Magic in generalized Rokhsar-Kivelson wavefunctions

Poetri Sonya Tarabunga1,2,3 and Claudio Castelnovo4

1The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy
2International School for Advanced Studies (SISSA), via Bonomea 265, 34136 Trieste, Italy
3INFN, Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy
4TCM Group, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, UK

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Abstract

Magic is a property of a quantum state that characterizes its deviation from a stabilizer state, serving as a useful resource for achieving universal quantum computation e.g., within schemes that use Clifford operations. In this work, we study magic, as quantified by the stabilizer Renyi entropy, in a class of models known as generalized Rokhsar-Kivelson systems, i.e., Hamiltonians that allow a stochastic matrix form (SMF) decomposition. The ground state wavefunctions of these systems can be written explicitly throughout their phase diagram, and their properties can be related to associated classical statistical mechanics problems, thereby allowing powerful analytical and numerical approaches that are not usually available in conventional quantum many body settings. As a result, we are able to express the SRE in terms of wave function coefficients that can be understood as a free energy difference of related classical problems. We apply this insight to a range of quantum many body SMF Hamiltonians, which affords us to study numerically the SRE of large high-dimensional systems, and in some cases to obtain analytical results. We observe that the behaviour of the SRE is relatively featureless across quantum phase transitions in these systems, although it is indeed singular (in its first or higher order derivative, depending on the nature of the transition). On the contrary, we find that the maximum of the SRE generically occurs at a cusp away from the quantum critical point, where the derivative suddenly changes sign. Furthermore, we compare the SRE and the logarithm of overlaps with specific stabilizer states, asymptotically realised in the ground state phase diagrams of these systems. We find that they display strikingly similar behaviors, which in turn establish rigorous bounds on the min-relative entropy of magic.

Scientists are on a quest to build quantum computers that can solve problems impossible for classical computers. One key ingredient for this is a property called "non-stabilizerness" or also known as "magic". This research explores how magic behaves in a subclass of quantum many-body models. The cool thing is that these systems can be described using powerful tools from classical physics, allowing to analyze them in new ways, and beyond the state-of-the-art methods.

The study reveals some surprising findings. While magic seems to be present throughout these systems, it doesn't necessarily peak at critical points where we would think that the system becomes particularly complex. Instead, it reaches a maximum at a different point, that doesn't seem to be related to any critical behavior.

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[3] Xhek Turkeshi, Anatoly Dymarsky, and Piotr Sierant, "Pauli Spectrum and Magic of Typical Quantum Many-Body States", arXiv:2312.11631, (2023).

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[5] Gianluca Passarelli, Rosario Fazio, and Procolo Lucignano, "Nonstabilizerness of Permutationally Invariant Systems", arXiv:2402.08551, (2024).

[6] ChunJun Cao, Gong Cheng, Alioscia Hamma, Lorenzo Leone, William Munizzi, and Savatore F. E. Oliviero, "Gravitational back-reaction is magical", arXiv:2403.07056, (2024).

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