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

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 behaviour 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 colour-neutrality constraints. This work pioneers the quantum simulation of QCD at finite density and temperature for two and three colours, laying the foundation to explore QCD phenomena on quantum platforms.

Introduction

The phase diagram of quantum chromodynamics (QCD) underpins our understanding of possible phases of matter in nature and addresses foundational questions in nuclear physics, particle physics, and cosmology. The QCD phase diagram maps out quarks and gluons across various temperatures and densities (non-zero fermionic chemical potentials). Despite intense scientific interest in exploring the QCD phase diagram^1,2, most current numerical approaches, which are based on Monte Carlo methods, are hindered by sign problems³.

Several strategies have been explored for addressing the sign problem in order to probe different regions of the QCD phase diagram^1,3,4,5,6. One promising approach to avoid sign problems at non-zero fermionic chemical potentials is to adopt the Hamiltonian formalism^7,8. Initial studies using tensor-network states within this framework show encouraging results^9,10,11. Another path forward is the use of quantum computing, which offers a fundamentally sign-problem-free approach for future lattice gauge theory (LGT) calculations at non-zero densities^12,13,14.

While quantum computing represents an enormous scientific opportunity, realising its potential requires both experimental and theoretical advances. Progress has been made in experimental demonstrations of lattice gauge theories in one and two spatial dimensions (1D and 2D)^10,12,13,15. These lower-dimensional models can capture interesting physics with reduced resource requirements and serve as a pathway to quantum simulations in higher dimensions. However, even in 1D, quantum simulations of phase diagrams face significant hurdles that suggest the need for new approaches. The first one is the preparation of mixed states on quantum computers^{16,17,18,19,20,21,22,23,24}. This is difficult since it requires non-unitary evolution or thermal sampling of multiple eigenstates. The second one is ensuring the colour-neutrality constraints imposed by the boundary conditions and the gauge symmetry of the model.

In this article, we overcome these challenges by introducing (i) motional ancillae to efficiently create thermal probability distributions on a trapped ion device, and (ii) charge-singlet measurements that use group-theoretical projector techniques.

To address the first challenge, we leverage a subset of the 3N vibrational modes that are naturally available in a system of N trapped-ion qubits as motional ancillae²⁵. By employing these motional modes beyond their typical role as intermediaries of entangling qubit operations, we add an independent ancilla register without increasing the system size. Our approach paves the way for the applications of motional modes in the study of thermal states in quantum many-body systems.

The second challenge requires enforcing the non-Abelian gauge symmetry constraints²⁶. In the absence of background charges, the thermal state should be a probabilistic mixture of colour-neutral states, i.e., states with zero global colour charge. Instead of requiring the thermal state to respect these symmetry constraints, we incorporate them into the measurement process. This strategy uses a group-theoretical projection technique²⁷ that projects a state onto the colour-neutral or singlet subspace and enhances our protocol’s flexibility, facilitating an effective implementation of the colour-neutrality condition.

In combination, the above methods allow us to perform the first experimental study of the phase diagram of one-dimensional QCD on a quantum computer. Our work provides an important initial step for understanding fundamental particle interactions at non-zero temperatures and densities.

Results

Thermal states in gauge theories

Gauge theories are the backbone of the Standard Model of particle physics, with QCD using the SU(3) gauge group to describe the interaction between quarks (fermions) and gluons (gauge bosons). We study the gauge groups SU(2) and SU(3)—with two and three colours, respectively—in 1D with open boundary conditions, with dynamical fermionic matter at non-zero temperature T, and chemical potential μ.

We use here the Kogut-Susskind Hamiltonian approach²⁶ to lattice gauge theory (LGT), that describes the interaction between (fermionic) matter fields and (bosonic) gauge fields defined on the vertices of the lattice and on the links between vertices, respectively (see Fig. 1a). In natural units (ℏ = c = k_B = 1), the Hamiltonian consists of the following terms

$$\widehat{H}={\widehat{H}}_{{{{\rm{kin}}}}}+am{\widehat{H}}_{{{{\rm{mass}}}}}+\frac{1}{2x}{\widehat{H}}_{{{{\rm{elec}}}}}-a\mu {\widehat{H}}_{{{{\rm{chem}}}}}\,,$$

(1)

where the first term is the kinetic energy and describes how matter fields interact with gauge fields as they move between lattice sites. The second term encodes the mass contribution of the matter fields, where m denotes the bare fermion mass and a is the lattice spacing. The third term is the colour electric field energy contribution, where x = 1/(ga)² is related to the gauge-matter coupling strength g. Finally, the last term describes the matter-antimatter imbalance in the system and accounts for non-zero chemical potential μ. Throughout the rest of this work, we will adopt the conventional lattice units where a = 1, making the temperature and chemical potential dimensionless. The explicit form of this Hamiltonian in terms of fermionic and gauge fields is given in the Supplementary Note 3 and 4).

Fig. 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 μ. 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 colour 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 In the variational quantum eigensolver (VQE) protocol, a parametrised circuit ${\hat{U}}_{A}({{{\boldsymbol{\theta }}}})$ prepares a probability distribution p_n(θ), used to calculate the entropy S(θ) of the thermal state. The resulting distribution of initial states in the system register is subject to a second parametrised 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 (θ^*, φ^*) 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.

The theory’s non-Abelian local gauge symmetry leads to a set of non-commuting conserved charges that give rise to constraints known as Gauss laws (see Supplementary Note 3 and 4). Physical states satisfy these Gauss laws and are referred to as the gauge-invariant states. We use the Gauss constraints to eliminate the gauge fields^28,29,30, resulting in a purely fermionic Hamiltonian that is then mapped onto qubits by a Jordan-Wigner (JW) transformation (see Supplementary Note 3 and 4). Elimination of the gauge fields using JW transformations results in long-range four-body interactions for SU(2) and six-body interactions for SU(3). In this work, we focus on the phase diagram of a single unit cell, where such long-range interactions are absent. Nonetheless, it is possible to implement these interactions on a trapped-ion platform due to the intrinsic all-to-all connectivity between the ions^28,31.

After eliminating the local gauge degrees of freedom, we restrict our analysis to the sector with zero global colour charge, as required by the boundary condition that assumes the absence of background charges. This choice is motivated by the physical observation that hadrons with non-zero colour charge are not observed in nature. States that satisfy this zero global charge condition are referred to as charge-singlet states (see Fig. 1d). Our goal is for the thermal state to be a probabilistic mixture of such charge-singlet states at a given temperature.

To study the phase diagram (see Fig. 1a), we prepare the Gibbs thermal states at temperature β = T⁻¹,

$${\widehat{\rho }}_{G}=\frac{{e}^{-\beta \widehat{H}}}{Z}=\sum\limits_{n}{p}_{n}\left\vert {E}_{n}\right\rangle \left\langle {E}_{n}\right\vert,\,\,Z=\,{\mbox{Tr}}\,\,\left({e}^{-\beta \widehat{H}}\right).$$

(2)

Here, Z is the partition function and ensures proper normalisation. The thermal density matrix ${\widehat{\rho }}_{G}$ is a probabilistic mixture of pure states as can be seen from the expansion in the eigenstate basis $\left\vert {E}_{n}\right\rangle$ of the Hamiltonian $\widehat{H}$, where ${p}_{n}={e}^{-\beta {E}_{n}}/Z$ are the Boltzmann weights and E_n are the eigenvalues of the Hamiltonian. The Gibbs state minimises the free energy F = E − TS, where $E=\,{\mbox{Tr}}\,({\widehat{\rho }}_{G}\widehat{H})$ is the internal energy of the system and $S=-\,{\mbox{Tr}}\,({\widehat{\rho }}_{G}\ln {\widehat{\rho }}_{G})$ is the entropy.

Importantly, to construct the charge-singlet thermal state from ${\widehat{\rho }}_{G}$, the trace in the partition function and the eigenstates $\left\vert {E}_{n}\right\rangle$ must be restricted to the singlet subspace. This requirement adds significant complexity to the process of preparing thermal states for LGTs.

A natural choice of order parameter for exploring the phase diagram is the chiral condensate $\widehat{\chi }$, which is proportional to the mass term ${\widehat{H}}_{{{{\rm{mass}}}}}$. The explicit form of this operator is provided in Methods for the SU(2) and SU(3) cases, respectively. In (3 + 1)-D QCD, the chiral condensate serves as the order parameter for the T − μ phase diagram. A change from a negative chiral condensate value (corresponding to condensate formation) to zero chiral condensate (corresponding to quark-gluon plasma formation) as μ is varied in (3 + 1)-D QCD signifies a phase transition from chiral symmetry broken phase to chiral symmetry restored phase. However, the precise location and nature of the critical points in the QCD phase diagram at finite density remain unknown due to the sign problem. Motivated by this open question in fundamental physics, we adopt $\widehat{\chi }$ as the order parameter in our (1 + 1)-D lattice QCD simulation. Our goal is to observe the same transition from a negative to zero chiral condensate, which in our context reflects a change in the dominant eigenstate contributions to the thermal density matrix, an interpretation that we elaborate upon in the discussion of the experimental results below.

Thermal state preparation with motional ancillae

We describe here our protocol for preparing thermal states in gauge theories on a quantum computer and present the two key components that enable it: the introduction of motional ancillae and charge-singlet measurements.

Experimental setup:—A chain of ¹⁷¹Yb⁺ ions is held in a linear Paul trap (Fig. 1b). Each ion provides a qubit in its internal degrees of freedom, encoded in the hyperfine-split electronic ground level, with $\left\vert 0\right\rangle={| }^{2}{S}_{1/2},\left.F=0,{m}_{F}=0\right\rangle$ and $\left\vert 1\right\rangle={| }^{2}{S}_{1/2},F=1,\left.{m}_{F}=0\right\rangle$. An array of N ions in the trap also provides three sets of N orthogonal harmonic motional modes, one in each spatial direction. We utilise the modes in the radial x-direction as intermediaries to realise standard entangling gate operations between qubits with a variant of the Mølmer-Sørensen (MS) gate scheme (see Methods for details). The modes in the y-direction are typically unused, but here we leverage them as an independent ancilla register for preparing the probabilities in the Gibbs ensemble.

Protocol:—To prepare the Gibbs state, we use a variational quantum eigensolver (VQE)³² to find the state that minimises the free energy²³. Our parametrised VQE circuit (Fig. 1c) consists of two parts. The first unitary circuit ${\widehat{U}}_{A}({{{\boldsymbol{\theta }}}})$ couples the motional ancillae with the system register. When the ancilla register is traced out, it creates a tunable probability distribution ${\tilde{p}}_{j}({{{\boldsymbol{\theta }}}})$. For a general probability distribution, the unitary ${\widehat{U}}_{A}({{{\boldsymbol{\theta }}}})$ contains a sequence of parametrised single-qubit and entangling gates acting on the motional ancillae, followed by gates entangling the motional ancillae with the system qubits. For the system considered in this work, it is sufficient to create the pairwise qubit-motion entangled state $\cos ({\theta }_{i}/2)\left\vert 0,0\right\rangle+\sin ({\theta }_{i}/2)\left\vert 1,1\right\rangle$, where the computational basis is denoted by $\left\vert \,{\mbox{spin,motion}}\,\right\rangle$. In a trapped-ion system, this qubit-motion entangled state is created using a partial sideband rotation (see Methods) with a laser drive detuned by the motional frequency. By tuning θ_i for each qubit-motional mode pair, and by tracing out the motional ancillae, the entire system register is prepared in a mixture of bit strings $\left\vert \, j\right\rangle \equiv \left\vert \, {j}_{1}\,{j}_{2}\cdots {j}_{N}\right\rangle$, with a probability ${\tilde{p}}_{j}({{{\boldsymbol{\theta }}}})={\prod }_{i=1}^{N}{\tilde{p}}_{{j}_{i}}({\theta }_{i})$. A more complex set of unitary operations on the motional ancillae³³ will lead to more general probabilities.

A second unitary circuit ${\widehat{U}}_{S}({{{\boldsymbol{\varphi }}}})$ is applied on the qubit register to create the eigenstates of the density matrix. In combination with the motional ancillae, this produces the following variational ansatz for the density matrix

$$\widehat{\rho }({{{\boldsymbol{\theta }}}},{{{\boldsymbol{\varphi }}}})={\sum }_{j}{\widetilde{p}}_{j}({{{\boldsymbol{\theta }}}})\,{\widehat{U}}_{S}({{{\boldsymbol{\varphi }}}})\left\vert \, j\right\rangle \left\langle j\right\vert {\widehat{U}}_{S}^{{{\dagger}} }({{{\boldsymbol{\varphi }}}}).$$

(3)

The variational parameters (θ, φ) are then updated through a feedback loop between a classical optimiser and the ion trap to minimise the cost function $F[\widehat{\rho }({{{\boldsymbol{\theta }}}},{{{\boldsymbol{\varphi }}}})]={{{\rm{Tr}}}}(\widehat{\rho }({{{\boldsymbol{\theta }}}},{{{\boldsymbol{\varphi }}}})\widehat{H})-T\langle \widehat{S}\rangle ({{{\boldsymbol{\theta }}}})$. The average of the Hamiltonian in the cost function is measured on the system register. For a general construction of the unitary ${\widehat{U}}_{A}({{{\boldsymbol{\theta }}}})$, the entropy can be obtained by measuring the ancilla register $\langle \widehat{S}\rangle ({{{\boldsymbol{\theta }}}})=-{\sum }_{j}{\tilde{p}}_{j}({{{\boldsymbol{\theta }}}})\log {\tilde{p}}_{j}({{{\boldsymbol{\theta }}}})$. For the design of ${\widehat{U}}_{A}$ in our experiment, the entropy can be calculated analytically (see Methods for more details) from the partial sideband rotation angles θ_i in the ancilla using

$$\begin{array}{rc}S({{{\boldsymbol{\theta }}}})=-\sum_{i}&\left[{\cos }^{2}({\theta }_{i}/2)\log ({\cos }^{2}({\theta }_{i}/2))\right.\\ &+\left.{\sin }^{2}({\theta }_{i}/2)\log ({\sin }^{2}({\theta }_{i}/2))\right]\,.\end{array}$$

(4)

Consequently, measurements of the motional ancillae are not required in our protocol. Furthermore, due to the structure of our ancilla circuit, the probability distribution is insensitive to the relative phases of the motional modes. This makes our experiment resilient to motional decoherence and imperfect cooling, bypassing the typical hurdles in using motional modes for computational tasks³³.

At the end of the variational search, using the optimal parameters (θ^*, φ^*), we prepare the Gibbs state in Eq. (2) on our trapped-ion device. If the VQE converges successfully, the probabilities ${\tilde{p}}_{j}({{{{\boldsymbol{\theta }}}}}^{*})$ will match the Boltzmann weights p_n in Eq. (2) and the states ${\widehat{U}}_{S}({{{{\boldsymbol{\varphi }}}}}^{*})\left\vert \, j\right\rangle$ will approximate the basis vectors $\left\vert {\psi }_{n}\right\rangle$ of the thermal density matrix. For non-degenerate eigenvalues p_n, these $\left\vert {\psi }_{n}\right\rangle$ s are energy eigenstates $\left\vert {E}_{n}\right\rangle$, and otherwise they form orthonormal linear combinations of the energy eigenstates within the degenerate subspace. Importantly, the basis states $\left\vert {\psi }_{n}\right\rangle$ are obtained through variational optimization of the cost function without requiring any prior knowledge of the energy eigenstates and the underlying spectrum.

So far, we have not incorporated the global charge constraints in our protocol. Hence, the eigenstates of this density matrix are not restricted to the charge-singlet subspace. To distinguish the prepared thermal state from the density matrix restricted to the singlet subspace, we refer to it as the unconstrained density matrix.

Charge-singlet measurements:—To implement the global charge constraint, we modify the measurement of physical observables to yield the same expectation values as those obtained from the thermal state restricted to the singlet subspace. For a given physical observable $\widehat{O}$, we use a group-theoretical projection method^27,34 to define the observable expectation value ${\langle \widehat{O}\rangle }_{0}$ on the singlet subspace as

$${\langle \widehat{O}\rangle }_{0}=\frac{\langle \widehat{O}\widehat{K}\rangle }{\langle \widehat{K}\rangle }\,.$$

(5)

Here, the averages on the right-hand side are measured with respect to the unconstrained density matrix $\widehat{\rho }({{{{\boldsymbol{\theta }}}}}^{*},{{{{\boldsymbol{\varphi }}}}}^{*})$ prepared on the device at the end of the variational optimisation and $\widehat{K}$ is a projection operator specific to the group under consideration. In Methods we explain in more detail how the projection operator $\widehat{K}$ can be calculated explicitly for SU(2) and SU(3) groups. In our case, the order parameter ${\langle \widehat{\chi }\rangle }_{0}$ is evaluated using Eq. (5). Both $\widehat{\chi }\widehat{K}$ and $\widehat{K}$ are diagonal operators (see Methods and Supplementary Note 2), and can thus be measured using ${\widehat{\sigma }}^{z}-$ basis measurements only. Eq. (5) can be used for determining the expectation value of non-diagonal observables (e.g., the Hamiltonian) from the unconstrained density matrix, but would require non-diagonal Pauli measurements to be performed on the system due to $\widehat{O}\widehat{K}$ consisting of non-diagonal Pauli strings.

By determining ${\langle \widehat{\chi }\rangle }_{0}$ using the charge-singlet measurement technique and leveraging motional ancillae, we can now investigate the phase diagram on a trapped-ion device.

SU(2) and SU(3) phase diagram on an ion-trap quantum computer

We implement our protocol for the LGT unit cell (Fig. 1a), which hosts red and green quarks and anti-quarks for SU(2), and additional blue quarks and anti-quarks for SU(3). The Hamiltonians for each case are given explicitly in Methods. For the experiment, we choose the Hamiltonian parameters x = 1 and m = 0.5 in Eq. (1) for both models, placing the system in the intermediate coupling strength regime, where neither the electric field nor the mass term dominates.

Experimental realisation:—The variational circuit ${\widehat{U}}_{S}({{{\boldsymbol{\varphi }}}})$ in Fig. 1c consists of gates that implement different terms in the target Hamiltonian in Eq. (1), as shown in Methods. The SU(2) and SU(3) LGT circuit contain three- and four-qubit gates, respectively, which can be expressed in terms of the native two-qubit Mølmer-Sørensen (MS) gates (see Supplementary Note 5). This circuit, combined with the parametrised partial sideband rotations that couple the motional ancillae with the system register, completes our ansatz. The VQE parameters (10 for SU(2) and 21 for SU(3)) are then classically optimised using a Bayesian direct search algorithm³⁵, where the cost function is evaluated on the quantum computer.

The VQE cost function F = E − TS consists of two components: energy E and entropy S. The entropy term is computed analytically using Eq. (4) based on the gate angles of the partial sideband rotations. The energy is measured on the system register using the Pauli-decomposition of the Hamiltonian. In both SU(2) and SU(3) LGT, the Hamiltonian decomposes into two families of commuting Pauli strings: one comprising the diagonal terms and the other comprising the non-diagonal terms (Methods). The diagonal terms can be measured directly in the ${\widehat{\sigma }}^{z}-$ basis. We design a measurement circuit ${\widehat{M}}_{H}$ using the formalism developed in ref. ³⁶ to evaluate the expectation value of all the non-diagonal Pauli strings simultaneously. This reduces the number of required measurements at the cost of adding a small number of entangling gates (see Methods). Each ${\widehat{\sigma }}^{z}-$ basis measurement is repeated 2000 and 3000 times for SU(2) and SU(3), respectively. As the chemical potential μ varies, the structure of the VQE ansatz remains unchanged; however, the optimised parameters (θ^*, φ^*) change accordingly, resulting in a different composition of the optimised density matrix for each value of μ.

To validate the performance of our VQE scheme, we run ideal simulations of our protocol for different temperature and chemical potential values, with the results shown in Figs. 2b and 3b. For the experiment, we fix the temperature at T = 0.5, where the chiral condensate shows a phase transition with μ. In this intermediate temperature regime, conventional action-based methods face difficulties in probing the phase diagram for larger lattice sizes.

Fig. 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 μ limit. Chiral symmetry ${\langle \hat{\chi }\rangle }_{0}$= 0 is restored at high μ and T → ∞. b Classical simulation results for our variational quantum eigensolver (VQE) protocol (Fig. 1) for the noise-free case. c Experimental data for T = 0.5 (dashed line in 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. Error bars for the experimental VQE points show one standard deviation for repeated trials with parameters obtained from the VQE run. 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. f shows the composition of the physical eigenstates in terms of the strong coupling (x ≪ 1) eigenstates (e). The heights of the various bar-segments represent the contributions of the strong-coupling states.

Fig. 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. 2a, but differs quantitatively, with the transition point at zero temperature occurring at a distinct μ-value compared to SU(2). b Classical simulation results for our variational quantum eigensolver (VQE) protocol (Fig. 1) in the noiseless case. c The VQE experiment is run for μ = 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 μ 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. Error bars for the experimental VQE and direct implementation points show one standard deviation for repeated trials with parameters obtained from the VQE run. 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. e, f show the strong coupling (x ≪ 1) and physical eigenstates. Due to the presence of three colours, the unit cell allows for more gauge-invariant states than the SU(2) model in Fig. 2, which did not include the tetraquark state.

SU(2) thermal states:—The full variational protocol is executed on the trapped-ion system for five different values of the chemical potential, shown in Fig. 2c. The results show excellent agreement with exact diagonalization. At very low chemical potential, the chiral condensate is strongly negative, gradually rising towards zero as the chemical potential is increased. This behaviour resembles the phase transition from chiral symmetry broken phase to chiral symmetry restored phase in (3 + 1)-D QCD. The large negative values of the chiral condensate indicates a thermal mixture dominated by the physical vacuum state, whereas a zero value of chiral condensate corresponds to a mixture dominated by the baryon state. This can be explained from the expression of the Boltzmann weights ${e}^{-\beta {E}_{n}}={e}^{-\beta ({\tilde{E}}_{n}-\mu B)}$, where ${\tilde{E}}_{n}$ are the energy eigenvalues corresponding to the eigenstates $\left\vert {E}_{n}\right\rangle$ at zero chemical potential and B is the baryon number (expectation value of ${\widehat{H}}_{{{{\rm{chem}}}}}$). At low chemical potential, the energy contribution ${\tilde{E}}_{n}$ dominates in the Boltzmann weight, leading to a thermal mixture where the lowest energy eigenstate, i.e., the physical vacuum is energetically favoured. On the contrary, at high μ, the baryon contribution μB dominates over the energy ${\tilde{E}}_{n}$, leading to a baryon-dominated density matrix.

Figure 2d shows the probabilities (the Boltzmann weights) of the physical singlet eigenstates of the Hamiltonian that appear in the thermal state mixture. This phase transition from vacuum to baryon-dominated density matrix is effectively captured by our experiment. Additionally, we employ a noise model that simulates the effects of the dominant noise sources (see Methods) in our experimental device and conduct multiple noisy VQE trials to assess our ansatz’s performance. The distribution of chiral condensate values for different μ in Fig. 2c (represented by the grey boxes) underscores the reliability of the protocol.

The physical colour-neutral eigenstates in Fig. 2d are a linear superposition of the strong-coupling colour-neutral eigenstates (Fig. 2e), which form a convenient basis for the subspace. For example, the physical vacuum state is different from the strong-coupling state $\left\vert vac\right\rangle$, which corresponds to an all-empty lattice configuration. Figure 2f shows the composition of the physical eigenstates in terms of the strong coupling basis. The contributions of these basis states to the physical eigenstates are governed by the choice of the Hamiltonian parameters x and m, but do not depend on T and μ. For fixed x and m values, the temperature and chemical potential determine the Boltzmann weight of each physical eigenstate in the mixture, which determines the shape of the phase curve in Fig. 2c. In particular, as temperature increases, the transition from the vacuum-dominated to baryon-dominated thermal density matrix becomes smoother, eventually disappearing at high T. This produces a localized region of negative chiral condensate values in the lower left corner of the phase diagram in Fig. 2a.

SU(3) thermal states:—Including an additional colour (blue) allows for richer physics compared to SU(2). In particular, an SU(3) baryon is composed of three coloured quarks and exhibits true fermionic behaviour, unlike the SU(2) model, where a baryon consists of only two quarks and follows bosonic statistics. As a result, the SU(3) model encounters the sign problem, while the SU(2) model does not. This makes SU(3) LGT an ideal candidate for leveraging quantum computers. Additionally, the enlarged Hilbert space for SU(3) allows more degrees of freedom in constructing charge-singlet states on a lattice.

The increased non-locality of the interactions present in the Hamiltonian, as well as the larger size of the Hilbert space, makes the experimental realisation of the VQE more demanding. Again, we chose intermediate parameter values, in this case μ = 2 and T = 0.5, at the phase transition from a vacuum-dominated phase to a baryon-dominated phase, see Fig. 3d. Our experimental VQE successfully prepares the thermal mixture, which combined with charge-singlet measurements for the chiral condensate, agrees with the exact diagonalization value of ${\langle \widehat{\chi }\rangle }_{0}$ (Fig. 3c).

In SU(3), constructing the physical vacuum is more involved than SU(2) (Fig. 3f) due to the presence of an additional strong-coupling singlet eigenstate in the superposition, which we call the tetraquark state³⁷. Absent in the unit cell of SU(2), the tetraquark state consists of a pair of quarks and a pair of antiquarks (Fig. 3e). Our successful VQE optimisation for the phase transition point, where contributions from the physical vacuum and baryon state become of similar magnitude, demonstrates the capability to capture both vacuum- and baryon-dominated phases effectively. Demonstrating the successful VQE performance for the transition point provides strong evidence that the approach is expected to work at other values of the chemical potential as well. It is thus a significant first step towards simulating the whole phase diagram experimentally for larger lattice sizes.

We use the same device-aware noise model as before for SU(3), and run multiple independent numerical trials of the noisy VQE for different values of the chemical potential. Figure 3c shows that the noisy VQE successfully traces the path from a chiral symmetry-broken phase to a symmetry-restored phase. The larger spread of the error bars from the noisy VQE indicates that although it is more error-prone compared to SU(2), the VQE still performs well in estimating the chiral condensate value. The larger fluctuations in optimization are a consequence of increased parameter space, register size, and circuit depth. Additionally, we observe that the fluctuations are most pronounced for intermediate values of μ, near the phase transition point. This aligns with our expectation that this region is the most challenging to capture reliably in experiments, motivating our choice of μ for the experiment. Furthermore, we use the optimised parameters obtained from our noisy VQE simulations to directly prepare the unconstrained thermal state on our trapped-ion experiment and evaluate the chiral condensate. Our measurements yield excellent agreement with the simulated VQE and exact diagonalization results, validating our noise model.

Discussion

Quantum simulations of particle physics have so far mostly focused on pure states at T = 0^12,13,15. However, to describe nature, it is crucial to understand states of matter at finite temperature. Our work opens the door to resource-efficient quantum simulations of thermal states in gauge theories.

Our charge-singlet measurement technique is broadly applicable for different gauge theories and can be easily extended to studying dynamics using, e.g., trotterized time evolutions. The projection-based technique can be extended to two or three spatial dimensions, when it is no longer possible to integrate out all gauge degrees of freedom. In these higher-dimensional settings, the charge-singlet projection method can be generalised to enforce both the local Gauss law at each vertex and the global charge constraints, offering an alternative to explicitly imposing gauge invariance. This generalised projection is also applied during the measurement of observables.

Beyond quantum computing, the projector technique used here is equally valuable for classical Hamiltonian-based computations, such as tensor network state calculations. For both quantum- and classical simulations, charge-singlet measurements could be employed to explore thermodynamic quantities like entropy and work within charge-singlet subspaces, offering exciting links to quantum thermodynamics in gauge theories^38,39.

An important next step toward simulating particle physics is the extension to two and ultimately three dimensions. One advantage of using quantum computers for LGT simulations is that the computational framework can be generalised without significant theoretical roadblocks. While the concepts from previous works are transferable (see also^{40,41,42,43,44,45}), scaling up the lattice size will be crucial, increasing the need for resource-efficient methods. Moreover, incorporating quark flavours and designing protocols to connect future quantum simulations with observables in particle physics are also interesting topics for future work.

Our resource-efficient motional ancillae approach leverages otherwise unused degrees of freedom and can be further developed into a fully functional qubit register. Utilising motional modes in ions for certain special applications is already well underway^{46,47,48,49,50,51,52,53,54}, with ongoing efforts realising motional qubits capable of readout, performing general unitary³³ and state-dependent operations⁵⁵. Similar to the all-to-all connectivity available with entangling gates, these ancillary states can couple with nearly any qubit in the system register. This capability holds the potential to create arbitrary probability distributions for general thermal states-using, e.g., circuits inspired by autoregressive models⁵⁶. Our proof-of-concept can also be adapted to bosonic modes that remain otherwise idle in other quantum systems, such as cavity quantum electrodynamics and superconducting circuit platforms.

The use of motional ancillae for the generation of thermal states provides a practical toolbox for studying quantum many-body systems at finite temperature. Applicable to fields like condensed matter physics, chemistry, and particle physics, our results pave the way for leveraging quantum computing to explore phase diagrams and thermodynamic properties in gauge theories and beyond.

Methods

Trapped-ion platform setup

The experiment described here significantly expands the capabilities of our fully programmable ion-trap quantum computer⁵⁷. The system is based on a chain of ¹⁷¹Yb⁺ ions confined in a linear Paul trap. Each ion hosts a pseudo-spin qubit encoded in the hyperfine splitting of the electronic ground state, with $\left\vert 0\right\rangle={| }^{2}{S}_{1/2},F=0,\left.{m}_{F}=0\right\rangle$ and $\left\vert 1\right\rangle={| }^{2}{S}_{1/2},F=1,\left.{m}_{F}=0\right\rangle$. The qubit splitting is approximately 12.643 GHz. Laser beams resonant with the ²S_1/2 → ²P_1/2 transition are used to initialize the qubit into $\left\vert 0\right\rangle$ through optical pumping and to perform projective measurements through state-specific fluorescence⁵⁸. The state of each qubit in the chain is measured individually by focusing the scattered light for each ion onto a distinct photomultiplier tube (PMT). Single qubit measurement fidelities are greater than 99%, limited by off resonant coupling, a fundamental limitation to fluorescence-based state detection. Detector cross talk further contributes to lower multi-qubit measurement fidelities ranging from 92 to 99%, depending on the state. To mitigate these two errors, we perform an independent characterization of state measurement. Using a single ion to eliminate cross talk, we synthetically create representative measurement signals of each multi-qubit state and determine the probability that such a state would be measured correctly or incorrectly. This process allows us to eliminate the effect of measurement cross talk and off-resonant coupling from the qubit probability measurements, which underpin the energy measurements made in this work.

Coherent manipulation of the qubit state is driven by off-resonant Raman transitions using two counter-propagating pulsed laser beams at 355 nm⁵⁹. These operations include a universal gate set consisting of arbitrary single qubit rotations and all-to-all connected ${R}_{XX}^{i,j}(\theta )=\exp (-i\theta \,{\widehat{\sigma }}_{i}^{x}\otimes {\widehat{\sigma }}_{j}^{x}/2)$ entangling gates, utilising the Mølmer-Sørensen (MS) interaction⁶⁰. Single- and two-qubit gate fidelities are greater than 99.9 and 98%, respectively. Of the two beams needed to drive the Raman transition, one beam is split into an array of individual addressing beams, such that each unique beam has independent frequency, phase, and amplitude control and is focused on one ion. The second beam illuminates the chain as a whole for simplicity. The MS gates are implemented using pulse shaping techniques in ref. ⁶¹.

The radial modes along the x-axis are chosen to mediate gates while a subset of the radial modes along the y-axis are used as ancillae (see Fig. 1), so there is no interference between them.

Thermal state preparation using motional ancillae

The first step to preparing each system qubit-ancilla pair in a suitable arbitrary superposition is to initialize to $\left\vert {{{\rm{spin,\; motion}}}}\right\rangle=\left\vert 0,0\right\rangle$. After the motion is laser cooled to near the Doppler limit, all ions in the chain are subject to an optical pumping beam, which places them in $\left\vert {{{\rm{spin}}}}\right\rangle=\left\vert 0\right\rangle$ with a fidelity greater than 99.5% in 5 μs. Subsequently, we perform resolved sideband cooling on all motional modes, with each mode requiring approximately 200 μs to reach the ground state⁶².

After all modes are prepared in the ground state, the ion chain’s motion is in the Lamb-Dicke regime with respect to the Raman laser beams, so the laser may be frequency tuned to drive a resonant blue sideband (BSB) transition on any motional mode²⁵. Hence, for each qubit-mode pair, population can be coherently transferred between $\left\vert 0,0\right\rangle$ and $\left\vert 1,1\right\rangle$, such that the final state is $\cos (\theta /2)\left\vert 0,0\right\rangle+\sin (\theta /2)\left\vert 1,1\right\rangle$ (see Fig. 4). Here, θ = Ω_BSBτ = η_i,mΩ₀τ with τ being the gate time, Ω₀ being the Rabi frequency of the qubit transition, and η_i,m being the Lamb-Dicke parameter for ion i and mode m. Because these BSB pulses are used to create incoherent superpositions, a strict determination of their fidelity is not necessary. In fact, for the purpose of thermal state preparation, only the relative population in $\left\vert 0,0\right\rangle$ and $\left\vert 1,1\right\rangle$ is relevant. This ratio can be prepared with about 98% accuracy.

Fig. 4: Two methods for thermal state preparation.

Bottom left: The protocol suitable for a qubit ancilla using a CNOT gate. Right: 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 $\left\vert {{{\rm{qubit,\; motion}}}}\right\rangle=\left\vert 0,0\right\rangle$ to $\left\vert 1,1\right\rangle$, resulting in the final state $\cos (\theta /2)\left\vert 0,0\right\rangle+\sin (\theta /2)\left\vert 1,1\right\rangle$.

Each motional ancilla may be assigned arbitrarily to almost any system qubit, as long as η_i,m is not impractically small⁶³. First, we describe the choice of qubit-mode pairs for the experiment presented in Fig. 2. Although this experiment strictly requires only as many ions as there are system qubits, we choose the number of ions in the chain to not only support the required number of qubits, but also to optimise alignment of each ion to its addressing optics. For the experiments represented in Fig. 2, we use a chain of seven ions, with four hosting system qubits. Of the seven radial modes along the y-axis, the second, fourth, and sixth modes are not used as ancillae, in order to limit deleterious off-resonant driving arising from imperfect mode resolution. Here, we index the ions from one to seven according to their position in the chain and the modes from one to seven, with mode one being the highest energy (centre of mass) mode.

On top of the mode resolution consideration, we choose the set of qubit-motional mode pairs with generally higher values of η_i,m to minimise the time needed for state preparation. Specifically, we choose the pairs designated by η_i,m = {η_2,1, η_3,3, η_4,5, η_5,7}. The motional modes used as ancillae for ions {2 − 5} have radial frequencies ω = 2π × {2.893, 2.863, 2.830, 2.786} MHz.

For the experiment shown in Fig. 3, we perform our experiment with nine ions in the trap. Ions 3 − 8 host pseudo-spin system qubits. These are paired with modes designated by η_i,m = {η_3,6, η_4,5, η_5,2, η_6,9, η_7,8, η_8,7}. The motional modes used as ancillae have radial frequencies ω = 2π × {2.840, 2.854, 2.889, 2.788, 2.807, 2.824} MHz.

Noise model for VQE

To evaluate the performance of our VQE ansatz in the presence of noise, we used a device-aware noise model. The two primary noise sources in our trapped-ion quantum computer are: (i) random over- or under-rotations in the angles of the partial sideband gate coupling the ancillae with the system, and (ii) imperfections in the implementation of the MS gate on the system register.

Given a target rotation ${R}_{X}({\theta }_{i})=\exp (-i\theta \,{\widehat{\sigma }}_{i}^{x}/2)$ on a ancilla qubit, we model the rotation error by applying ${R}_{X}({\theta }_{i}^{{\prime} })$ on the ancilla mode, where ${\theta }_{i}^{{\prime} }$ is sampled from a normal distribution ${{{\mathcal{N}}}}({\theta }_{i},0.03\times {\theta }_{i})$ for each cost function evaluation. The factor of 0.03 in the standard deviation is specific to our device’s characteristics. Consequently, we substitute ${\theta }_{i}^{{\prime} }$ for θ_i in our density matrix and entropy calculations.

The noisy MS gate in the system register is simulated using a two-qubit depolarizing noise channel applied after each MS gate. The channel strength is chosen to ensure an MS gate fidelity of 98%, consistent with the experimental fidelity achieved in our system. We note that an MS gate fidelity greater than 95% captures the phase transition shown in Figs. 2c and 3c well.

The statistical error introduced by the projective measurement is modelled by sampling the eigenvalues of our observables with N_meas = 2000 shots for SU(2) and N_meas = 3000 shots for SU(3), respectively. The variational search was conducted using the PyBADS optimiser^35,64, with a maximum of 230 function evaluations for SU(2) gauge theory and 350 for SU(3).

For each chemical potential μ, we conducted 20 independent noisy simulations. In each run, we evaluated 10 instances of the thermal expectation value of the chiral condensate using our charge-singlet measurement protocol. The average of these 10 measurements provided a single data point, resulting in 20 averages per μ value. To quantify the accuracy of the measurements, we used a box plot and calculated the interquartile range (IQR), shown as the grey boxes in Fig. 2c. The error bar represents the spread of data points, and in the presence of outliers denotes the interval (Q₁ − 1.5 IQR, Q₃ + 1.5 IQR), where Q₁ and Q₃ represents the first and third quartile of the data set. This comprehensive analysis allowed us to evaluate the robustness of the VQE circuit and accuracy of our gauge-invariant measurement protocol under realistic noise conditions.

Derivation of Eq. 5

The projection operator enables charge-singlet measurements of observables in our protocol on states that are not necessarily charge-singlet. We use it to compute averages of operators within the singlet subspace, defined by the colour-neutrality constraint. Our approach adapts and extends the ideas presented in refs. ^27,34, tailoring them for use and implementation on a quantum computer.

We start with the general formalism for an SU(N_c) gauge group, where N_c is the number of colours. The Hilbert space of our system can be decomposed as ${{{\mathcal{H}}}}{=\bigoplus }_{{{{\mathbf{\alpha }}}}}{{{{\mathcal{H}}}}}_{{{{\mathbf{\alpha }}}}}$, where the direct sum runs over the irreducible representations α of the colour group. Any gauge-invariant operator, such as the Hamiltonian $\widehat{H}$ or the Gibbs density operator ${\widehat{\rho }}_{G}$ can be decomposed into a sum over irreducible representations and acts on the Hilbert space without mixing different representations. In particular, the colour singlet subspace with α = 0 is of interest in this work.

Following²⁷, we can express the trace over the whole Hilbert space of the product of a gauge-invariant observable $\widehat{\Omega }$ with a general group element $\widehat{U}\in$ SU(N_c) as

$${{{\rm{Tr}}}}(\widehat{\Omega }\widehat{U})=\sum\limits_{\alpha }\frac{{{{{\rm{Tr}}}}}_{\alpha }(\widehat{\Omega }){{{{\rm{Tr}}}}}_{\alpha }(\widehat{U})}{{d}_{\alpha }},$$

(6)

where the sum runs over irreducible representations of the colour group, ${{{{\rm{Tr}}}}}_{\alpha }$ are traces restricted to states that transforms under the representation α, and d_α is the dimension of the representation. In order to extract the trace over a particular representation, we make use of the orthogonality relation between the irreducible character functions ${\chi }_{\alpha }({{{\mathbf{\eta }}}})={{{{\rm{Tr}}}}}_{\alpha }\,\widehat{U}(\eta )$ with respect to the Haar measure dμ(η) of the group ${\int}_{SU({N}_{c})}d\mu (\eta )\,{\chi }_{\alpha }^{*}({{{\mathbf{\eta }}}}){\chi }_{\beta }({{{\mathbf{\eta }}}})={\delta }_{\alpha \beta }$, where η are variables parametrising group elements. Focusing on the charge-singlet subspace, for which the character function is given by χ₀ = 1, we obtain the restricted trace

$${{{{\rm{Tr}}}}}_{0}(\widehat{\Omega })={{{\rm{Tr}}}}(\widehat{\Omega }\widehat{K}),$$

(7)

where the general expression of the charge-singlet projector is

$$\widehat{K}=\int\,d\mu \,\widehat{U}.$$

(8)

Our charge-singlet measurement protocol is based on Eq. (7) and allows us to evaluate thermal averages restricted to the singlet subspace from averages on the full Hilbert space. To see this, we first replace $\widehat{\Omega }={\widehat{\rho }}_{G}$ with the Gibbs state ${\widehat{\rho }}_{G}$ given in Eq. (2) and find that the thermal average of the operator $\widehat{K}$ is equal to

$$\langle \widehat{K}\rangle={{{\rm{Tr}}}}(\,{\widehat{\rho }}_{G}\widehat{K})=\frac{{Z}_{0}}{Z}\,,$$

(9)

where ${Z}_{0}={{{{\rm{Tr}}}}}_{0}({e}^{-\beta \widehat{H}})$ is the gauge-single partition function, and Z the one over the full Hilbert space. By then choosing $\widehat{\Omega }=\widehat{O}{\widehat{\rho }}_{G}$, where $\widehat{O}$ is a physical observable, we recover the charge-singlet measurement formula (5) given in the main text with ${\langle \widehat{O}\rangle }_{0}={{{{\rm{Tr}}}}}_{0}(\widehat{O}{e}^{-\beta \widehat{H}})/{Z}_{0}=\langle \widehat{O}\widehat{K}\rangle /\langle \widehat{K}\rangle$.

Our formula is particularly well-suited for implementation on a quantum computer, as it requires only the measurement of two observables, $\widehat{O}\widehat{K}$ and $\widehat{K}$, to recover the charge-singlet thermal average of the observable $\widehat{O}$. To realise our charge-singlet measurement protocol, it is necessary to evaluate the projection operator $\widehat{K}$. The group integral in Eq. (8) is explicitly computed for the SU(2) and SU(3) gauge groups in Supplementary Note 2.

In our protocol, we relegate the projection to the end of the VQE process rather than incorporating it into the VQE loop. Evaluating the projected energy during the VQE is resource-intensive. Additionally, it would require knowledge of the entropy within the singlet subspace to compute the cost function.

Details of SU(2) for experimental realisation

SU(2) gauge group basic building block N = 2

The general expression of the SU(2) Hamiltonian for N sites can be found in Supplementary Note 3. Our experimental demonstration focuses on the unit cell with N = 2 lattice sites consisting of two antimatter and matter fermions with (anti-)red and (anti-)green colours, which is mapped to a system of 4 qubits. The Hamiltonian $\widehat{H}={\widehat{H}}_{1}+{\widehat{H}}_{2}$ decomposes into two non-commuting families

$$\begin{array}{rc}{\widehat{H}}_{1}=&\left(2m+\frac{3}{16x}\right)+\left(\frac{m}{2}-\frac{\mu }{4}\right)({\widehat{\sigma }}_{3}^{z}+{\widehat{\sigma }}_{4}^{z})\\ &-\left(\frac{m}{2}+\frac{\mu }{4}\right)({\widehat{\sigma }}_{1}^{z}+{\widehat{\sigma }}_{2}^{z})-\frac{3}{16x}{\widehat{\sigma }}_{1}^{z}{\widehat{\sigma }}_{2}^{z}\,,\end{array}$$

(10)

and

$${\widehat{H}}_{2}=-\frac{1}{4}({\widehat{\sigma }}_{1}^{x}{\widehat{\sigma }}_{2}^{z}{\widehat{\sigma }}_{3}^{x}+{\widehat{\sigma }}_{1}^{\,y}{\widehat{\sigma }}_{2}^{z}{\widehat{\sigma }}_{3}^{\,y}+{\widehat{\sigma }}_{2}^{x}{\widehat{\sigma }}_{3}^{z}{\widehat{\sigma }}_{4}^{x}+{\widehat{\sigma }}_{2}^{\,y}{\widehat{\sigma }}_{3}^{z}{\widehat{\sigma }}_{4}^{\,y})\,,$$

(11)

where m, μ and x = 1/g² are the mass, chemical potential and inverse coupling constant, respectively. ${\widehat{\sigma }}_{n}^{i}$ with i = x, y, z denotes the usual single qubit Pauli matrices at site n. ${\widehat{H}}_{1}$ consists of exclusively diagonal Pauli strings and ${\widehat{H}}_{2}$ contains only non-diagonal Pauli strings. Pauli operators within the same family commute with each other and can therefore be measured simultaneously. However, since ${\widehat{H}}_{2}$ is non-diagonal, a measurement circuit is required in practice to rotate it to the diagonal basis before performing measurements in the ${\widehat{\sigma }}^{z}-$ basis (see Fig. 5). For our target plot shown in Fig. 2, we fix m = 0.5, x = 1, while the chemical potential μ varies from 0 to 4. The coefficients of the Pauli strings for ${\widehat{H}}_{1}$ thus vary with the chemical potential μ.

Fig. 5: Circuit for the variational preparation of SU(2) LGT thermal states.

The circuit includes parameterised R_X(θ_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.

SU(2) VQE circuit

For the basic building block studied here, we need 4 system qubits and 4 motional ancilla modes. The circuit employed in the VQE protocol consists of two main parts. First, a parametrised unitary ${\widehat{U}}_{A}({{{\boldsymbol{\theta }}}})$ is applied to couple the ancillae with the system qubits

$${\widehat{U}}_{A}({{{\boldsymbol{\theta }}}})={\bigotimes}_{i=1}^{4}{R}_{X}({\theta }_{i})\,{{{{\rm{CNOT}}}}}_{{A}_{i},{S}_{i}}\,,$$

(12)

where ${R}_{X}({\theta }_{i})=\exp (-i{\theta }_{i}\,{\widehat{\sigma }}_{i}^{x}/2)$ denotes the rotation around the x-axis by an angle θ_i on the ancilla mode i, and ${{{{\rm{CNOT}}}}}_{{A}_{i},{S}_{i}}$ entangle each mode i in the ancilla register ${{{\mathcal{A}}}}$ with the qubit i in the system register ${{{\mathcal{S}}}}$ (see Fig. 5). From here on, we will use the notation ${R}_{P}(\theta )=\exp (-i\theta \,\widehat{P}/2)$ for a rotation gate, where $\widehat{P}$ is a Pauli string. By tracing out the ancilla modes, we obtain the system’s density matrix

$${\widehat{\rho }}_{S}={{{{\rm{Tr}}}}}_{A}({\widehat{\rho }}_{AS})={\bigotimes}_{i=1}^{4}\left(\begin{array}{cc}{\cos }^{2}({\theta }_{i}/2)&0\\ 0&{\sin }^{2}({\theta }_{i}/2)\end{array}\right).$$

(13)

Expanding this in the computational basis, the density matrix of the system is

$${\widehat{\rho }}_{S}=\sum\limits_{j}{\tilde{p}}_{j}\left\vert \; j\right\rangle \left\langle j\right\vert,$$

(14)

where j denotes the computational basis vectors $\left\vert j\right\rangle=\left\vert {j}_{1}{j}_{2}{j}_{3}{j}_{4}\right\rangle$ with each j_i ∈ {0, 1}. The entropy of the system is then analytically obtained from the probabilities ${\tilde{p}}_{j}$ of the bit string j using Eq. (4).

In the second part of the circuit, the state in Eq. (14) is evolved by the unitary ${\widehat{U}}_{S}({{{\boldsymbol{\varphi }}}})$ acting only on the system qubits to get the desired thermal state. The unitary gates in ${\widehat{U}}_{S}$ are inspired by the Pauli strings appearing in the decomposition of the Hamiltonian. Specifically, we use the three-body gate

$${R}_{YZX}({\varphi }_{i})\equiv \exp \left(-i{\varphi }_{i}({\widehat{\sigma }}^{y}\otimes {\widehat{\sigma }}^{z}\otimes {\widehat{\sigma }}^{x})/2\right).$$

(15)

Here, we specifically choose R_YZX gates instead of R_XZX, as their commutation with Pauli strings in ${\widehat{H}}_{1}$ accurately reproduces the terms of the Hamiltonian in ${\widehat{H}}_{2}$. This three-body gate can be decomposed into native entangling MS gates (see Supplementary Note 5). In total, we need three entangling R_XX gates to implement the three-body gates. In our circuit design, we employ a shifted block structure, where we first apply two consecutive R_YZX gates sharing the same variational parameters on consecutive three qubits, then apply a layer of single-qubit R_Z with independent variational parameters, and finally apply two additional parameter-sharing three-body gates. Using gate identities, the system circuit can be reduced to have only 8 MS gates compared to the initial naive counting of 18 MS gates. The reduced circuit in terms of native gates is shown in Supplementary Fig. 3 in Supplementary Note 5.

The circuit design outlined above is scalable and can be readily extended to larger lattice sizes by increasing the number of qubits in the ancilla and system registers. Because the many-body nature of the interactions in the Hamiltonian remains fixed across different lattice sizes, the type of the gates in the circuit also remains unchanged. The number of two-qubit gates scales polynomially with system size, as the number of Pauli strings in the Hamiltonian grows polynomially with N. The ancilla circuit is similarly straightforward to generalise for larger systems. However, at general values of temperature and chemical potential, preparing the thermal state may necessitate entangling operations among the motional ancilla modes. This, in turn, would require measurements on the motional ancillae to determine the entropy. Multiple techniques for measuring trapped-ion motional states have already been demonstrated in small systems, with their application to larger systems being limited by low motional coherence times^33,65,66,67. Coherence time improvements driven by growing interest in quantum technology based on qumodes will render motional mode measurements feasible on larger devices⁵⁰.

After the circuit execution, a measurement in the computational basis allows us to determine and measure the diagonal contribution of the Hamiltonian ${\widehat{H}}_{1}$. Since the Hamiltonian decomposition also contains non-diagonal Pauli strings given by ${\widehat{H}}_{2}$, we need to integrate an additional circuit ${\widehat{M}}_{H}$ to the unitary ${\widehat{U}}_{S}$ (indicated as measurement circuit in Fig. 5) in order to measure ${\widehat{H}}_{2}$. To find the measurement circuit ${\widehat{M}}_{H}$, we used the stabilizer approach to transform the stabilizer matrix associated with the commuting family of Pauli strings in ${\widehat{H}}_{2}$ into its representation in the computational basis³⁶. The circuit ${\widehat{M}}_{H}$ diagonalizes the Hamiltonian ${\widehat{H}}_{2}$.

Charge-singlet measurement of $\widehat{\chi }$ for SU(2)

The observable of interest in our study is the chiral condensate

$$\widehat{\chi }=\sum\limits_{n=1}^{N}\frac{{(-1)}^{n}}{2}\left({\widehat{\sigma }}_{2n-1}^{z}+{\widehat{\sigma }}_{2n}^{z}\right)$$

(16)

and serves as an order parameter to probe the phase diagram at finite temperature and chemical potential. In order to evaluate its thermal average in the charge-singlet subspace, we use Eq. (5) with $\widehat{O}=\widehat{\chi }$

$${\langle \widehat{\chi }\rangle }_{0}=\frac{\langle \widehat{\chi }\widehat{K}\rangle }{\langle \widehat{K}\rangle },$$

(17)

where ${\langle \widehat{\chi }\rangle }_{0}={{{{\rm{Tr}}}}}_{0}\{{e}^{-\beta \widehat{H}}\widehat{\chi }\}/{Z}_{0}$ and ${Z}_{0}={{{{\rm{Tr}}}}}_{0}({e}^{-\beta \widehat{H}})$ is the singlet partition function. The thermal averages on the right hand side are expressed in the full Hilbert space or the unconstrained space as $\langle \widehat{O}\rangle={{{\rm{Tr}}}}(\widehat{\rho }\widehat{O})$.

The group integral defining our projector in Eq. (8) can be evaluated exactly for SU(2). The general expression of the operator $\widehat{K}$ in terms of the diagonal charge ${\widehat{Q}}_{{{{\rm{tot}}}}}^{z}$ can be found in Supplementary Note 2 (Eq. (5)). Since ${\widehat{Q}}_{{{{\rm{tot}}}}}^{z}$ is a diagonal operator, the projection operator $\widehat{K}$ is also diagonal in the computational basis. In particular, for N = 2, the Pauli decomposition of the operator $\widehat{K}$ reads

$$\begin{array}{rc}\widehat{K}=&\frac{3}{16}({\widehat{\sigma }}_{3}^{z}{\widehat{\sigma }}_{4}^{z}+{\widehat{\sigma }}_{2}^{z}{\widehat{\sigma }}_{3}^{z}+{\widehat{\sigma }}_{1}^{z}{\widehat{\sigma }}_{4}^{z}+{\widehat{\sigma }}_{1}^{z}{\widehat{\sigma }}_{2}^{z}-{\widehat{\sigma }}_{2}^{z}{\widehat{\sigma }}_{4}^{z}-{\widehat{\sigma }}_{1}^{z}{\widehat{\sigma }}_{3}^{z})\\ &+\frac{5}{16}(1+{\widehat{\sigma }}_{1}^{z}{\widehat{\sigma }}_{2}^{z}{\widehat{\sigma }}_{3}^{z}{\widehat{\sigma }}_{4}^{z}),\end{array}$$

(18)

and

$$\begin{array}{l}\widehat{\chi }\widehat{K}=-\frac{1}{4}\left({\widehat{\sigma }}_{1}^{z}+{\widehat{\sigma }}_{2}^{z}+{\widehat{\sigma }}_{2}^{z}{\widehat{\sigma }}_{3}^{z}{\widehat{\sigma }}_{4}^{z}+{\widehat{\sigma }}_{1}^{z}{\widehat{\sigma }}_{3}^{z}{\widehat{\sigma }}_{4}^{z}\right.\\ \left.-{\widehat{\sigma }}_{3}^{z}-{\widehat{\sigma }}_{4}^{z}-{\widehat{\sigma }}_{1}^{z}{\widehat{\sigma }}_{2}^{z}{\widehat{\sigma }}_{3}^{z}-{\widehat{\sigma }}_{1}^{z}{\widehat{\sigma }}_{2}^{z}{\widehat{\sigma }}_{4}^{z}\right).\end{array}$$

(19)

In practice, after the VQE optimisation concludes and the optimal parameters (θ^⋆, φ^⋆) are found, they are used to evaluate the expectation values of the observables $\widehat{\chi }\widehat{K}$ and $\widehat{K}$ on the quantum hardware. Since both observables are diagonal in the computational basis, no additional quantum resources are required for their measurement.

Details of SU(3) for experimental implementation

SU(3) basic building block for N = 2

In this work, we perform a finite temperature VQE for the basic building block (N = 2) of SU(3). The general expression of the Hamiltonian for N > 2 can be found in the Supplementary Note 4. For N = 2, the Hamiltonian describes a system of six qubits. Written in terms of the Pauli operators, the Hamiltonian $\widehat{H}={\widehat{H}}_{1}+{\widehat{H}}_{2}$ decomposes into two non-commuting families given by

$$\begin{array}{rcl}{\widehat{H}}_{1}=&&\left(\frac{m}{2}-\frac{\mu }{6}\right)({\widehat{\sigma }}_{4}^{z}+{\widehat{\sigma }}_{5}^{z}+{\widehat{\sigma }}_{6}^{z})-\left(\frac{m}{2}+\frac{\mu }{6}\right)\left({\widehat{\sigma }}_{1}^{z}+{\widehat{\sigma }}_{2}^{z}\right.\\ &&\left.+{\widehat{\sigma }}_{3}^{z}\right)-\frac{1}{6x}({\widehat{\sigma }}_{1}^{z}{\widehat{\sigma }}_{2}^{z}+{\widehat{\sigma }}_{1}^{z}{\widehat{\sigma }}_{3}^{z}+{\widehat{\sigma }}_{2}^{z}{\widehat{\sigma }}_{3}^{z})+\left(3m+\frac{1}{2x}\right),\end{array}$$

(20)

$$\begin{array}{l}{\widehat{H}}_{2}=\frac{1}{4}\left({\widehat{\sigma }}_{2}^{x}{\widehat{\sigma }}_{3}^{z}{\widehat{\sigma }}_{4}^{z}{\widehat{\sigma }}_{5}^{x}+{\widehat{\sigma }}_{2}^{y}{\widehat{\sigma }}_{3}^{z}{\widehat{\sigma }}_{4}^{z}{\widehat{\sigma }}_{5}^{y}-{\widehat{\sigma }}_{1}^{x}{\widehat{\sigma }}_{2}^{z}{\widehat{\sigma }}_{3}^{z}{\widehat{\sigma }}_{4}^{x}-{\widehat{\sigma }}_{1}^{y}{\widehat{\sigma }}_{2}^{z}{\widehat{\sigma }}_{3}^{z}{\widehat{\sigma }}_{4}^{y}\right.\\ \left.-{\widehat{\sigma }}_{3}^{x}{\widehat{\sigma }}_{4}^{z}{\widehat{\sigma }}_{5}^{z}{\widehat{\sigma }}_{6}^{x}-{\widehat{\sigma }}_{3}^{y}{\widehat{\sigma }}_{4}^{z}{\widehat{\sigma }}_{5}^{z}{\widehat{\sigma }}_{6}^{y}\right).\end{array}$$

(21)

SU(3) VQE circuit

The SU(3) circuit used has the same structure as the one devised for the SU(2) gauge group. First, a layer of single qubit R_X(θ_i) rotations with i = 1, 2, …, 6 is applied to the ancilla modes, which are then independently coupled to the system qubit by a series of CNOT gates (see Fig. 6). The system qubits are then acted on with four-body R_YZZX gates inspired by the Pauli strings decomposition of the Hamiltonian

$${R}_{YZZX}({\varphi }_{i})\equiv \exp \left(-i{\varphi }_{i}({\widehat{\sigma }}^{y}\otimes {\widehat{\sigma }}^{z}\otimes {\widehat{\sigma }}^{z}\otimes {\widehat{\sigma }}^{x})/2\right).$$

(22)

Again, here we use R_YZZX gates instead of R_YZZY to recover the different terms of the Hamiltonian through the commutation algebra. The layer of four-body gates is followed by a series of two-body R_ZZ gates. Next, a layer of single-qubit parametrised rotation gates R_Z(θ_i) is applied. The circuit concludes with another series of three four-body gates. Each four-body gate can be decomposed into five two-qubit gates. To measure the expectation value of the non-diagonal family of Pauli strings appearing in ${\widehat{H}}_{2}$, a measurement circuit ${\widehat{M}}_{H}$ is added to the system register (see Fig. 6).

Fig. 6: Circuit for preparing the SU(3) thermal state of a unit cell with N = 2.

The circuit includes parameterised R_X(θ) rotations applied to the ancilla qubits, followed by a series of CNOT gates that entangle the ancilla qubits with the system qubits. Post-entanglement, a series of three four-body gates R_YZZX(φ) with independent variational parameters is applied, followed by three two-body R_ZZ gates and a layer of parametrised R_Z rotations. The unitary ${\hat{U}}_{S}({{{\boldsymbol{\varphi }}}})$ concludes with another series of three four-body gates. A measurement circuit ${\hat{M}}_{H}$, shown in the inset, is required for measuring the non-diagonal contribution ${\hat{H}}_{2}$ in the Hamiltonian. In total, 21 variational parameters are needed for the simulation.

The naive transpilation of this circuit into native gates results in 39 entangling MS gates in the system circuit (including the measurement circuit). However, this number can be reduced to 9 by using gate identities and circuit simplification techniques (see Supplementary Fig. 4 in Supplementary Note 5). The circuit implemented on the quantum hardware and used in the numerical simulation is the optimised version obtained after reduction.

SU(3) chiral condensate charge-singlet measurement

A general group element $\widehat{U}\in {{{\rm{SU}}}}(3)$ can be parametrised using eight variables ${\{{\eta }_{a}\}}_{a=1,\ldots,8}$ and the eight non-Abelian charges ${\widehat{Q}}_{{{{\rm{tot}}}}}^{a}$ (see Supplementary Note 4)

$$\widehat{U}=\exp \left(i\sum\limits_{a=1}^{8}{\eta }_{a}{\widehat{Q}}_{{{{\rm{tot}}}}}^{a}\right).$$

(23)

The SU(3) group possesses two diagonal charge generators ${\widehat{Q}}_{{{{\rm{tot}}}}}^{3}$ and ${\widehat{Q}}_{{{{\rm{tot}}}}}^{8}$. We can thus calculate the projection operator $\widehat{K}$ by computing a double integral over the diagonal charges (see Supplementary Note 2 for more details). In particular, for the unit cell with N = 2, the Pauli decomposition of the projector $\widehat{K}$ reads

$$\begin{array}{l}\widehat{K}=\frac{5}{96}\sum\limits_{i < j}{\widehat{\sigma }}_{i}^{z}{\widehat{\sigma }}_{j}^{z}+\frac{1}{96}\sum\limits_{i < j < k < l}{\widehat{\sigma }}_{i}^{z}{\widehat{\sigma }}_{j}^{z}{\widehat{\sigma }}_{k}^{z}{\widehat{\sigma }}_{l}^{z}+\frac{3}{32}\,\widehat{{\mathbb{1}}}\\+\frac{1}{6}\left({\widehat{\sigma }}_{1}^{z}{\widehat{\sigma }}_{2}^{z}{\widehat{\sigma }}_{4}^{z}{\widehat{\sigma }}_{5}^{z}+{\widehat{\sigma }}_{1}^{z}{\widehat{\sigma }}_{3}^{z}{\widehat{\sigma }}_{4}^{z}{\widehat{\sigma }}_{6}^{z}+{\widehat{\sigma }}_{2}^{z}{\widehat{\sigma }}_{3}^{z}{\widehat{\sigma }}_{5}^{z}{\widehat{\sigma }}_{6}^{z}-{\widehat{\sigma }}_{1}^{z}{\widehat{\sigma }}_{4}^{z}\right.\\ \left.-{\widehat{\sigma }}_{2}^{z}{\widehat{\sigma }}_{5}^{z}-{\widehat{\sigma }}_{3}^{z}{\widehat{\sigma }}_{6}^{z}\right)-\frac{1}{32}\,{\widehat{\sigma }}_{1}^{z}{\widehat{\sigma }}_{2}^{z}{\widehat{\sigma }}_{3}^{z}{\widehat{\sigma }}_{4}^{z}{\widehat{\sigma }}_{5}^{z}{\widehat{\sigma }}_{6}^{z},\end{array}$$

(24)

where i, j, k, l = 1, 2, …, 6. The general formula for the SU(3) chiral condensate is given by

$$\widehat{\chi }=\sum\limits_{n=1}^{N}\frac{{(-1)}^{n}}{2}\left({\widehat{\sigma }}_{3n-2}^{z}+{\widehat{\sigma }}_{3n-1}^{z}+{\widehat{\sigma }}_{3n}^{z}\right).$$

(25)

For the unit cell, N = 2, the chiral condensate decomposes as $\widehat{\chi }=(-{\widehat{\sigma }}_{1}^{z}-{\widehat{\sigma }}_{2}^{z}-{\widehat{\sigma }}_{3}^{z}+{\widehat{\sigma }}_{4}^{z}+{\widehat{\sigma }}_{5}^{z}+{\widehat{\sigma }}_{6}^{z})/2$. To obtain the decomposition of $\widehat{\chi }\widehat{K}$ necessary for the evaluation of ${\langle \widehat{\chi }\rangle }_{0}$, we can multiply the two Pauli decompositions above. Similar to the SU(2) case, the operators $\widehat{\chi }\widehat{K}$, and $\widehat{K}$ are also diagonal, and can be measured on the quantum device without requiring an additional measurement circuit.