Simulating the Same Physics with Two Distinct Hamiltonians

We develop a framework and give an example for situations where two distinct Hamiltonians living in the same Hilbert space can be used to simulate the same physics. As an example of an analog simulation, we first discuss how one can simulate an infinite-range-interaction one-axis twisting Hamiltonian using a short-range nearest-neighbor-interaction Heisenberg XXX model with a staggered field. Based on this, we show how one can build an alternative version of a digital quantum simulator. As a by-product, we present a method for creating many-body maximally entangled states using only short-range nearest-neighbor interactions.

Figure 1

The set of all initial quantum states is given by $S$. Even if a quantum simulator Hamiltonian ${\widehat{H}}_{\mathrm{QS}}$ is not the same as the target Hamiltonian ${\widehat{H}}_T$, there still exists some set of states $S_Q$ that can simulate the physics of ${\widehat{H}}_T$. If the overlap $S_S$ between the set of experimentally accessible states $S_A$ and $S_Q$ is not zero, the concept of using the connector can be used in analog quantum simulators. The same concept can be also extended to the digital version of a quantum simulator by applying “quantum kicks” using ${\widehat{H}}_{\mathrm{QS}}$. The size of $S_Q$ depends on the form of ${\widehat{H}}_{\mathrm{QS}}$. If ${\widehat{H}}_{\mathrm{QS}}={\widehat{H}}_T$, then $S=S_Q$ and $S_A=S_S$ is always a subset of $S_Q$.

Figure 2

In order to calculate the time evolution of $⟨{\widehat{S}}_x{⟩}_T$ in the target system, it is necessary to measure how $⟨{\widehat{S}}_x{⟩}_{\mathrm{QS}}$ and $⟨{\widehat{S}}_y{⟩}_{\mathrm{QS}}$ depend on time in the quantum simulator system. In the numerical simulations, we have set $\chi =1$, $\beta /\alpha \approx 40.0$ ($\alpha \approx 1.299\sqrt{N-1}\sqrt{{\chi }^2+\chi \beta }$, see the main text for details), and $N=5$ spins.

Figure 3

Fidelity between states generated by one-axis twisting [Eq. (6)] and Heisenberg spin chain [Eq. (7)] as a function of time and $\beta /\alpha$ ($\alpha \sim \sqrt{\beta }$) for a system with (a) an even number of spins ($N=6$) and (b) an odd number of spins ($N=5$). In (c), the data for the odd number of spins is replotted using a frame of reference rotating with frequency $\omega =\alpha /N$ around the $z$ axis. In (b), one cannot only observe one-axis twisting but also rotation of the state (due to the lack of discrete translational symmetry), which can be removed by moving to a proper frame of reference. In the numerical simulations, we have set $\chi =1$. Note that for $\chi t=\pi /2$ the state of the system is the maximally entangled Greenberger-Horne-Zeilinger state.