Low depth quantum circuits for Ising models

Iblisdir, Sofyan; Cirio, Mauro; Boada, O.; Brennen, Gavin K.

doi:10.1016/j.aop.2013.11.001

Cited by 15 publications

(23 citation statements)

References 45 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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“…where the exponentiated sum is over the complete graph on n vertices, w ij and v k are real edge and vertex weights, and ω ∈ C. Then, for any ω = e iθ , Z(ω) arises straightforwardly as an amplitude of some IQP circuit C I (ω): 0| ⊗n C I (ω)|0 ⊗n = Z(ω)/2 n (see Appendix A and [7][8][9][10][11]). For our purposes it is sufficient to restrict to the case where ω = e iπ/8 and the weights are picked by choosing uniformly at random from the set {0, .…”

mentioning

confidence: 99%

Average-Case Complexity Versus Approximate Simulation of Commuting Quantum Computations

2016

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We use the class of commuting quantum computations known as IQP (Instantaneous Quantum Polynomial time) to strengthen the conjecture that quantum computers are hard to simulate classically. We show that, if either of two plausible average-case hardness conjectures holds, then IQP computations are hard to simulate classically up to constant additive error. One conjecture relates to the hardness of estimating the complextemperature partition function for random instances of the Ising model; the other concerns approximating the number of zeroes of random low-degree polynomials. We observe that both conjectures can be shown to be valid in the setting of worst-case complexity. We arrive at these conjectures by deriving spin-based generalisations of the Boson Sampling problem that avoid the so-called permanent anticoncentration conjecture.Quantum computers are conjectured to outperform classical computers for a variety of important tasks ranging from integer factorisation [1] to the simulation of quantum mechanics [2]. However, to date there is relatively little rigorous evidence for this conjecture. It is well established that quantum computers can yield an exponential advantage in the query and communication complexity models. But in the more physically meaningful model of time complexity, there are no proven separations known between quantum and classical computation.This can be seen as a consequence of the extreme difficulty of proving bounds on the power of classical computing models, such as the famous P vs. NP problem. Given this difficulty, the most we can reasonably hope for is to show that quantum computations cannot be simulated efficiently classically, assuming some widely believed complexity-theoretic conjecture. For example, any set of quantum circuits that can implement Shor's algorithm [1] provides a canonical example, with the unlikely consequence of efficient classical simulation of this class of quantum circuits being the existence of an efficient classical factoring algorithm. However, one could hope for the existence of other examples that have wider-reaching complexity-theoretic consequences.With this in mind, in both [3] and [4] it was shown that the existence of an efficient classical sampler from a distribution that is close to the output distribution of an arbitrary quantum circuit, to within a small multiplicative error in each output probability, would imply that post-selected classical computation is equivalent to post-selected quantum computation. This consequence is considered very unlikely as it would collapse the infinite tower of complexity classes known as the Polynomial Hierarchy [5] to its third level. In both works this was proven even for non-universal quantum circuit families: commuting quantum circuits in the case of [3], and linear-optical networks in [4]. These non-universal families are of physical interest because they are simpler to implement, and easier to analyse because of the elegant mathematical structures on which they are based. * michael.bremner@uts.edu.au Unfortu...

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“…The accuracy of the proposed algorithm is comparable to that in Ref. [25] (at least in the size of the lattice mentioned), which utilizes an analytical continuation in order to estimate the partition function with real parameters. In the ferromagnetic case without magnetic fields, the scheme in Ref.…”

Section: B Extended Quantum Algorithmmentioning

confidence: 71%

“…Unfortunately, the approximation scale ∆ still depends on the size n of the system. Thus an approximation of free energy per site with an additive error 1/poly(n) cannot be achieved, although this is also the case for other quantum algorithms approximating the Ising partition functions [11,21,25,37]. The accuracy of the proposed algorithm is comparable to that in Ref.…”

Section: B Extended Quantum Algorithmmentioning

confidence: 86%

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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Low depth quantum circuits for Ising models

Cited by 15 publications

References 45 publications

Commuting quantum circuits and complexity of Ising partition functions

Commuting quantum circuits and complexity of Ising partition functions

Average-Case Complexity Versus Approximate Simulation of Commuting Quantum Computations

Quantum algorithm for an additive approximation of Ising partition functions

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