Abstract
Topological insulators and superconductors at finite temperature can be characterized by the topological Uhlmann phase. However, a direct experimental measurement of this invariant has remained elusive in condensed matter systems. Here, we report a measurement of the topological Uhlmann phase for a topological insulator simulated by a system of entangled qubits in the IBM Quantum Experience platform. By making use of ancilla states, otherwise unobservable phases carrying topological information about the system become accessible, enabling the experimental determination of a complete phase diagram including environmental effects. We employ a stateindependent measurement protocol which does not involve prior knowledge of the system state. The proposed measurement scheme is extensible to interacting particles and topological models with a large number of bands.
Introduction
The search for topological phases in condensed matter^{1,2,3,4,5,6,7,8} 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 properties^{9,11,16} 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 phase^{19} is a special instance of quantum phase, that is purely geometrical^{20} 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 phases^{21} 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 onedimensional systems^{35} and extended to twodimensions 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 wellsuited 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 class^{51,52} of TIs (those with chiral symmetry) in the presence of disturbing external noise. Other cases of twodimensional 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 twoband 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,
where r stands for the mixedness parameter between the θdependent 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 socalled purification of \(\rho _\theta = {\mathrm{Tr}}_{\mathrm{A}}\left( {\left {{\mathrm{\Psi }}_\theta } \right\rangle \left\langle {{\mathrm{\Psi }}_\theta } \right} \right)\), where Tr_{A} 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 U_{A} 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
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 U_{S}(t) is a unitary matrix determined by the θdependence. Moreover the arbitrary unitaries U_{A}(t) can be selected to fulfill the socalled 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 socalled Uhlmann phase Φ_{U}) 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 twoband TI in the AIII chiralunitary 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 oneparticle Hamiltonian is
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 TI^{1,2,3} to a tunable timedependent 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 nontrivial 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,
Then, using Eqs. (3) and (4), the system is topological (ω_{1} = 1) when the hopping amplitude is less than unity (M < 1) and trivial (ω_{1} = 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 U_{S} in Eq. (2) for a transportation in time of θ according to the Hamiltonian (3) yields
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 U_{A} in Eq. (2),
where the parameter p_{a} ∈ [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 Φ_{M} 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
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 Φ_{M} becomes the Uhlmann phase Φ_{U} associated to the trajectory. For a closed path t_{f} = 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
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 ω_{1} = 0, and Φ_{U} = 0 (trivial phase) for every mixedness parameter r. If M < 1 then ω_{1} = 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 r_{c2} = 1 − r_{c1} 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 nontrivial 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 r_{c2} 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 yaxis for an angle \(\gamma = 2\,{\mathrm{arcos}}\sqrt {1  r}\) and a twoqubit controlled not gate. For superconducting qubits, the latter can be performed, e.g., by implementing a controlled phase gate for frequencytunable transmons^{54} or by a crossresonance gate.^{55}
Step 2. We apply the bilocal unitary U_{S}(t) ⊗ U_{A}(t) on S ⊗ A conditional to the state of the probe P. This is accomplished by single qubit rotations about an angle β_{1} or β_{2}, determined by h(t) and p_{ a } (blue block of Fig. 2), and twoqubit gates. This decomposition is based on the fact that any controlled unitary gate can be always decomposed as a product of unitary singlequbit gates and twoqubit 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
Step 3. After the holonomic evolution has been completed, we read out Φ_{M} from the state of the probe qubit. Tracing out the system and ancilla in Eq. (9), the reduced state for the probe qubit is
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 Φ_{M} in the form
In Fig. 3 we present the results of phase measurements performed on the IBM Quantum Experience platform,^{56} using three transmon qubits coupled through coplanar waveguide resonators (see Methods). In Fig. 3a, we show the measurement of the Uhlmann phase Φ_{U} 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 Φ_{U} = π (topological) to Φ_{U} = 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.
Stateindependent protocol
The application of U_{S}(t) and U_{A}(t) with p_{ a } = p_{ r } to the purification \(\left {{\mathrm{\Psi }}_{\theta (t)}} \right\rangle\) implements the Uhlmann parallel transport and hence Φ_{M} = Φ_{U}. 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 Φ_{M} 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 rindependent value for p_{ a }, such that the observed phase Φ_{M} takes on the same value as the Uhlmann phase for a Hamiltonian with the form of (3). By studying Φ_{M} as a function of the applied p_{ a }, we conclude that if we tune the ancillary weight
the value of the observed phase Φ_{M}(p_{ a } = p_{ T }) coincides with the topological Uhlmann phase Φ_{U}. 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 nontrivial 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 U_{A}(t). In Fig. 3c, we present the results for the stateindependent protocol recovering the topological Uhlmann phase without prior knowledge of the state, for M = 0.6 and t_{f} = 0.6. These are qualitatively the same as in Fig. 3a, but the stateindependent protocol is more sensitive to errors mainly around the transition point. The mismatch between experiment and simulations is most likely caused by small calibrationdependent systematic errors in the crossresonance gates.
Discussion
We have successfully measured the topological Uhlmann phase, originally proposed in the context of TIs and superconductors, making use of ancillabased 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 stateindependent 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/temperatureinduced 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 useraccess based on five fixedfrequency transmontype qubits coupled via coplanar 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 T_{1} and T_{2} times of this set of qubits when compared to the set {Q2, Q3, Q4} at the time of the experiment. We have used the opensource 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 β_{1} = I_{f}(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 stateindependent protocol [Fig. 3c, main text] we set M = 0.6 and the final time t_{f} = 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 statedependent 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 twoqubit 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 U_{S/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.
Stateindependent 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 Φ_{M} goes abruptly from π to 0 as a function of p_{ r } and I_{f},
If we set \(\frac{1}{2} < t_{\mathrm{f}} < 1\), then \(p_a^c\) is a monotonically decreasing function of p_{ r },
If M > 1, then \( \pi {\mathrm{/}}2 < I_0^{t_{\mathrm{f}}} < \pi {\mathrm{/}}2\), which from Eq. (7) implies that Φ_{M} = 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 Φ_{M} = 0 always. This maps Φ_{M} to the Uhlmann phase Φ_{U} 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 Φ_{U} = 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 U_{A} with p_{ a } = p_{T} and measuring the associated phase Φ_{M} we can extract the following conclusions:

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

If we measure Φ_{M}(p_{T}) = π, the system is in a topological phase (Φ_{U} = π). Because in that case \(p_a^c > p_{\mathrm{T}}\) and then p_{ r } < 0.5 (Φ_{U} = π).
Hence, we have just proven that Φ_{M}(p_{T}) = Φ_{U}.
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 H_{0}, i.e., \(U_{{\mathrm{gate}}} = e^{  iH_0\tau }\), we substitute
This error Liouvillian includes typical sources of imperfections: a) a residual IX term during the crossresonance ZX90 gate in the implementation of the CNOTs, H_{ZX} = 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 T_{1} = 51 μs and T_{2} = 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 τ_{2π} ~ 200 ns and τ_{ZX90} ~ 600 ns as characteristic times for a 2πrotation 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 Φ_{U}. Despite the errors, the topological transition is clearly noticed.
Data availability
All relevant data are available from the authors on reasonable request.
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Acknowledgements
M.A.M.D., A.R. and O.V. thank the Spanish MINECO grant FIS201233152, a “Juan de la CiervaIncorporación” reseach contract, the CAM research consortium QUITEMAD+ S2013/ICE2801, the U.S. Army Research Office through grant W911NF1410103, Fundación Rafael del Pino, Fundación Ramón Areces, and RCC Harvard. S.G., A.W. and S.F. acknowledge support by the Swiss National Science Foundation (SNF, Project 150046).
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O.V., A.R. and M.A.M.D. conceived and developed the theoretical project. S.G., A.W. and S.F. devised the experiment. S.F. carried out the experimental measurements and analyzed the data with input from the theoretical teams. All authors wrote the manuscript.
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Viyuela, O., Rivas, A., Gasparinetti, S. et al. Observation of topological Uhlmann phases with superconducting qubits. npj Quantum Inf 4, 10 (2018). https://doi.org/10.1038/s4153401700569
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DOI: https://doi.org/10.1038/s4153401700569
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