The phase diagram of quantum chromodynamics in one dimension on a quantum computer

TL;DR

The paper tackles the challenge of mapping the QCD phase diagram at finite temperature and density onto a quantum computer by presenting a practical scheme to prepare thermal states of one-dimensional SU(2) and SU(3) lattice gauge theories with dynamical matter. It introduces motional ancillae to generate Gibbs distributions and a charge-singlet projection to enforce color neutrality, enabling experiments on a trapped-ion quantum computer. The authors demonstrate SU(2) and SU(3) thermal states and observe a transition from vacuum- to baryon-dominated mixtures as the chemical potential grows, providing experimental access to finite-T, finite-density gauge-theory physics previously hindered by the sign problem. The work delivers a resource-efficient framework for thermal-state quantum simulations in gauge theories and lays the groundwork for scaling to higher dimensions, more colors, and richer observables. Overall, this constitutes a significant advance in quantum simulations of particle physics, enabling thermodynamic studies of gauge theories beyond classical capabilities, with broad implications for QCD phenomenology and quantum-information-inspired methods in high-energy physics.

Abstract

The quantum chromodynamics (QCD) phase diagram, which reveals the state of strongly interacting matter at different temperatures and densities, is key to answering open questions in physics, ranging from the behavior of particles in neutron stars to the conditions of the early universe. However, classical simulations of QCD face significant computational barriers, such as the sign problem at finite matter densities. Quantum computing offers a promising solution to overcome these challenges. Here, we take an important step toward exploring the QCD phase diagram with quantum devices by preparing thermal states in one-dimensional non-Abelian gauge theories. We experimentally simulate the thermal states of SU(2) and SU(3) gauge theories at finite densities on a trapped-ion quantum computer using a variational method. This is achieved by introducing two features: Firstly, we add motional ancillae to the existing qubit register to efficiently prepare thermal probability distributions. Secondly, we introduce charge-singlet measurements to enforce color-neutrality constraints. This work marks the first lattice gauge theory quantum simulation of QCD at finite density and temperature for two and three colors, laying the foundation to explore QCD phenomena on quantum platforms.

Figures (10)

• Figure 1: Particle physics phase diagram on a quantum computer. (a) We study the SU$(2)$ and SU$(3)$ phase diagram on a 1D lattice by preparing thermal states at finite chemical potential $\mu$. A unit cell consists of an antimatter site (striped circles) and a matter site (solid circles), connected by a gauge field (wiggly line). (b) In our experiment, each ion acts as a qubit, encoding quark color components in its internal states. For $N$ ions, $N$ motional modes in the $y$-direction serve as an ancilla register (purple), and $N$ motional modes in the $x$-direction mediate entangling gates between qubits (orange). Qubit and motional operations are driven by a set of addressed laser beams, and the qubit states are measured by collecting fluorescence on a photo-multiplier tube (PMT) array. (c) A parametrized circuit $\hat{U}_A(\boldsymbol{\theta})$ prepares a probability distribution $p_{n}(\boldsymbol{\theta})$, used to calculate the entropy $S(\boldsymbol{\theta})$ of the thermal state. The resulting distribution of initial states in the system register is subject to a second parametrized circuit $\hat{U}_S(\boldsymbol{\varphi})$. The energy $E(\boldsymbol{\theta}, \boldsymbol{\varphi})=\langle \hat{H}\rangle$, is measured by suitably rotating the measurement basis using additional unitaries $\hat{M}_H$. Using the measured energy and entropy values, the free energy is calculated and classically minimised to find optimal parameters $(\boldsymbol{\theta}^*,\boldsymbol{\varphi}^*)$ for a given temperature and chemical potential. (d) Our unconstrained variational search (dashed path) explores the model Hilbert space (large oval). A projection method retrieves the expectation value $\langle \hat{O}\rangle_{0}$ of an observable $\hat{O}$ within the charge-singlet subspace as the ratio of $\langle\hat{O}\hat{K}\rangle$ and $\langle\hat{K}\rangle$, where $\hat{K}$ is a projection operator specific to the underlying gauge group.
• Figure 2: SU(2) thermal states for a unit cell with trapped ions. (a) Exact diagonalization (ED) results for the SU(2) unit cell for $x=1$ and $m=0.5$. The order parameter $\langle\hat{\chi}\rangle_{0}$ (chiral condensate) takes large negative values in the low $T$ and $\mu$ limit. Chiral symmetry $\langle\hat{\chi}\rangle_{0}$=0 is restored at high $\mu$ and $T\rightarrow \infty$. (b) Classical simulation results for our variational protocol (Fig. \ref{['fig:protocol']}) for the noise-free case. (c) Experimental data for $T=0.5$ (dashed line in panel (a)). Our motional ancillae based protocol uses up to 230 cost function evaluations per point, determining the chiral condensate for five distinct chemical potential values. The experimental VQE results (red diamonds) are in good agreement with both the ED (black curve) and noisy simulation results (grey boxes). The grey boxes show the spread of mean chiral condensate values from twenty noisy VQE runs (represented by the error bar with the box denoting the inter-quartile range) for each chemical potential, highlighting the protocol’s high success rate. (d) Composition of the charge-singlet thermal state at varying chemical potentials. The mixtures of SU(2) physical eigenstates show the transition from a vacuum-dominated to a baryon-dominated phase. Panel (f) shows the composition of the physical eigenstates in terms of the strong coupling ($x \ll 1$) eigenstates (panel (e)). The heights of the various bar-segments represent the contributions of the strong-coupling states.
• Figure 3: SU(3) thermal states for a unit cell with trapped ions. (a) Chiral condensate for a unit cell obtained from exact diagonalization (ED) for $x=1.0, m = 0.5$. The phase diagram is qualitatively similar to Fig. \ref{['fig:targetplot']}a, but differs quantitatively, with the transition point at zero temperature occurring at a distinct $\mu$-value compared to SU(2). (b) Classical simulation results for our variational protocol (Fig. \ref{['fig:protocol']}) in the noiseless case. (c) The VQE experiment is run for $\mu = 2$ close to the phase transition, allowing up to 350 cost function evaluations. The experimental result matches well with the noisy VQE simulation, showing the effectiveness of the ansatz in preparing the thermal state near the transition. Additionally, the VQE circuit is run using the optimised ideal VQE parameters for $T = 0.5$ for a range of $\mu$ values, confirming our noise model. The spread of the noisy VQE simulation collected over twenty trials (represented by the error bar with the box denoting the interquartile range) highlights the reliability of our protocol. (d) Boltzmann weights of eigenstates of the Hamiltonian in the charge-singlet thermal state are shown at three different chemical potentials, highlighting the transition from vacuum-dominated density matrix to baryon-dominated density matrix. Panels (e) and (f) show the strong coupling ($x\ll 1$) and physical eigenstates. Due to the presence of three colors, the unit cell allows for more gauge-invariant states than the SU(2) model in Fig. \ref{['fig:targetplot']}, which did not include the tetraquark state.
• Figure 4: Two methods for thermal state preparation. Bottom: The protocol suitable for a qubit ancilla using a CNOT gate. Top: Alternative protocol utilising a motional ancilla. The system qubit and ancillary motional mode are both in their ground states. A blue sideband transition on the qubit resonance coherently transfers population from $\ket{\mathrm{qubit,motion}}=\ket{0,0}$ to $\ket{1,1}$, resulting in the final state $\cos(\theta/2)\ket{0,0}+\sin(\theta/2)\ket{1,1}$.
• Figure 5: Circuit for the variational preparation of SU(2) LGT thermal states. The circuit includes parameterized $R_{X}(\theta_i)$ rotations applied to the ancillae, followed by CNOT gates coupling the motional ancillae with the system qubits. This first group of operations forms $\hat{U}_{A}(\theta)$. Then, a layer of $R_{Z}$ rotations sandwiched between blocks of three-body $R_{YZX}$ gates is applied. The gates acting on the system qubits form the unitary operation $\hat{U}_{S}(\varphi)$. The circuit has 10 variational parameters. Additionally, a measurement circuit is required for measuring the non-diagonal contribution $\hat{H}_{2}$ in the Hamiltonian.