Time reversal and charge conjugation in an embedding quantum simulator

A quantum simulator is an important device that may soon outperform current classical computations. A basic arithmetic operation, the complex conjugate, however, is considered to be impossible to be implemented in such a quantum system due to the linear character of quantum mechanics. Here, we present the experimental quantum simulation of such an unphysical operation beyond the regime of unitary and dissipative evolutions through the embedding of a quantum dynamics in the electronic multilevels of a 171Yb+ ion. We perform time reversal and charge conjugation, which are paradigmatic examples of antiunitary symmetry operators, in the evolution of a Majorana equation without the tomographic knowledge of the evolving state. Thus, these operations can be applied regardless of the system size. Our approach offers the possibility to add unphysical operations to the toolbox of quantum simulation, and provides a route to efficiently compute otherwise intractable quantities, such as entanglement monotones.

C n onto an enlarged 2n-dimensional real Hilbert space R 2n , . . .
In the 1 + 1 dimension case, we consider the specific mapping M : C 2 → R 4 . In the following, we use a plane-wave initial state ψ p (x) as an example of the encoding of states in the enlarged Hilbert space, where Ψ ± p (x) corresponds to plane-wave states (unnormalized) with momentum ±p. Here we want to emphasize two points: (i) although Ψ p (x) is real, the components Ψ ± p (x) are usually composed of complex functions; (ii) there are always +p and −p components in the enlarged space to guarantee Ψ p (x) is real.
in the enlarged space, which takes the following form, In the momentum space, the bispinor Ψ (p, t) is obtained via the Fourier transformation, and the equation of motion becomes where the momentum operatorp x is substituted by its eigenvalue p. It is clear that the dynamics governed byĤ p is ready to be implemented in a quantum four-level system.

Supplementary Note 3: Quantum State Tomography
We measure a state's density matrix by performing qudit state tomography [3]. The density matrix of a state in the 4-level system can be written aŝ whereλ j are the SU(4) generators, which are referred as four dimensional generalized Gell-Mann matrices [4]. They have following properties: Tr(λ j ) = 0, Tr(λ jλk ) = 2δ jk .
Thus we can have r j = Tr(ρ 4λj ) = λ j , and the density matrixρ 4 can be reconstructed by measure all expectation values λ j .
In our trapped-ion system, the florescence detecting method can only directly measure population on state |1 . But it can be easily extended to measure the population on other state |n by state flipping with a π pulse resonant at |1 ↔ |n transition. For a diagonal operatorÂ = diag{a 1 , a 2 , a 3 , a 4 }, its expectation value can be calculated by where ρ jj is the population on state |j . For a off-diagonal operatorB, we can diagonalizê B toB d =ÛBÛ † . The expectation value ofB on state ψ can be written as which is just the expectation value of diagonal operatorB d on stateÛ ψ. Sinceλ j is one of Pauli matrices extending to a higher dimension, the corresponding unitary operationÛ is always pretty simple. So we can first perform the unitary operationÛ on state ψ, then measure populations and calculate the expectation value ofB d . Each population measurement is repeated by 1000 times. For the uncertainty of the measurements, we mainly consider the quantum projection noise and use the standard deviation of projection outcomes. The measurement result of the final physical observable can be always written as a function of expectation values ofλ j operators. We finally find the error bar by using the standard error propagating method.

Supplementary Note 4: Physical Observables
In this part, we will describe in detail the procedure to extract the information of various physical observables from experimental data. The time-dependent enlarged four-component spinor can be formally written in the momentum space as follows, with |p being the momentum basis, i.e. the plane-wave states, and |χ p (t) describing the internal state |χ p (t) = 4 j=1 χ p,j (t) |j . Note that the wave function in the momentum space Ψ (p) does not depend on time and is fully determined by the initial condition |ψ (0) in the original space, because the effective HamiltonianĤ commutes with the momentum operatorp x . The only time-dependent part in Eq. (6) is the internal state |χ p (t) , whose dynamics is determined by the enlarged space Hamiltonian given by i ∂ t |χ p (t) =Ĥ p |χ p (t) , can be simulated in a quantum four-level system. Using quantum state tomography, we experimentally obtain the density matrixˆ p (t) correspond- The general form of a diagonal operatorÔ dg in the momentum space can be written as withΣ = c 0Î +c 1σx +c 2σy +c 3σz and f (·) being an arbitrary algebraic function, . The expectation value of this operator at arbitrary time t can be obtained as follows, We may take the average momentum as a simple example, The quantum simulation for each |χ p (t) will be as follows.
2. Implement the HamiltonianĤ p and let the system evolve for certain time duration t, 3. Perform the quantum state tomography and obtainˆ p (t) = |χ p (t) χ p (t)|.
Then the matrix element mentioned above can be obtained straightforwardly, Then we turn to investigate the method to obtain the expectation value of some offdiagonal operatorsÔ od in the momentum space. We will take position-dependent operators as examples, i.e.,Ô As mentioned above, the expectation value can be written as Since f (x) is not diagonal in the momentum space, the above expression will involve offdiagonal matrix-element as χ p (t) M †Σ M χ p (t) . If we stick to the previous scheme, we will obtain two independent density matricesˆ p (t) andˆ p (t), from which we can not construct the off-diagonal matrix element between two distinct momenta.
Inspired by the effective HamiltonianĤ p for some definite momentum p, we notice that the first qubit can be diagonalized in theσ where |± ≡ 1 √ 2 (|0 ± |1 ) are eigenstates ofσ x . In other words, the equation of motion for |χ p (t) can be written in the new basis as follows, and where χ ± p (t) are column vectors with two entries andĤ ± p are 2×2 matrices in the new basis. As shown in Eq. (16), we note that the dynamics for χ ± p (t) are totally decoupled from each other, and can be separately simulated in quantum two-level systems. In order to obtain off-diagonal matrix elements between two distinct momenta p and p , we have to simulate χ p (t) and χ p (t) coherently. We obtain the following equations of motion by rearranging Eq. (16), which can be simulated in quantum four-level systems.
In the following investigation, we will use the average position x ≡ ψ (t) |x| ψ (t) as an example. The detailed derivation is as follows, The last line in the above equation is valid because of the following identity, (20) which can be verified using ρ E (p) = ρ E (−p) and χ ± p (t) = χ ± −p (t) * .
The experiment procedure would be as follows.
1. Prepare the initial state determined by the initial condition 2. ImplementĤ ± p,p and let the system evolve for some time period t, 3. Perform the quantum state tomography and obtain ρ ± p,p , Sweeping the momenta p and p over all possible values, we would obtain all of the information that is needed to calculate the expectation value x (t) ≡ ψ (t) |x| ψ (t) . The number of separate simulations for different (p, p ) pairs for both signs will be N 2 P , where N P is the number of points with which we discretize the momentum axis.

Supplementary Note 5: Charge conservation and charge conjugation
The non-Hermitian Majorana Hamiltonian does not have eigenstates. However, we can define the concepts of particle and antiparticle from the eigenstates of the corresponding Dirac Hamiltonian, which is obtained by substituting the Majorana mass term with the Dirac mass term. Under the same convention, the 1 + 1 Dirac equation takes the following dimensionless form, with the eigenvalues ± p 2 + m 2 and the corresponding eigenstates Starting from an initial Majorana spinor |ψ (0) = u 1 (0) Majorana spinor can be formally written as follows, Note that the appearance of the negative momentum component is originated from the charge conjugation in the Majorana mass term. By definition, the time-dependent charge is obtained as follows, By setting u 1 (0) In addition to the results in the main text, here we show the theoretical result for the dynamics of the internal degree of freedom in Fig. 1. In Fig. 1 B, we can clearly see that the populations of the particle and antiparticle components are interchanged right after the implementation of the charge-conjugation operator. Fig. 1 C shows the momentum distributions for the particle and antiparticle components at different times above and below the base lines, respectively. We clearly see from the Majorana dynamics of the internal degree of freedom that the evolution is continued after the implementation of the charge conjugation, although the roles of the particle and antiparticle are interchanged.

Supplementary Note 6: Microwave Implementation
In our 171 Yb + ion system, we use the microwaves for the transitions between |1 and |2 , |3 , |4 levels. We use a microwave Raman scheme similar to the widely used Raman laser scheme for the transitions between |2 to |3 and |3 to |4 transitions. We cannot apply a radio frequency for the operations of these transitions, since the energy gap between |2 ↔ |3 and |3 ↔ |4 is very close, which is (2π)31 kHz for our experimental condition.
As shown in Fig. 1 of the main text, 6 different frequencies of microwaves are combined and simultaneously applied to the trap. For the control of 6 microwaves, we use a PCI-board arbitrary waveform generator (AWG) with 1 GHz sampling rate, which is mixed with a 12442.8213 MHz microwave. The AWG generates the signal of 6 frequencies from 186 ∼ 214 MHz.
The system is described by the HamiltonianĤ =Ĥ A +Ĥ AL , with atomic partĤ A beinĝ and the interaction partĤ AL beinĝ respectively. We set the 6 frequencies in the microwave as follows, where ∆ is the detuning for the stimulated Raman transitions and δ i are the frequency shifts used to compensate the AC Stark effect. Using the method in Ref. [ In order to make the final rotating frame Hamiltonian time-independent, the additional detunings should satisfy the following relations, st , st , st .
One problem in this scheme is the slowing down of operations. The Raman transition is 10 times slower than normal Rabi flopping, even with full power. But now we need 6 microwaves together. Decoherence occurs when the whole microwave duration is longer than 600µs. This decoherence problem is later solved by applying a line trigger [2] to the pulse sequencer.