Quantum generative model for sampling many-body spectral functions

Quantum phase estimation is at the heart of most quantum algorithms with exponential speedup. In this paper we demonstrate how to utilize it to compute the dynamical response functions of many-body quantum systems. Specifically, we design a circuit that acts as an efficient quantum generative model, providing samples out of the spectral function of high rank observables in polynomial time. This includes many experimentally relevant spectra such as the dynamic structure factor, the optical conductivity, or the NMR spectrum. Experimental realization of the algorithm, apart from logarithmic overhead, requires doubling the number of qubits as compared to a simple analog simulator.

Figure 1

Quantum circuit. Quantum phase estimation is performed on a purified operator. The purified state can be prepared by entangling two copies with an ancilla control qubit and postselecting the result on outcomes $\left|1\right.〉$, see Fig. 2. A phase difference between the two copies appears because each phase estimation bit propagates one copy according to $U$ and the other as $U^{†}$. The output distribution after quantum Fourier transform is the spectral function.

Figure 2

State preparation scheme. An initial entangled pair states is created between two $N$-qubit registers. Next, one of the two copies is connected to an ancilla control qubit, which is placed in an equal superposition of $z$ states, i.e., both are evolved for some time $\varphi$ under the Hamiltonian, $H=O({\sigma }_a^z+1)/2$. Performing another Hadamard gate on the ancilla and postselecting the outcome on $\left|1\right.〉$, the entangled pair state will be transformed into the desired $\left|O\right.〉$ state. The success probability of the procedure is determined by the ratio of the typical value of $O^2$ to its maximal value $O_{\text{max}}^2$.

Figure 3

Preparation efficiency fidelity between the post-selected state $\left|{\psi }_1\right.〉$ and the target state $\left|O\right.〉$ decays with the rotation angle $\varphi$ of the controlled unitary rotation $U\left(\varphi \right)$ (full lines). Similarly, the success probability increases from 0 to $1/2$ when the angle increases (dashed lines). Different curves show expressions (15) and (14) for different eigenvalue distributions of $O$, i.e., results are shown for Wigner semicircle, uniform, arcsine, and Gaussian eigenvalue distributions. Each of these distributions has a success probability $P=c(1-F)$ in a broad region of $\varphi$'s around zero. The constant $c=O\left(1\right)$ for all distributions, i.e., $1/2, 5/9, 2/3$, and $1/3$ for the semicircle, uniform, arcsine, and Gaussian, respectively.