Qubit discretizations of d=3 conformal field theories

Abstract

We propose that scaling dimensions of d=3 conformal field theories can be studied on a system of qubits with near term quantum simulation platforms. Our proposal chooses couplings of quantum many-body problems on a polyhedral lattice at which the conformal state-operator correspondence can be observed most accurately in the spectrum. We validate our protocol on the Ising model where we extract the scaling dimensions of a number of scaling operators with a few percent accuracy from the spectrum of a system of just 20 qubits. The procedure makes only minor assumptions beyond general conformal invariance -- it may hence be applied widely. Requirements and challenges to realize this proposal on quantum computers are discussed. Our results demonstrate that for current or near term qubit platforms, three dimensional conformal field theories present a unique opportunity -- a forefront problem that is difficult on classical computers but may be solved through quantum simulation.

Figures (5)

• Figure 1: In the manuscript we validate our proposal on two polyhedral solids: the icosahedron and the dodecahedron. The dodecahedron is the dual of the icosahedron with faces and vertices interchanged. We place qubits on the vertices (resulting in a Hilbert space of $2^{12}$ and $2^{20}$) and choose interactions to realize quantum models with an Ising $\mathbb{Z}_2^{\rm sf}$ symmetry with a conformal spectrum. Even though both structures share the same icosahedral $I_h$ symmetry (the largest discrete subgroup of $O(3)$), we present evidence that the dodecahedron has a more accurate conformal spectrum than the icosahedron, which we attribute to the increase of Hilbert space dimension. Increasing the Hilbert space dimension further preserving $I_h$ should yield even more accuracy -- although out of reach of exact diagonalization on a classical computer, it is proposed as an ideal project for near term quantum simulators.
• Figure 2: The spectrum of the nearest neighbor transverse field Ising model Eq. \ref{['eq:tfim']} on the icosahedron (12 qubits) and dodecahedron (20 qubits) as a function of $h_\perp$. The spectrum of the TFIM on the icosahedron (left) and on the dodecahedron (right) has appeared previously Lao_2023cruz2026:yl. We divide $h_\perp$ by the coordination number (3 and 5), so that we may view both spectra on the same scale. We have subtracted the ground state energy from each level at each coupling. The ground state is hence by definition at exactly zero energy. It is in the $\mathbf{1}_g$E irrep consistent with the identity operator, $I$. We have restricted ourselves to the 7 irreps of the TFIM on these $I_h$ symmetric graphs shown in the legend, because these sectors contain all the low lying energy levels of interest here. In light gray we show for reference energy levels from other sectors. At each $h_\perp$ we have rescaled the energy so the lowest level in the $\mathbf{5}_g$E sector is 3, since we identify this level with the stress-tensor $T_{\mu\nu}$. Also shown as straight lines and labeled on the right edge of the right panel are the bootstrap values for the $\Delta$ of each of the low-lying scaling fields. The colors of the bootstrap lines are chosen to match the expected representations of the $I_h\times$$\mathbb{Z}_2^{\rm sf}$ for these levels. A line with alternating colors indicates that CFT predicts two operators with different quantum numbers that have the same $\Delta$. Clearly there is a range of $h_\perp$ at which the bootstrap and icosahedral spectrum results seem close for both solids. A careful look at this picture already seems to show that this matching is better on the dodecahedron than the icosahedron (even though they share the same symmetry) apparently because the dodecahedron has a larger Hilbert space. We explore this question quantitatively further in this section.
• Figure 3: Constraint curves satisfying $c_{i}(h_{\perp},\lambda)=0$ with each $c_i$ defined in Eqn. \ref{['eq:constrain_eqns']}. As the parameters have been rescaled to be comparable across graphs, the observation that the constraint curves have less spread for the dodecahedron is evidence that the conformal point is better defined for a system with more qubits. Crosshairs mark the minimization of $|c|$ for the $\sigma$-tower of constraints.
• Figure 4: Comparison of scaling dimensions and spin inferred from our study on the icosahedron (open circles) and dodecahedron (solid circles) with the conformal bootstrap (solid lines) for the $\mathbb{Z}_2^{\rm sf}$ even (left) and odd (right) sectors. The spectrum here is shown for lattice couplings $(h_\perp,\lambda)$ at which the $\sigma$-tower constraints ($c_1,c_2,c_3$) are best satisfied according to the Gauss-Newton method discussed in the text.
• Figure 5: Classical simulation of the quantum simulator protocol we have proposed to extract the spectrum and $I_h$ quantum numbers for icoshedral spectroscopy. (Top) An example of the time series matrix element $D_{ij}(t,\alpha)$ discussed in Eq. \ref{['eq:correlator']} for $\mathbb{Z}_{2}$ even operator $\mathcal{O}=X$, $\alpha=\pi / 4$ and separation of graph distance $|i-j|=1$ for qubits on an icosahedron. We use $(h_\perp,\lambda)=(3.33,0.57)$. (Bottom) $I_{h}$ resolved dynamical structure factor $S_{\Gamma}(\omega)$ computed by processing signal $D_{ij}$ (see Appendix \ref{['app:spectro']}). For comparison we have shown as vertical lines the ED energy levels at the same coupling with the same irrep color coding -- they line up well with the peaks in $S_{\Gamma}(\omega)$. Increasing the duration of the quantum simulation sharpens the peaks. To resolve peaks in this case, we need the time series for $T \sim 14$. All energies and times are in units of the Ising exchange constant.