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Direct Application of the Phase Estimation Algorithm to Find The Chemical Physics xxx (2018) xxx–xxx Contents lists available at ScienceDirect Chemical Physics journal homepage: www.elsevier.com/locate/chemphys Direct application of the phase estimation algorithm to find the eigenvalues of the Hamiltonians ⇑ Ammar Daskin a, Sabre Kais b, a Department of Computer Engineering, Istanbul Medeniyet University, Kadikoy, Istanbul, Turkey b Department of Chemistry, Department of Physics and Birck Nanotechnology Center, Purdue University, West Lafayette, IN, USA article info abstract Article history: The eigenvalue of a Hamiltonian, H, can be estimated through the phase estimation algorithm given the Available online xxxx matrix exponential of the Hamiltonian, expðiHÞ. The difficulty of this exponentiation impedes the appli- cations of the phase estimation algorithm particularly when H is composed of non-commuting terms. In this paper, we present a method to use the Hamiltonian matrix directly in the phase estimation algorithm by using an ancilla based framework: In this framework, we also show how to find the power of the Hamiltonian matrix-which is necessary in the phase estimation algorithm-through the successive appli- cations. This may eliminate the necessity of matrix exponential for the phase estimation algorithm and therefore provide an efficient way to estimate the eigenvalues of particular Hamiltonians. The classical and quantum algorithmic complexities of the framework are analyzed for the Hamiltonians which can be written as a sum of simple unitary matrices and shown that a Hamiltonian of order 2n written as a sum of L number of simple terms can be used in the phase estimation algorithm with ðn þ 1 þ logLÞ num- ber of qubits and Oð2anLÞ number of quantum operations, where a is the number of iterations in the phase estimation. In addition, we use the Hamiltonian of the hydrogen molecule as an example system and pre- sent the simulation results for finding its ground state energy. Ó 2018 Elsevier B.V. All rights reserved. 1. Introduction in terms of unitary matrices, then a possible implementation may yield a computational efficiency over the classical algorithms: e.g. With the recent efforts such as digitalizing adiabatic quantum Shor’s integer factoring algorithm [7]. Many quantum algorithms computers [1], a digital quantum simulation of the real-time use the exponential expðihHÞ to map the description of the orig- dynamic of a lattice gauge theory [2], a simulation of the Hubbard inal problem H to a unitary matrix. An exact mapping through model [3], and the race to build universal quantum computers matrix exponential generally requires eigendecomposition of H: among the big companies [4], quantum computers become closer In spite of the existence of various numerical methods, the eigen- to be used in unsolved real-world applications. decomposition in the cases where H is a large-dense matrix is still As Feynman suggested [5], one of the breakthroughs of quan- a big computational challenge for classical computers. Because of tum computers is expected to be in the simulation of quantum sys- this computational difficulty, it is common to use some order of tems, a known difficult problem for classical computers because of Trotter-Suzuki approximation [8] by writing H as a sum of terms the exponential growth of the system size with the number of whose exponentials are known or easy to compute. In general case, qubits (e.g., see Ref. [6]). In the standard formalism of quantum the order of the approximation affects the correctness and the high mechanics, the Hamiltonian of a quantum system is considered order approximations dramatically increases the computational as a Hermitian matrix. Since the exponential of a skew-Hermitian complexity [9]. Therefore, the complexity of this exponentiation matrix is a unitary matrix, the time evolution, expðihHÞ,ofa should be also embodied in the consideration of the computational quantum system represented by the Hamiltonian H describes a complexity of a quantum algorithm (see Ref. [10] for a relevant dis- unitary transformation. In general, a computation or an algorithm cussion of the topic). should be represented by a unitary matrix in order to be imple- It is known that using additional qubits eases the implemen- mented on a quantum computer. If an algorithm can be formalized tation difficulty of a quantum circuit. A quantum operator (or a dynamic of a system) can be implemented inside a larger system, ⇑ Corresponding author. where some reduced part of the system represents the action of E-mail address: [email protected] (S. Kais). the original system. This idea is used in different contexts such https://doi.org/10.1016/j.chemphys.2018.01.002 0301-0104/Ó 2018 Elsevier B.V. All rights reserved. Please cite this article in press as: A. Daskin, S. Kais, Chem. Phys. (2018), https://doi.org/10.1016/j.chemphys.2018.01.002 2 A. Daskin, S. Kais / Chemical Physics xxx (2018) xxx–xxx as designing circuits from matrix elements [11,12], and generat- iH He ¼ I À ð1Þ ing exponential of a matrix written as a sum of unitary matrices j through Taylor series [13]. Moreover, in Ref. [14], to use a matrix where I is an identity matrix, and j is a coefficient. The eigenvalues on a quantum computer directly, we considered converting a ek k =j k non-unitary matrix, H, to a unitary by using its square root. In of this matrix are in the form j ¼ð1 À i j Þ, where j is the jth a more general fashion, Ref. [15] described a qubitization eigenvalue of H. When j is sufficiently large; in the polar form of e approach to complete a non-unitary matrix to a unitary matrix kj, the value of the angle becomes kj=j since sinðkj=jÞkj=j: whose multiple applications generate Chebyshev polynomials of k k k j H ek j j ij : . j ¼ 1 À i j ¼ 1 þ i j e ð2Þ In this paper, we follow a different approach: for a Hamiltonian n H j H of order 2 with real eigenvalues; we first put the matrix in the Also note that when j is large, He gives an approximation to ei .To e form H ¼ ðÞI À iH=j , where I is an identity matrix and j is a coef- be able to use the matrix in Eq. (1) directly in the phase estimation, ficient. For a sufficiently large j; the value of the angle, the phase, we will initially assume that we have an efficient mechanism to seen in the polar form of the eigenvalue of He becomes approxi- produce the following unitary matrix: ! mately equal to kj=j, where kj is an eigenvalue of H. Using this fact He and assuming H is a sum of some simple terms which can be easily Uð1Þ ¼ ; ð3Þ mapped to quantum circuits; we present a phase estimation framework for finding k =j. In this framework, we use He directly j where the first part of the matrix is equal to He with/without some and show that necessary powers of He can be estimated through normalization, and a indicates another part of the matrix. As also successive applications after the application of the oblivious ampli- mentioned in the introduction, these types of matrices are used in tude amplification. This provides a way to estimate the eigenvalues various contexts: either to design quantum circuits from matrix ele- H a of by using ðn þ 1 þ logLÞ qubits and Oð2 nLÞ number of quantum ments [11,12] or to be able to use it with the oblivious amplitude operations. Here, a is the number of iterations in the phase estima- amplification[14], or to estimate unitary dynamic of a Hamiltonian H tion and L is the number of terms in the sum yielding . We also through truncated Taylor series [13] and its generalized form [15]. e show that the powers of H can be more accurately estimated by In the next section we will discuss how the above matrix can be using additional qubits and a permutation operator in the phase generated by using an ancilla register. Now let us consider the input estimation algorithm, which requires also the same number of state ji0 jiu , where jiu represents an abstract quantum state on the quantum gates and Oða þ n þ logLÞ number of qubits. As an exam- system register and ji0 is the first vector in the standard basis and ple system, we use the Hamiltonian matrix of the hydrogen mole- the state on the ancilla register. The application of Uð1Þ to this input cule and estimated its ground state energy within the presented yields the following state [13]: framework. ffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1Þ u p He u u : In the following sections, after first describing the proposed U ji0 ji¼ p0ji0 jiþ 1 À p0ji1 ji ð4Þ framework, we will present the complexity analysis. Then, in Sec- He u tion 5, the simulation results for the hydrogen molecule are pre- The application of to the input jion the system register can be sented and discussed. And in the final section the paper is obtained from the above output with the probability given by He u : He concluded with a summary. p0k k ik, where jj jj represents a vector norm. For a unitary , the probability simply becomes p0. The desired output can be sin- gled out through the application of a projector P ¼j0ih0j to the 2. Notes on notations ancilla register. Throughout this paper, we use the following notations: j 3.1. Estimation of He 2 While the Hamiltonian is represented by H, bold capital letters For the phase estimation, we need to be able to efficiently gen- are used for other matrices. ð20Þ ð2aÞ n is used to represent the number of qubits in the system.
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