Testing measurement-based computational phases of quantum matter on a quantum processor

TL;DR

This work experimentally validates theoretical predictions about computational phases of quantum matter using measurement-based quantum computation on a superconducting IBM device. It demonstrates the uniformity of MBQC power across the 1D $ ext{Z}_2 imes ext{Z}_2$ cluster phase and verifies that the physical string order parameter $oldsymbol{\sigma}$ coincides with the computational order parameter $\nu$, linking phase order to computational capability. The study introduces and tests splitting as a resource-efficient method to counter logical decoherence, confirms the $\epsilon_m \propto 1/m$ scaling, and, in the correlated regime, supports the counterintuitive finding that densest packing can maximize efficiency under specified conditions. These results establish practical strategies for MBQC in symmetry-protected and symmetry-enriched quantum phases and point toward broader exploration of computational phase diagrams on larger or higher-dimensional systems.

Abstract

Many symmetry protected or symmetry enriched phases of quantum matter have the property that every ground state in a given such phase endows measurement based quantum computation with the same computational power. Such phases are called computational phases of quantum matter. Here, we experimentally verify four theoretical predictions for them on an IBM superconducting quantum device. We comprehensively investigate how symmetric imperfections of the resource states translate into logical decoherence, and how this decoherence is mitigated. In particular, the central experiment probes the scaling law from which the uniformity of computational power follows. We also analyze the correlated regime, where local measurements give rise to logical operations collectively. We test the prediction that densest packing of a measurement-based algorithms remains the most efficient, in spite of the correlations. Our experiments corroborate the operational stability of measurement based quantum computation in quantum phases of matter with symmetry.

Figures (12)

• Figure 1: MBQC resource state and measurement pattern. (a) MPS representation of MBQC. (b) Implementation of a logical operation, consisting of a symmetry-breaking measurement and trailing oblivious wire.
• Figure 2: Experimental results for VQE (dots) with theory comparison (lines). (a) Expectation values of the energy terms. Individual points are obtained by measurements of the 1/2-qubit circuits of Eqs. \ref{['eq:Xmeascircuit']} - \ref{['eq:Kbulkmeascircuit']}, with 5000 shots per point. Error bars represent the standard error of the mean. (b) Optimization of the variational parameter depending on the Hamiltonian interpolation $\alpha$. We plot the experimental optimization for a 5-site chain, as well as theory results for the 5 and infinite-size chains (obtained by optimizing over the analytical results of Eqs. \ref{['eq:Xi']}, \ref{['eq:Ki']}). Error bars are estimated from the range of $\theta$ which fall within the minimum energy up to uncertainty.
• Figure 3: (a) 5-qubit MBQC scheme with a single symmetry breaking measurement of angle $\beta$, used to verify the relation between logical gate action and decoherence. (b) 5-qubit quantum circuit with equivalent logical action, which is run on the IBM device. This circuit (and all others) was drawn using the Quantikz package kay_tutorialquantikzpackage_2023.
• Figure 4: Experimental results for the logical expectation values $\langle X\rangle, \langle Y \rangle$, as a function of the Hamiltonian interpolation parameter $\alpha$ (which specifies the resource state) and the implemented rotation angle $\beta \in [0, \pi]$. Each point corresponds to 10000 shots for $\langle X \rangle, \langle Y \rangle$. Error bars represent the standard error of the mean. Solid lines represent best-fit ellipses to the data.
• Figure 5: Experimental results for demonstration of string order equals computational order. (a) Ratio of the logical $Y, X$ expectation values plotted against the tangent of the rotation angle $\beta$. Solid lines represent best-fit linest to the data. (b) The linear fits of (a) yield values of the computational order parameter (COP, blue crosses). The string order parameter (SOP, pink dots) is obtained from the appropriate cluster state stabilizer value measured in Experiment #0. The theory curve (green) is obtained from the analytical expression in Eq. (\ref{['eq:Ki']}). Error bars for the COPs are obtained from the variance in the optimal paramaeters of the least squares fit. Error bars for the SOP are obtained by combining the statistical error in the stabilizer measurement with the variance in $\langle K_i\rangle_\theta$ arising from determining the optimal $\theta_{\text{min}}$ for a given $\alpha$.