Introduction

The search for topological phases in condensed matter1,2,3,4,5,6,7,8 has triggered an experimental race to detect and measure topological phenomena in a wide variety of quantum simulation experiments.9,10,11,12,13,14,15 In quantum simulators the phase of the wave function can be accessed directly, opening a whole new way to observe topological properties9,11,16 beyond the realm of traditional condensed matter scenarios. These quantum phases are very fragile, but when controlled and mastered, they can produce very powerful computational systems like a quantum computer.17,18 The Berry phase19 is a special instance of quantum phase, that is purely geometrical20 and independent of dynamical contributions during the time evolution of a quantum system. In addition, if that phase is invariant under deformations of the path traced out by the system during its evolution, it becomes topological. Topological Berry phases have also acquired a great relevance in condensed matter systems. The now very active field of topological insulators (TIs) and superconductors (TSCs)1,2,3 ultimately owes its topological character to Berry phases21 associated to the special band structure of these exotic materials.

However, if the interaction of a TI or a TSC with its environment is not negligible, the effect of the external noise in the form of, e.g., thermal fluctuations, makes these quantum phases very fragile,22,23,24,25,26,27,28,29,30,31,32,33,34 and they may not even be well defined. For the Berry phase acquired by a pure state, this problem has been successfully adressed for one-dimensional systems35 and extended to two-dimensions later.36,37,38 The key concept behind this theoretical characterization is the notion of “Uhlmann phase”,39,40,41,42,43,44,45,46,47 a natural extension of the Berry phase for density matrices. In analogy to the Berry phase, when the Uhlmann phase for mixed states remains invariant under deformations, it becomes topological.

Although this phase is gauge invariant and thus, in principle, observable, a fundamental question remains: how to measure a topological Uhlmann phase in a physical system? To this end, we employ an ancillary system as a part of the measurement apparatus. By encoding the temperature (or mixedness) of the system in the entanglement with the ancilla, we find that the Uhlmann phase appears as a relative phase that can be retrieved by interferometric techniques. The difficulty with this type of measurement is that it requires a high level of control over the environmental degrees of freedom, beyond the reach of condensed matter experiments. On the contrary, this situation is especially well-suited for a quantum simulation scenario.

Specifically, in this work we report: (i) the measurement of the topological Uhlmann phase on a quantum simulator based on superconducting qubits,48,49,50 in which we have direct control over both system and ancilla, and (ii) the computation of the topological phase diagram for qubits with an arbitrary noise degree. A summary and a comparison with pure state topological measures are shown in Fig. 1. In addition, we construct a state independent protocol that detects whether a given mixed state is topological in the Uhlmann sense. Our proposal also provides a quantum simulation of the AIII class51,52 of TIs (those with chiral symmetry) in the presence of disturbing external noise. Other cases of two-dimensional TIs, TSCs and interacting systems can also be addressed by appropriate modifications as mentioned in the conclusions.

Results

Topological Uhlmann phase for qubits

We briefly present the main ideas of the Uhlmann approach for a two-band model of TIs and TSCs simulated with a qubit. Let $$\left. {\theta (t)} \right|_{t = 0}^1$$ define a closed trajectory along a family of single qubit density matrices parametrized by θ,

$$\rho _\theta = \left( {1 - r} \right)\left| {0_\theta } \right\rangle \left\langle {0_\theta } \right| + r\left| {1_\theta } \right\rangle \left\langle {1_\theta } \right|,$$
(1)

where r stands for the mixedness parameter between the θ-dependent eigenstates $$\left| {1_\theta } \right\rangle$$ and $$\left| {0_\theta } \right\rangle$$, e.g., that of a transmon qubit.53 The mixed state ρ θ can be seen as a “part” of a state vector $$\left| {{\mathrm{\Psi }}_\theta } \right\rangle$$ in an enlarged Hilbert space $${\cal H} = {\cal H}_{\mathrm{S}} \otimes {\cal H}_{\mathrm{A}}$$, where S stands for system and A for the ancilla degrees of freedom with $${\mathrm{dim}}\,{\cal H}_{\mathrm{A}} \ge {\mathrm{dim}}\,{\cal H}_{\mathrm{S}}$$. The state vector $$\left| {{\mathrm{\Psi }}_\theta } \right\rangle$$ is a so-called purification of $$\rho _\theta = {\mathrm{Tr}}_{\mathrm{A}}\left( {\left| {{\mathrm{\Psi }}_\theta } \right\rangle \left\langle {{\mathrm{\Psi }}_\theta } \right|} \right)$$, where TrA performs the partial trace over the ancilla. There is an infinite number of purifications for every single density matrix, specifically $$\left({\mathbb{I}} \otimes U_{\mathrm{A}} \right)\left| {{\mathrm{\Psi }}_\theta } \right\rangle$$ for any unitary UA acting on the ancilla purifies the same mixed state as $$\left| {{\mathrm{\Psi }}_\theta } \right\rangle$$. Hence, for a family of density matrices ρ θ , there are several sets of purifications $$\left| {{\mathrm{\Psi }}_\theta } \right\rangle$$ according to a U(n) gauge freedom. This generalizes the standard U(1) gauge (phase) freedom of state vectors describing quantum pure states to the general case of density matrices.

Along a trajectory $$\left. {\theta (t)} \right|_{t = 0}^1$$ for ρ θ the induced purification evolution (system qubit S and ancilla qubit A) can be written as

$$\begin{array}{*{20}{l}} {\left| {{\mathrm{\Psi }}_{\theta (t)}} \right\rangle } \hfill & = \hfill & {\sqrt {1 - r} U_{\mathrm{S}}(t)\left| 0 \right\rangle _{\mathrm{S}} \otimes U_{\mathrm{A}}(t)\left| 0 \right\rangle _{\mathrm{A}} + } \hfill \\ {} \hfill & + \hfill & {\sqrt r U_{\mathrm{S}}(t)\left| 1 \right\rangle _{\mathrm{S}} \otimes U_{\mathrm{A}}(t)\left| 1 \right\rangle _{\mathrm{A}},} \hfill \end{array}$$
(2)

where $$\left| 0 \right\rangle = \left( {\begin{array}{*{20}{c}} 1 \\ 0 \end{array}} \right)$$ and $$\left| 1 \right\rangle = \left( {\begin{array}{*{20}{c}} 0 \\ 1 \end{array}} \right)$$ is the standard qubit basis, and US(t) is a unitary matrix determined by the θ-dependence. Moreover the arbitrary unitaries UA(t) can be selected to fulfill the so-called Uhlmann parallel transport condition. Namely, analogously to the standard Berry case, the Uhlmann parallel transport requires that the distance between two infinitesimally close purifications $$\left\| {\left| {{\mathrm{\Psi }}_{\theta (t + dt)}} \right\rangle - \left| {{\mathrm{\Psi }}_{\theta (t)}} \right\rangle } \right\|^2$$ reaches a minimum value (which leads to removing the relative infinitesimal “phase” between purifications).39 Physically, this condition ensures that the accumulated quantum phase (the so-called Uhlmann phase ΦU) along the trajectory is purely geometrical, that is, without dynamical contributions. This is a source of robustness, since variations on the transport velocity will not change the resulting phase.

Next, we consider the Hamiltonian of a two-band TI in the AIII chiral-unitary class,51,52 $$H = \mathop {\sum}\nolimits_k {\mathrm{\Psi }}_k^\dagger H_k{\mathrm{\Psi }}_k$$, in the spinor representation $${\mathrm{\Psi }}_k = \left( {\hat a_k,\hat b_k} \right)^{\mathrm{t}}$$ where $$\hat a_k$$ and $$\hat b_k$$ stands for two species of fermionic operators. The one-particle Hamiltonian is

$$\begin{array}{l}H_k = \frac{{G_k}}{2}{\boldsymbol{n}}_k \cdot \sigma ,\\ {\boldsymbol{n}}_k = \frac{2}{{G_k}}\left( {{\mathrm{sin}}\,k,0,M + {\mathrm{cos}}\,k} \right),\\ G_k = 2\sqrt {1 + M^2 + 2M\,{\mathrm{cos}}\,k} .\end{array}$$
(3)

where G k represents the actual gap between the valence and conduction bands in the TI, and n k is a unit vector called winding vector.35 We now map the crystalline momentum k of the TI1,2,3 to a tunable time-dependent paramenter θ of the quantum simulator. When invoking the rotating wave approximation this model also describes, e.g., the dynamics of a driven transmon qubit.14,16 The detuning $${\mathrm{\Delta }} = 2\left( {{\mathrm{cos}}\,\theta + M} \right)$$ between qubit and drive is parametrized in terms of θ and a hopping amplitude M, whereas the coupling strength between the qubit and the incident microwave field is given by $${\mathrm{\Omega }} = 2\,{\mathrm{sin}}\,\theta$$.

The non-trivial topology of pure quantum states (r {0, 1}) of this class of topological materials can be witnessed by the winding number. This is defined as the angle swept out by n θ as θ varies from 0 to 2π, namely,

$$\omega _1: = \frac{1}{{2\pi }}{\oint} \left( {\frac{{\partial _\theta n_\theta ^x}}{{n_\theta ^z}}} \right)d\theta .$$
(4)

Then, using Eqs. (3) and (4), the system is topological (ω1 = 1) when the hopping amplitude is less than unity (M < 1) and trivial (ω1 = 0) if M > 1. In fact, the topological phase diagram coincides with the one given by the Berry phase acquired by the “ground” state $$\left| 0 \right\rangle _\theta$$ (or the “excited” state $$\left| 1 \right\rangle _\theta$$) of Hamiltonian (3) when θ varies from 0 to 2π, (see Supplementary Note 2).

The computation of the unitary US in Eq. (2) for a transportation in time of θ according to the Hamiltonian (3) yields

$$U_{\mathrm{S}}(t) = {\mathrm{e}}^{ - {\mathrm{i}}{\int}_0^t h\left( {t^{\prime}} \right)dt^{\prime}{\boldsymbol{\sigma }}_y},$$
(5)

with $$h(t): = \frac{{\partial _tn_t^x}}{{2n_t^z}}$$. This implements the eigenstate transport $$\left| {1_{\theta (t)}} \right\rangle = U_{\mathrm{S}}(t)\left| 1 \right\rangle$$ and $$\left| {0_{\theta (t)}} \right\rangle = U_{\mathrm{S}}(t)\left| 0 \right\rangle$$. In addition, we can consider a similar form for the unitary UA in Eq. (2),

$$U_{\mathrm{A}}(t) = \left[ {U_{\mathrm{S}}(t)} \right]^{p_a} = {\mathrm{e}}^{ - {\mathrm{i}}{\int}_0^t p_ah(t^{\prime})dt^{\prime}{\boldsymbol{\sigma }}_y},$$
(6)

where the parameter pa [0, 1] is defined as an ancillary “weight”. We find that the Uhlmann parallel transport condition is satisfied for $$p_a = p_r: = 2\sqrt {r(1 - r)}$$. The detailed technical derivation is provided in Supplementary Notes 1 and 2.

Now, from Eq. (2) it is possible to define the relative phase ΦM between the initial $$\left| {{\mathrm{\Psi }}_{\theta (0)}} \right\rangle$$ and the final state, i.e., $$\left| {{\mathrm{\Psi }}_{\theta \left( {t_{\mathrm{f}}} \right)}} \right\rangle$$. For Hamiltonian (3), density matrix (1) and purification (2), we find

$$\begin{array}{*{20}{l}} {{\mathrm{\Phi }}_{\mathrm{M}}} \hfill & {: = } \hfill & {{\mathrm{arg}}\left[ {\left\langle {{\mathrm{\Psi }}_{\theta (0)}|{\mathrm{\Psi }}_{\theta \left( {t_{\mathrm{f}}} \right)}} \right\rangle } \right] = } \hfill \\ {} \hfill & = \hfill & {{\mathrm{arg}}\left[ {{\mathrm{cos}}\left( {I_0^{t_{\mathrm{f}}}} \right){\mathrm{cos}}\left( {p_aI_0^{t_{\mathrm{f}}}} \right) + p_r\,{\mathrm{sin}}\left( {I_0^{t_{\mathrm{f}}}} \right){\mathrm{sin}}\left( {p_aI_0^{t_{\mathrm{f}}}} \right)} \right],} \hfill \end{array}$$
(7)

where $$I_{t_0}^{t_{\mathrm{f}}}: = {\int}_{t_0}^{t_{\mathrm{f}}} h\left( {t^{\prime}} \right)dt^{\prime}$$. As commented before, by assuming $$p_a = p_r: = 2\sqrt {r(1 - r)}$$, the purification precisely follows Uhlmann parallel transport and the relative phase ΦM becomes the Uhlmann phase ΦU associated to the trajectory. For a closed path tf = 1, the integral $$I_0^1 = \pi \omega _1 = {\mathrm{\Phi }}_B$$ becomes the topological Berry phase. In that case, the Uhlmann phase simplifies to

$${\mathrm{\Phi }}_{\mathrm{U}} = {\mathrm{arg}}\left\{ {{\mathrm{cos}}\left[ {\left( {1 - 2p_r} \right)\pi \omega _1} \right]} \right\}.$$
(8)

We can now deduce the topological properties of these phases in the presence of external noise, as measured by the parameter r [Eq. (1)]. This is depicted in Fig. 1. Namely, if M > 1 then ω1 = 0, and ΦU = 0 (trivial phase) for every mixedness parameter r. If M < 1 then ω1 = 1 and one obtains $${\mathrm{\Phi }}_{\mathrm{U}} = {\mathrm{arg}}\left[ { - {\mathrm{cos}}\left( {2\pi \sqrt {r(1 - r)} } \right)} \right]$$. If the state is pure (r = 0), then $${\mathrm{\Phi }}_{\mathrm{U}}^0 = \pi$$, recovering the same topological phase given by the winding number and the Berry phase. However, for r ≠ 0 there are critical values of the mixedness r c at which the Uhlmann phase, according to Eq. (8), jumps from π to zero (see Fig. 1). The first $$r_{c1} = \frac{1}{4}\left( {2 - \sqrt 3 } \right) \approx 0.067$$ signals the mixedness at which the system loses the topological character of the ground state. Moreover, there exists another rc2 = 1 − rc1 at which the system becomes topological again due to the topological character of the excited state (r → 1). Notice that at r = 1 the system becomes a pure state again (the excited state), which is also topologically non-trivial according to the Berry phase. Actually, provided that the weight $$p_r < p_{r = r_{c1(2)}} = 0.5$$, the system is topological in the Uhlmann sense as long as M < 1. This reentrance in the topological phase at rc2 was absent in previous works.35,36,37

Experimental realization

Measuring the topological Uhlmann phase is a very challenging task since its definition in terms of purifications implies precise control over auxiliary/environmental degrees of freedom (the ancilla). In an experiment, we therefore include an extra ancilla qubit representing the environment. We also include a third qubit acting as a probe system P, such that by measuring qubit P we retrieve the accumulated phase by means of interferometric techniques. The measurement protocol is described in Fig. 2:

Step 1. Following Eq. (2), we prepare the initial state $$\left| {{\mathrm{\Psi }}_{\theta (0)}} \right\rangle \otimes \left| 0 \right\rangle _{\mathrm{P}}$$ (red block of Fig. 2) using single qubit rotations $$R_y^\gamma$$ about the y-axis for an angle $$\gamma = 2\,{\mathrm{arcos}}\sqrt {1 - r}$$ and a two-qubit controlled not gate. For superconducting qubits, the latter can be performed, e.g., by implementing a controlled phase gate for frequency-tunable transmons54 or by a cross-resonance gate.55

Step 2. We apply the bi-local unitary US(t) UA(t) on SA conditional to the state of the probe P. This is accomplished by single qubit rotations about an angle β1 or β2, determined by h(t) and p a (blue block of Fig. 2), and two-qubit gates. This decomposition is based on the fact that any controlled unitary gate can be always decomposed as a product of unitary single-qubit gates and two-qubit CNOT gates.18 Figure 2 shows the final result after the decomposition of the Uhlmann transport, conditional to the probe qubit P, is performed. As a result, the three qubits {S, A, P} are in the superposition

$$\left| {\mathrm{\Phi }} \right\rangle _{{\mathrm{SAP}}} = \frac{1}{{\sqrt 2 }}\left( {\left| {{\mathrm{\Psi }}_{\theta (0)}} \right\rangle \otimes \left| 0 \right\rangle _{\mathrm{P}} + \left| {{\mathrm{\Psi }}_{\theta \left( {t_{\mathrm{f}}} \right)}} \right\rangle \otimes \left| 1 \right\rangle _{\mathrm{P}}} \right).$$
(9)

Step 3. After the holonomic evolution has been completed, we read out ΦM from the state of the probe qubit. Tracing out the system and ancilla in Eq. (9), the reduced state for the probe qubit is

$$\rho _{\mathrm{P}} = \frac{1}{2}\left( { \cdot + {\mathrm{Re}}\left( {\left\langle {{\mathrm{\Psi }}_{\theta (0)}\left| {{\mathrm{\Psi }}_{\theta \left( {t_{\mathrm{f}}} \right)}} \right.} \right\rangle } \right)\sigma _x + {\mathrm{Im}}\left( {\left\langle {{\mathrm{\Psi }}_{\theta (0)}\left| {{\mathrm{\Psi }}_{\theta \left( {t_{\mathrm{f}}} \right)}} \right.} \right\rangle } \right)\sigma _y} \right).$$
(10)

Thus, by measuring the expectation values $$\left\langle {\sigma _x} \right\rangle$$ and $$\left\langle {\sigma _y} \right\rangle$$ (green block of Fig. 2), we can retrieve ΦM in the form

$$\begin{array}{*{20}{l}} {{\mathrm{\Phi }}_{\mathrm{M}}} \hfill & = \hfill & {{\mathrm{arg}}\left[ {\left\langle {\sigma _x} \right\rangle + {\mathrm{i}}\left\langle {\sigma _y} \right\rangle } \right] = } \hfill \\ {} \hfill & = \hfill & {\arg \left[ {\left\langle {{\mathrm{\Psi }}_{\theta (0)}} \right|U_{\mathrm{S}}\left( {t_{\mathrm{f}}} \right) \otimes U_{\mathrm{A}}\left( {t_{\mathrm{f}}} \right)\left| {{\mathrm{\Psi }}_{\theta (0)}} \right\rangle } \right].} \hfill \end{array}$$
(11)

In Fig. 3 we present the results of phase measurements performed on the IBM Quantum Experience platform,56 using three transmon qubits coupled through co-planar waveguide resonators (see Methods). In Fig. 3a, we show the measurement of the Uhlmann phase ΦU for different values of the mixedness parameter r, where we set M = 0.2 and p a  = p r , i.e., fulfilling the parallel transport condition. The critical jump from ΦU = π (topological) to ΦU = 0 (trivial) is clearly observed following the previous protocol.

Additionally, we can check whether the Uhlmann parallel transport condition is satisfied at every time interval during the experiment. By partitioning the closed trajectory in small time steps δt, the relative phase between the state at time nδt and at (n + 1)δt must be close to zero if the condition is fulfilled. This is the case in the experiment as shown in Fig. 3b. During the state preparation (Step 1), we need to include two additional single qubit rotations $$R_y^{\alpha _1^{n\delta t}}$$ and $$R_y^{\alpha _2^{n\delta t}}$$ acting on the system and ancilla qubits respectively, where $$\alpha _1^{n\delta t} = 2I_0^{n\delta t}$$ and $$\alpha _2^{n\delta t} = p_r\alpha _1^{n\delta t}$$. These two unitaries make the entangled state between system and ancilla evolve until the state $$\left| {{\mathrm{\Psi }}_{n\delta t}} \right\rangle$$ is reached. In Step 2, the state evolves to $$\left| {{\mathrm{\Psi }}_{(n + 1)\delta t}} \right\rangle$$ conditional to the state of the probe P. The measurement scheme (Step 3) to retrieve the relative phase in Fig. 3b remains the same. Technical details are described in the Supplementary Note 3. We have included a simulation—green solid line in Fig. 3—based on experimental imperfections, mainly finite coherence time (~50 μs) and spurious terms accounting for certain type of electromagnetic crosstalk between qubits. A more detailed description of the error model is given in Methods.

State-independent protocol

The application of US(t) and UA(t) with p a  = p r to the purification $$\left| {{\mathrm{\Psi }}_{\theta (t)}} \right\rangle$$ implements the Uhlmann parallel transport and hence ΦM = ΦU. However, this would imply some knowledge about the mixedness parameter r beforehand, which is not always possible. Hence, we present a modification of the previous protocol to measure the topological Uhlmann phase without prior knowledge of the state ρ and its mixedness parameter r.

Firstly, we fix θ(t) = 2πt and consider open holonomies $$\frac{1}{2} < t_{\mathrm{f}} < 1$$ covering more than one half of the complete path. No previous knowledge of the state is assumed to perform the evolution. Hence, the ancillary weight p a can be different than p r in Eq. (2), but still satisfying 0 ≤ p a  ≤ 1. From Eq. (7), the overlap $$\left\langle {{\mathrm{\Psi }}_{\theta = 0}|{\mathrm{\Psi }}_{\theta = 2\pi t_{\mathrm{f}}}} \right\rangle$$ is always real and thus the phase ΦM is either 0 or π, depending on both the weight p r associated to the state ρ θ [Eq. (1)] and the ancillary weight p a .

We aim to find an r-independent value for p a , such that the observed phase ΦM takes on the same value as the Uhlmann phase for a Hamiltonian with the form of (3). By studying ΦM as a function of the applied p a , we conclude that if we tune the ancillary weight

$$p_a = p_T: = \frac{{ - 1}}{{I_0^{t_{\mathrm{f}}}}}{\mathrm{arctan}}\left( {\frac{2}{{{\mathrm{tan}}\left( {I_0^{t_{\mathrm{f}}}} \right)}}} \right),$$
(12)

the value of the observed phase ΦM(p a  = p T ) coincides with the topological Uhlmann phase ΦU. Algebraic details are provided in Methods.

Note that there is an intuitive reason why we can get topological information out of a phase associated to a open path longer than one half of a non-trivial topological loop. Indeed, h(t) is symmetric around $$t = \frac{1}{2}$$. Then, once we have covered one half of the path, we know about the topology of the whole system thanks to this symmetry. Therefore, even an open path for $$\frac{1}{2} < t_{\mathrm{f}} < 1$$ can be considered as global.

In terms of the experimental protocol, we only need to modify Step 2 by fixing p a  = p T for the unitary UA(t). In Fig. 3c, we present the results for the state-independent protocol recovering the topological Uhlmann phase without prior knowledge of the state, for M = 0.6 and tf = 0.6. These are qualitatively the same as in Fig. 3a, but the state-independent protocol is more sensitive to errors mainly around the transition point. The mismatch between experiment and simulations is most likely caused by small calibration-dependent systematic errors in the cross-resonance gates.

Discussion

We have successfully measured the topological Uhlmann phase, originally proposed in the context of TIs and superconductors, making use of ancilla-based protocols. The experiment is realized within a minimal quantum simulator consisting of three superconducting qubits. We have exploited the quantum simulator to realize a controlled coupling of the system to an environment represented by the ancilla degrees of freedom. Moreover, we have proposed and tested a state-independent protocol that allows us to classify states of topological systems according to the Uhlmann measure. To our knowledge, this is the first time that a noise/temperature-induced topological transition in a quantum phase is observed. Recently, these transitions have been addressed in connection to new thermodynamical properties of these systems.57 The fact that these effects can be experimentally observed opens the possibility for the search of warm topological matter in the lab. Due to the intrinsic geometric character of the Uhlmann phase, our results may find application in generalizations of holonomic quantum protocols for general, possibly mixed, states.

In addition, an increase of experimental resources such as the number of qubits, the speed and fidelity of the quantum gates, etc. will allow us to study additional topological phenomena with superconducting qubits. In particular, by including interactions in the model Hamiltonian we can test different features: quantum simulations of thermal topological transitions in 2D TIs and TSCs, the interplay between noise and interactions within a topological phase, etc. These effects can be achieved since a system with more interacting qubits can be mapped onto models for interacting fermions with spin.15 Further details can be found in the Supplementary Note 5. Although such a proposal would be experimentally more demanding, it represents a clear outlook that would need precise controllability of more qubits and the ability to perform more gates with high fidelity.

Methods

Superconducting qubit realization of a controllable uhlmann phase

The experiments on the topological Uhlmann phase have been realized on the IBM Quantum Experience (ibmqx2),56 a quantum computing platform with online user-access based on five fixed-frequency transmon-type qubits coupled via co-planar waveguide resonators. We have used three qubits, qubit Q0 as the probe qubit, Q1 as the system qubit and Q2 as the ancilla qubit. This choice is motivated by the connectivity required for the measurement protocol and the superior T1 and T2 times of this set of qubits when compared to the set {Q2, Q3, Q4} at the time of the experiment. We have used the open-source python SDK QISKit (https://www.qiskit.org) to program the quantum computer and retrieve the data. The explicit quantum algorithm to measure the expectation values of σ x and σ y is provided in Supplementary Note 4 using the OPENQASM intermediate representation (https://github.com/QISKit/openqasm). The phase is then extracted from the measured data by evaluating $${\mathrm{\Phi }}_{\mathrm{M}} = {\mathrm{arg}}\left( {\left\langle {\sigma _x} \right\rangle - i\left\langle {\sigma _y} \right\rangle } \right)$$.

For all experiments we have measured 8192 repetitions providing a single value for the phase. For the measurement of the topological Uhlmann phase (Fig. 3a) we vary the initial mixedness of the system state r by setting the rotation angle $$\gamma = 2\,{\mathrm{arccos}}\left( {\sqrt {1 - r} } \right)$$. The transport of the state according to Uhlmann’s parallel transport condition is set by the value β1 = If(0, 1) = π for M < 1 and $$\beta _2 = p_aI_{\mathrm{f}}(0,1) = 2\pi \sqrt {r(1 - r)}$$, as defined in Eq. (7). The energy relaxation times of the qubits are $$\left\{ {T_1^{Q0},T_1^{Q1},T_1^{Q2}} \right\} = \left\{ {45\,\mu {\mathrm{s}},31\,\mu {\mathrm{s}},46\,\mu {\mathrm{s}}} \right\}$$ and the decoherence times $$\left\{ {T_2^{Q0},T_2^{Q1},T_2^{Q2}} \right\} = \left\{ {40\,\mu {\mathrm{s}},27\,\mu {\mathrm{s}},80\,\mu {\mathrm{s}}} \right\}$$ as stated in the calibration data.

For the state-independent protocol [Fig. 3c, main text] we set M = 0.6 and the final time tf = 0.6. The system is rotated about $$\beta _1 = I_0^{0.6} = 2.18537$$ and $$\beta _2 = p_TI_0^{0.6} = 0.954407$$. In this measurement energy relaxation and decoherence times are $$\left\{ {T_1^{Q0},T_1^{Q1},T_1^{Q2}} \right\} = \left\{ {41\,\mu {\mathrm{s}},52\,\mu {\mathrm{s}},62\,\mu {\mathrm{s}}} \right\}$$ and $$\left\{ {T_2^{Q0},T_2^{Q1},T_2^{Q2}} \right\} = \left\{ {31\,\mu {\mathrm{s}},37\,\mu {\mathrm{s}},87\,\mu {\mathrm{s}}} \right\}$$. Note, that here the error bars are larger as compared to the state-dependent measurement described above, because the expectation values $$\left\langle {\sigma _x} \right\rangle$$ and $$\left\langle {\sigma _y} \right\rangle$$ are closer to zero leading to larger statistical errors in the phase. Also, we notice a systematic offset of $$\bar \sigma _y = 0.098 \pm 0.014$$ from the expected value $$\left\langle {\sigma _y} \right\rangle _{{\mathrm{th}}} = 0$$. Here, $$\bar \sigma _y$$ is the average over all r values and repetitions. This offset is subtracted from the phase data $${\mathrm{\Phi }}_{\mathrm{M}} = {\mathrm{arg}}\left[ {\left\langle {\sigma _x} \right\rangle - i\left\langle {\left\langle {\sigma _y} \right\rangle - \bar \sigma _y} \right\rangle } \right]$$ and the result is plotted in Fig. 3c. We consider accumulated phase shifts during two-qubit operations as the main reason for this mismatch. We have also noticed that this value changes for different calibrations of the IBM Quantum Experience and when taking different sets of qubits.

Finally, for the measurement of the parallel transport condition we modify the algorithm to prepare the intermediate state $$\left| {{\mathrm{\Psi }}_{\theta (n\delta t)}} \right\rangle$$ by applying US/A(nδt) to system and ancilla qubit. For the measurement of the Uhlmann phase, the same circuit as above is used to obtain a state evolution $$\left| {{\mathrm{\Psi }}_{\theta (n\delta t)}} \right\rangle \to \left| {{\mathrm{\Psi }}_{\theta ((n + 1)\delta t)}} \right\rangle$$. The complete protocol to measure the parallel transport condition is shown in the Supplementary Fig. 1. In the experiment, we choose M = 0.2 and r = 0.02 to stay within the topological sector. The mixedness angle evaluates to $$\gamma = 2\,{\mathrm{arccos}}\left( {\sqrt {0.95} } \right) = 0.2838$$. The angles for the intermediate state preparation are determined by $$\alpha _1(n) = I_0^{n\delta t}$$ and $$\alpha _2(n) = p_rI_0^{n\delta t}$$ = $$2\sqrt {r(1 - r)} I_0^{n\delta t} = 0.28I_0^{n\delta t}$$, the evolution from nδt to (n + 1)δt is determined by the angles $$\beta _1(n) = I_{n\delta t}^{(n + 1)\delta t}$$ and $$\beta _2(n) = p_rI_{n\delta t}^{(n + 1)\delta t} = 0.28I_{n\delta t}^{(n + 1)\delta t}$$. The recorded data shown in Fig. 3b, main text, shows that the measured phase difference $$\left\langle {{\mathrm{\Phi }}_{\mathrm{M}}(n\delta t)} \right\rangle = - 0.07 \pm 0.2$$ is zero within the statistics. However, the residuals do not follow a normal distribution which hints at systematic gate errors instead of stochastic errors.

State-independent derivation

The derivation of the value for p T [Eq. (12)] is as follows. From Eq. (7) we find the value $$p_a = p_a^c$$ (where the superindex c stands for critical) at which ΦM goes abruptly from π to 0 as a function of p r and If,

$$p_a^c = \frac{{ - 1}}{{I_0^{t_{\mathrm{f}}}}}{\mathrm{arctan}}\left( {\frac{1}{{p_r\,{\mathrm{tan}}\left( {I_0^{t_{\mathrm{f}}}} \right)}}} \right).$$
(13)

If we set $$\frac{1}{2} < t_{\mathrm{f}} < 1$$, then $$p_a^c$$ is a monotonically decreasing function of p r ,

$$\frac{{\partial p_a^c}}{{\partial p_r}} = \frac{{{\mathrm{tan}}\left( {I_0^{t_{\mathrm{f}}}} \right)}}{{I_0^{t_{\mathrm{f}}}\left[ {1 + p_r^2\,{\mathrm{tan}}^2\left( {I_0^{t_{\mathrm{f}}}} \right)} \right]}} < 0.$$
(14)

If M > 1, then $$- \pi {\mathrm{/}}2 < I_0^{t_{\mathrm{f}}} < \pi {\mathrm{/}}2$$, which from Eq. (7) implies that ΦM = 0 for any value of p r and p a . Hence, for the trivial case M > 1, there is no critical value $$p_a^c$$ and ΦM = 0 always. This maps ΦM to the Uhlmann phase ΦU at least for this case. On the contrary, if M < 1, then $$\pi {\mathrm{/}}2 < I_0^{t_{\mathrm{f}}} < \pi$$ which implies $${\mathrm{tan}}\left( {I_0^{t_{\mathrm{f}}}} \right) < 0$$. Since 0 < p r  < 1, then $$- {\mathrm{arctan}}\left( {\frac{1}{{p_r\,{\mathrm{tan}}\left( {I_0^{t_{\mathrm{f}}}} \right)}}} \right) < \pi {\mathrm{/}}2$$. Thus, there is always a solution of Eq. (13) with $$0 < p_a^c < 1$$ for any p r . As discussed in the main text, the state ρ θ in Eq. (1) is topological in the Uhlmann sense ΦU = π, only if M < 1 and p r  < 0.5.

Now, we define $$p_{\mathrm{T}}: = p_a^c\left( {p_r = 0.5} \right)$$ using Eq. (13). Note that the true p r of the system is unknown as we have assumed no knowledge of the state. Nevertheless, if p r  > 0.5, then its associated critical value [from Eq. (13)] is $$p_a^c < p_{\mathrm{T}}$$. This means that by applying UA with p a  = pT and measuring the associated phase ΦM we can extract the following conclusions:

• If we measure ΦM(pT) = 0, the system is within a trivial phase (ΦU = 0). Because this implies $$p_a^c < p_{\mathrm{T}}$$ and hence p r  > 0.5 (ΦU = 0), as we have proven that $$p_a^c$$ always decreases with p r .

• If we measure ΦM(pT) = π, the system is in a topological phase (ΦU = π). Because in that case $$p_a^c > p_{\mathrm{T}}$$ and then p r  < 0.5 (ΦU = π).

Hence, we have just proven that ΦM(pT) = ΦU.

Error simulation

The detrimental effect of experimental errors is modeled by means of a Liouvillian term $${\cal L}_{{\mathrm{error}}}$$, so that the Liouvillian $${\cal L}_0$$, accounting for the idealized dynamics, is in fact substituted by $${\cal L}_0 + {\cal L}_{{\mathrm{error}}}$$. Specifically, if a gate is performed during a time τ via a Hamiltonian H0, i.e., $$U_{{\mathrm{gate}}} = e^{ - iH_0\tau }$$, we substitute

$$e^{ - iH_0\tau }\rho e^{iH_0\tau } \equiv e^{{\cal L}_0\tau }\rho \to e^{\left( {{\cal L}_0 + {\cal L}_{{\mathrm{error}}}} \right)\tau }\rho .$$
(15)

This error Liouvillian includes typical sources of imperfections: a) a residual IX term during the cross-resonance ZX90 gate in the implementation of the CNOTs, HZX = mIX + μZX;58,59,60 b) spontaneous emission and dephasing terms $${\cal L}_ - (\rho ) = \gamma _ - \left( {\sigma _ - \rho \sigma _ + - {\textstyle{1 \over 2}}\left\{ {\sigma _ + \sigma _ - ,\rho } \right\}} \right)$$ and $${\cal L}_z(\rho ) = \gamma _z\left( {\sigma _z\rho \sigma _z - \rho } \right)$$, respectively.

We have accommodated the values of γ and γ z to the characteristic longitudinal and transverse relaxation times of T1 = 51 μs and T2 = 51 μs reported by the IBM Quantum Experience calibration team the day of the measurements. The residual IX strength has been taken to be about m ~ 0.4 MHz. In addition, we consider τ2π ~ 200 ns and τZX90 ~ 600 ns as characteristic times for a 2π-rotation on a single qubit and the ZX90 gates, respectively. Waiting times of 5 ns after a single qubit gate and 40 ns after a ZX90 gate are also included.

In Fig. 3, we plot the result of the simulation including these experimental imperfections together with the experimental measurements of the topological Uhlmann phase ΦU. Despite the errors, the topological transition is clearly noticed.

Data availability

All relevant data are available from the authors on reasonable request.