Probing the Full Distribution of Many-Body Observables By Single-Qubit Interferometry

We present an experimental scheme to measure the full distribution of many-body observables in spin systems, both in and out of equilibrium, using an auxiliary qubit as a probe. We focus on the determination of the magnetization and the kink number statistics at thermal equilibrium. The corresponding characteristic functions are related to the analytically continued partition function. Thus, both distributions can be directly extracted from experimental measurements of the coherence of a probe qubit that is coupled to an Ising-type bath, as reported in [X. Peng et al., Phys. Rev. Lett. 114, 010601 (2015)] for the detection of Lee-Yang zeros.

Figure 1

Scheme for probing the characteristic function of statistical quantity X by a quantum simulator. (a) Initial state preparations. The classical spin (bit) systems (CBs) and the auxiliary qubit (AQ) are prepared respectively in probability distribution $\rho$ (or a thermal equilibrium state ${\rho }_{\mathrm{th}}$ discussed in the main text), and a superposition state $|+⟩=(|\downarrow ⟩+|\uparrow ⟩)/\sqrt{2}$ by a Hadamard gate. (b) The information of the characteristic function is encoded into the auxiliary qubit by a simulated unitary operator $U_t^S$ (see, e.g., Figs. 2 and 3, respectively, for the detection of magnetization order parameter and kink number). (c) The real and imaginary parts of the characteristic function are detected by $⟨{\widehat{\sigma }}_x⟩$ and $⟨{\widehat{\sigma }}_y⟩$, respectively.

Figure 2

Probing the distribution of the magnetization in a nearest-neighbor Ising chain. (a) A schematic quantum circuit for simulating the evolution of the composite system $U_t^S=\prod _{n=1}^N\exp (-it\epsilon {\widehat{\sigma }}_z{\sigma }_n)$ [Eq. (14)]. The kernel of $U_t^S$ is realized by a controlled-$R_z^{(\gamma )}$ gate ($\gamma =2\epsilon t$), with $R_z^{(\alpha )}=\exp [-(i\alpha /2){\widehat{\sigma }}_z]$. Note that for the controlled-not gate, the target qubit is flipped if the control classical bit is set to ${\sigma }_n=-1$. The characteristic function of the magnetization distribution is detected by the coherence function of an auxiliary qubit. For an Ising chain of $N=50$ spins at inverse temperature $\beta =1$ with magnetic field (b) $h=0$ and (c) $h=0.2$, the real ($⟨{\widehat{\sigma }}_x⟩$) and imaginary ($⟨{\widehat{\sigma }}_y⟩$) parts of the coherence function are displayed as a function of time $t\in [0,\pi /\epsilon ]$, with the spin-qubit coupling $\epsilon =0.01$. The right panels of (b) and (c) correspond to the probability distribution of the magnetization obtained by a Fourier transform (FT) of the characteristic function.

Figure 3

Probing the distribution of the kink number in a nearest-neighbor Ising chain. (a) A schematic quantum circuit for simulating the evolution of the composite system $U_t^S=e^{-it(N/2)\epsilon {\widehat{\sigma }}_z}{\prod }_{n=1}^N\exp [it(\epsilon /2){\widehat{\sigma }}_z{\sigma }_n{\sigma }_{n+1}]$ [Eq. (19)], with ${\gamma }_1=N\epsilon t$ and ${\gamma }_2=-\epsilon t$. The real ($⟨{\widehat{\sigma }}_x⟩$) and imaginary ($⟨{\widehat{\sigma }}_y⟩$) parts of the coherence function are displayed as a function of time $t\in [0,\pi /\epsilon ]$, with the spin-qubit coupling $\epsilon =0.01$, $N=50$, $\beta =0.1$ and magnetic field $h=0$ in (b) and $h=10$ in (c). The right parts of (b) and (c) are the probability distribution of the kink number.