, byproduct handling, and flow/gflow–style constraints into a unified picture of when adaptive measurement schedules yield deterministic computation. Together, these contributions clarify when physically motivated ground states serve as universal MBQC resources under realistic measurement noise, and how temporal constraints fundamentally shape the space of admissible computations in MBQC.