Quantum computing applied to calculations of molecular energies: CH2 benchmark

Veis, Libor; Pittner, Jiřı́

doi:10.1063/1.3503767

Cited by 58 publications

(105 citation statements)

References 34 publications

(66 reference statements)

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“…1a). Reducing the number of qubits in quantum simulations and quantum chemistry has been achieved with recursive phase estimation [13][14][15][16] , while ground state projections have been demonstrated by exploiting similar techniques in NMR 17 .…”

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Experimental realization of Shor's quantum factoring algorithm using qubit recycling

et al. 2012

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Quantum computational algorithms exploit quantum mechanics to solve problems exponentially faster than the best classical algorithms 1-3 . Shor's quantum algorithm 4 for fast number factoring is a key example and the prime motivator in the international effort to realise a quantum computer 5 . However, due to the substantial resource requirement, to date, there have been only four small-scale demonstrations 6-9 . Here we address this resource demand and demonstrate a scalable version of Shor's algorithm in which the n qubit control register is replaced by a single qubit that is recycled n times: the total number of qubits is one third of that required in the standard protocol [10][11][12] . Encoding the work register in higher-dimensional states, we implement a two-photon compiled algorithm to factor N = 21. The algorithmic output is distinguishable from noise, in contrast to previous demonstrations. These results point to larger-scale implementations of Shor's algorithm by harnessing scalable resource reductions applicable to all physical architectures.Shor's factoring algorithm consists of a quantum order finding algorithm, preceded and succeeded by various classical routines. While the classical tasks are known to be efficient on a classical computer, order finding is understood to be intractable classically. However, it is known that this part of the algorithm can be performed efficiently on a quantum computer. To determine the prime factors of an odd integer N , one chooses a coprime of N , x. The order r relates x to N according to x r mod N = 1, and can be used to obtain the factors, given by the greatest common divisor gcd(x r 2 ± 1, N ). The quantum order finding circuit involves two registers: a work register and a control register. In the standard protocol, the work register performs modular arithmetic with m = log 2 N qubits, enough to encode the number N , and the n qubit control register provides the algorithmic output, with n bits of precision.Measuring the control register in the computational basis will yield a result of k2 n /r where k is an integer between 0 and r − 1, with the value of k occurring probabilistically. Dividing the result by 2 n gives the first n bits of k/r and r may be found with classical processing, using the continued fraction algorithm. For large n, and a perfectly functioning circuit, the output probability distribution of the control register is a series of well defined peaks at values of k2 n /r (Fig. 1b). (See Appendix for details.)Here we implement an iterative version of the order finding algorithm 10,11 , in which the control register contains only a single qubit which is recycled n times, using a sequence of measurement and feed-forward operations, with each step providing an additional bit of precision (Fig. 1a). Reducing the number of qubits in quantum simulations and quantum chemistry has been achieved with recursive phase estimation 13-16 , while ground state projections have been demonstrated by exploiting similar techniques in NMR 17 .The iterative version of...

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Experimental realization of Shor's quantum factoring algorithm using qubit recycling

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“…An efficient (polynomially scaling) algorithm for calculations of non-relativistic molecular energies, that employs the phase estimation algorithm (PEA) of Abrams and Lloyd [19], was proposed in the pioneering work by Aspuru-Guzik, et al [6]. When the ideas of measurement based quantum computing are adopted [20], the phase estimation algorithm can be formulated in an iterative manner [iterative phase estimation (IPEA)] with only one read-out qubit [8,9]. If the phase φ (0 ≤ φ < 1), which is directly related to the desired energy [9], is expressed in the binary form: φ = 0.φ 1 φ 2 .…”

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“…The PEA always needs an initial guess of the wave function corresponding to the desired energy. This can be either the result of some approximate, polynomially scaling ab initio method [7,9], or as originally proposed by Aspuru-Guzik, et al[6] the exact state or its approximation prepared by the adiabatic state preparation (ASP) method. Figure 1.…”

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“…When the ideas of measurement based quantum computing are adopted [20], the phase estimation algorithm can be formulated in an iterative manner [iterative phase estimation (IPEA)] with only one read-out qubit [8,9]. If the phase φ (0 ≤ φ < 1), which is directly related to the desired energy [9], is expressed in the binary form: φ = 0.φ 1 φ 2 .…”

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confidence: 99%

“…These cover: calculations of thermal rate constants of chemical reactions [5], non-relativistic energy calculations [6][7][8][9], quantum chemical dynamics [10], calculations of molecular properties [11], initial state preparation [12,13], and also first proof-of-principle experimental realizations [14][15][16][17]. An interested reader can find a comprehensive review in [18].…”

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Relativistic quantum chemistry on quantum computers

et al. 2012

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Last years witnessed a remarkable interest in application of quantum computing for solving problems in quantum chemistry more efficiently than classical computers allow. Very recently, even first proof-of-principle experimental realizations have been reported. However, so far only the nonrelativistic regime (i.e. Schroedinger equation) has been explored, while it is well known that relativistic effects can be very important in chemistry. In this communication we present the first quantum algorithm for relativistic computations of molecular energies. We show how to efficiently solve the eigenproblem of the Dirac-Coulomb Hamiltonian on a quantum computer and demonstrate the functionality of the proposed procedure by numerical simulations of computations of the spinorbit splitting in the SbH molecule. Finally, we propose quantum circuits with 3 qubits and 9 or 10 CNOTs, which implement a proof-of-principle relativistic quantum chemical calculation for this molecule and might be suitable for an experimental realization.Quantum computing [1] is one of the fastest growing fields of computer science nowadays. Recent huge interest in this interdisciplinary field has been fostered by the prospects of solving certain types of problems more effectively than in the classical setting [2,3]. The prominent example is the integer factorization problem where quantum computing offers an exponential speedup over its classical counterpart [2]. But it is not only cryptography that can benefit from quantum computers. As was first proposed by R. Feynman [4], quantum computers could in principle be used for efficient simulation of another quantum system. This idea, which employs mapping of the Hilbert space of a studied system onto the Hilbert space of a register of quantum bits (qubits), both of them being exponentially large, can in fact be adopted also in quantum chemistry.Several papers using this idea and dealing with the interconnection of quantum chemistry and quantum computing have appeared in recent years. An efficient (polynomially scaling) algorithm for calculations of non-relativistic molecular energies, that employs the phase estimation algorithm (PEA) of Abrams and Lloyd [19], was proposed in the pioneering work by Aspuru-Guzik, et al. [6]. When the ideas of measurement based quantum computing are adopted [20], the phase estimation algorithm can be formulated in an iterative manner [iterative phase estimation (IPEA)] with only one read-out qubit [8,9]. If the phase φ (0 ≤ φ < 1), which is directly related to the desired energy [9], is expressed in the binary form: φ = 0.φ 1 φ 2 . . ., φ i = {0, 1}, one bit of φ is measured on the read-out qubit at each iteration step. The algorithm is iterated backwards from the least significant bits of φ to the most significant ones, where the k-th iteration is shown in Figure 1. Not to confuse the reader, H in the exponential denotes the Hamiltonian operator, whereas H (in a box) denotes the standard single-qubit Hadamard gate. |ψ system represents the part of a quantum register that e...

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Introduction to Quantum Information and Computation for Chemistry

Kais

2014

Advances in Chemical Physics

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Quantum computing applied to calculations of molecular energies: CH2 benchmark

Cited by 58 publications

References 34 publications

Experimental realization of Shor's quantum factoring algorithm using qubit recycling

Experimental realization of Shor's quantum factoring algorithm using qubit recycling

Relativistic quantum chemistry on quantum computers

Introduction to Quantum Information and Computation for Chemistry

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