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arXiv:2408.06342

Measuring central charge on a universal quantum processor

Nazlı Uğur Köylüoğlu, Swarndeep Majumder, Mirko Amico, Sarah Mostame, E. van den Berg, M. A. Rajabpour, Zlatko Minev, Khadijeh Najafi

Published Aug 12, 2024• 1 citation
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Abstract

Central charge is a fundamental quantity in conformal field theories (CFT), and plays a crucial role in determining universality classes of critical points in two-dimensional systems. Despite its significance, the measurement of central charge has remained elusive thus far. In this work, we present the first experimental determination of the central charge using a universal quantum processor. Using a classically optimized variational quantum circuit and employing advanced error mitigation techniques, we successfully prepare ground states of various $1+1D$ quantum spin chain models at their critical point. Leveraging the heavy-hex structure of IBM quantum processors, we are able to implement periodic boundary conditions and mitigate boundary effects. We then extract the central charge from the scaling behavior of the sub-leading term of R{é}nyi generalizations of classical Shannon entropy, computed for local Pauli measurements in the conformal bases ($σ^{z}$ and $σ^x$). The experimental results are consistent with the known central charge values for the transverse field Ising (TFI) chain ($c=0.5$) and the XXZ chain ($c=1$), achieving relative errors as low as 5 percent.

DOI
10.48550/arxiv.2408.06342

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Central charge is a fundamental quantity in conformal field theories (CFT), and plays a crucial role in determining universality classes of critical points in two-dimensional systems. Despite its significance, the measurement of central charge has remained elusive thus far. In this work, we present the first experimental determination of the central charge using a universal quantum processor. Using a classically optimized variational quantum circuit and employing advanced error mitigation techniques, we successfully prepare ground states of various 1+1D1+1D1+1D quantum spin chain models at their critical point. Leveraging the heavy-hex structure of IBM quantum processors, we are able to implement periodic boundary conditions and mitigate boundary effects. We then extract the central charge from the scaling behavior of the sub-leading term of R{é}nyi generalizations of classical Shannon entropy, computed for local Pauli measurements in the conformal bases (σzσ^{z}σz and σxσ^xσx). The experimental results are consistent with the known central charge values for the transverse field Ising (TFI) chain (c=0.5c=0.5c=0.5) and the XXZ chain (c=1c=1c=1), achieving relative errors as low as 5 percent.
Central charge is a fundamental quantity in conformal field theories (CFT), and plays a crucial role in determining universality classes of critical points in two-dimensional systems. Despite its significance, the measurement of central charge has remained elusive thus far. In this work, we present the experimental determination of the central charge using a universal quantum processor. Using a classically optimized variational quantum circuit and employing advanced error mitigation techniques, we successfully prepare ground states of various 1 + 1D quantum spin chain models at their critical point. Leveraging the heavy-hex structure of IBM quantum processors, we are able to implement periodic boundary conditions and mitigate boundary effects. We then extract the central charge from the scaling behavior of the sub-leading term of Rényi generalizations of classical Shannon entropy, computed for local Pauli measurements in the conformal bases (σz and σx). The experimental results are consistent with the known central charge values for the transverse field Ising (TFI) chain (c = 0.5) and the XXZ chain (c = 1), achieving relative errors as low as 5 percent. Central charge, a key quantity in conformal field theories, is crucial in the study of critical phenomena, yet its measurement has remained elusive. Here, the authors extract the central charge of several quantum critical models by accurately preparing their ground states on a superconducting quantum processor.
Conformal invariance powerfully constrains the critical behavior of two-dimensional classical systems with short-range interactions and the critical theories in two-dimensions are parametrized by a dimensional number, termed central charge or conformal anomaly ccc. However, experimental determination of the central charge of a conformal field theory has not been done before. Here we propose to extract the central charge of the conformal field theory corresponding to a critical point of a two-dimensional lattice models from the quantum decoherence measurement of a probe spin which is coupled to the two-dimensional lattice models. Conformal invariance predicts that the leading finite-size correction to the free energy for a two-dimensional system at a conformal invariant critical point is linearly related to the conformal anomaly for various boundary conditions. A recent discovery of thermodynamic holography allows us to obtain the free energy of many-body system from central spin decoherence measurement. Thus the central charge of the conformal field theory could be extracted from central spin decoherence measurement. We have applied the method to the two-dimensional Ising model and extracted the central charge of two-dimensional Ising model with good precision. This work provides a useful approach to extracting the central charge of conformal field theory in various two-dimensional lattice systems.
Universality classes of non-unitary critical theories in two-dimensions are characterized by the central charge. However, experimental determination of the central charge of a non-unitary critical theory has not been done before because of the intrinsic difficulty that complex parameters usually occur in non-unitary theory, which is not physical. Here we propose to extract the effective central charge of the non-unitary critical point of a two-dimensional lattice model from the quantum coherence measurement of a probe spin which is coupled to the lattice model. A recent discovery shows that quantum coherence of a probe spin which is coupled to a bath is proportional to the partition function of the bath with a complex parameter. Thus the effective central charge of a non-unitary conformal field theory may be extracted from quantum coherence measurement of a probe spin which is coupled to a bath. We have applied the method to the Yang-Lee edge singularity of the two-dimensional Ising model and extracted the effective central charge of the Yang-Lee edge singularity with good precision and tested other predictions of non-unitary conformal field theory. This work paves the way for the first experimental observation of the effective central charge of non-unitary conformal field theory.
We study the Shannon and R\'enyi mutual information (MI) in the ground state (GS) of different critical quantum spin chains. Despite the apparent basis dependence of these quantities we show the existence of some particular basis (we will call them conformal basis) whose finite-size scaling function is related to the central charge ccc of the underlying conformal field theory of the model. In particular, we verified that for large index nnn, the MI of a subsystem of size $\ensuremath{\ell}$ in a periodic chain with LLL sites behaves as $\frac{c}{4}\frac{n}{n\ensuremath{-}1}ln[\frac{L}{\ensuremath{\pi}}sin(\frac{\ensuremath{\pi}\ensuremath{\ell}}{L})]$, when the ground-state wave function is expressed in these special conformal basis. This is in agreement with recent predictions. For generic local basis, we will show that, although in some cases ${b}_{n}ln[\frac{L}{\ensuremath{\pi}}sin(\frac{\ensuremath{\pi}\ensuremath{\ell}}{L})]$ is a good fit to our numerical data, in general, there is no direct relation between bn{b}_{n}bn​ and the central charge of the system. We will support our findings with detailed numerical calculations for the transverse field Ising model, Q=3,4Q=3,4Q=3,4 quantum Potts chain, quantum Ashkin-Teller chain, and the XXZ quantum chain. We will also present some additional results of the Shannon mutual information (n=1n=1n=1), for the parafermionic ZQ{Z}_{Q}ZQ​ quantum chains with Q=5,6,7Q=5,6,7Q=5,6,7, and 8.
We study the entanglement entropy scaling of the XXZ chain. While in the critical XY phase of the XXZ chain the entanglement entropy scales logarithmically with a coefficient that is determined by the associated conformal field theory, at the ferromagnetic point, however, the system is not conformally invariant yet the entanglement entropy still scales logarithmically albeit with a different coefficient. We investigate how such an nontrivial scaling at the ferromagnetic point influences the estimation of the central charge ccc in the critical XY phase. In particular we use the entanglement scaling of the finite or infinite system, as well as the finite-size scaling of the ground state energy to estimate the value of ccc. In addition, the spin-wave velocity and the scaling dimension are also estimated. We show that in all methods the evaluations are influenced by the nearby ferromagnetic point and result in crossover behavior. Finally we discuss how to determine whether the central charge estimation is strongly influenced by the crossover behavior and how to properly evaluate the central charge.
The scaling of the entanglement entropy at a quantum critical point allows us to extract universal properties of the state, e.g., the central charge of a conformal field theory. With the rapid improvement of noisy intermediate-scale quantum (NISQ) devices, these quantum computers present themselves as a powerful tool to study critical many-body systems. We use finite-depth quantum circuits suitable for NISQ devices as a variational ansatz to represent ground states of critical, infinite systems. We find universal finite-depth scaling relations for these circuits and verify them numerically at two different critical points, i.e., the critical Ising model with an additional symmetry-preserving term and the critical XXZ model.