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.

Introduction.-The concept behind quantum simulators is fairly straightforward to understand [1,2] but extremely challenging from an experimental point of view [3]. Imagine that one has a target HamiltonianĤ T and wants to study its properties or the dynamics governed by it. However, the system is either too large to perform numerical and analytical calculations on or is intractable from an experimental point of view. In this case, one can either come up with some other physical system that has a HamiltonianĤ QS that is identical toĤ T and therefore possesses the same system properties and leads to the same dynamics. Or one can perform the desired evolution based onĤ T using an approximative stroboscopic time evolution through "quantum kicks." The first case describes so-called analog quantum simulators and the second case digital quantum simulators [4]. The idea of quantum simulators is commonly attributed to Richard Feynman, who proposed it in 1982 [5]; however, due to the experimental difficulties, in particular, controlling and tuning Hamiltonian parameters with high fidelities, the first viable ideas for quantum simulators were only proposed and realized very recently [6][7][8][9][10][11][12][13][14][15][16][17][18][19] on a number of experimental platforms, including ultracold quantum gases [20,21], trapped ions [22], photonic systems [23], and superconducting circuits [24].
The requirement forĤ QS to be a suitable Hamiltonian of a quantum simulator can be formulated in the following way (for the sake of brevity, we set ℏ ¼ 1 throughout the entire manuscript): where ξðtÞ is a mostly real-valued function of time. If the imaginary part of ξ is zero, i.e., ℑðξÞ ¼ 0, thenĤ QS is an ideal simulator, whereas if ℑðξÞ ≠ 0, the simulator is only suitable for times during which ℑ½ξðtÞ ≪ 1. In the ideal case, the original idea of a quantum simulator considered either ξ ¼ 0, so thatĤ QS ¼Ĥ T , or ξ to be some real-valued number c multiplied by time, so thatĤ QS ¼Ĥ T þ cÎ, whereÎ is the identity operator. Making use of the Baker-Campbell-Hausdorff formula, it is straightforward to write whereĥðtÞ is some, in general, time-dependent Hermitian operator where ½•; • stands for the commutator and … indicates terms involving higher order commutators ofĤ QS andĤ T . The interpretation ofĥðtÞ is then straightforward. It is nothing else but an operator that transforms dynamics governed byĤ T to dynamics governed byĤ QS . If one knows theĥðtÞ that relates the two HamiltoniansĤ T and H QS , it is possible to simulate dynamics generated byĤ T byĤ QS by using the transformation for any observableÔðtÞ. For brevity, we will callĥðtÞ a connector operator or simply connector. Unfortunately, due to their construction, connectors are likely to be rather complicated, time-dependent, or even nonlocal, and therefore most often of no practical help. However, we will show in the following that under certain conditions one can connect the dynamics governed by two substantially different Hamiltonians in a valuable way. One such condition is given in situations where two Hamiltonians commute, i.e.,ĥðtÞ ¼ĥ · t. This means thatĥ,Ĥ QS , andĤ T share the same eigenbasis but have different eigenspectra.
Then, ifĥ has a degenerate eigenspectrum, it might happen that a state jψi composed of degenerate eigenstates ofĥ 1 H QS −Ĥ T will not be an eigenstate ofĤ QS orĤ T , but the two Hamiltonians will yield the same quantum dynamics with respect to that state (see Fig. 1). Of course, finding two different Hamiltonians that commute so that one of them can act as a quantum simulator is not easy and potentially a vast limitation. However, in the following we will discuss two interesting cases and in particular show how to make use of the knowledge ofĥðtÞ in order to simulate infiniterange interactions with a system that exhibits only shortrange nearest-neighbor interactions. Analog quantum simulators.-As a first example, we will consider how to simulate the well-known one-axis twisting Hamiltonian [25] H oat ¼ χ z i is the collective spin operator. Despite its simplicity, this Hamiltonian is known to generate a wide spectrum of many-body entangled states such as spinsqueezed, twin Fock, and Greenberger-Horne-Zeilinger states if the initial state is an eigenstate of theŜ x operator with maximal eigenvalue, i.e.,Ŝ x jψi ¼ N=2jψi [26]. It can also be realized experimentally with ultracold gases [27] and trapped ions [28]. On the other hand, due to its formal simplicity, we can easily find a nontrivial and interesting Hamiltonian that commutes with the one-axis twisting Hamiltonian. It is straightforward to show that the Heisenberg XX model commutes withĤ oat and therefore any eigenstate of H oat −Ĥ XX will give the same dynamics under the action of the two different Hamiltonians. However, asĤ oat −Ĥ XX possesses a nondegenerate eigenspectrum, such a simulator is fundamentally not very interesting as it can only simulate the dynamics of eigenstates. Nevertheless, one can add an arbitrary function ofσ z i to the Heisenberg XX model and it will still commute with theĤ oat since ½σ z i ;Ĥ oat ¼ 0. This then allows one to manipulate the form ofĥ in such a way that the initial stateŜ x jψi ¼ N=2jψi is also the eigenstate ofĥ but not of the Hamiltonians building it. As an example, we show that the Heisenberg XXX model with a staggered fieldĤ can simulate a one-axis twisting Hamiltonian in the limit and for an even number of spins, i.e., N ¼ 2k with k ∈ N. Even thoughĤ QS and H oat are completely different, they realize the same dynamics. Most strikingly,Ĥ QS contains only short-range nearestneighbor interactions, whileĤ oat contains infinite-range interactions. Interestingly, we find that, for an odd number of spins, N ¼ 2k þ 1, and similar conditions, i.e., β ≫ α and α ≈ 1.299 , the Heisenberg XXX model with staggered field realizes both one-axis twisting and an effective rotation around the z axis with the frequency given by α=N. The rotation can be easily eliminated by moving to a frame that rotates around the z axis with the same frequency but in the opposite direction, i.e., performing the transformation jψi →Ûjψi withÛ ¼ exp½itðαŜ z =NÞ (note, however, thatĥ does not have to be proportional toŜ z ). This idea is similar to moving to a frame of reference rotating with the frequency of a pumping laser, which is a typical situation in quantum optics. We can therefore identify another interesting condition for a quantum simulator using the connector. This is, even if the initial state is not an eigenstate ofĥ butĥ happens to trivially transform jψi (as in the case of a collective rotation or a translation), measuring an observable in the quantum simulator allows for measuring it by performing a straightforward manipulation on the measured data, in this case given by The results of the numerical simulation and calculation of hŜ x ðtÞi T are presented in Fig. 2.
In order to investigate the robustness of simulating the one-axis twisting dynamics with a Heisenberg XXX chain with a staggered field, we plot the fidelity between the FIG. 1. The set of all initial quantum states is given by S. Even if a quantum simulator HamiltonianĤ QS is not the same as the target HamiltonianĤ T , there still exists some set of states S Q that can simulate the physics ofĤ 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" usingĤ QS . The size of S Q depends on the form ofĤ QS . If H QS ¼Ĥ T , then S ¼ S Q and S A ¼ S S is always a subset of S Q .
states generated with these two Hamiltonians for two cases N ¼ 6 and N ¼ 5 as a function of time and β=α in Fig. 3. One can see why the condition given by Eq. (1) does not require ℑ½ξðtÞ ¼ 0. For times such that ℑ½ξðtÞ ≪ 1, the dynamics governed by the simulator still very much resembles the dynamics governed by the target Hamiltonian.
It can be also shown that Heisenberg XXX model with an arbitrary transverse field in the z direction commutes with the special case of the Lipkin-Meshkov-Glick model H LMG ¼Ŝ 2 x þŜ 2 y þ ΩŜ z [29] or withĤ ¼ P ∞ n¼1 γ nŜ n z , and the Heisenberg XXX model without a transverse field commutes with a generalized two-axis countertwisting HamiltonianĤ tact ¼ χðŜ xŜy þŜ yŜx Þ þ αŜ x þ βŜ y þ γŜ z . However, the question of whether one can simulate the nontrivial physics of these Hamiltonians using the connector approach remains open at this time. An interesting situation arises when the two Hamiltonians do not commute. In such a case, the connector can be expressed asĥ ¼ P ∞ n¼1 t nÂ n , whereÂ n are operators that can be found according to the Baker-Campbell-Hausdorff formula. Also, when at least one of the Hamiltonians is time-dependent, it might lead to interesting possibilities of quantum simulation. All of these possibilities may relax constraints imposed on the universal analog quantum simulator, but we defer all of them to future investigations. Instead, we will focus now on the possibility of using the connector operator in the digital quantum simulator.
Digital quantum simulator.-A digital quantum simulator [4] works by evolving a system forward using small and discrete time steps according to e iĤ T t ≈ ðe iĤ 1 t=n …e iĤ l t=n Þ n : ð9Þ By making t=n small enough and using error correction protocols, this allows us to simulateĤ T with an arbitrary precision. This concept can also be applied to perform digital quantum simulation using the connector operator. If the time evolution interval is short enough, we can neglect the higher order commutators in Eq. (3), i.e., In contrast to the situation where the HamiltoniansĤ QS andĤ T commute, here the eigenstates ofĥ are different from the eigenstates ofĤ QS andĤ T . While this is in general a simplification, the price to be paid for it is that the eigenstates ofĤ QS −Ĥ T are only approximate eigenstates ofĥ for short time intervals, while ½iðδtÞ 2 =2½Ĥ QS ; −Ĥ T ≈ 0. However, since during these the two Hamiltonians will yield the same dynamics, one can perform stroboscopic dynamics by changingĤ QS tô H 0 QS after every quantum kick. If the new Hamiltonian is chosen such that the state after the last quantum kick FIG. 2. In order to calculate the time evolution of hŜ x i T in the target system, it is necessary to measure how hŜ x i QS and hŜ y i QS depend on time in the quantum simulator system. In the numerical simulations, we have set , see the main text for details), and N ¼ 5 spins.  5). In (c), the data for the odd number of spins is replotted using a frame of reference rotating with frequency ω ¼ α=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 χ ¼ 1. Note that for χt ¼ π=2 the state of the system is the maximally entangled Greenberger-Horne-Zeilinger state.
is the eigenstate of the operatorĤ 0 QS −Ĥ T , one can then simulateĤ T with the quantum kicks generated by fĤ QS ;Ĥ 0 QS ; …;Ĥ ðnÞ QS g, where n labels the nth quantum kick. Naturally, the smaller the commutator, the longer each quantum kick can be applied for, and in the limit of the commutator going to 0, we recover the analog quantum simulator discussed in the previous section. In this sense, the analog quantum simulation is a special case of digital quantum simulation where the length of the quantum kick can be infinitely long.
As in the original idea of the digital quantum simulator, the digital quantum simulator using the connector operator has to be first accordingly prepared. In the former case, one has to use the so-called Trotter expansion, and in the latter case one has to ensure that jψi is an eigenstate ofĤ ðnÞ QS −Ĥ T after each quantum kick. However, the digital quantum simulator using the connector requires many fewer steps as the sequence of kicks has to applied only once instead of n times [see Eq. (9)]. The price to be paid for this simplicity in relation to the standard digital quantum simulator is the fact that, for every initial state, one has to come up with a unique set of quantum kicks. Nevertheless, given the fact that in the experiment only a tiny fraction of all possible quantum states can be addressed, it should not be viewed as a major obstacle (see Fig. 1). Also, depending on the particular target Hamiltonian, some quantum simulators will be better than others since some of them will minimize the commutator ½Ĥ QS ; −Ĥ T , allowing thus for increasing the length of a single time step δt.
Last but not least, one can think about combining the Trotter decomposition with the connector approach. Imagine that one has an operatorÔ that commutes with the target HamiltonianĤ T or thatĥ can be easily calculated. Then, as we have shown, for the eigenstates of O −Ĥ T , the unitary evolution operators expð−itÔÞ and expð−itĤ T Þ will yield the same dynamics. As a consequence, ifÔ is much simpler thanĤ T , decomposing expð−itÔÞ should become much easier than decomposing expð−itĤ T Þ.
Conclusions and outlook.-By using the knowledge of a connector operator of two Hamiltonians residing in the same Hilbert space, we have proposed a way of simulating the dynamics governed by one Hamiltonian using a different one. As an example of an analog quantum simulation, we have shown how to implement the one-axis twisting Hamiltonian in the Heisenberg XXX model with a staggered field. Using the connector, we have also proposed an alternative approach to digital quantum simulators. Instead of trying to build the target HamiltonianĤ T out of many small steps, one has to apply short quantum kicks with a quantum simulator HamiltonianĤ QS such that after each quantum kick the state is an eigenstate of theĤ QS −Ĥ T operator. This can significantly reduce the complexity of a digital quantum simulator. The price being paid is the fact that not all initial states can be easily used in the simulator (see Fig. 1). However, given the fact that not all initial states can be prepared in an experiment, by appropriately tuning the parameters of the simulator one should be able to simulate nontrivial physics of other systems. We have also identified interesting possibilities for future research, including analog quantum simulation in the case where two Hamiltonians,Ĥ QS andĤ T , do not commute or when the target Hamiltonian is time-dependent. A fascinating question that remains to be addressed in future research is whether the presented framework can be used with dissipative time evolution.
The results presented in this Letter might have direct implications in many branches of modern physics, as well as quantum chemistry [30][31][32] and quantum biology [33,34], and can be tested in most of the current quantum simulator experimental setups. However, the most striking consequences pave a way toward an approach to simulating dynamics, not only with other systems but with other Hamiltonians. This might relax the constraints on the universal quantum simulator as it is not necessary to use exactly the same Hamiltonian to simulate the physics of some other Hamiltonians. On the downside, even though in certain situations it might be easier to perform quantum simulations exploiting the connector operator, in general it might be more challenging to find proper quantum simulators that allow for taking advantage of this framework of connectors.
Additionally, we have proposed a method for creating many-body entangled states, including the spin-squeezed and the maximally entangled Greenberger-Horne-Zeilinger state, in a system exhibiting exclusively nearest-neighbor interactions. This might become extremely useful for the quantum computer architectures based on superconducting qubits as they typically exhibit only nearest or next-nearest neighbor interactions [35].
Simulations were performed using the open-source QUANTUMOPTICS.JL framework in JULIA [36]. K. G. would like to acknowledge discussions with Tomasz Maciążek, Mohamed Boubakour, Friederike Metz, Lewis Ruks, Hiroki Takahashi, and Jan Kołodyński. This work was supported by the Okinawa Institute of Science and Technology Graduate University. K. G. acknowledges support from the Japanese Society for the Promotion of Science (JSPS) Grant No. P19792. A. U. acknowledges a Research Fellowship of JSPS for Young Scientists. K. G. would like to thank Linda Gietka and Simon Hellemans for support and Michał Jachura for reading the manuscript.