Experimental Verification of Fluctuation Relations with a Quantum Computer

Inspired by the idea that quantum computers can be useful in advancing basic science, we use a quantum processor to experimentally validate a number of theoretical results in non-equilibrium quantum thermodynamics, that were not (or were very little) corroborated so far. In order to do so, we first put forward a novel method to implement the so called two-point measurement scheme, which is at the basis of the study of nonequilibrium energetic exchanges in quantum systems. Like previously established methods, our method uses an ancillary system, but at variance with them, it provides direct access to the energy exchange statistics, and is, accordingly more effective, at least when applied to small quantum systems. Using a quantum computer as a remotely programmable experimental platform, we first validate our ancilla-assisted two-point measurement scheme, and then apply it to (i) experimentally verify that fluctuation theorems are robust against projective measurements, a theoretical prediction, which was not validated so far; (ii) experimentally verify the so-called heat-engine fluctuation relation, by implementing a swap quantum heat engine; (iii) experimentally verify that the heat-engine fluctuation relation holds for measurement-fueled quantum heat engines, by implementing the design at the basis of the so-called quantum-measurement-cooling concept. For both engines, we report the measured average heat and work exchanged and single out their operation mode. Our experiments constitute an experimental basis for the understanding of the nonequilibrium energetics of quantum computation and for the implementation of energy-management devices on quantum processors.

Popular Summary

Quantum computers hold the promise of revolutionizing our information-based society. As of now that goal is far away from reach, because current quantum computers do not quite work perfectly yet: they are still subject to uncontrollable heating. Understanding and controlling the energetics of quantum devices is thus of crucial importance for improving them. For this reason a new field of research, dubbed “quantum thermodynamics” has recently emerged leading to a great number of theoretical results, many of which still await an experimental verification. Our idea is that quantum computers themselves can be used to experimentally validate those theoretical results. After all, a quantum computer is nothing but a remotely controllable experimental platform.

Specifically, we focus on the so-called quantum fluctuation relations. These are exact relations in nonequilibrium thermodynamics that pose strict conditions on the fluctuations of energy exchanges at the quantum level. In order to address them we put forward and demonstrated a new method to measure energy exchanges, and use it to verify, for example, the prediction that fluctuation theorems are robust to invasive quantum projective measurements. We also use it to validate a number of methods to manipulate and transform energy at the quantum level (specifically, methods to implement quantum heat engines and refrigerators).

These, in turn, could be used in the future to improve the performance of the quantum computers themselves, thus giving origin to a virtuous cycle that contributes to the advancement of quantum science and technology.

Figure 1

Quantum-circuit representation of the AATPM scheme. The two-qubit operation in the blue box creates the purified state, Eq. (9), that both prepares the main qubit $q_0$ in a thermal mixture, and imprints the information about its state in the ancilla qubit $q_1$. The main qubit evolves according to a generic map $\mathcal{E}$, pink box. Finally, both qubits are read, providing information about the state of the main qubit before and after the evolution.

Figure 2

(a),(c) Work statistics as a function of nominal rescaled inverse temperature for the standard TPM scheme and the AATPM scheme, respectively, for a qubit driven by a Hadamard gate. (b),(d) Same as (a),(c) but as a function of measured rescaled inverse temperature. Dots, experimental data; solid lines, theory.

Figure 3

Average exponentiated work $⟨e^{-{\beta }_{m}W}⟩$ for a qubit driven by a Hadamard gate, as measured using the standard TPM (red crosses) and the AATPM (blue triangles), for various values of measured rescaled inverse temperature ${\beta }_m\omega$. Inset: the relative statistical uncertainty for some values of ${\beta }_m\omega$ obtained as the standard deviations of averages as detailed in the text.

Figure 4

Quantum circuit employed for the experimental validation of the Jarzynski equality, Eq. (12), in the presence of $N-1$ intermediate projective measurements (the case $N=4$ is displayed). The green box operations are projective measurements in the ${\sigma }_x$ basis. The main qubit $q_0$ initial state is both read off directly, according to the standard TPM, and indirectly by reading the state of the ancilla, according to the AATPM scheme. The outcome of the intermediate measurements is ignored.

Figure 5

(a) Work statistics with $N-1$ intermediate projective measurements. (b) Robustness of the Jarzynski identity in the presence of projective measurements. The inset shows the statistical uncertainty for a few values of $N$: $N=1,25,50,75$. The blue triangles refer to the ancilla-assisted method while the red crosses are obtained with the standard two-point measurement scheme.

Figure 6

Expectation value of ${\sigma }_z$ for a single qubit initialized in $|0⟩$ after the application of a $R_Y(\alpha )$ gate. Red dots, experimental result; dashed line, theoretical prediction.

Figure 7

Quantum-circuit implementation of the swap quantum heat engine. The main qubits $q_1,q_2$ are first prepared in thermal states at different temperatures with the aid of their ancillas $q_0,q_3$ and then undergo a swap evolution. Finally, their initial states are read off the ancillas, and their final states are read directly.

Figure 8

Joint probabilities $P_{ab}$ with $a,b=+,0,-$, Eq. ((30a)), for the swap engine in Fig. 7, as a function of the rescaled nominal inverse temperatures ${\beta }_1{\omega }_1$ and ${\beta }_2{\omega }_2$.

Figure 9

Measured values of $⟨e^{-{\beta }_{m,1}\mathrm{\Delta }{E}_1-{\beta }_{m,2}\mathrm{\Delta }{E}_2}⟩$ for the swap engine in Fig. 7 as a function of nominal inverse rescaled temperatures ${\beta }_1{\omega }_1$, ${\beta }_2{\omega }_2$.

Figure 10

(a),(c),(e) Measured rescaled energy exchanges $⟨\mathrm{\Delta }{E}_1⟩/{\omega }_1$, $⟨\mathrm{\Delta }{E}_2⟩/{\omega }_2$, $⟨W⟩/\overline{\omega }$, for the swap engine in Fig. 7 as a function of nominal inverse rescaled temperatures ${\beta }_1{\omega }_1$, ${\beta }_2{\omega }_2$. (b),(d),(f) Their relative experimental error.

Figure 11

(a) Experimentally observed modes of operation of the swap engine. (b) Theoretical prediction.

Figure 12

Quantum-circuit implementation of the quantum-measurement-cooling heat-engine design. The main qubits $q_1,q_2$ are first prepared in thermal states at different temperatures with the aid of their ancillas $q_0,q_3$ and then undergo a measurement in the singlet-triplet basis, Eq. (37), purple box. Finally, their initial states are read off the ancillas, and their final states are read directly. The outcome of the singlet-triplet basis measurement is discarded.

Figure 13

Measured values of $⟨e^{-{\beta }_{m,1}\mathrm{\Delta }{E}_1-{\beta }_{m,2}\mathrm{\Delta }{E}_2}⟩$ for the QMC engine design in Fig. 12 as a function of nominal inverse rescaled temperatures ${\beta }_1{\omega }_1$, ${\beta }_2{\omega }_2$.

Figure 14

(a),(c),(e) Measured rescaled energy exchanges $⟨\mathrm{\Delta }{E}_1⟩/{\omega }_1$, $⟨\mathrm{\Delta }{E}_2⟩/{\omega }_2$, $⟨W⟩/\overline{\omega }$, for the QMC engine design in Fig. 12 as a function of nominal inverse rescaled temperatures ${\beta }_1{\omega }_1$, ${\beta }_2{\omega }_2$. (b),(d),(f) Their relative experimental error.