Efficiency of free-energy calculations of spin lattices by spectral quantum algorithms

Master, Cyrus P.; Yamaguchi, Fumiko; Yamamoto, Yoshihisa

doi:10.1103/physreva.67.032311

Cited by 7 publications

(10 citation statements)

References 17 publications

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“…Specifically, certain sets of instances are shown to be BQP-complete, which means that such algorithms can actually do a nontrivial task, which would be intractable on a classical computer. In [6], a quantum algorithm for an additive approximation of real Ising partition functions on square lattices has been proposed by using an analytic continuation (see also a Fourier sampling scheme for spin models for estimating free energy [71]). In [7], another quantum algorithm for an additive approximation of square-lattice Ising partition functions with completely general parameters including real physical ones has been constructed based on a linear operator simulation by a unitary circuit with ancilla qubits (see also a linear operator simulation for an additive approximation of Tutte polynomials [4]).…”

Section: Related Workmentioning

confidence: 99%

Commuting quantum circuits and complexity of Ising partition functions

Fujii

Morimae

2017

New J. Phys.

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Instantaneous quantum polynomial-time (IQP) computation is a class of quantum computation consisting only of commuting two-qubit gates and is not universal. Nevertheless, it has been shown that if there is a classical algorithm that can simulate IQP efficiently, the polynomial hierarchy collapses to the third level, which is highly implausible. However, the origin of the classical intractability is still less understood. Here we establish a relationship between IQP and computational complexity of calculating the imaginary-valued partition functions of Ising models. We apply the established relationship in two opposite directions. One direction is to find subclasses of IQP that are classically efficiently simulatable by using exact solvability of certain types of Ising models. Another direction is applying quantum computational complexity of IQP to investigate (im)possibility of efficient classical approximations of Ising partition functions with imaginary coupling constants. Specifically, we show that a multiplicative approximation of Ising partition functions is #P-hard for almost all imaginary coupling constants even on planar lattices of a bounded degree.Aaronson and Arkhipov introduced BOSONSAMPLING [15], a sampling problem according to the probability distribution of n bosons scattered by linear optical unitary operations. The probability distribution is given by the permanent of a complex matrix, which is determined by the linear optical unitary operations. Calculation of the permanent of complex matrices is known to be #P-hard [16,17]. Since a polynomial-time machine with an oracle for #P can solve all problems in the PH according to Toda's theorem [18], an exact classical simulation (in the strong sense [19,20] meaning a calculation of the probability distribution of the output) of BOSONSAMPLING is highly intractable in a classical computer. They showed under assumptions of plausible conjectures that if there exists an efficient classical approximation of BOSONSAMPLING (classical simulation in the weak sense [19, 20] meaning a sampling according the probability distribution of the output), the PH collapses to the third level, which is unlikely to occur. (The detailed notions of classical simulation are provided in section 3.) This result brings a novel perspective on linear optical quantum computation and drives many researchers into the recent proof-of-principle experiments [21][22][23][24][25][26][27][28].Another subclass of quantum computation of this kind is instantaneous quantum polynomial-time computation (IQP) proposed by Shepherd and Bremner [29]. IQP consists only of commuting unitary gates, such as q  Î [ ] Z exp i k S k . Here q p Î [ ) 0, 2 is a rotational angle, Z k indicates the Pauli operator on the kth qubit, and S indicates a set of qubits on which the commuting gate acts. (A detailed definition will be provided in the next section.) The input is given by +ñ Ä | n with +ñ º ñ + ñ | (| | ) 0 1 2, and the output qubits are measured in the X-basis. Since all unitary operations are commutabl...

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Section: Related Workmentioning

confidence: 99%

Commuting quantum circuits and complexity of Ising partition functions

Fujii

Morimae

2017

New J. Phys.

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“…Furthermore, even an efficient (multiplicative) approximation of antiferromagnetic Ising partition functions on general lattices does not exist unless RP = NP [7,8], which is highly implausible to occur [9,10]. It is a natural question how quantum computer is useful in this context [11,12].…”

Section: Introductionmentioning

confidence: 99%

“…The construction of the quantum algorithm in this work can be regarded as a measurement-based version of these works, which would be simpler for people who are familiar with MBQC. They have also considered approximation of the Ising partition functions with real coupling strengths and magnetic fields, while a rather different approach, analytic continuation, was taken (see also a related work [11]). Instead of analytic continuation, we here straightforwardly simulate linear operators by using unitary circuits.…”

Section: Introductionmentioning

confidence: 99%

Quantum algorithm for an additive approximation of Ising partition functions

2014

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We investigate quantum computational complexity of calculating partition functions of Ising models. We construct a quantum algorithm for an additive approximation of Ising partition functions on square lattices. To this end, we utilize the overlap mapping developed by Van den Nest, Dür, and Briegel [Phys. Rev. Lett. 98, 117207 (2007)] and its interpretation through measurementbased quantum computation (MBQC). We specify an algorithmic domain, on which the proposed algorithm works, and an approximation scale, which determines the accuracy of the approximation. We show that the proposed algorithm does a nontrivial task, which would be intractable on any classical computer, by showing the problem solvable by the proposed quantum algorithm are BQPcomplete. In the construction of the BQP-complete problem coupling strengths and magnetic fields take complex values. However, the Ising models that are of central interest in statistical physics and computer science consist of real coupling strengths and magnetic fields. Thus we extend the algorithmic domain of the proposed algorithm to such a real physical parameter region and calculate the approximation scale explicitly. We found that the overlap mapping and its MBQC interpretation improves the approximation scale exponentially compared to a straightforward constant depth quantum algorithm. On the other hand, the proposed quantum algorithm also provides us a partial evidence that there exist no efficient classical algorithm for a multiplicative approximation of the Ising partition functions even on the square lattice. This result supports that the proposed quantum algorithm does a nontrivial task also in the physical parameter region.

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“…† aspuru@chemistry.harvard.edu ‡ enrique.solano@ehu.es is a very active field of study and various methods have been developed. Quantum simulation methods have been proposed for preparing specific states such as ground [8][9][10][11][12][13] and thermal states [14][15][16][17][18][19][20], simulating time evolution [21][22][23][24][25][26][27], and the measurement of physical observables [28][29][30][31]. Trapped-ion systems (see Fig.…”

mentioning

confidence: 99%

From transistor to trapped-ion computers for quantum chemistry

Yung

Casanova

Mezzacapo

et al. 2014

Sci Rep

245

239

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Over the last few decades, quantum chemistry has progressed through the development of computational methods based on modern digital computers. However, these methods can hardly fulfill the exponentially-growing resource requirements when applied to large quantum systems. As pointed out by Feynman, this restriction is intrinsic to all computational models based on classical physics. Recently, the rapid advancement of trapped-ion technologies has opened new possibilities for quantum control and quantum simulations. Here, we present an efficient toolkit that exploits both the internal and motional degrees of freedom of trapped ions for solving problems in quantum chemistry, including molecular electronic structure, molecular dynamics, and vibronic coupling. We focus on applications that go beyond the capacity of classical computers, but may be realizable on state-of-the-art trapped-ion systems. These results allow us to envision a new paradigm of quantum chemistry that shifts from the current transistor to a near-future trapped-ion-based technology.

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Efficiency of free-energy calculations of spin lattices by spectral quantum algorithms

Cited by 7 publications

References 17 publications

Commuting quantum circuits and complexity of Ising partition functions

Commuting quantum circuits and complexity of Ising partition functions

Quantum algorithm for an additive approximation of Ising partition functions

From transistor to trapped-ion computers for quantum chemistry

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