Observation of exceptional point in a PT broken non-Hermitian system simulated using a quantum circuit

Exceptional points (EPs), the degeneracy points of non-Hermitian systems, have recently attracted great attention because of their potential of enhancing the sensitivity of quantum sensors. Unlike the usual degeneracies in Hermitian systems, at EPs, both the eigenenergies and eigenvectors coalesce. Although EPs have been widely explored, the range of EPs studied is largely limited by the underlying systems, for instance, higher-order EPs are hard to achieve. Here we propose an extendable method to simulate non-Hermitian systems and study EPs with quantum circuits. The system is inherently parity-time (PT) broken due to the non-symmetric controlling effects of the circuit. Inspired by the quantum Zeno effect, the circuit structure guarantees the success rate of the post-selection. A sample circuit is implemented in a quantum programming framework, and the phase transition at EP is demonstrated. Considering the scalable and flexible nature of quantum circuits, our model is capable of simulating large-scale systems with higher-order EPs.


I. INTRODUCTION
Quantum computation is long believed to be faster than the classical counterpart for many tasks. The advantages of the quantum computation in various applications, such as factoring and searching 1-3 , have been shown theoretically years ago. However, the quantum supremacy, or advantage 4 is only experimentally achieved by Google on their Sycamore processor recently 5 . These newly available devices attracts considerable attention. Among all researches on such noisy intermediate quantum chips, the simulation of the quantum systems may be one of the most practical and promising applications [6][7][8] . Most existing simulations 9,10 are designed for Hermitian systems. This could be a natural choice considering the energy conservation of physical systems. However, it is common that a system may be entangled and exchange energy with the environment. After tracing out the environment, the evolution of the system follows an effective non-Hermitian Hamiltonian (i.e. H = H † ) [11][12][13][14][15] . Therefore, the simulation of physical systems should not be limited to Hermitian systems.
Due to the unique properties of the exceptional point (EP), the degeneracy points of the non-Hermitian Hamiltonian, and the parity-time (PT ) phase transition [16][17][18] , the non-Hermitian physics has also attracted intensive interest recently. In contrast to the conventional level degeneracy, at the EPs, not only the eigenenergies but also the corresponding eigenstates merge to be identical (coalesce) 16,18 . This coalescence leads to many distinctive phenomena around EPs, such as the 1/n dependence of the level-splitting on the perturbations around the nth order EP 19 and some nontrivial topological properties in the complex plane 20,21 . Such properties raised vast new topics in the study of quantum sensing and system control 22,23 . For instance, though with some doubts 24-28 , the last theoretical research and experimental evidence suggest that, EPs may be utilized for dramatically improve the sensitivity of level-splitting detection 22,23,29,30 .
After first demonstrated in microwave cavities 31 , the non-Hermitian effects were also soon observed in optical microcavities 32,33 , atomic systems 34,35 , electronics 36,37 , acoustics 38,39 , transmon circuits 40 and most recently nitrogen-vacancy centers in diamonds 41 . However, the power of the fast developing quantum computing is largely ignored in the study of simulating non-Hermitian systems and investigating EPs. Here we propose a realization of non-Hermitian system to study EPs using the quantum circuits, which is applicable to NISQ devices. Similar to the heralded entanglement protocols 42 , the effective non-Hermitian model is heralded by measuring the ancillas to the |0 states. For demonstration, a single qubit non-Hermitian system is simulated, where a phase transition at EP is observed. This simulation is implemented with Huawei HiQ 43 , a quantum programming framework based on the open-source python package ProjectQ 44,45 . It is straightforward to generalized the method to multi-qubit systems and higher-order EPs, where an example is shown in Appendix. We expect that the quantum chips in the near future could outperform the classical simulators for large non-Hermitian systems. Once the quantum chips are ready, the code can be migrated to the real device with minor modifications. We believe this work paves the way for simulating non-Hermitian physics and investigating EPs with quantum computers.

II. MODEL
The motivation of our design of quantum circuit comes from the fact that, non-Hermiticity of physical systems are generated from the entanglement with the environments. To imitate the real-world scenarios, the "system" qubits are entangled with the ancillas in the quantum circuit. By measuring the ancilla qubits and postselecting specific measurement results we can design the non-Hermiticity of the system qubits.
For simulations of two-dimensional non-Hermitian system, we take the circuit in Fig. 1(a) as a concrete instance (see Appendix B for circuit that simulates higher dimensional system). The non-Hermiticity remains if the gates are replaced by other one-or two-qubits gates. We take the first qubit as the "system" and the second qubit as an ancilla. The non-Hermitian unit is repeated only if the measurement result of the ancilla is |0 . Since the measurement on ancilla is repeated in the same basis, similar to the quantum Zeno effect, the success rate can be boosted by dividing each unit to smaller units. Starting from an initial state |ψ , after n cycles the final state of the system |ψ(n) is close to exp(−iH eff n)|ψ (φ 1). The H eff here is an effective non-Hermitian Hamiltonian (see Appendix. A) where σ x and σ z are the Pauli operators, and Γ = φ 2 /8. This approximation is similar to Trotterization 46,47 , whose error is O(Γ 2 )+O(Γθ). The non-Hermiticity of the system comes from the post-selection on the ancilla qubit. This process is similar to the non-Hermitian Hamiltonian in some quantum simulation experiments, such as the one heralded by the absence of a spontaneous decay in coldatom experiments 42 . It should be noted that the wavefunction evolved under the non-Hermitian Hamiltonian is unnormalized. It requires renormalization for further analysis.
The eigenenergies and the corresponding eigenstates of this Hamiltonian are where N is a normalization constant.
The real parts and the imaginary parts of the eigenergies are shown in Fig. 1(b) and (c) respectively. Since the imaginary part is nonzero except for the point Γ/θ = 0, this effective non-Hermitian system always lies in the PT-broken phase. This is a result of the shared imaginary part − iΓ 2 in both eigenvalues.This term is essentially caused by the non-symmetric controlling effect (see Appendix. A).

III. IMPLEMENTATION OF THE CIRCUIT
The quantum circuit in Fig. 1(a) can be implemented on any quantum devices that support the circuit-based quantum computing. To show that the simulated system is indeed non-Hermitian, we implemented the circuit on the HiQ simulator. Once the quantum chips are ready and connected to the HiQ, we expect that the same algorithm can be run on the quantum chips with minor modifications (for instance, by changing the backend from simulator to quantum chips).
The non-Hermitian unit in Fig. 1(a) can be intuitively translated to the HiQ/ProjectQ language as: # R_x r o t a t i o n on the qubit Rx ( theta ) | qubit # CR_x with qubit as control and ancilla as target C ( Rx ( phi )) | ( qubit , ancilla ) # m e a s u r e m e n t on the ancilla Measure | ancilla where the standard Python |(or) operator is reloaded and used to apply the gates to qubits here. qubit and ancilla are the qubits allocated in the MainEngine representing the "system" and the ancilla respectively. Rx(theta) and C(Rx(phi)) are the rotational and the controlled rotational operators with respect to Pauli-X, and Measure represents the quantum measurement in computational basis.
In order to implement the n-cycle non-Hermitian circuit in Fig. 1(a), we first allocate a qubit and an ancilla. The qubit is initialized to an arbitrarily chosen state |ψ , and the ancilla is initialized to |0 . After applying one non-Hermitian unit, it successes if the measurement result is |0 . If success, we allocate another ancilla which is also initialized to the |0 state, and repeat the first step. Otherwise, we start all over again. The process is repeated until we achieve n successes in a row, and the final state of the "system" should be proportional to |ψ(n) . In a trail, the whole process is repeated many times to estimate the probabilities P 0 = 0|ψ(n) . Several trails are used to get the mean and the standard deviation.
Obtaining an accurate estimation by sampling is resource consuming.
In order to quickly verify if the circuit simulates an effective non-Hermitian system, we utilize the collapse_wavefunction and cheat functions that is only available to the simulator backend (see Appendix. C). By using collapse_wavefunction(ancilla, [0]), the postselected wavefunction of the "system" with the ancilla at |0 is directly achieved. Further, by using cheat() the  1. (a) The circuit for simulating a non-Hermitian system on quantum computers. |ψ is an arbitrary initial state of the system. For each cycle, the ancilla qubit is reinitialized to the |0 state. We only post-select the results with ancilla measured to be |0 . full information of the wavefunction can also be directly obtained.
In Fig. 2, the simulated result is compared to the analytical solution to the effective non-Hermitian Hamiltonian. The initial state is set to |ψ(0) = |0 . It shows that, as long as φ 1 (important for both the Trotterization and the success rate as shown in Appendix. A), and θ is small (so it is not far away from EP, see Sec. IV), the circuit simulates the desired non-Hermitian system well.

IV. EP AND PHASE TRANSITION
As marked in Fig. 1(b), Γ/θ = 1 is corresponding to the exceptional point (EP), where, unlike the Hermitian system, not only the eigenvalues but also the eigenvectors coalesce. This degeneracy at EP leads to non-analytic behavior of the system 42 , which can be easily observed by compute and plot M z ≡ σ z around EP.
When Γ/θ < 1, we can rewrite the eigenvalues and the eigenvectors as λ ± = (−iΓ ± θ cos α)/2 and |v ± = [±e ±iα , 1] T with α = sin −1 (Γ/θ). The two eigenvalues has the same imaginary parts, and therefore the two eigenvectors are equally stationary. As shown in Appendix. D, begin with the fully mixed state ρ(0) = I/2, M z will always oscillate in this regime. For instance, when θ Γ and at the long time limit, M z (t) ≈ sin(2α) sin(θt), from which it is not hard to see that, the lone time average vanishes, i.e., M z = 0. Therefore, the expectation value M z takes distinctive behaviors on each side of the EP point, which shows a phase transition even for a single qubit. This is very different from the usual phase transition, which only happens when the number of particles goes to infinity. In Fig. 3(b) we show the good agreement between the simulations from HiQ and the theoretical results for Γ > θ, which confirms this phase transition. However, in the region of Γ < θ, the huge number of cycles required for taking the time average, especially when Γ is in the same order of θ, is beyond our current scope.

V. CONCLUSION AND DISCUSSION
In summary, we've proposed a scheme of simulating non-Hermitian systems with quantum circuits, and numerically demonstrated the phase transition at EP of such system. This is achieved by imitating the effect of environment with the post-selection of the measurement results on the ancilla qubits. The codes for our numerical experiment is based on the simulator backend of HiQ, which can be cast to programs on physical quantum chips once they are available in the near future. The non-Hermiticity of the quantum circuit have been shown and the phase transitions at EPs are also demonstrated. Although the number of cycles (> 5000) required to show the phase transitions is hard to be achieved for quantum chips at this stage, the non-Hermiticity of the circuit may be demonstrated experimentally on existing quantum chips (∼ 100 cycles). Compared to previous implementations, which utilize the specific properties of the underlying systems, our method benefits from the univer-sality and scalability of the quantum circuits. The idea of this work can be generalized to multi-qubit circuits and higher-order EPs, such as the one in Appendix B, where the advantage over other methods can be foreseen. Our results could open a new path to the applications of quantum computers beyond the usual simulation paradigms that confined to Hermitian systems. where Γ = φ 2 /8 and σ z i is the Pauli operator on the ith qubit.
With the effective multi-qubits non-Hermitian system, physics of higher order EPs can be investigated. For instance, assume the Hermitian Hamiltonian H = a(σ x 0 σ z 1 + σ y 0 σ z 1 ) + b(σ z 0 σ x 1 + σ z 0 σ y 1 )and without loss of generality take Γ = 1, we have four eigenenergies When a = b = 1/2 √ 2, the four eigenenergies coalesce, as shown in the Fig. 5, and we have a 4th order EP. Appendix C: Useful functions of the simulator backend Several functions in projectQ/HiQ are utilized in the simulator backend, which simplified the simulation of the circuit. But, to be noticed, they are only available to the simulator backend and cannot be used for simulations on real quantum chips.
The cheat function can be used to directly access and manipulate the full wavefunction. This function returns a list of two elements. cheat()[0] is the mapping of the qubits with the bitlocations, which may depend on how the compiler is optimized. cheat() [1] is the amplitudes of the wavefunction. they are stored as a numpy array of length 2 n with n being the number of qubits.
The set_wavefunction function is used to set the qubits to a specific state. This can be used a debugging tool, for instance, to verify the correctness the non-Hermitian units.
The collapase_wavefunction can be used to directly obtain the desired post-selected wavefunction by specifying a specific measurement outcome (unless the probability is 0), e.g., |0 .
The following code fragment illustrates the usage of these functions: where the flush() function is required to push all of above gates to the simulator and execute.