Duality quantum algorithm efficiently simulates open quantum systems

Because of inevitable coupling with the environment, nearly all practical quantum systems are open system, where the evolution is not necessarily unitary. In this paper, we propose a duality quantum algorithm for simulating Hamiltonian evolution of an open quantum system. In contrast to unitary evolution in a usual quantum computer, the evolution operator in a duality quantum computer is a linear combination of unitary operators. In this duality quantum algorithm, the time evolution of the open quantum system is realized by using Kraus operators which is naturally implemented in duality quantum computer. This duality quantum algorithm has two distinct advantages compared to existing quantum simulation algorithms with unitary evolution operations. Firstly, the query complexity of the algorithm is O(d3) in contrast to O(d4) in existing unitary simulation algorithm, where d is the dimension of the open quantum system. Secondly, By using a truncated Taylor series of the evolution operators, this duality quantum algorithm provides an exponential improvement in precision compared with previous unitary simulation algorithm.


env env
where ρ′ is the density matrix of the final state of principal system and U is time evolution operator imposed on the total system. The corresponding Hamiltonian H of U is in the space   ⊗ p e . For convenience, we assume that the dimensions of principal system and environment are the same, namely, d = d p = d e . The dimension of total Hamiltonian H is d 2 . Lloyd firstly proposed a quantum algorithm to simulate open quantum system efficiently 7 . In this algorithm, by enlarging the system to include the environment, the total system Hamiltonian is decomposed in the form = ∑ = H H j l j 1 where each ∈ × H C j d d 2 2 is Hermitian and satisfies ≤ ‖ ‖ H h j for a given constant h. The query complexity of simulating time evolution of the open quantum system in an accuracy ε over time t is approximated to O(ld 4 ht 2 /ε). By regarding the total system as a bigger closed quantum system, this algorithm performs unitary transformation as same as in the closed system.
The concept of duality quantum computers is first proposed by Long in 2002 based on the general principle of quantum interference 17,18 , which draw many attentions  . It is shown that any bounded linear operator can be expressed as a linear combinations of unitary operators in a duality quantum computer 21 . Thus, duality quantum computers can perform non-unitary transformation and provide novel way to design quantum algorithms, which can adapt the techniques in classical algorithm design to quantum algorithms, already showing flexibility and good performance in precision for closed quantum systems. Recently, several duality quantum algorithms have been proposed, which simulate Hamiltonian dynamics by linear combinations of unitary operations in a closed quantum system [46][47][48][49] . In the algorithms in refs 47-49, the performance has exponential improvement in the dependence on precision.
Alternatively, in an open quantum system coupled with surrounding environment, the dynamics can also be described by a completely positive linear map ε(ρ). The quantum operations can be represented in operator-sum representation by Kraus operators. Suppose the initial state of environment is a pure state, denoted as ρ env = |e 0 〉 〈 e 0 |. Equation (1) can be rewritten as 6 . The complete set of † E E k k is known as a "Positive Operator-Valued Measure". It should be noted that the operator E k is only acted on the principle system. So, if we can realize the Kraus operator, the complexity of evolution simulation that is dependent on dimensions will be decreased. Generally speaking, Kraus operator E k is non-unitary and can not be realized in quantum computer directly. However, the Kraus operator can be realized in a duality quantum computer 40,41 . In our method, the query complexity is , which exponentially improved the performance of quantum algorithm in ref. 7.
In this paper, we present a duality quantum algorithm to simulate Hamiltonian evolution for an open quantum system. There are two stages in our method. The first stage realizes Kraus operators in the duality quantum computer. The second stage of the algorithm is based upon a truncated Taylor series to approximate the evolution operators. The query complexity of the algorithm is significant decreased compared with Lloyd's algorithm 7 . We demonstrate this algorithm by a single quibit open quantum system as an example.

Results
Realization of Kraus operators in duality quantum computer. A duality quantum computer is a moving quantum computer passing through a d-slit which exploits the wave-particle duality of quantum systems 17 . The physic picture is : a quantum system passing through a d-slits with its wave function being divided into d sub-waves, the dividing operation denoted as the quantum wave divider (QWD) operation. Different unitary operations are performed simultaneously on the sub-waves at different slits. This is called the duality parallelism, and it enables the duality quantum computer to perform non-unitary gate operations. Conversely, the quantum wave combiner (QWC) operation adds up all the sub-waves into one wave function. Compared to ordinary quantum computers in which only unitary operators are allowed, One can perform different gate operations on the sub-wave functions at different slits in the duality quantum computer 17 . Generally, we only measure the final wave functions on 0-slit to realize a duality quantum gate, which is called single output duality quantum computing. Furthermore, we make measurements of the final wave functions on all d-slits, which is called complete measurements. After detecting, through QWD operation and QWC operation, every path on each-slit realized a duality quantum gate. It means that d duality quantum gates are performed in one process. The process is denoted as multi-output duality quantum computing. Duality quantum gates are generally non-unitary and naturally suitable to perform non-unitary evolutions. A three-slits duality quantum computer is shown in Fig. 1. The input is from the 0-th slit, and it is divided into three sub-waves by the middle screen with three slits. After the middle screen, different operations are performed on the different sub-waves, and three outputs of duality quantum computing are collected from three-slits on the right wall 18 .
It has been proven that a moving n-qubit duality computer passing through a d-slit can be perfectly simulated by an ordinary quantum computer with n-qubit and an extra qudit resource [18][19][20] , which is called duality quantum computing mode. For the convenience, we use the expressions from duality quantum computing mode 19,20,32,33 in this article.
The n-qubit ordinary quantum computer is represented by n work qubit and an auxiliary qudit represents a d-slits. The QWD operation can be represented by a general unitary operation V and the QWC operation can be represented by a general unitary operation W. The two unitary operations act on an auxiliary qudit. There are d controlled unitary operations act on ordinary quantum computer between the operations V and W. The quantum circuit of duality quantum computer is given in Fig. 2.
It is convenient to divide the whole process into four steps to illustrate the multi-output duality computing in a quantum computer.
Step one. The quantum system is prepared with initial state |Ψ 〉 |0〉 firstly. The QWD operation is implemented by performing the operator V on the auxiliary qudit |0〉 , and this operation transforms the initial state into represents the divider structure and is the first column element of the unitary matrix V representing the coefficient in each slit. The closure condition ∑ |i〉 〈 i| = I in quantum mechanics has been used in the deviation. The final state |Ψ 〉 |i〉 represents the sub-wave at the i-th slit.
Step two. Performing the auxiliary qudit controlled operations U 0 , U 1 , U d−1 on the work qubits with initial state |Ψ 〉 which leads to the following transformation, The corresponding physical picture is that unitary operations are implemented simultaneously on the sub-waves at different slits.
Step three. Performing the unitary operation W on the auxiliary qudit |i〉 . Then the following state is obtained, Figure 2. The multi-output duality quantum computing circuit in a quantum computer. |Ψ 〉 denotes the initial state of work qubit, and |0〉 is the initial state of the controlling auxiliary qudit. The circles represent the state of the controlling qudit and the squares represent unitary operations. Unitary operations U 0 , U 1 , U d−1 are activated only when the qudit holds the respective values indicated in circles 18 . The "readout" part marked by yellow rectangle means that: When the auxiliary qudit in the |j〉 state, where j ∈ {0, 1 , d}, the corresponding output (final state) of the work qubit will be redout by corresponding detector.
Scientific RepoRts | 6:30727 | DOI: 10.1038/srep30727 In previous paper 17,18 , only L 0 is studied as a duality quantum gate. In this article, we discuss all the k number duality quantum gates.
Step four. After step three, the auxiliary qudit is in a superposition state. Making the complete measurements, namely, measuring the final wave function when the qudit is in state |j〉 by placing j detectors at j different slits. which described as "readout" in Fig. 2. The complete measurements are also clearly visualized by the detectors in Fig. 1.
The duality quantum gate, or generalized quantum gate is defined as follows where U i is unitary and c i is the complex coefficient and satisfies When c i is restricted to positive real, c i is denoted by r i , and satisfies the constrained condition of ∑ i r i ≤ 1. In this scenario, the duality quantum gate is called real duality gate which is denoted as L r . So, the form of real duality quantum gate can be expressed as This corresponds to a physical picture of an asymmetric d-slit, and r i is the probability that the duality computer system passes through the i-th slit.
Because unitary operators have the unclosed property under addition, the duality quantum gates are generally non-unitary. Moreover, Gudder has proved that all linear bounded operators in a finite dimensional Hilbert space can be expressed as an element in the positive cone of generalized quantum gates 21 . Many recent studies about the mathematical theory of duality quantum computer have been made 19  Then, the expression of ∼ U can be simplified as . According to Eq. (10), = ∼ L U s / is a quantum duality gate and s is the normalization constant. We give the quantum circuit for realizing the approximation ∼ U is given in Fig. 3, which is the same as that in ref. 35  The implementation of operation ∼ U needs an auxiliary system and a work system(target state). The auxiliary system is composed by K auxiliary qubits |0〉 K and K numbers of L level auxiliary qudits |0〉 L , for the implementations of two QWD operations and two QWC operations. We denote the initial state of the whole system as Ψ 0 0 K L K , where |Ψ 〉 is the work qubit state and 0 L K means K numbers of L level auxiliary qudits all in state |0〉 L 35 . The first QWD can be expressed as a 2 K × 2 K matrix, denoted as V F . Defining α = ∑ = N l L l 1 , the elements of the matrix is Similarly, we denote the second QWD operation as V S , which can be viewed as a L × L matrix. The elements of the matrix satisfy S ,0 Corresponding to K auxiliary qubits |0〉 K , K numbers of L level |0〉 L auxiliary controlling qudits should be transformed into K numbers of state by the same QWD operation V S . They can be denoted as Applying the two QWD operations V F and V S to the state 0 0 K L K produces the state of total auxiliary system where s = ∑ j∈J β j is the normalization constant. We perform the auxiliary system controlled operation U j on the work system. The state of the whole system is transformed into Then, we need to perform two QWC operations to combine the wave functions, denoted as W F = (V F ) † and W S = (V S ) † , respectively. Physically, the two QWC operations are the counterparts of the two QWD operations. We denote the state orthogonal to Ψ 0 0 K L K as |Φ 〉 , the total process can be described as: The results of the duality quantum computing are in the terms with the auxiliary system in state 0 0 K L K . Therefore, we only need to readout the output of the work system with auxiliary system in state 0 0 K L K , which corresponds to the single-output duality quantum computing. Namely, the initial state goes through the transformation we interested is Thus, we obtain Consequently, we have successfully realized the following process, Scientific RepoRts | 6:30727 | DOI: 10.1038/srep30727 If we make measurement directly, the probability of detecting the auxiliary state 0 0 Namely the probability of implementing U on the target state |Ψ 〉 successfully is P s . Amplifying the amplitude of the desired term before the measurement by applying the robust obvious amplitude amplification given in Res. 48 47 , the query complexity in each segment is

 
The query complexity for the full simulation algorithm is r times K. Consider all the operations performed on auxiliary system and work system, the total number of gates in the simulation for time t/r in each segment is 48 where T = (α 1 +  + α L )t. In last section, we have realized Kraus operator by the duality quantum gate E k = L k = ∑ i W ki V i0 U i . In this section, unitary operator U i is realised by BCCKS algorithm in duality quantum computing form with precision . So, we have successfully simulated the total evolution of an open quantum system, The complexity of performing U i with precision  is Kr. Consider the fact that the coefficients satisfy ∑ i |W ki V i0 | ≤ 1, the complexity of performing E k is as the same as the complexity of performing U i . The total complexity of the whole algorithm with d numbers of E k is 3 max Compared with the complexity of Llyod 's algorithm O(ld 4 ht 2 /), the dependence on dimension of principal system is decreased from O(d 4 ) to O(d 3 ) and the performance is exponential improved on precision . An example to show the implementation of this simulation algorithm is given in next section.
Application to a single quibit open quantum system. Suppose we have a principal system with single qubit, interacting with a single qubit environment. U is time evolution operator imposed on the total system 6 . The expression of U is where X represents the usual Pauli matrix acting on the environment, and P 0 = |0〉 〈 0|, P 1 = |1〉 〈 1| are projectors acting on system. The initial state of environment is |0〉 . In this special case, the number of state k is 2.
Equation (1) After obtaining the expression of U 0 and U 1 and finding the corresponding Hamiltonian, we are able to simulate the Hamiltonian by approximating the truncated Taylor series of the evolution operator in duality quantum computer. The process of realizing U 0 or U 1 has given in the last section.

Discussion
In the present paper, we have briefly described the dynamics of an open quantum system and the quantum operations can be elegantly represented in operator-sum representation. The dynamics in the principal system can be described by trace preserving Kraus operators. The duality quantum computing is a suitable way to realise Kraus operators with non-unitary feature. Duality quantum computer provides the capability to perform linear combinations of unitary operations in the computation, which is called the duality quantum gates or the generalized quantum gates. The duality quantum computer can be perfectly simulated by an ordinary quantum computer with n-qubit and an additional qudit resource. By realizing Kraus operators through duality quantum computing, and approximating Hamiltonian simulation by the truncated Taylor series of the evolution operator in duality quantum computer, we present an efficient quantum algorithm for simulating Hamiltonian in open quantum system. Consider the fact that all quantum system is inevitable coupled with its environment in the real world, our method can be applied in a class of general physical systems. By realizing Kraus operators, the query complexity is decrease from O(d 4 ) dimension dependence to O(d 3 ) of the open quantum system. Moreover, through the use of truncated Taylor series in duality computing, our algorithm can provide an exponential improvement in precision. |Ψ 〉 denotes the initial state of principal system, and environment is in the |0〉 state. The squares represent unitary operations and the circles represent the state of the controlling qubit. Unitary operations U 0 , U 1 are activated only when the auxiliary qubit is |0〉 and |1〉 respectively.