Abstract
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 ^{171}Yb^{+} 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.
Introduction
Quantum computers or quantum simulators are important quantum devices that may enable us to experimentally address intriguing phenomena that are not directly tractable in the laboratory^{1} or may outperform current classical computations for analysing complex quantum systems^{2,3}. In recent years, various physical platforms such as neutral atoms^{4}, ions^{5}, photons^{6} and superconducting circuits^{7} have been fruitfully developed for the purpose of quantum simulation. However, they are not yet able to perform some basic arithmetic calculations such as the complex conjugate, which changes the sign of the imaginary part of the coefficients of the state on a certain basis. Although we are used to computing the transformation with classical resources for useful scientific calculations, operations involving the complex conjugate require an antiunitary process, which is impossible to be implemented in a quantum system. Moreover, the complex conjugate is not scalable in classical calculation, since it requires full knowledge of the quantum state, and the number of measurements grows exponentially with the size of the system.
The complex conjugate is deeply inherent to the important concepts of discrete symmetries. Wigner^{8} proved that any symmetry operation acts as a unitary or antiunitary transformation in the Hilbert space, while an antiunitary transformation can always be decomposed into a unitary transformation together with the complex conjugate. The study of symmetries has profoundly shaped our comprehension of physical laws in the quantum field theory, which unifies quantum mechanics and special relativity. Charge conjugation and time reversal are paradigmatic examples of antiunitary discrete symmetry operations^{9,10}. The charge conjugation, together with the parity symmetry, is not conserved in the weak interaction^{11,12}, just as the timereversal symmetry. The discovery of the violation of these symmetries has been a decisive breakthrough of the quantum field theory, leading to the standard model. Recently, several important algorithms for the simulation of relativistic quantum mechanics and quantum field theory have been discovered^{13,14,15,16,17,18}. So far, however, quantum simulators of unitary and dissipative processes, the only physically allowed dynamics, have been realized^{19,20,21}.
Here, we perform the quantum simulation of the complex conjugate and these symmetry operations in our multilevel ^{171}Yb^{+} ion system through the use of the concept of embedding quantum simulator (EQS)^{22,23,24} beyond the boundary of the unitary operations. Our demonstration is scalable, where we can apply the timereversal or chargeconjugation operations in systems of any size, since they do not require the tomographical knowledge of the state. The essence of the EQS is based on the finding that antiunitary operations can be implemented in a physical system by doubling the associated Hilbert space^{22}. The scheme of the EQS enables us to efficiently compute entanglement monotones^{23} or multitime correlation functions^{24,25}. The reconstruction of these quantities would otherwise require a number of measurements that grows exponentially with the system size. We comment that the measurement of such quantities can be considered as an intractable task even for mediumsize systems composed by, for example, only a dozen of qubits, whereas the EQS scheme provides the solution at the expense of one additional qubit to double the Hilbert space.
Results
Majorana dynamics
We first simulate the Majorana dynamics to show the ‘unphysical’ capability of the EQS before implementing antiunitary symmetry operations. The Majorana equation^{26}, one of the representative relativistic equations,
where the Feynman slash notation with being the Dirac matrices^{27}, describes the dynamics of a nonHamiltonian system. Note that the spinor and its charge conjugation are present simultaneously in equation (1). Majorana envisioned that equation (1) together with the Majorana condition would be the fundamental equation describing neutrinos^{26}, which exhibit the novel phenomenon as ‘neutrino oscillation’^{28}. Besides, the Majorana equation (1) has its own theoretical importance in exploring physics beyond the standard model. Moreover, the utility of the relativistic equations is not limited to relativistic quantum mechanics and the quantum field theory. For example, electrons propagating through graphene are described by the (2+1)dimensional Dirac equation^{29}, and the symmetry breaking induced by tachyon condensation is described by a (1+1)dimensional Diraclike equation with imaginary mass^{30}, a nonHamiltonian system. Recently, a quantum simulation of the Majorana dynamics was performed in a photonic quantum platform, by decomposing its evolution in two Dirac equations^{31,32}. Through the quantum simulation of the inherently unphysical Majorana equation, we demonstrate various unique features, such as violation of charge and momentum conservation, broken orthogonality and nontrivial effect of the state’s global phase.
Embedding quantum simulator
The essential idea of an EQS is the mapping from the original Hilbert space to the enlarged one for spinors in 1+1 spacetime dimension, . In the position basis, as shown in Fig. 1, the EQS mapping is defined as
where are real functions satisfying the overall normalization condition . Inversely, as depicted in Fig. 1, the original spinor is retrieved through a matrix multiplication after evolving in the EQS for certain duration,
Through the EQS mapping (equation (3)), the complex conjugate operation, , is represented by a unitary operator in the enlarged Hilbert space, which can be implemented directly in a quantum system (see Methods and Supplementary Note 1).
With a certain choice of the Dirac matrices in the (1+1) dimension, and , the chargeconjugate spinor is properly defined as . The Majorana equation in 1+1 dimensions,
inherently contains the complex conjugate operator , which makes the Majorana dynamics prohibited by nature. For simplicity, we introduce a set of dimensionless units, that is, mc^{2} for the energy, mc for the momentum and for the time.
In the enlarged Hilbert space, the original nonHamiltonian system is mapped to a Hamiltonian one governed by an effective Hamiltonian,
Note that the equation of motion in the enlarged space, , keeps evolving inside . Because the effective Hamiltonian (equation (5)) does not contain a position operator, we perform the experimental implementation in momentum representation, where the dynamics of the Fouriertransformed spinor is governed by a simpler Hamiltonian obtained by substituting the momentum operator with its eigenvalue in equation (5) (see Supplementary Note 2).
Along the same line, some discrete symmetry operations, that is, the time reversal and the charge conjugation take form of unitary twoqubit gate operations in the enlarged Hilbert space: and , respectively.
Experimental setup
The EQS is built in an iontrap system, which is a leading platform for quantum simulation^{5}. The system consists of a single ^{171}Yb^{+} ion confined in a linear Paul trap^{33}, subjected to multifrequency microwaves. As shown in Fig. 1, the four internal states of the groundstate manifold ^{2}S_{1/2} are encoded as F=0, m_{F}=0〉≡1〉 and F=0, m_{F}=−1, 0, 1〉≡m_{F}+3〉, 1〉 and {2〉, 3〉, 4〉} are separated by the hyperfine splitting , and a uniform static magnetic field B=9.694 G is applied to define the quantization axis and causes Zeeman splitting among the upper states. As shown in the Fig. 1, the couplings between 1〉 and the upper states can be directly driven by microwave with frequencies as and , respectively. The couplings among the equally spaced upper states, that is, 2〉, 3〉 and 4〉, are implemented by the stimulated Raman process of microwaves (see Methods). On the basis of the multifold microwave technique, we achieve ultimate controllability over the Hilbert space spanned by all of the four internal states. In other words, we construct a ququad, an elementary unit of quantum information processing consisting of four basis states. In principle, largescale EQS can always be constructed by substituting one of the qubits in an array by a ququad, and the requisite microwave techniques involved in the control of the ququad have been developed in this work.
With the ability to perform any singleququad operation, we implement the effective Hamiltonian equation (5) in the momentum space,
on top of the EQS.
Experimental procedure
The experimental procedure is as follows. First, we map an initial Majorana spinor to a real bispinor in the enlarged space. The momentum representation of the bispinor evolves according to the enlarged space Hamiltonian . After encoding the initial condition into the EQS, we implement for a certain duration to simulate the Majorana dynamics. Then we perform quantumstate tomography (see Supplementary Note 3) to obtain the enlarged space density matrix , which can be mapped to the original space density matrix . The average value of a diagonal operator A_{d} in the momentum space can be directly obtained via integration over the momentum, . To obtain the average value of an offdiagonal operator in the momentum space, for example, the average position of the Majorana particle, we change the fourcomponent equation of motion in the enlarged Hilbert space into a pair of decoupled twocomponent equations by diagonalizing the first qubit (see Supplementary Note 4). By coherently evolving a couple of twodimensional equations with different momenta, we obtain the phase information between different momentum components. We repeat each measurement 1,000 times to get the expectation value. The statistical errors, which are mainly due to quantum projection, are estimated by the s.d. of mean value.
Majorana dynamics
Figure 2 shows our experimental results of the Majorana dynamics, where the initial spinors are chosen to be planewave states with , that is, . Figure 2a shows the momentumspace Zitterbewegung for a Majorana particle. Due to the existence of the complex conjugate operator in the Majorana equation, the momentum, which is conserved for free Dirac particles, is no longer a conserved quantity in the Majorana dynamics. Because the violation of momentum conservation is originated by the Majorana mass term, the amplitude of the oscillation is inversely proportional to the magnitude of the momentum of the initial state. Meanwhile, the frequency of the oscillation is determined by the relativistic dispersion relation , so the initial plane wave with larger momentum will oscillate faster. As shown in Fig. 2b, the Majorana dynamics also violates charge conservation, which may lead to physics beyond the standard model^{34}. In the rest frame, the charge operator measures the difference between the populations of the internal states, which is equivalent to the operator^{35}. For the nonzero momentum case, the particle and antiparticle basis is obtained by diagonalizing the corresponding Dirac equation with the same momentum, and the charge of a Majorana spinor is defined as the difference between the populations of the particle and antiparticle components (see Supplementary Note 5). For the same reason, the amplitude and frequency of the charge oscillation exhibits similar momentum dependence as that of the momentumspace Zitterbewegung.
Besides the above physical consequences, the dynamics governed by Majorana equation also shows unphysical phenomena. For example, the fidelity , where and are two Majorana spinors that evolve from initial states differing only in a global phase, , will not always be unity as shown in Fig. 2c. In other words, a Majorana spinor does not have the freedom to choose an arbitrary global phase. The reason for this surprising effect is the existence of the complex conjugate in the Majorana equation in equation (4). This effect can be more explicitly shown in the mapping in equation (2), that is, the global phase actually changes the initial fourcomponent spinor in the enlarged Hilbert space. Figure 2e,f shows an example of the experimental results of the density matrices in the enlarged and original Hilbert spaces, which are indeed different from each other. In Fig. 2d, we experimentally observe the nonconservation of the orthogonality defined as , with being the Majorana spinor evolved from an orthogonal initial state . During the evolution, the initial Majorana spinor will be coupled to through the Hermitian relativistic kinetic term , and through the nonHermitian Majorana mass term . The orthogonality , where is the Majorana spinor that evolves from the initial state , is always zero. This clearly indicates that the nonconservation of the orthogonality stems from the nonHermitian part of the Majorana Hamiltonian. As a result, given the same Majorana mass, we understand that the amplitude of the orthogonality oscillation is inversely proportional to the initial momentum.
Symmetry operations
Other than the plane waves, we also implement Majorana dynamics with realistic initial wave packets in our EQS. For example, the initial states for the Majorana dynamics in Figs 3 and 4 are moving Gaussian states with momentum distributions centred around P_{0}=1 with an internal state . The first part of the time axis (0≤t<4) in Figs 3 and 4 represent the Majorana dynamics of a moving wave packet, where we observe damping oscillation in the momentum space and Zitterbewegung in the position space. The reason of the damping in the momentum space is that a Gaussian wave packet has distribution over many different momentum components, and each momentum component oscillates with different frequency. To our surprise, although the average momentum of a Majorana particle behaves quite different from that of a Dirac particle, there is no visible difference in the behaviours of the average position as well as the probability distribution in position space. This is because a Majorana particle oscillates between the particle and antiparticle components with inverse momentum, but the positions as well as the velocities of the particle and antiparticle are exactly the same^{9}.
During the evolution of the Majorana equation, we implement the antiunitary timereversal and chargeconjugation operations. Figure 3 shows our experimental results of the timereversal operation during the Majorana time evolution. As shown in Fig. 3a, right after the timereversal operation, the momentum as well as the velocity changes sign. As a result, the direction of the wave packet is reversed as shown in Fig. 3c. The damped average momentum as well as the position centre of the wave packet is revived, which clearly shows that time is indeed reversed. Figure 4 demonstrates the experimental implementation of the chargeconjugation operation. The latter interchanges the particle and antiparticle components, which are defined from the corresponding Dirac equation with the same momentum as discussed in Fig. 2b. By definition, the particle and corresponding antiparticle have opposite momentum but the same velocity. As a result, right after the chargeconjugation operation, the average momentum is reversed but not the velocity. Therefore, the trajectory in position space remains intact, which is different from the timereversal operation.
Discussion
The demonstrated embedding scheme would potentially reduce the computational complexity of ordinary quantum simulations in the sense that it eliminates the requirements for tomographic information. By enlarging the EQS, the demonstrated symmetry operations, can be potentially scaled up to manyparticle systems in higher spacetime dimensions, in which the conventional quantumstate tomography is theoretically impossible. The EQS for multipartite systems only requires doubling the original Hilbert space dimension, which can be achieved by replacing a single qubit in an array of coupled qubits by a ququad (quantum fourlevel system). The proposed embedding scheme for the implementation of time reversal and chargeconjugation operations may be extended to parity symmetry operations^{24}. This enhanced toolbox for quantum simulators will be valuable for studying conservation laws and improving the computational capabilities of current quantum platforms.
Methods
Example of EQS mapping
In the following, we use a planewave initial state as an example of the encoding of states in the enlarged Hilbert space,
where corresponds to planewave states (unnormalized) with momentum ±p. Here we want to emphasize two points: (i) although is real, the components are usually composed of complex functions; (ii) there are always +p and −p components in the enlarged space to guarantee is real.
Stimulated Raman couplings
We implement a microwave Raman scheme for the transitions between 2〉 to 3〉 and 3〉 to 4〉. The strengths of effective Raman couplings are given by and , where are Rabi frequencies between 1〉 to 2〉, 1〉 to 3〉, and 1〉 to 4〉, respectively, are the detuning from 1〉 to 2〉(4〉) and δ_{23(34)} are frequency shifts for the compensation of ACStark effect between 1〉 to 3〉. The strengths of the Raman transitions are balanced to the direct transitions, which have around (2π)3 kHz. The cross talks between two transitions and are negligible because is produced by the combination of σ_{−}(σ_{+}) and π polarizations of microwave, which is impossible to couple to 3〉↔4〉 transition (2〉↔3〉). The ACStark shifts from all the microwave transitions are carefully compensated by properly adjusting the microwave frequencies (see Supplementary Note 6).
Additional information
How to cite this article: Zhang, X. et al. Time reversal and charge conjugation in an embedding quantum simulator. Nat. Commun. 6:7917 doi: 10.1038/ncomms8917 (2015).
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Acknowledgements
We acknowledge funding from the National Basic Research Program of China grants 2011CBA00300 (No. 2011CBA00301); the National Natural Science Foundation of China grant 11374178, 11405093 and 041303016; the Alexander von Humboldt grant; the Basque Government IT47210 Grant, MINECO FIS201236673C0302, Ramón y Cajal Grant RYC201211391, UPV/EHU UFI 11/55, CCQED, PROMISCE, SCALEQIT European projects and the UPV/EHU Project No. EHUA14/04. M.H.Y. and K.K. acknowledge the recruitment program of global youth experts of China.
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X.Z., Y.S., J.Z. developed the experimental system and performed the experiments. X.Z. analysed the results. J.C., L.L., E.S., M.H.Y. and J.N.Z. provided theoretical support. J.N.Z. developed the concrete models for the experiments. K.K. supervised the project. All authors contributed to writing the manuscript.
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Supplementary Information
Supplementary Figure 1, Supplementary Note 16 and Supplementary References (PDF 310 kb)
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Zhang, X., Shen, Y., Zhang, J. et al. Time reversal and charge conjugation in an embedding quantum simulator. Nat Commun 6, 7917 (2015). https://doi.org/10.1038/ncomms8917
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